{"group":{"id":1,"name":"Community","lockable":false,"created_at":"2012-01-18T18:02:15.000Z","updated_at":"2025-12-14T01:33:56.000Z","description":"Problems submitted by members of the MATLAB Central community.","is_default":true,"created_by":161519,"badge_id":null,"featured":false,"trending":false,"solution_count_in_trending_period":0,"trending_last_calculated":"2025-12-14T00:00:00.000Z","image_id":null,"published":true,"community_created":false,"status_id":2,"is_default_group_for_player":false,"deleted_by":null,"deleted_at":null,"restored_by":null,"restored_at":null,"description_opc":null,"description_html":null,"published_at":null},"problems":[{"id":58144,"title":"Given the mass and stiffness of an undamped SDOF system, find the natural frequency and the natural period of vibration","description":"Problem Statement\r\nGiven the mass  and stiffness  of an undamped SDOF system, find the system's natural frequency in both Hz and radians per second as well as the natural period of vibration. \r\nBackground\r\nAn undamped, linear single degree-of-freedom (SDOF) system can be visualized as a spring with stiffness , fixed at one end and having a mass  attached to the other:\r\n\r\nOnce set into motion, the system will oscillate at its natural frequency, or fundamental frequency: \r\n\r\nwhere the natural frequency  is in radians per second. \r\nThe time required for the undamped system to complete one cycle of free vibration is the natural period of vibration of the system, , which has units of seconds:\r\n\r\nA system executes  cycles in 1 second. This is the natural cyclic frequency of vibration:\r\n\r\nwhich has units of Hertz (Hz) or cycles per second. The term natural frequency of vibration applies to both  and . They have the relationship:\r\n\r\nExample\r\nm = 10;\r\nk = 2;\r\n[wn,fn,Tn] = udSDOF(m,k)\r\nwn = 5\r\nfn = 0.7958\r\nTn = 1.2566\r\n\r\n","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.44px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: none solid rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 878.625px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 439.312px; transform-origin: 407px 439.312px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 20px; border-block-end-color: rgb(60, 60, 60); border-block-start-color: rgb(60, 60, 60); border-bottom-color: rgb(60, 60, 60); border-inline-end-color: rgb(60, 60, 60); border-inline-start-color: rgb(60, 60, 60); border-left-color: rgb(60, 60, 60); border-right-color: rgb(60, 60, 60); border-top-color: rgb(60, 60, 60); caret-color: rgb(60, 60, 60); color: rgb(60, 60, 60); column-rule-color: rgb(60, 60, 60); font-family: Helvetica, Arial, sans-serif; font-size: 20px; font-weight: 700; line-height: 20px; margin-block-end: 5px; margin-block-start: 3px; margin-bottom: 5px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 3px; outline-color: rgb(60, 60, 60); perspective-origin: 384px 10px; text-align: left; text-decoration: none; text-decoration-color: rgb(60, 60, 60); text-emphasis-color: rgb(60, 60, 60); transform-origin: 384px 10px; white-space: pre-wrap; margin-left: 4px; margin-top: 3px; margin-bottom: 5px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eProblem Statement\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 21px; text-align: left; transform-origin: 384px 21px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eGiven the mass \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003em\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e and stiffness \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ek\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e of an undamped SDOF system, find the system's natural frequency in both Hz and radians per second as well as the natural period of vibration. \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 20px; border-block-end-color: rgb(60, 60, 60); border-block-start-color: rgb(60, 60, 60); border-bottom-color: rgb(60, 60, 60); border-inline-end-color: rgb(60, 60, 60); border-inline-start-color: rgb(60, 60, 60); border-left-color: rgb(60, 60, 60); border-right-color: rgb(60, 60, 60); border-top-color: rgb(60, 60, 60); caret-color: rgb(60, 60, 60); color: rgb(60, 60, 60); column-rule-color: rgb(60, 60, 60); font-family: Helvetica, Arial, sans-serif; font-size: 20px; font-weight: 700; line-height: 20px; margin-block-end: 5px; margin-block-start: 20px; margin-bottom: 5px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 20px; outline-color: rgb(60, 60, 60); perspective-origin: 384px 10px; text-align: left; text-decoration: none; text-decoration-color: rgb(60, 60, 60); text-emphasis-color: rgb(60, 60, 60); transform-origin: 384px 10px; white-space: pre-wrap; margin-left: 4px; margin-top: 20px; margin-bottom: 5px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eBackground\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 21px; text-align: left; transform-origin: 384px 21px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eAn undamped, linear single degree-of-freedom (SDOF) system can be visualized as a spring with stiffness \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ek\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e, fixed at one end and having a mass \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003em\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e attached to the other:\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 99px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 49.5px; text-align: left; transform-origin: 384px 49.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cimg class=\"imageNode\" width=\"90\" height=\"93\" style=\"vertical-align: baseline;width: 90px;height: 93px\" src=\"data:image/jpeg;base64,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\" data-image-state=\"image-loaded\"\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eOnce set into motion, the system will oscillate at its \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003enatural frequency\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e, or fundamental frequency: \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 41px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 20.5px; text-align: left; transform-origin: 384px 20.