frequency equation for multiple degree freedom system

syms omega;

M = 5×5

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 = 5×5

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 = 5×5

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

the matrices for mass,frequency and damping coeffcinets are as above is it corrrect to to obtain the frequency equation from the following code

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

and also how to obtain non zero natural frequencies of the system

[Rena Berman]

(Answers Dev) Restored edit

Answer 1

[Athanasios Paraskevopoulos]

% Define the mass matrix M
M = diag([1.8, 6.3, 5.4, 22.5, 54.0]);

% Define the damping matrix C
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];

% Define the stiffness matrix K
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];

% Initialize symbolic variable for frequency
syms omega;

% Define the system matrix for characteristic equation
A = -omega^2 * M + 1i * omega * C + K;

% Solve the characteristic equation det(A) = 0 for omega
% This provides the natural frequencies of the system
natural_frequencies = solve(det(A) == 0, omega, 'MaxDegree', 4);
% Display the computed natural frequencies
disp('Natural Frequencies:');

Natural Frequencies:

disp(vpa(natural_frequencies, 6));  % Display frequencies with 6 decimal places

[Athanasios Paraskevopoulos]

I used 𝑖(the imaginary unit) in the original explanation because it is often encountered in damping matrices in complex dynamics systems, particularly when dealing with complex eigenvalues. However, if your damping matrix and analysis are real, the imaginary unit 𝑖 is unnecessary and should be removed from the expression.

% Define the mass matrix M
M = diag([1.8, 6.3, 5.4, 22.5, 54.0]);

% Define the damping matrix C
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];

% Define the stiffness matrix K
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];

% Initialize symbolic variable for frequency
syms omega;

% Define the system matrix for the characteristic equation
A = -omega^2 * M + omega * C + K;

% Solve the characteristic equation det(A) = 0 for omega
% This provides the natural frequencies of the system
natural_frequencies = solve(det(A) == 0, omega);

% Display the computed natural frequencies
disp('Natural Frequencies:');

Natural Frequencies:

disp(vpa(natural_frequencies, 6));  % Display frequencies with 6 decimal places