Frequency response function of periodic-stiffness model of system

[marasciallo]

Hi,
I'm analyzing a 3 dof undamped system with discrete springs and masses. Three of the springs have time-dependent stiffness, following periodic law (with period T), they are modulated at the same frequency but with a phase difference of 120 deg one from the other.
So this is my system:

(M * d^2x/dt^2) + K(t)*x = f(t)

with:
- M diagonal mass matrix (3x3), constant

- x=[x1;x2;x3] displacements vector (3x1)

- f(t)= f0*[sen(w*t);0;0] forcing vector (3x1) acting only on the first dof

- K(t) symmetric stiffness matrix:

> k11= ko + 2*kc/cos(30) + km*(cos(wm*t));
> k22= ko + 2*kc/cos(30) + km*(cos(wm*t + 2*pi/3));
> k33= ko + 2*kc/cos(30) + km*(cos(wm*t + 4*pi/3));

> k(i,k)= -kc/cos(30) for i~=k

wm = modulation frequency
ko, kc = constantsHow can I calculate the frf for this system in order to obtain output amplitude at the second and third dof?
Any help is very appreciated!
Have a nice day.

Paolo

 

[Achy47]

Hi Paolo,

Thank you for sharing your system with us. It seems like a complex but interesting problem to analyze. To calculate the frequency response function (FRF) for this system, you will need to follow a few steps:

1. First, you will need to find the equations of motion for your system. This can be done by applying Newton's second law to each mass element, taking into account the time-dependent stiffness of the springs.

2. Next, you will need to linearize your equations of motion by assuming small displacements and using a Taylor series expansion. This will allow you to express your equations in terms of the displacement and velocity of each mass element, and the time-dependent stiffness.

3. Once you have linearized your equations, you can use the Laplace transform to convert them to the frequency domain. This will allow you to express the equations in terms of the complex frequency variable s.

4. With your equations now in the frequency domain, you can solve for the transfer function, which is the ratio of the output amplitude at each degree of freedom to the input amplitude at the first degree of freedom.

5. Finally, you can use the transfer function to calculate the frequency response function for your system. This will give you the output amplitude at each degree of freedom for a given input frequency.

I hope this helps you get started on calculating the FRF for your system. Best of luck with your analysis!