What is the value of the dampening constant?

[HiggsBrozon]

Homework Statement

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).

Homework Equations

x(t) = (A+Bt)e^-βt where β = ω₀

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.

 

[ehild]

Homework Statement

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).

Homework Equations

x(t) = (A+Bt)e^-βt .

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.

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

 

[HiggsBrozon]

Thanks for you reply!

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?

 

[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

 

[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?

 

[ehild]

Yes, it will be correct.

ehild