1. The problem statement, all variables and given/known data 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)? 2. Relevant 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] 3. 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 text book, 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!