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"vertical-align:-16px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"62.5\" height=\"41\" style=\"width: 62.5px; height: 41px;\"\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21.5px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.75px; text-align: left; transform-origin: 384px 10.75px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003ewhere the natural frequency \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAACUAAAAoCAYAAAB5ADPdAAAAAXNSR0IArs4c6QAAAulJREFUWEft1u/Ln2MYBvDPoimUJS+UthczYRRNilbai0XbC8lCMzNGsUZL8uMNWig/pq0wPxLGhIWW1bK/wAvFlF+lFJssaQ2R7Q0dOm/d3d3P89zP4+nbXnyv+tb3vq/rvs7jOs7jPI9rjuNwzDkOMRmDGpqVMVNjpoYyMHTdWFNjpoYyMHTdWFOzzdQJOAcX4kR8jO8ryFwsw+n4veYODwXQt26q9AXMjbgPP2JLBf2zZ7Oz8DBuwHN4FEdnAm4yUDl5QNxaQR7EH1MEORfvYBHuxwuzCSqAXsYqfIR1+LkCnIQncDu2I2D/rrnT8Dquwdu1po/VSbH2MZV3j9TvF6zHntYuG/AUTsVu3IJfe0B9Uoc6MF22+kAtwVs4D7sKVJO2NhOJFTbvnICpWQW1uQSboGHlxdZJLyrAF9S77vx8vI9L8WGl/cj/ZeoM7MRV+BJr8Hlr06sLVFL3Tc1/2ppfijewsIokVTvt0U1fUz0XYx9uQnTVjGuLiTyn7O/FsdZ8UpmK69PiYHBdUJfjTZxdVXRHJ2gD6gvcjM9akdJEXyrh78VazKiJdkG1NRMtRTPtcReenSA1TYHMw20IsBmNLqj0pzC1soep2MwOhM2k7u5WxPSupO06pFCeaVXkcqxAJJF4Yf+ycokFeKAY/m+7vpYQm9iGn0rIX+N8PI/fkL5zSVXWtzgFj9farQWoay/X493a4yCS6h/wGl7BxrZMJmqeV9YJEjCB42sf4NU6zj24ogAuxnd4GvsnyFeqMLaT4snaHCL2FXYjiRx4UqZmpINJPjq52FhdOkv6Yu6RQFpPCiK3jpGCatpM9LqprOnM8sa/Or76L7Cpri6zwVpMPSYdn0wfi2U1TfY9PNRpOyMBFa0FTLST6s3I/5h+BB4z/6r0ORKmGgNPM06zTSE0GsstNoaetvBYVfZIQDXNOA6QhprUpR08WXet3LlyWz00aqFPW5ejEPoY1LQZGPrBOH1DmfoHNsmbKapCDnoAAAAASUVORK5CYII=\" width=\"18.5\" height=\"20\" style=\"width: 18.5px; height: 20px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e is in radians per second. \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 42.5px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 21.25px; text-align: left; transform-origin: 384px 21.25px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eThe time required for the undamped system to complete one cycle of free vibration is the \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003enatural period\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e of vibration of the system, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"16.5\" height=\"20\" style=\"width: 16.5px; height: 20px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e, which has units of seconds:\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 37px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 18.5px; text-align: left; transform-origin: 384px 18.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"vertical-align:-17px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"51.5\" height=\"37\" style=\"width: 51.5px; height: 37px;\"\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21.5px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.75px; text-align: left; transform-origin: 384px 10.75px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eA system executes \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"31.5\" height=\"20\" style=\"width: 31.5px; height: 20px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e cycles in 1 second. This is the \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003enatural cyclic frequency\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e of vibration:\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 37px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 18.5px; text-align: left; transform-origin: 384px 18.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"vertical-align:-17px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"44\" height=\"37\" style=\"width: 44px; height: 37px;\"\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 42.5px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 21.25px; text-align: left; transform-origin: 384px 21.25px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003ewhich has units of Hertz (Hz) or cycles per second. The term natural frequency of vibration applies to both \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"18.5\" height=\"20\" style=\"width: 18.5px; height: 20px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"12\" height=\"20\" style=\"width: 12px; height: 20px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e. They have the relationship:\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 38px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 19px; text-align: left; transform-origin: 384px 19px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"vertical-align:-15px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"47.5\" height=\"38\" style=\"width: 47.5px; height: 38px;\"\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 20px; border-block-end-color: rgb(60, 60, 60); border-block-start-color: rgb(60, 60, 60); border-bottom-color: rgb(60, 60, 60); border-inline-end-color: rgb(60, 60, 60); border-inline-start-color: rgb(60, 60, 60); border-left-color: rgb(60, 60, 60); border-right-color: rgb(60, 60, 60); border-top-color: rgb(60, 60, 60); caret-color: rgb(60, 60, 60); color: rgb(60, 60, 60); column-rule-color: rgb(60, 60, 60); font-family: Helvetica, Arial, sans-serif; font-size: 20px; font-weight: 700; line-height: 20px; margin-block-end: 5px; margin-block-start: 20px; margin-bottom: 5px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 20px; outline-color: rgb(60, 60, 60); perspective-origin: 384px 10px; text-align: left; text-decoration: none; text-decoration-color: rgb(60, 60, 60); text-emphasis-color: rgb(60, 60, 60); transform-origin: 384px 10px; white-space: pre-wrap; margin-left: 4px; margin-top: 20px; margin-bottom: 5px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eExample\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgb(247, 247, 247); block-size: 122.625px; border-bottom-left-radius: 4px; border-bottom-right-radius: 4px; border-end-end-radius: 4px; border-end-start-radius: 4px; border-start-end-radius: 4px; border-start-start-radius: 4px; border-top-left-radius: 4px; border-top-right-radius: 4px; margin-block-end: 10px; margin-block-start: 10px; margin-bottom: 10px; margin-inline-end: 3px; margin-inline-start: 3px; margin-left: 3px; margin-right: 3px; margin-top: 10px; perspective-origin: 404px 61.3125px; transform-origin: 404px 61.3125px; margin-left: 3px; margin-top: 10px; margin-bottom: 10px; margin-right: 3px; \"\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4375px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10.2188px; transform-origin: 404px 10.2188px; white-space: nowrap; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; white-space: pre; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003em = 10;\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4375px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10.2188px; transform-origin: 404px 10.2188px; white-space: nowrap; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; white-space: pre; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003ek = 2;\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4375px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10.2188px; transform-origin: 404px 10.2188px; white-space: nowrap; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; white-space: pre; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e[wn,fn,Tn] = udSDOF(m,k)\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4375px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10.2188px; transform-origin: 404px 10.2188px; white-space: nowrap; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; white-space: pre; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003ewn = 5\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4375px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10.2188px; transform-origin: 404px 10.2188px; white-space: nowrap; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; white-space: pre; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003efn = 0.7958\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4375px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10.2188px; transform-origin: 