Getting unrealistic result in Newmakrs Beta method while solving structural dynamic problem

I am finding response of multi degree of freedom system by using Newmarks Beta method. But i am getting some unrealistic value of response.Follwoing is my code.Please help me . Thank you in advance.

u=zeros(24,1); % assume displacemnt

du=zeros(24,1); % assume velocity

M_m=transpose(V_linear)*M*(V_linear); % Modal mass

C_m=transpose(V_linear)*C*(V_linear); % Modal damping

K_m=transpose(V_linear)*K_linear_1*(V_linear); % Modal stiffaness

q=(transpose(V_linear)*M*(V_linear))\transpose(V_linear)*M*u; % Modal displacement ,A\B == inv(A)*B

dq=(transpose(V_linear)*M*(V_linear))\transpose(V_linear)*M*du; % Modal velocity

P(:,1)=transpose(V_linear)*F(:,1); % Modal force

ddq=(M_m)\(P(:,1) - C_m*dq(:,1) - K_m*q(:,1)); % Modal acceleration

gamma=1/2;

beta=1/4;

K_cap=K_linear_1+ ((gamma)/(beta*dt))*C_m+(1/(beta*dt^2))*M_m;

a_n=(1/(beta*dt))*M_m +(gamma)/(beta)*C_m;

b=(1)/(2*beta)*M_m+dt*(gamma/(2*beta)-1)*C_m;

for i=1:(length(time)-1)

P(:,i+1)=transpose(V_linear)*F(:,i+1);% Modal force

P_cap=P(:,i+1)-P(:,i)+(a_n)*dq(:,i)+b*ddq(:,i);

delq=(K_cap)\P_cap; % A\B == inv(A)*B

deldq=(gamma)/(beta*dt)*delq-(gamma/beta)*dq(:,i)+dt*(1-gamma/(2*beta))*ddq(:,i);

delddq=1/(beta*dt^2)*delq -1/(beta*dt)*ddq(:,i)-1/(2*beta)*ddq(:,i);

q(:,i+1)=q(:,i)+delq;

dq(:,i+1)=dq(:,i)+deldq;

ddq(:,i+1)=ddq(:,i)+delddq;

u(:,i+1)=(V_linear)*q(:,i+1);

du(:,i+1)=(V_linear)*dq(:,i+1);

ddu(:,i+1)=(V_linear)*ddq(:,i+1);

tu=u(:,i+1);

tq=q(:,i+1);

tdq=dq(:,i+1);

tddq=ddq(:,i+1);

error=1;

[K_nonlinear]= K_nonlinear_coeeficient_calulcation(u,K_linear_1);

while (error>=0.001)

ttu=tu;

ttq=tq;

ttdq=tdq;

ttddq=tddq;

Pti(:,i+1)=transpose(V_linear)*F(:,i+1) - transpose(V_linear)*(K_nonlinear)*(V_linear)*ttq;

Pcapi=Pti(:,i+1)-P(:,i)+(a_n)*ttdq + b*ttddq;

delqi=K_cap\Pcapi;

deldqi=(gamma/(beta*dt))*delqi-(gamma/beta)*dq(:,i)+dt*(1-gamma/(2*beta))*ddq(:,i);

delddqi=1/(beta*dt^2)*delqi -1/(beta*dt)*ddq(:,i)-1/(2*beta)*ddq(:,i);

tq=q(:,i)+delqi;

tdq=dq(:,i)+deldqi;

tddq=ddq(:,i)+delddqi;

tu=(V_linear)*tq;

error=((tu-ttu)/ttu)*100;

end

u(:,i+1)=tu;

du(:,i+1)=(V_linear)*tdq;

ddu(:,i+1)=(V_linear)*tddq;

end

Answer 1

[SOUMNATH PAUL]

Hi,

I understand you are getting undersired results for Newmark-beta method.

Without more details about your variables used in the code it is bit difficult to diagnose the exact cause of the issue.

To begin with you may try implementing the below points:

1. Verify initial conditions are set correctly for displacement, velocity, and acceleration.
2. Confirm that "V_linear", "M", "C", and "K_linear_1" matrices are correctly defined and normalized.
3. Check that the time step "dt" is small enough to accurately capture the system's dynamics.
4. Review the convergence criteria and ensure the error calculation is robust, possibly using absolute error to avoid division by zero.
5. Debug by stepping through the code to examine where unrealistic values originate.

You may also refer to the following page which will give you an idea about Newmark-beta method :

https://in.mathworks.com/matlabcentral/fileexchange/71007-newmark-beta-method-for-nonlinear-single-dof-systems

Hope it helps!

Regards,

Soumnath