Uses two bouncing ball Simulink® models to show different approaches to modeling hybrid dynamic systems with Zeno behavior. The example also provides the benefits of each approach.

Use Simulink® to model a hydraulic cylinder. You can apply these concepts to applications where you need to model hydraulic behavior. See two related examples that use the same basic components: four cylinder model and two cylinder model with load constraints.

Use Simulink® to create the thermal model of a house. This system models the outdoor environment, the thermal characteristics of the house, and the house heating system.

Approximate nonlinear relationships of a type S thermocouple.

Some of the main steps needed to design and evaluate a sine wave data table for use in digital waveform synthesis applications in embedded systems and arbitrary waveform generation instruments.

How zero crossings work in Simulink®. In this model, three shifted sine waves are fed into an absolute value block and saturation block. At exactly t = 5, the output of the switch block changes from the absolute value to the saturation block. Zero crossings in Simulink will automatically detect exactly when the switch block changes its output, and the solver will step to the exact time that the event happens. This can be seen by examining the output in the scope.

This model was inspired by the classic paper "Galactic Bridges and Tails" (Toomre & Toomre 1972). The original paper explained how disc shaped galaxies could develop spiral arms. Two disc shape galaxies originally are far apart. They then fly by each other and almost collide. Once the galaxies are close enough, mutual gravitational forces cause spiral arms to form.

The contrast between enabled subsystems and triggered subsystems for the same control signal, through the use of counter circuits. After running the simulation, the scope shows three plots.

Model friction one way in Simulink®. The two integrators in the model calculate the velocity and position of the system, which is then used in the Friction Model to calculate the friction force.

Handle state events. Run the simulation and see the phase plane plot, where the state x1 is along the X-axis and the state x2 is along the Y-axis.

Use Stateflow® to model a bang-bang temperature control system for a boiler. The boiler dynamics are modeled in Simulink®.

Model an inverted pendulum. The animation is created using MATLAB® Handle Graphics®. The animation block is a masked S-function. For more information, use the context menu to look under the Animation block's mask and open the S-function for editing.

Model a double spring-mass-damper system with a periodically varying forcing function. The model uses an S-Function block to animate the mass system during simulation. In the system, the only sensor is attached to the mass on the left, and the actuator is attached to the mass on the left. The example uses state estimation and linear-quadratic regulator (LQR) control.

Model the dynamics of liquid in a tank. The associated animation provides a graphical display of the tank as it empties and refills, based on user-defined tank parameters. The tank empties at the start of the simulation and again part way through the simulation. When the simulation is stopped, a plot is generated showing the liquid height and the states of the two valves.

Two cases where you can use Simulink® to model variable transport delay phenomena.

Model a Foucault pendulum. The Foucault pendulum was the brainchild of the French physicist Leon Foucault. It was intended to prove that Earth rotates around its axis. The oscillation plane of a Foucault pendulum rotates throughout the day as a result of axial rotation of the Earth. The plane of oscillation completes a whole circle in a time interval T, which depends on the geographical latitude.

Solve the differential equations for the Foucault Pendulum problem and displays the pendulum bob movement in the VRML scene. You can modify the Pendulum location by changing the Latitude / Longitude constant values in the model and other parameters (g, Omega, L and initial conditions) in MATLAB® workspace.

The behavior of variable-step solvers in a Foucault pendulum model. Simulink® solvers ode45, ode15s, ode23, and ode23t are used as test cases. Stiff differential equations are used to solve this problem. There is no exact definition of stiffness for equations. Some numerical methods are unstable when used to solve stiff equations and very small step sizes are required to obtain a numerically stable solution to a stiff problem. A stiff problem may have a fast changing component and a slow changing component.

The example shows how to use Simulink® to explore the solver Jacobian sparsity pattern, and the connection between the solver Jacobian sparsity pattern and the dependency between components of a physical system. A Simulink model that models the synchronization of three metronomes placed on a free moving base are used.

Choose the correct zero-crossing location algorithm, based on the system dynamics. For Zeno dynamic systems, or systems with strong chattering, you can select the adaptive zero-crossing detection algorithm through the Configure pane:

Use Simulink® to create a model with four hydraulic cylinders. See two related examples that use the same basic components: single cylinder model and model with two cylinders and load constraints.

Model a rigid rod supporting a large mass interconnecting two hydraulic actuators. The model eliminates the springs as it applies the piston forces directly to the load. These forces balance the gravitational force and result in both linear and rotational displacement.

Model transport delay in a variable speed conveyor belt.

Analyze mechanical power of mass-spring damper system.

Model the second-order Van der Pol (VDP) differential equation in Simulink®. In dynamics, the VDP oscillator is non-conservative and has nonlinear damping. At high amplitudes, the oscillator dissipates energy. At low amplitudes, the oscillator generates energy. The oscillator is given by this second-order differential equation: