# Linear Damping Model

1. Jun 25, 2008

### moolimanj

Hi All

This is my first post and I was hoping that someone could help. I have the following attached question regarding a linear damping model (in this case a seismograph).

(1) Draw a force diagram.

If W=weight, R= air resistance, and H= spring force, then I get W pointing down and both R and H pointing up. Is this right? Should there be another force as well (i.e. at the pointer and in the damper)

(2) Express forces on particle in terms of given variables and parameters and unit vector i.

I'm confused on this one. Can someone help?

(3) Express displacement vector p of mass relative to origin in terms of x, y, d and i. Write down equation of motion of particle, and hence show that x(t) satisfies the differential equation md^2x/dt^2 +rdx/dt+kx=mg+kl+md^2y/dt^2

For (3) I get the following (can someone check):

We are applying Newton's second law here, but there is one extra feature: the acceleration of the box comes in as relative motion. So whatever expression we get for the net force, we need to add a m d^2y / dt^2 to it. I will simply put that into the equation "by hand."

I have a coordinate system with the positive direction upward. So in the Free Body diagram of the mass there is a weight (w) acting downward, a spring force (f) acting upward, and a resistive force (R) from the piston acting downward.

f = kl = k(x - l_0)

w = mg

R = r \frac{dx}{dt}

So Newton's second in the vertical direction gives:
m \frac{d^2x}{dt^2} = -mg + k(x - l_0) - r \frac{dx}{dt} + m \frac{d^2y}{dt^2}

(4) Suppose displacement vector y is given by y=acos(ft), where a is input amplitude, and f is omega (forcing angular frequency). Show that M, the given ratio of the amplitude A of the steady state output oscillations to a is given by:

M=A/a=mf^2/SQRT((k-mf^2)^2+r^2f^2)

I havent got a clue with this one. Any help would be greatly appreciated

Many thanks

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2. Jun 26, 2008

### moolimanj

For part (2), am I right in thinking that the following are the forces acting on the particle:

W=-mgi
R=rxi
H=k(l-l0)s

note the x term in R has a dot on the top (ie function of velocity I think

3. Jun 26, 2008

### billymystery

This block really sucks.

The dot means the its the derivative of x(t)...

I think somebody might be more inclined to help if you didnt present the problems in Latex code.