Impulse effects on an overdamped vibrating system derivation

[amr55533]

Homework Statement

For a zero initial deflection and for a finite initial velocity, the time dependence of the vibration response of an overdamped system to an impulse is given by:

(1) θ(t)=[I/(mω)√(ζ^2-1)][e^-(ζωt)][cosh(ω√(ζ^2-1))t]

which for large values of time becomes:

(2) θ(t)=[I/(2mω)√(ζ^2-1)][e^-(ζ+√(ζ^2-1))t]

How could equation (2) be derived from equation (1)?

Homework Equations

I=FΔt
θ(t)=[I/(mω)√(ζ^2-1)][e^-(ζωt)][cosh(ω√(ζ^2-1))t]
θ(t)=[I/(2mω)√(ζ^2-1)][e^-(ζ+√(ζ^2-1))t]

The Attempt at a Solution

I am trying to derive equation (2) from equation (1).

As t becomes large, [e^-(ζωt)] approaches 0.

I checked the chapter on impulse response functions in my vibrations textbook, but couldn't seem to find either of these equations.

Also, I tried a few arbitrary values with a large values for time in each equation, but was coming up with completely different answers. Are these equations viable?

Thanks!

 

[bossman27]

I'm too brain dead to be of much help at this point, but have you tried writing cosh in terms of exponential functions and seeing if anything resolves itself that way? http://en.wikipedia.org/wiki/Hyperbolic_cosine

I haven't seen this problem before, but that'd be my first try, and the 2 in the denominator of those expressions for cosh seems like it might be a friendly sign.

 

[amr55533]

Thanks! I didn't even think of converting the cosh to an algebraic function. I think I got it now!