404px 10.2188px; white-space: nowrap; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; white-space: pre; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003eTn = 1.2566\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 10px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 10px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 10px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function [wn,fn,Tn] = udSDOF(m,k)\r\n  % put yo answer here ^(o.O)^\r\nend","test_suite":"%%\r\nm = 2;\r\nk = 50;\r\ny_correct = [5 0.7958 1.2566];\r\n[wn,fn,Tn] = udSDOF(m,k);\r\nassert(sum(round([wn,fn,Tn]./y_correct,2))==3)\r\n\r\n%%\r\nm = 0.2;\r\nk = 2000;\r\ny_correct = [100 15.9 0.0628];\r\n[wn,fn,Tn] = udSDOF(m,k);\r\nassert(sum(round([wn,fn,Tn]./y_correct,2))==3)\r\n\r\n%%\r\nm = 0.2533; \r\nk = 10; \r\ny_correct = [6.2832 1 1];\r\n[wn,fn,Tn] = udSDOF(m,k);\r\nassert(sum(round([wn,fn,Tn]./y_correct,2))==3)\r\n\r\n%%\r\nfiletext = fileread('udSDOF.m');\r\nnot_allowed = contains(filetext, 'regex');\r\nassert(~not_allowed)\r\n","published":true,"deleted":false,"likes_count":1,"comments_count":1,"created_by":91130,"edited_by":91130,"edited_at":"2023-04-28T00:16:14.000Z","deleted_by":null,"deleted_at":null,"solvers_count":58,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2023-04-27T23:21:12.000Z","updated_at":"2026-02-05T13:39:50.000Z","published_at":"2023-04-28T00:16:14.000Z","restored_at":null,"restored_by":null,"spam":null,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"heading\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eProblem Statement\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eGiven the mass \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003em\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and stiffness \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ek\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e of an undamped SDOF system, find the system's natural frequency in both Hz and radians per second as well as the natural period of vibration. \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"heading\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eBackground\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eAn undamped, linear single degree-of-freedom (SDOF) system can be visualized as a spring with stiffness \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ek\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, fixed at one end and having a mass \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003em\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e attached to the other:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"image\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"height\\\" w:val=\\\"93\\\"/\u003e\u003cw:attr w:name=\\\"width\\\" w:val=\\\"90\\\"/\u003e\u003cw:attr w:name=\\\"verticalAlign\\\" w:val=\\\"baseline\\\"/\u003e\u003cw:attr w:name=\\\"altText\\\" w:val=\\\"\\\"/\u003e\u003cw:attr w:name=\\\"relationshipId\\\" w:val=\\\"rId1\\\"/\u003e\u003c/w:customXmlPr\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eOnce set into motion, the system will oscillate at its \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003enatural frequency\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e, or fundamental frequency: \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"true\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e$\\\\omega_n =\\\\sqrt{\\\\frac{k}{m}}$\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003ewhere the natural frequency \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e$\\\\omega_n$\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e is in radians per second. \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe time required for the undamped system to complete one cycle of free vibration is the \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003enatural period\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e of vibration of the system, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eT_n\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, which has units of seconds:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"true\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e$T_n =\\\\frac{2\\\\pi }{\\\\omega_n }$\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eA system executes \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e1/T_n\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e cycles in 1 second. This is the \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003enatural cyclic frequency\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e of vibration:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"true\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e$f_n =\\\\frac{1}{T_n }$\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003ewhich has units of Hertz (Hz) or cycles per second. The term natural frequency of vibration applies to both \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e$\\\\omega_n$\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ef_n\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. They have the relationship:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"true\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e$f_n =\\\\frac{\\\\omega_n }{2\\\\pi }$\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"heading\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eExample\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"code\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c![CDATA[m = 10;\\nk = 2;\\n[wn,fn,Tn] = udSDOF(m,k)\\nwn = 5\\nfn = 0.7958\\nTn = 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Properties - 5","description":"Similar to the previous problem, materials may be characterized by their stiffness-to-weight ratio, which is the elastic modulus divided by density. Write a function to calculate this ratio for a material provided its elastic modulus and density.\r\n\r\nPrevious problem: 4 - \u003chttp://www.mathworks.com/matlabcentral/cody/problems/8051-stress-strain-properties-4 strength-to-weight ratio\u003e. Next problem: 6 - \u003chttp://www.mathworks.com/matlabcentral/cody/problems/8053-stress-strain-properties-6 absorbed strain energy\u003e.","description_html":"\u003cp\u003eSimilar to the previous problem, materials may be characterized by their stiffness-to-weight ratio, which is the elastic modulus divided by density. Write a function to calculate this ratio for a material provided its elastic modulus and density.\u003c/p\u003e\u003cp\u003ePrevious problem: 4 - \u003ca href = \"http://www.mathworks.com/matlabcentral/cody/problems/8051-stress-strain-properties-4\"\u003estrength-to-weight ratio\u003c/a\u003e. Next problem: 6 - \u003ca href = \"http://www.mathworks.com/matlabcentral/cody/problems/8053-stress-strain-properties-6\"\u003eabsorbed strain energy\u003c/a\u003e.