- #1
amr55533
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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!