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# Matrix equation of motion

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1) For the two degree of freedom structure shown:

i) what is the matrix equation of motion?. Assume that m1 and m2 oscillate harmonically with the same frequency but with different amplitudes X1 and X2 of x1(t) and x2(t).
ii) what are the values of the amplitude ratios r1 and r2
if m1 = m2 = m, k1 = k, and k2 =3k?

See attached file for full problem description.

https://brainmass.com/engineering/mechanical-engineering/matrix-equation-of-motion-109261

#### Solution Summary

Using a provided degree of freedom structure, the solution explains what the matrix equation of motion is and what the value of the amplitude rations r1 and r2 are.

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## Matrix Form: Inhomogeneous Differential Equations

How do I express the following inhomogeneous system of first-order differential equations for x(t) and y(t) in matrix form?
(see the attachment for the full question)
x = -2x - y + 12t + 12,
y = 2x - 5y - 5

How do I express the corresponding homogeneous system of differential equations, also in matrix form?

How do I find the eigenvalues of the matrix of coefficients and an eigenvector corresponding to each eigenvalue. From this how would I write down the complementary function for the system of differential equations?

How would you calculate a particular integral for the inhomogeneous system, and then find the general solution?

How would I determine the particular solution of the initial-value problem with the initial conditions x(0) = 3 and y(0) = 2?

I have another problem below but on a similar topic:

If an object moves in the plane in such a way that its Cartesian coordinates (x, y) at time t satisfy the following homogeneous system of second-order differential equations:
x = -2x - y,
y = 2x - 5y.

How would I:
Express the system in matrix form?
Find the general solution of the system?
I think this system undergoes simple harmonic motion in a straight line in two distinct ways but why?
And for each such simple harmonic motion how do I determine the angular frequency and the vector giving the direction of motion?

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