\u003c/p\u003e","function_template":"function [EtWR] = stress_strain5(E,density)\r\n\r\nEtWR = 1\r\n\r\nend\r\n","test_suite":"%% Note\r\n% The following properties are measured at room temperature and are tensile\r\n% in a single direction. Some materials, such as metals are generally\r\n% isotropic, whereas others, like composite are highly anisotropic\r\n% (different properties in different directions). Also, property values can\r\n% range depending on the material grade. Finally, thermal or environmental\r\n% changes can alter these properties, sometimes drastically.\r\n\r\n%% steel alloy (ASTM A36)\r\nS_y = 250e6; %Pa\r\nS_u = 400e6; %Pa\r\ne_y = 0.00125;\r\ne_u = 0.35;\r\nnu = 0.26;\r\nG = 79.3e9; %Pa\r\nE = 200e9; %Pa\r\ndensity = 7.85; %g/cm^3\r\nsh_exp = 0.14; %strain-hardening exponent\r\nsh_coeff = 0.463; %strain-hardening coefficient\r\nEtWR_corr = 2.548e10;\r\nassert(abs(stress_strain5(E,density)-EtWR_corr)/EtWR_corr\u003c1e-2)\r\n\r\n%% titanium (Ti-6Al-4V)\r\nS_y = 830e6; %Pa\r\nS_u = 900e6; %Pa\r\ne_y = 0.00728;\r\ne_u = 0.14;\r\nnu = 0.342;\r\nG = 44e9; %Pa\r\nE = 114e9; %Pa\r\ndensity = 4.51; %g/cm^3\r\nsh_exp = 0.04; %strain-hardening exponent\r\nsh_coeff = 0.974; %strain-hardening coefficient\r\nEtWR_corr = 2.528e10;\r\nassert(abs(stress_strain5(E,density)-EtWR_corr)/EtWR_corr\u003c1e-2)\r\n\r\n%% Inconel 718\r\nS_y = 1172e6; %Pa\r\nS_u = 1407e6; %Pa\r\ne_y = 0.00563;\r\ne_u = 0.027;\r\nnu = 0.29;\r\nG = 11.6e9; %Pa\r\nE = 208e9; %Pa\r\ndensity = 8.19; %g/cm^3\r\nsh_exp = 0.075; %strain-hardening exponent\r\nsh_coeff = 1.845; %strain-hardening coefficient\r\nEtWR_corr = 2.540e10;\r\nassert(abs(stress_strain5(E,density)-EtWR_corr)/EtWR_corr\u003c1e-2)\r\n\r\n%% aluminum alloy (6061-T6)%^\u0026\r\nS_y = 241e6; %Pa\r\nS_u = 300e6; %Pa\r\ne_y = 0.0035;\r\ne_u = 0.15;\r\nnu = 0.33;\r\nG = 26e9; %Pa\r\nE = 68.9e9; %Pa\r\ndensity = 2.7; %g/cm^3\r\nsh_exp = 0.042; %strain-hardening exponent\r\nsh_coeff = 0.325; %strain-hardening coefficient\r\nEtWR_corr = 2.552e10;\r\nassert(abs(stress_strain5(E,density)-EtWR_corr)/EtWR_corr\u003c1e-2)\r\n\r\n%% copper\r\nS_y = 70e6; %Pa\r\nS_u = 220e6; %Pa\r\ne_y = 0.00054;\r\ne_u = 0.48;\r\nnu = 0.34;\r\nG = 48e9; %Pa\r\nE = 130e9; %Pa\r\ndensity = 8.92; %g/cm^3\r\nsh_exp = 0.44; %strain-hardening exponent\r\nsh_coeff = 0.304; %strain-hardening coefficient\r\nEtWR_corr = 1.457e10;\r\nassert(abs(stress_strain5(E,density)-EtWR_corr)/EtWR_corr\u003c1e-2)\r\n\r\n%% rhenium\r\nS_y = 317e6; %Pa\r\nS_u = 1130e6; %Pa\r\ne_y = 0.000685;\r\ne_u = 0.24;\r\nnu = 0.3;\r\nG = 178e9; %Pa\r\nE = 463e9; %Pa\r\ndensity = 21.02; %g/cm^3\r\nsh_exp = 0.353; %strain-hardening exponent\r\nsh_coeff = 1.870; %strain-hardening coefficient\r\nEtWR_corr = 2.203e10;\r\nassert(abs(stress_strain5(E,density)-EtWR_corr)/EtWR_corr\u003c1e-2)\r\n\r\n%% polymer (nylon, 6/6)\r\nS_y = 82e6; %Pa\r\nS_u = 82e6; %Pa\r\ne_y = 0.0265;\r\ne_u = 0.45;\r\nnu = 0.41;\r\nG = 2.8e9; %Pa\r\nE = 3.1e9; %Pa\r\ndensity = 1.14; %g/cm^3\r\nEtWR_corr = 0.272e10;\r\nassert(abs(stress_strain5(E,density)-EtWR_corr)/EtWR_corr\u003c1e-2)\r\n\r\n%% polymer (nylon, 6/6) reinforced with 45wt.% glass fiber\r\nS_y = 230e6; %Pa\r\nS_u = 230e6; %Pa\r\ne_y = 0.016;\r\ne_u = 0.016;\r\nnu = 0.35;\r\nG = 13.0e9; %Pa\r\nE = 14.5e9; %Pa\r\ndensity = 1.51; %g/cm^3\r\nEtWR_corr = 0.960e10;\r\nassert(abs(stress_strain5(E,density)-EtWR_corr)/EtWR_corr\u003c1e-2)\r\n\r\n%% diamond\r\nS_y = 1200e6; %Pa\r\nS_u = 1200e6; %Pa\r\ne_y = 0.001;\r\ne_u = 0.001;\r\nnu = 0.20;\r\nG = 478e9; %Pa\r\nE = 1200e9; %Pa\r\ndensity = 3.51; %g/cm^3\r\nEtWR_corr = 34.19e10;\r\nassert(abs(stress_strain5(E,density)-EtWR_corr)/EtWR_corr\u003c1e-2)\r\n\r\n%%\r\nind = randi(4);\r\nswitch ind\r\n\tcase 1\r\n\t\tE = 114e9; %Pa\r\n\t\tdensity = 4.51; %g/cm^3\r\n\t\tEtWR_corr = 2.528e10;\r\n\tcase 2\r\n\t\tE = 68.9e9; %Pa\r\n\t\tdensity = 2.7; %g/cm^3\r\n\t\tEtWR_corr = 2.552e10;\r\n\tcase 3\r\n\t\tE = 200e9; %Pa\r\n\t\tdensity = 7.85; %g/cm^3\r\n\t\tEtWR_corr = 2.548e10;\r\n\tcase 4\r\n\t\tE = 1200e9; %Pa\r\n\t\tdensity = 3.51; %g/cm^3\r\n\t\tEtWR_corr = 34.19e10;\r\nend\r\nassert(abs(stress_strain5(E,density)-EtWR_corr)/EtWR_corr\u003c1e-2)\r\n\r\n%%\r\nind = randi(4);\r\nswitch ind\r\n\tcase 1\r\n\t\tE = 68.9e9; %Pa\r\n\t\tdensity = 2.7; %g/cm^3\r\n\t\tEtWR_corr = 2.552e10;\r\n\tcase 2\r\n\t\tE = 3.1e9; %Pa\r\n\t\tdensity = 1.14; %g/cm^3\r\n\t\tEtWR_corr = 0.272e10;\r\n\tcase 3\r\n\t\tE = 14.5e9; %Pa\r\n\t\tdensity = 1.51; %g/cm^3\r\n\t\tEtWR_corr = 0.960e10;\r\n\tcase 4\r\n\t\tE = 208e9; %Pa\r\n\t\tdensity = 8.19; %g/cm^3\r\n\t\tEtWR_corr = 2.540e10;\r\nend\r\nassert(abs(stress_strain5(E,density)-EtWR_corr)/EtWR_corr\u003c1e-2)\r\n\r\n%%\r\nind = randi(4);\r\nswitch ind\r\n\tcase 1\r\n\t\tE = 208e9; %Pa\r\n\t\tdensity = 8.19; %g/cm^3\r\n\t\tEtWR_corr = 2.540e10;\r\n\tcase 2\r\n\t\tE = 463e9; %Pa\r\n\t\tdensity = 21.02; %g/cm^3\r\n\t\tEtWR_corr = 2.203e10;\r\n\tcase 3\r\n\t\tE = 130e9; %Pa\r\n\t\tdensity = 8.92; %g/cm^3\r\n\t\tEtWR_corr = 1.457e10;\r\n\tcase 4\r\n\t\tE = 3.1e9; %Pa\r\n\t\tdensity = 1.14; %g/cm^3\r\n\t\tEtWR_corr = 0.272e10;\r\nend\r\nassert(abs(stress_strain5(E,density)-EtWR_corr)/EtWR_corr\u003c1e-2)\r\n","published":true,"deleted":false,"likes_count":2,"comments_count":0,"created_by":26769,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":212,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2015-03-30T19:40:12.000Z","updated_at":"2026-03-10T20:42:38.000Z","published_at":"2015-03-30T19:40:12.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"targetMode\":\"\",\"relationshipId\":\"rId1\",\"target\":\"/matlab/document.xml\"},{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/output\",\"targetMode\":\"\",\"relationshipId\":\"rId2\",\"target\":\"/matlab/output.xml\"}],\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"relationship\":[],\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\\n\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eSimilar to the previous problem, materials may be characterized by their stiffness-to-weight ratio, which is the elastic modulus divided by density. Write a function to calculate this ratio for a material provided its elastic modulus and density.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003ePrevious problem: 4 -\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"http://www.mathworks.com/matlabcentral/cody/problems/8051-stress-strain-properties-4\\\"\u003e\u003cw:r\u003e\u003cw:t\u003estrength-to-weight ratio\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e. Next problem: 6 -\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"http://www.mathworks.com/matlabcentral/cody/problems/8053-stress-strain-properties-6\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eabsorbed strain energy\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\"},{\"partUri\":\"/matlab/output.xml\",\"contentType\":\"text/xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\" standalone=\\\"no\\\" ?