Find natural equation w with 5 dof

% Using matrix to determine natural frequency w of 5 DOF

clc; % Clear command window

clear; % Clear workspace

syms w; % Define w as a symbolic variable

% Define masses, damping coefficients, and stiffness coefficients

m1 = 1.8; m2 = 6.3; m3 = 5.4; m4 = 22.5; m5 = 54;

c2 = 10000; c3 = 500; c4 = 1500; c5 = 1100;

k2 = 1*10^8; k3 = 50*10^3; k4 = 75*10^3; k5 = 10*10^3;

[Sam Chak]

@Trong Nhan Tran, can you at least type out all the equations?

syms w omega X1 X2; % Define w as a symbolic variable

% Define masses, damping coefficients, and stiffness coefficients
m1 = 1.8; m2 = 6.3; m3 = 5.4; m4 = 22.5; m5 = 54; c2 = 10000; c3 = 500; c4 = 1500; c5 = 1100; k2 = 1*10^8; k3 = 50*10^3; k4 = 75*10^3; k5 = 10*10^3;

eq1 = - k2*(X1 - X2) - c2*omega*(X1 - X2) == - m1*omega^2*X1

eq1 = 

eq2 = ...
eq3 = ...

[Trong Nhan Tran]

eq2 = -k3*(X2-X3) - c3*(X2-X3) + k2*(X1-X2) + c2*omega*(X1-X2) == - m2*omega^2*X2

eq3 = -k4*(X3-X4) - c4*(X3-X4) + k3*(X2-X3) + c3*omega*(X2-X3) == - m3*omega^2*X3

eq4 = k5*(X5-X4) - c5*(X5-X4) + k4*(X3-X4) + c4*omega*(X3-X4) == -m4*omega^2*X4

eq1 = -k5*(X5-X4) - c5*(X5-X4) == -m5*omega^2*X5

[Sam Chak]

Hi @Trong Nhan Tran

Could you please review the diagram and equations provided? I noticed a few discrepancies, such as the absence of in the diagram and the direction of the acceleration. Additionally, it would be helpful if you could group the following matrices for clarity:

1. Mass matrix
2. Damping matrix
3. Stiffness matrix

Answer 1

[Sam Chak]

Hi @Trong Nhan Tran

Based on the diagram and your equations, the natural frequencies of the system can be computed as follows:

%% Parameters
m1 = 1.8; m2 = 6.3; m3 = 5.4; m4 = 22.5; m5 = 54; c2 = 10000; c3 = 500; c4 = 1500; c5 = 1100; k2 = 1*10^8; k3 = 50*10^3; k4 = 75*10^3; k5 = 10*10^3;

%% Mass matrix
M       = diag([m1, m2, m3, m4, m5])

M = 5x5

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

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%% 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]

K = 5x5

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

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%% Eigenvalues
lambda  = eig(M\K)

lambda = 5x1

1.0e+07 * 7.1430 0.0028 0.0005 0.0000 -0.0000

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

omega = 
   1.0e+03 *

   8.4516 + 0.0000i
   0.1665 + 0.0000i
   0.0714 + 0.0000i
   0.0212 + 0.0000i
   0.0000 + 0.0000i

[Trong Nhan Tran]

Can you do a favour? Can you do it without Eigenvalues method? Please I had solution for Eigenvalues method already. And need another one to compare results.

[Sam Chak]

Hi @Trong Nhan Tran

Finding the frequencies for 5th-order and above systems has to be calculated numerically. Can you provide some non-eigenvalue methods or computational algorithms that were taught by your Professor? We can review how to apply those techniques to your problem.

For higher-order systems, the analytical solutions become increasingly complex, so numerical approaches are often necessary. Your Professor likely covered some efficient computational methods that avoid relying solely on eigenvalue decomposition.

If you can share those non-eigenvalue techniques, I'd be happy to walk through how we can leverage them to solve your specific system. That way, we can explore a more practical and scalable approach, rather than getting bogged down in complex analytical derivations.