What is the value of the dampening constant?

1. Feb 25, 2013

HiggsBrozon

1. The problem statement, all variables and given/known data

The shock absorber of a car has elastic constant k. When the car is empty, the design is such that the absorber is critically damped. At time t=0 the absorber is compressed by an amount A from its equilibrium position and released.
a) If after 1 second the absorber compression is reduced to A/2, what is the value of the damping coefficient? (Note you will have to solve numerically an implicit equation).

2. Relevant equations

x(t) = (A+Bt)e-βt where β = ω0

3. The attempt at a solution

A(t) = Ae-βt
1/2A = Ae-βt where t = 1s
ln(1/2) = -β

I know this can't be right because I'm suppose to arrive at an implicit equation. I can't seem to figure out where I'm going wrong at. Any help would be appreciated.

2. Feb 25, 2013

ehild

The general equation for the displacement is

x(t) = (A+Bt)e-βt .

At t=0, x=A. What is the value of B if the velocity is 0 at t=0 (that is, the derivative of x(t) has to be zero at t=0)?

ehild

3. Feb 26, 2013

HiggsBrozon

x' = (-Aβ - Btβ + β)e-βt
The velocity is 0 at t=0 so,

0 = (-Aβ + β)(1)
β=0 and A = 1

Would I then go on to use t = 1 sec at A0 = A0/2
where A0 = A+βt, the initial amplitude, or is A0 equal to the value I just solved for, A = 1?

4. Feb 27, 2013

ehild

β is the damping coefficient. It is a parameter of the problem. It does not depend on the initial condition.

You have to fit the constant B to the initial condition V=0. There is a mistake in your derivation. Check the derivative of x(t) = (A+Bt)e-βt.

ehild

5. Feb 27, 2013

HiggsBrozon

After correcting my derivative I got,
x' = (-Aβ - Btβ + B)e-βt
Then using my initial conditions v = 0 at t = 0,

0 = (-Aβ + B)(1)
→ B = Aβ

so, x(t) = (A+Bt)e-βt where B = Aβ
and then I can use A = A/2 when t = 1s correct?

6. Feb 27, 2013

ehild

Yes, it will be correct.

ehild