Question

A structural model consists of a mass matrix M (kg):

| 1 0|

|0 1|

and a spring stiffness matrix K (N/m):

|200 -100 |

|-100 100|

What are the natural frequencies in rad/s? Use ss and lsim on MATLAB to simulate the displacements of the 2DOF model with the initial conditions: x(0) = [1 1] x_dot(0) = [1 -1] with no force acting on the system. Use 2^13 samples with a delta t of .01s for the simulation. Make a plot of the displacements vs time and make a plot of the coefficients of the fast Fourier transform vs frequency for both outputs.

Answer 1

The natural frequencies of a 2DOF system with given mass and stiffness matrices can be found by solving the eigenvalue problem and simulating the system's response with specific initial conditions in MATLAB.

Step-by-step explanation:

To find the natural frequencies of a 2 Degree Of Freedom (DOF) model with a given mass matrix M and stiffness matrix K, one can solve the eigenvalue problem for the system. The equations for the system without damping and external forces are Mx'' + Kx = 0, where M is the mass matrix and K is the stiffness matrix. By assuming a solution of the form x = A exp(iωt), where ω is the angular frequency, we can find ω as the square roots of the eigenvalues of the matrix M-1K.

For the given matrices:

• Mass matrix M (kg):
  | 1 0|
  | 0 1|
• Spring stiffness matrix K (N/m):
  | 200 -100 |
  | -100 100|

The MATLAB code to calculate the natural frequencies would involve the eig function to compute the eigenvalues. To simulate the system response with initial conditions x(0) = [1 1] and x_dot(0) = [1 -1], one would use the state-space representation and lsim function in MATLAB. The simulation would be performed using 213 samples and a Δt of 0.01s.

The displacements vs. time can be plotted using the MATLAB plot function, whereas the coefficients of the Fast Fourier Transform (FFT) vs frequency for both outputs can be obtained using the fft function and visualized with a corresponding plot.