\u003e\u003cembeddedOutputs\u003e\u003cmetaData\u003e\u003cevaluationState\u003emanual\u003c/evaluationState\u003e\u003clayoutState\u003ecode\u003c/layoutState\u003e\u003coutputStatus\u003eready\u003c/outputStatus\u003e\u003c/metaData\u003e\u003coutputArray type=\\\"array\\\"/\u003e\u003cregionArray type=\\\"array\\\"/\u003e\u003c/embeddedOutputs\u003e\"}]}"}],"problem_search":{"errors":[],"problems":[{"id":58144,"title":"Given the mass and stiffness of an undamped SDOF system, find the natural frequency and the natural period of vibration","description":"Problem Statement\r\nGiven the mass  and stiffness  of an undamped SDOF system, find the system's natural frequency in both Hz and radians per second as well as the natural period of vibration. \r\nBackground\r\nAn undamped, linear single degree-of-freedom (SDOF) system can be visualized as a spring with stiffness , fixed at one end and having a mass  attached to the other:\r\n\r\nOnce set into motion, the system will oscillate at its natural frequency, or fundamental frequency: \r\n\r\nwhere the natural frequency  is in radians per second. \r\nThe time required for the undamped system to complete one cycle of free vibration is the natural period of vibration of the system, , which has units of seconds:\r\n\r\nA system executes  cycles in 1 second. This is the natural cyclic frequency of vibration:\r\n\r\nwhich has units of Hertz (Hz) or cycles per second. The term natural frequency of vibration applies to both  and . They have the relationship:\r\n\r\nExample\r\nm = 10;\r\nk = 2;\r\n[wn,fn,Tn] = udSDOF(m,k)\r\nwn = 5\r\nfn = 0.7958\r\nTn = 1.2566\r\n\r\n","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.44px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: none solid rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 878.625px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 439.312px; transform-origin: 407px 439.312px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 20px; border-block-end-color: rgb(60, 60, 60); border-block-start-color: rgb(60, 60, 60); border-bottom-color: rgb(60, 60, 60); border-inline-end-color: rgb(60, 60, 60); border-inline-start-color: rgb(60, 60, 60); border-left-color: rgb(60, 60, 60); border-right-color: rgb(60, 60, 60); border-top-color: rgb(60, 60, 60); caret-color: rgb(60, 60, 60); color: rgb(60, 60, 60); column-rule-color: rgb(60, 60, 60); font-family: Helvetica, Arial, sans-serif; font-size: 20px; font-weight: 700; line-height: 20px; margin-block-end: 5px; margin-block-start: 3px; margin-bottom: 5px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 3px; outline-color: rgb(60, 60, 60); perspective-origin: 384px 10px; text-align: left; text-decoration: none; text-decoration-color: rgb(60, 60, 60); text-emphasis-color: rgb(60, 60, 60); transform-origin: 384px 10px; white-space: pre-wrap; margin-left: 4px; margin-top: 3px; margin-bottom: 5px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eProblem Statement\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 21px; text-align: left; transform-origin: 384px 21px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eGiven the mass \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003em\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e and stiffness \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ek\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e of an undamped SDOF system, find the system's natural frequency in both Hz and radians per second as well as the natural period of vibration. \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 20px; border-block-end-color: rgb(60, 60, 60); border-block-start-color: rgb(60, 60, 60); border-bottom-color: rgb(60, 60, 60); border-inline-end-color: rgb(60, 60, 60); border-inline-start-color: rgb(60, 60, 60); border-left-color: rgb(60, 60, 60); border-right-color: rgb(60, 60, 60); border-top-color: rgb(60, 60, 60); caret-color: rgb(60, 60, 60); color: rgb(60, 60, 60); column-rule-color: rgb(60, 60, 60); font-family: Helvetica, Arial, sans-serif; font-size: 20px; font-weight: 700; line-height: 20px; margin-block-end: 5px; margin-block-start: 20px; margin-bottom: 5px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 20px; outline-color: rgb(60, 60, 60); perspective-origin: 384px 10px; text-align: left; text-decoration: none; text-decoration-color: rgb(60, 60, 60); text-emphasis-color: rgb(60, 60, 60); transform-origin: 384px 10px; white-space: pre-wrap; margin-left: 4px; margin-top: 20px; margin-bottom: 5px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eBackground\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 21px; text-align: left; transform-origin: 384px 21px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eAn undamped, linear single degree-of-freedom (SDOF) system can be visualized as a spring with stiffness \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ek\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e, fixed at one end and having a mass \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003em\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e attached to the other:\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 99px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 49.5px; text-align: left; transform-origin: 384px 49.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cimg class=\"imageNode\" width=\"90\" height=\"93\" style=\"vertical-align: baseline;width: 90px;height: 93px\" src=\"data:image/jpeg;base64,/9j/4AAQSkZJRgABAQEAAAAAAAD/4QAuRXhpZgAATU0AKgAAAAgAAkAAAAMAAAABAAAAAEABAAEAAAABAAAAAAAAAAD/2wBDAAoHBwkHBgoJCAkLCwoMDxkQDw4ODx4WFxIZJCAmJSMgIyIoLTkwKCo2KyIjMkQyNjs9QEBAJjBGS0U+Sjk/QD3/2wBDAQsLCw8NDx0QEB09KSMpPT09PT09PT09PT09PT09PT09PT09PT09PT09PT09PT09PT09PT09PT09PT09PT09PT3/wAARCABdAFoDASIAAhEBAxEB/8QAHwAAAQUBAQEBAQEAAAAAAAAAAAECAwQFBgcICQoL/8QAtRAAAgEDAwIEAwUFBAQAAAF9AQIDAAQRBRIhMUEGE1FhByJxFDKBkaEII0KxwRVS0fAkM2JyggkKFhcYGRolJicoKSo0NTY3ODk6Q0RFRkdISUpTVFVWV1hZWmNkZWZnaGlqc3R1dnd4eXqDhIWGh4iJipKTlJWWl5iZmqKjpKWmp6ipqrKztLW2t7i5usLDxMXGx8jJytLT1NXW19jZ2uHi4+Tl5ufo6erx8vP09fb3+Pn6/8QAHwEAAwEBAQEBAQEBAQAAAAAAAAECAwQFBgcICQoL/8QAtREAAgECBAQDBAcFBAQAAQJ3AAECAxEEBSExBhJBUQdhcRMiMoEIFEKRobHBCSMzUvAVYnLRChYkNOEl8RcYGRomJygpKjU2Nzg5OkNERUZHSElKU1RVVldYWVpjZGVmZ2hpanN0dXZ3eHl6goOEhYaHiImKkpOUlZaXmJmaoqOkpaanqKmqsrO0tba3uLm6wsPExcbHyMnK0tPU1dbX2Nna4uPk5ebn6Onq8vP09fb3+Pn6/9oADAMBAAIRAxEAPwD2WiiigAqB722jcpJcQqw6q0gBFTnoa838JeEND8TaGdS1C2NxPLdXBM4kILjzGA/DAAoA9GjkWVA6MrKehU5FKWA6kCuWHw28OqAFtZlA6ATuMfrSn4ceHj1tpj9Z3/xoA6jcCSARkdRmlrlf+Fb+Hsk/ZpsnqfPf/Gl/4Vv4f/595/8AwIf/ABoA6miuE8Q+AdFtPDepXNtHcRTQW0ksbid8hlUkd/UV2dgpTT7cMxZhEuSTknigCxRRRQAUUUUAB6GuO+GU7SeHrqIhQsN9OqYHYuT/ADJrsT0P0rjPhSzS+CIriRcSz3E7vz1PmN27dOlAHZ1UvNUsbB0S8vLe3eT7qyyhS30z1q3XAeOfDdzrerXEkeni6UaRNHAxAOJi6bcZ6HGefrQB29rfWt8jPaXEU6KxVmicMAR1HHerFcn4X0F9F8RaqIbM2tg0cCwBCNjEIAxx654yeuK6ygDK8Vf8ijrP/XjP/wCi2q7Ybzp9v5gUN5a5AOR0ql4q/wCRR1n/AK8Z/wD0W1XdPULptsoYuBEoDE5J4oAsUUUUAFFFFAAehrlvh06nwsVUjK3dyCB2/etXU1xlj4L1jSGuk0vxGtvbzzvOImsQ+0sc4zv5oA7OiucbTvFUduVj13T5JFHytJYEbj74k4/AVjXMvxKtpFxHoVzHnkwI+4D6OyjP4/jQB3lFeczeMr+1mFvrOrro0zH5ftWkMA2PRhKynt371u2o8Q3umxXGmeINKvI5BlZWsWAYfVZOv4UAaniv/kUdZ/68Z/8A0W1WNFga20WyhbG5IUU4Oe1c7qGk+L9Rsbmym1LRxBcRtEzLayBgrAg4+frg109jbGzsYLYyNIYowm9urYGM0AWKKKKACiiigAJABJ4AorB8cF18EawyKWZbWQjBxjjr+HX8Kg8D+JZPFGmXF48KwIsxjjiJy6qAMb/c9foRQBq6zaaheWezStR+wTg58zyVk49MGuPk0LxjA8gvL5dYt3BLCK5ezf6Dbkc/hXU+IrXWbzTxHoN7b2dyW+Z5oyw2+2Oh964+PTNft7dl8SW+qampJ3NZXY2Hg9Ixhv8A9dAENzeeFNIkZ/EnhdrWViP3tyqXAOOmGJz+ld5oWo6bqmkxT6NJE9n91PLGAuO2O2K5q2l8EWrgzwW1lMzDCXylHz9H711unyWctosmnGE27cgw42n8qALNFFFABRRRQAUUyWWOGMyTOsaDqzHAFRLqFo6hlu4GUjgiQGgBmq6bBrGmXFhdbvJuEKPsODg+9Q6botnpUs0lpGUaZUWTLddqhR+OAKtfbrX/AJ+Yf+/go+3Wv/PzD/38FAFTXY9Vl0qVNEmtob0/ce4Usg/L/wCv9K4lbHxgTIPEQvryBTwNJuYoxjB7MFY9ujfhXY6zcXMumuui6hZQXnVWmw6/ln9a4mSTxFbIG1++vLyLo8mlXcce0A5zsCq2T06mgCxC3gOK5QanbS292x4GrLN1HfMmV/Ku706Kyhs0Gmx26WrfMgt1AQ57jHFcDY3Hw71O9CXE0VxdhjldSkeRs98l8iu6s5NNs7OKGzktYrZBiNY2UKB7YoAvUVB9utf+fmH/AL+Cj7da/wDPzD/38FAE9FNR1kUMjBlPQg5FOoA53xpqunadohi1G2+2tdHyobMDLTv2A/xrEsfAvh3RtEmvtcsoAxBmlXc2yD/YUZ6Dp9a7d4IZXSWSGN5IidjMoJX1we1SSRpKhWRFdT2YZFAHAeH/AAPpOr3L6vc6SttZSpttLNi2SvXzH56nsOwqGLwhofiTVJIrDS/s2l2km2S5jLK1xIOqIc/cHc9+gr0ekVFRQqgKB2AwKAOG13w74Z0oQ21toMd3qNyStvbqWy2OrE54UdSalsPh94e0nRjLq1rDNIimW4ldm2p3IXnhR0HsK7MxoZBIUXeBgNjkD60OiupVgGUjBBGQRQB5vongPQPEt2ustoyWumgEW8ByDcc/6x++D2H407XfDWgS348O6Jo8I1GRN0ko3BbSM8b+vX0Hc16RUS28MczzJFGssmN7hQC2OmT3xQBw+oeE/CfhTQInvbD7XKCIYy5ZpLiU9BwepP5CmaP8KNKS2abVleW6nbe0cUjLHFn+BR6DpzXePDHKVMiKxRty7hnB9R6Hk1JQBS0nSrbRbBLKyVlgjztDMWI/OrtFFAH/2Q==\" data-image-state=\"image-loaded\"\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eOnce