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Vibration Problem


Solution of a system of second-order homogeneous ordinary differential equations with constant coefficients of the form

 (d^2x)/(dt^2)+bx=0,

where b is a positive definite matrix. To solve the vibration problem,

1. Solve the characteristic equation of b to get eigenvalues lambda_1, ..., lambda_n. Define omega_i=sqrt(lambda_i).

2. Compute the corresponding eigenvectors e_1, ..., e_n.

3. The normal modes of oscillation are given by x_1=A_1sin(omega_1t+alpha_1)e_1, ..., x_n=A_nsin(omega_nt+alpha_n)e_n, where A_1, ..., A_n and alpha_1, ..., alpha_n are arbitrary constants.

4. The general solution is x=sum_(i=1)^(n)x_i.


See also

Second-Order Ordinary Differential Equation

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Cite this as:

Weisstein, Eric W. "Vibration Problem." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/VibrationProblem.html

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