# Spring-damper system with sine input

#### gomerpyle

1. Homework Statement

Variable y(t) is the position of the block of mass m, and u(t) is the position of the plate on the right. The spring is unstretched when y = u. We can think of u(t) as the input and y(t) as the output.

Derive the equation of motion for the block. You should get a second order differential equation
relating the output y(t) and the input u(t).

2. Homework Equations

u(t) = sin3t

Fd = cy'

Fk = ky

F=ma

3. The Attempt at a Solution

The thing that confuses me about this problem is that the spring/damper combo is not fixed to something, but attached to a plate which has a sine input. I know the two forces acting on the mass are the spring and damper, but the sine force acts through those two components so I'm not sure how to represent this in a free body diagram to obtain the necessary equations.

my'' = cy'(t)*u(t) + ky(t)*u(t)
y'' = u(t)/m*[cy'(t) + ky(t)]

However, I'm not sure if multiplying the input to both the force of the damper and spring is correct. Could someone help me?

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#### vela

Staff Emeritus
Homework Helper
The spring exerts a force proportional to how much it's stretched. How do you represent how much it's stretched in terms of u(t) and y(t)?

#### gomerpyle

The spring exerts a force proportional to how much it's stretched. How do you represent how much it's stretched in terms of u(t) and y(t)?
Well the spring would be stretched based upon the relative distance between the mass and the plate right? so would it be:

Fk = k*[u(t) - y(t)]

?

#### vela

Staff Emeritus
Homework Helper
Yup, and you have a similar situation for the damping term.

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