set into motion, the system will oscillate at its \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003enatural frequency\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e, or fundamental frequency: \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 41px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 20.5px; text-align: left; transform-origin: 384px 20.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"vertical-align:-16px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"62.5\" height=\"41\" style=\"width: 62.5px; height: 41px;\"\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21.5px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.75px; text-align: left; transform-origin: 384px 10.75px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003ewhere the natural frequency \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"18.5\" height=\"20\" style=\"width: 18.5px; height: 20px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e is in radians per second. \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 42.5px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 21.25px; text-align: left; transform-origin: 384px 21.25px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eThe time required for the undamped system to complete one cycle of free vibration is the \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003enatural period\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e of vibration of the system, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"16.5\" height=\"20\" style=\"width: 16.5px; height: 20px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e, which has units of seconds:\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 37px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 18.5px; text-align: left; transform-origin: 384px 18.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"vertical-align:-17px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"51.5\" height=\"37\" style=\"width: 51.5px; height: 37px;\"\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21.5px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.75px; text-align: left; transform-origin: 384px 10.75px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eA system executes \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"31.5\" height=\"20\" style=\"width: 31.5px; height: 20px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e cycles in 1 second. This is the \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003enatural cyclic frequency\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e of vibration:\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 37px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 18.5px; text-align: left; transform-origin: 384px 18.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"vertical-align:-17px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"44\" height=\"37\" style=\"width: 44px; height: 37px;\"\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 42.5px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 21.25px; text-align: left; transform-origin: 384px 21.25px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003ewhich has units of Hertz (Hz) or cycles per second. The term natural frequency of vibration applies to both \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"18.5\" height=\"20\" style=\"width: 18.5px; height: 20px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"12\" height=\"20\" style=\"width: 12px; height: 20px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e. They have the relationship:\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 38px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 19px; text-align: left; transform-origin: 384px 19px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"vertical-align:-15px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"47.5\" height=\"38\" style=\"width: 47.5px; height: 38px;\"\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 20px; border-block-end-color: rgb(60, 60, 60); border-block-start-color: rgb(60, 60, 60); border-bottom-color: rgb(60, 60, 60); border-inline-end-color: rgb(60, 60, 60); border-inline-start-color: rgb(60, 60, 60); border-left-color: rgb(60, 60, 60); border-right-color: rgb(60, 60, 60); border-top-color: rgb(60, 60, 60); caret-color: rgb(60, 60, 60); color: rgb(60, 60, 60); column-rule-color: rgb(60, 60, 60); font-family: Helvetica, Arial, sans-serif; font-size: 20px; font-weight: 700; line-height: 20px; margin-block-end: 5px; margin-block-start: 20px; margin-bottom: 5px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 20px; outline-color: rgb(60, 60, 60); perspective-origin: 384px 10px; text-align: left; text-decoration: none; text-decoration-color: rgb(60, 60, 60); text-emphasis-color: rgb(60, 60, 60); transform-origin: 384px 10px; white-space: pre-wrap; margin-left: 4px; margin-top: 20px; margin-bottom: 5px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eExample\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgb(247, 247, 247); block-size: 122.625px; border-bottom-left-radius: 4px; border-bottom-right-radius: 4px; border-end-end-radius: 4px; border-end-start-radius: 4px; border-start-end-radius: 4px; border-start-start-radius: 4px; border-top-left-radius: 4px; border-top-right-radius: 4px; margin-block-end: 10px; margin-block-start: 10px; margin-bottom: 10px; margin-inline-end: 3px; margin-inline-start: 3px; margin-left: 3px; margin-right: 3px; margin-top: 10px; perspective-origin: 404px 61.3125px; transform-origin: 404px 61.3125px; margin-left: 3px; margin-top: 10px; margin-bottom: 10px; margin-right: 3px; \"\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4375px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10.2188px; transform-origin: 404px 10.2188px; white-space: nowrap; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; white-space: pre; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003em = 10;\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4375px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10.2188px; transform-origin: 