# Differential Equations : Equations of Motion, Laplace Transforms and Transfer Functions

Refer to figure 2 in attachment.

a) Write the equations of motion for the mechanical system.****PLEASE SHOW YOUR FREE-BODY FOR EACH MASS****THANKS

B) Take the Laplace transform of these equations, arrange them in matrix form, solve for the displacement x2(t), and find the transfer function T(s)= X1(s)/F(s)

C) Using the concept of the transfer function, find X1(t) if f(t) = 4u(t) N.

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Posting 66903

We start by drawing our force diagrams for m1 and m2: ( I will use x for position, v for velocity, and a for acceleration. They are all functions of time, so I won't keep writing "(t)".)

On m1, we have the K1 spring force (negative, because if m1 moves to the right, so x1 > 0, then the force on m1 is to the left.

We also have the viscous force, which for positive v1 (moving to the right) provides a leftward (negative) drag force, with a coefficient of -2.

We also have the K2 spring force, which will pull m1 to the right if x2 > x1.

And we have the viscous damper force, which will pull m1 to the right if v2 > v1.

On m2, we have the K2 spring force, which will pull m2 to the left (negative) if x2 > x1.

We also have the vicous force, which will drag m2 to the left if it is moving to the right, with a coefficient of -1.

We also have the viscous damper force, which provides on m2 the negative of the force it provide on m1.

And we have the externally applied force f, to the right (positive).

Using Newton's Second Law we set the masses times their accelerations equal to the the sum of the forces:

m1 a1 = -K1 x1 - 2 v1 + K2 (x2 - x1) + (v2 - v1)

m2 a2 = -K2 (x2 - x1) - v2 - (v2 - v1) + f

Let's put in our ...

#### Solution Summary

Equations of Motion, Laplace Transforms and Transfer Functions are investigated. The solution is detailed and well presented. The response received a rating of "5/5" from the student who originally posted the question.