find frequency eqaution and then compute for the non zero natural frequencies

CODE

Answer 1

[Sam Chak]

Hi @DP

A 5-coupled mass-spring-damper network constitutes a 10th-order system. If you are referring to the 'frequency equation' as the Laplace transfer function, we can utilize the 'charpoly' and 'tf' functions. Additionally, the 'damp()' command can provide you with information about the system's natural frequencies.

Update: Check if you want to find the natural frequencies of the mass-damper-spring system using this approach:

m1  = 1;
m2  = m1;
m3  = m1;
m4  = m1;
m5  = m1;

c2  = 2*m1/100;
c3  = c2;
c4  = c2;
c5  = c2;

k2  = m1/1.1486;
k3  = k2;
k4  = k2;
k5  = k2;

% Construct mass matrix
M = [m1,  0,  0,  0,  0;
      0, m2,  0,  0,  0;
      0,  0, m3,  0,  0;
      0,  0,  0, m4,  0;
      0,  0,  0,  0, m5];

% Construct damping matrix
C =  [c2,   -c2,     0,     0,   0;
     -c2, c2+c3,   -c3,     0,   0;
       0,   -c3, c3+c4,   -c4,   0;
       0,     0,   -c4, c4+c5, -c5;
       0,     0,     0,   -c5,  c5];
C = 1000*C;

% Construct stiffness matrix
K = [k2,   -k2,     0,     0,   0;
    -k2, k2+k3,   -k3,     0,   0;
      0,   -k3, k3+k4,    k4,   0;
      0,     0,   -k4, k4+k5, -k5;
      0,     0,     0,   -k5,  k5];

%% state matrix
% A = [zeros(size(M)), eye(size(M));
%                -M\K,         -M\C]
% CE  = charpoly(A);
% Gp  = tf(1, CE)
% damp(Gp)

%% Eigenvalues
lambda  = eig(M\K)

lambda = 
   2.2793 + 0.0000i
   2.2793 - 0.0000i
   1.7413 + 0.0000i
   0.3325 + 0.0000i
   0.3325 + 0.0000i

%% Natural frequencies of the system
omega   = sqrt(lambda)

omega = 
   1.5097 + 0.0000i
   1.5097 - 0.0000i
   1.3196 + 0.0000i
   0.5767 + 0.0000i
   0.5767 + 0.0000i

[DP]

i m refering to the frequency eqaution as deriving the deteminant function as below for the spring damper system

where

syms omega;

M = [
    1.8000         0         0         0         0
    0    6.3000         0         0         0
    0         0    5.4000         0         0
    0         0         0   22.5000         0
    0         0         0         0   54.0000];

C = [
    10000      -10000           0           0           0
    -10000       10500        -500           0           0
    0        -500        2000       -1500           0
    0           0       -1500        2600       -1100
    0           0           0       -1100        1100];

K = [
    100000000  -100000000           0           0           0
    -100000000   100050000      -50000           0           0
    0      -50000      125000       75000           0
    0           0      -75000       85000      -10000
    0           0           0      -10000       10000];

det(K - omega^2 * M - omega * C)

ans = 

and then getting the ouputs for non zero natural frequencies , is my approach correct?

[Sam Chak]

Hi @DP

I am not aware of another term called the "determinant function." In my understanding, the general term for such an equation is referred to as the "characteristic polynomial" or "characteristic equation." That's why I initially used the 'charpoly()' function. However, numerically solving the 10th-degree polynomial will indeed provide you with ten eigenfrequencies, as demonstrated in my example.

syms omega;

M = [
    1.8000         0         0         0         0
    0    6.3000         0         0         0
    0         0    5.4000         0         0
    0         0         0   22.5000         0
    0         0         0         0   54.0000];

C = [
    10000      -10000           0           0           0
    -10000       10500        -500           0           0
    0        -500        2000       -1500           0
    0           0       -1500        2600       -1100
    0           0           0       -1100        1100];

K = [
    100000000  -100000000           0           0           0
    -100000000   100050000      -50000           0           0
    0      -50000      125000       75000           0
    0           0      -75000       85000      -10000
    0           0           0      -10000       10000];

eqn = det(K - omega^2 * M - omega * C)

eqn = 

lambda  = vpasolve(eqn == 0)

lambda =