404px 10.2188px; white-space: nowrap; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; white-space: pre; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003ek = 2;\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4375px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10.2188px; transform-origin: 404px 10.2188px; white-space: nowrap; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; white-space: pre; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e[wn,fn,Tn] = udSDOF(m,k)\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4375px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10.2188px; transform-origin: 404px 10.2188px; white-space: nowrap; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; white-space: pre; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003ewn = 5\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4375px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10.2188px; transform-origin: 404px 10.2188px; white-space: nowrap; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; white-space: pre; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003efn = 0.7958\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4375px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10.2188px; transform-origin: 404px 10.2188px; white-space: nowrap; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; white-space: pre; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003eTn = 1.2566\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 10px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 10px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 10px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function [wn,fn,Tn] = udSDOF(m,k)\r\n  % put yo answer here ^(o.O)^\r\nend","test_suite":"%%\r\nm = 2;\r\nk = 50;\r\ny_correct = [5 0.7958 1.2566];\r\n[wn,fn,Tn] = udSDOF(m,k);\r\nassert(sum(round([wn,fn,Tn]./y_correct,2))==3)\r\n\r\n%%\r\nm = 0.2;\r\nk = 2000;\r\ny_correct = [100 15.9 0.0628];\r\n[wn,fn,Tn] = udSDOF(m,k);\r\nassert(sum(round([wn,fn,Tn]./y_correct,2))==3)\r\n\r\n%%\r\nm = 0.2533; \r\nk = 10; \r\ny_correct = [6.2832 1 1];\r\n[wn,fn,Tn] = udSDOF(m,k);\r\nassert(sum(round([wn,fn,Tn]./y_correct,2))==3)\r\n\r\n%%\r\nfiletext = fileread('udSDOF.m');\r\nnot_allowed = contains(filetext, 'regex');\r\nassert(~not_allowed)\r\n","published":true,"deleted":false,"likes_count":1,"comments_count":1,"created_by":91130,"edited_by":91130,"edited_at":"2023-04-28T00:16:14.000Z","deleted_by":null,"deleted_at":null,"solvers_count":58,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2023-04-27T23:21:12.000Z","updated_at":"2026-02-05T13:39:50.000Z","published_at":"2023-04-28T00:16:14.000Z","restored_at":null,"restored_by":null,"spam":null,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"heading\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eProblem Statement\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eGiven the mass \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003em\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and stiffness \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ek\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e of an undamped SDOF system, find the system's natural frequency in both Hz and radians per second as well as the natural period of vibration. \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"heading\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eBackground\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eAn undamped, linear single degree-of-freedom (SDOF) system can be visualized as a spring with stiffness \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ek\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, fixed at one end and having a mass \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003em\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e attached to the other:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"image\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"height\\\" w:val=\\\"93\\\"/\u003e\u003cw:attr w:name=\\\"width\\\" w:val=\\\"90\\\"/\u003e\u003cw:attr w:name=\\\"verticalAlign\\\" w:val=\\\"baseline\\\"/\u003e\u003cw:attr w:name=\\\"altText\\\" w:val=\\\"\\\"/\u003e\u003cw:attr w:name=\\\"relationshipId\\\" w:val=\\\"rId1\\\"/\u003e\u003c/w:customXmlPr\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eOnce set into motion, the system will oscillate at its \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003enatural frequency\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e, or fundamental frequency: \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"true\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e$\\\\omega_n =\\\\sqrt{\\\\frac{k}{m}}$\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003ewhere the natural frequency \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e$\\\\omega_n$\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e is in radians per second. \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe time required for the undamped system to complete one cycle of free vibration is the \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003enatural period\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e of vibration of the system, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eT_n\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, which has units of seconds:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"true\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e$T_n =\\\\frac{2\\\\pi }{\\\\omega_n }$\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eA system executes \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e1/T_n\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e cycles in 1 second. This is the \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003enatural cyclic frequency\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e of vibration:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"true\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e$f_n =\\\\frac{1}{T_n }$\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003ewhich has units of Hertz (Hz) or cycles per second. The term natural frequency of vibration applies to both \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e$\\\\omega_n$\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ef_n\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. They have the relationship:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"true\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e$f_n =\\\\frac{\\\\omega_n }{2\\\\pi }$\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"heading\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eExample\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"code\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c![CDATA[m = 10;\\nk = 2;\\n[wn,fn,Tn] = udSDOF(m,k)\\nwn = 5\\nfn = 0.7958\\nTn = 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Properties - 5","description":"Similar to the previous problem, materials may be characterized by their stiffness-to-weight ratio, which is the elastic modulus divided by density. Write a function to calculate this ratio for a material provided its elastic modulus and density.\r\n\r\nPrevious problem: 4 - \u003chttp://www.mathworks.com/matlabcentral/cody/problems/8051-stress-strain-properties-4 strength-to-weight ratio\u003e. Next problem: 6 - \u003chttp://www.mathworks.com/matlabcentral/cody/problems/8053-stress-strain-properties-6 absorbed strain energy\u003e.","description_html":"\u003cp\u003eSimilar to the previous problem, materials may be characterized by their stiffness-to-weight ratio, which is the elastic modulus divided by density. Write a function to calculate this ratio for a material provided its elastic modulus and density.\u003c/p\u003e\u003cp\u003ePrevious problem: 4 - \u003ca href = \"http://www.mathworks.com/matlabcentral/cody/problems/8051-stress-strain-properties-4\"\u003estrength-to-weight ratio\u003c/a\u003e. Next problem: 6 - \u003ca href = \"http://www.mathworks.com/matlabcentral/cody/problems/8053-stress-strain-properties-6\"\u003eabsorbed strain energy\u003c/a\u003e.\u003c/p\u003e","function_template":"function [EtWR] = stress_strain5(E,density)\r\n\r\nEtWR = 1\r\n\r\nend\r\n","test_suite":"%% Note\r\n% The following properties are measured at room temperature and are tensile\r\n% in a single direction. Some materials, such as metals are generally\r\n% isotropic, whereas others, like composite are highly anisotropic\r\n% (different properties in different directions). Also, property values can\r\n% range depending on the material grade. Finally, thermal or environmental\r\n% changes can alter these properties, sometimes drastically.\r\n\r\n%% steel alloy (ASTM A36)\r\nS_y = 250e6; %Pa\r\nS_u = 400e6; %Pa\r\ne_y = 0.00125;\r\ne_u = 0.35;\r\nnu = 0.26;\r\nG = 79.3e9; %Pa\r\nE = 200e9; %Pa\r\ndensity = 7.85; %g/cm^3\r\nsh_exp = 0.14; %strain-hardening exponent\r\nsh_coeff = 0.463; %strain-hardening coefficient\r\nEtWR_corr = 2.548e10;\r\nassert(abs(stress_strain5(E,density)-EtWR_corr)/EtWR_corr\u003c1e-2)\r\n\r\n%% titanium (Ti-6Al-4V)\r\nS_y = 830e6; %Pa\r\nS_u = 900e6; %Pa\r\ne_y = 0.00728;\r\ne_u = 0.14;\r\nnu = 0.342;\r\nG = 44e9; %Pa\r\nE = 114e9; %Pa\r\ndensity = 4.51; %g/cm^3\r\nsh_exp = 0.04; %strain-hardening exponent\r\nsh_coeff = 0.974; %strain-hardening coefficient\r\nEtWR_corr = 2.528e10;\r\nassert(abs(stress_strain5(E,density)-EtWR_corr)/EtWR_corr\u003c1e-2)\r\n\r\n%% Inconel 718\r\nS_y = 1172e6; %Pa\r\nS_u = 1407e6; %Pa\r\ne_y = 0.00563;\r\ne_u = 0.027;\r\nnu = 0.29;\r\nG = 11.6e9; %Pa\r\nE = 208e9; %Pa\r\ndensity = 8.19; %g/cm^3\r\nsh_exp = 0.075; %strain-hardening exponent\r\nsh_coeff = 1.845; %strain-hardening coefficient\r\nEtWR_corr = 2.540e10;\r\nassert(abs(stress_strain5(E,density)-EtWR_corr)/EtWR_corr\u003c1e-2)\r\n\r\n%% aluminum alloy (6061-T6)%^\u0026\r\nS_y = 241e6; %Pa\r\nS_u = 300e6; %Pa\r\ne_y = 0.0035;\r\ne_u = 0.15;\r\nnu = 0.33;\r\nG = 26e9; %Pa\r\nE = 68.9e9; %Pa\r\ndensity = 2.7; %g/cm^3\r\nsh_exp = 0.042; %strain-hardening exponent\r\nsh_coeff = 0.325; %strain-hardening coefficient\r\nEtWR_corr = 2.552e10;\r\nassert(abs(stress_strain5(E,density)-EtWR_corr)/EtWR_corr\u003c1e-2)\r\n\r\n%% copper\r\nS_y = 70e6; %Pa\r\nS_u = 220e6; %Pa\r\ne_y = 0.00054;\r\ne_u = 0.48;\r\nnu = 0.34;\r\nG = 48e9; %Pa\r\nE = 130e9; %Pa\r\ndensity = 8.92; %g/cm^3\r\nsh_exp = 0.44; %strain-hardening exponent\r\nsh_coeff = 0.304; %strain-hardening coefficient\r\nEtWR_corr = 1.457e10;\r\nassert(abs(stress_strain5(E,density)-EtWR_corr)/EtWR_corr\u003c1e-2)\r\n\r\n%% rhenium\r\nS_y = 317e6; %Pa\r\nS_u = 1130e6; %Pa\r\ne_y = 0.000685;\r\ne_u = 0.24;\r\nnu = 0.3;\r\nG = 178e9; %Pa\r\nE = 463e9; %Pa\r\ndensity = 21.02; %g/cm^3\r\nsh_exp = 0.353; %strain-hardening exponent\r\nsh_coeff = 1.870; %strain-hardening coefficient\r\nEtWR_corr = 2.203e10;\r\nassert(abs(stress_strain5(E,density)-EtWR_corr)/EtWR_corr\u003c1e-2)\r\n\r\n%% polymer (nylon, 6/6)\r\nS_y = 82e6; %Pa\r\nS_u = 82e6; %Pa\r\ne_y = 0.0265;\r\ne_u = 0.45;\r\nnu = 0.41;\r\nG = 2.8e9; %Pa\r\nE = 3.1e9; %Pa\r\ndensity = 1.14; %g/cm^3\r\nEtWR_corr = 0.272e10;\r\nassert(abs(stress_strain5(E,density)-EtWR_corr)/EtWR_corr\u003c1e-2)\r\n\r\n%% polymer (nylon, 6/6) reinforced with 45wt.% glass fiber\r\nS_y = 230e6; %Pa\r\nS_u = 230e6; %Pa\r\ne_y = 0.016;\r\ne_u = 0.016;\r\nnu = 0.35;\r\nG = 13.0e9; %Pa\r\nE = 14.5e9; %Pa\r\ndensity = 1.51; %g/cm^3\r\nEtWR_corr = 0.960e10;\r\nassert(abs(stress_strain5(E,density)-EtWR_corr)/EtWR_corr\u003c1e-2)\r\n\r\n%% diamond\r\nS_y = 1200e6; %Pa\r\nS_u = 1200e6; %Pa\r\ne_y = 0.001;\r\ne_u = 0.001;\r\nnu = 0.20;\r\nG = 478e9; %Pa\r\nE = 1200e9; %Pa\r\ndensity = 3.51; %g/cm^3\r\nEtWR_corr = 34.19e10;\r\nassert(abs(stress_strain5(E,density)-EtWR_corr)/EtWR_corr\u003c1e-2)\r\n\r\n%%\r\nind = randi(4);\r\nswitch ind\r\n\tcase 1\r\n\t\tE = 114e9; %Pa\r\n\t\tdensity = 4.51; %g/cm^3\r\n\t\tEtWR_corr = 2.528e10;\r\n\tcase 2\r\n\t\tE = 68.9e9; %Pa\r\n\t\tdensity = 2.7; %g/cm^3\r\n\t\tEtWR_corr = 2.552e10;\r\n\tcase 3\r\n\t\tE = 200e9; %Pa\r\n\t\tdensity = 7.85; %g/cm^3\r\n\t\tEtWR_corr = 2.548e10;\r\n\tcase 4\r\n\t\tE = 1200e9; %Pa\r\n\t\tdensity = 3.51; %g/cm^3\r\n\t\tEtWR_corr = 34.19e10;\r\nend\r\nassert(abs(stress_strain5(E,density)-EtWR_corr)/EtWR_corr\u003c1e-2)\r\n\r\n%%\r\nind = randi(4);\r\nswitch ind\r\n\tcase 1\r\n\t\tE = 68.9e9; %Pa\r\n\t\tdensity = 2.7; %g/cm^3\r\n\t\tEtWR_corr = 2.552e10;\r\n\tcase 2\r\n\t\tE = 3.1e9; %Pa\r\n\t\tdensity = 1.14; %g/cm^3\r\n\t\tEtWR_corr = 0.272e10;\r\n\tcase 3\r\n\t\tE = 14.5e9; %Pa\r\n\t\tdensity = 1.51; %g/cm^3\r\n\t\tEtWR_corr = 0.960e10;\r\n\tcase 4\r\n\t\tE = 208e9; %Pa\r\n\t\tdensity = 8.19; %g/cm^3\r\n\t\tEtWR_corr = 2.540e10;\r\nend\r\nassert(abs(stress_strain5(E,density)-EtWR_corr)/EtWR_corr\u003c1e-2)\r\n\r\n%%\r\nind = randi(4);\r\nswitch ind\r\n\tcase 1\r\n\t\tE = 208e9; %Pa\r\n\t\tdensity = 8.19; %g/cm^3\r\n\t\tEtWR_corr = 2.540e10;\r\n\tcase 2\r\n\t\tE = 463e9; %Pa\r\n\t\tdensity = 21.02; %g/cm^3\r\n\t\tEtWR_corr = 2.203e10;\r\n\tcase 3\r\n\t\tE = 130e9; %Pa\r\n\t\tdensity = 8.92; %g/cm^3\r\n\t\tEtWR_corr = 1.457e10;\r\n\tcase 4\r\n\t\tE = 3.1e9; %Pa\r\n\t\tdensity = 1.14; %g/cm^3\r\n\t\tEtWR_corr = 0.272e10;\r\nend\r\nassert(abs(stress_strain5(E,density)-EtWR_corr)/EtWR_corr\u003c1e-2)\r\n","published":true,"deleted":false,"likes_count":2,"comments_count":0,"created_by":26769,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":212,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2015-03-30T19:40:12.000Z","updated_at":"2026-03-10T20:42:38.000Z","published_at":"2015-03-30T19:40:12.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"targetMode\":\"\",\"relationshipId\":\"rId1\",\"target\":\"/matlab/document.xml\"},{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/output\",\"targetMode\":\"\",\"relationshipId\":\"rId2\",\"target\":\"/matlab/output.xml\"}],\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"relationship\":[],\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\\n\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eSimilar to the previous problem, materials may be characterized by their stiffness-to-weight ratio, which is the elastic modulus divided by density. 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