Answer to Question #150323 in Biology for no me

Question

A damped mass-spring oscillator is subjected to a forcing F=F_0 cos⁡〖(ω_d t)〗. The steady-state solution is given by
x=A cos⁡〖(ω_d t-δ)〗,
where
A=A(ω_d )=(F_0/m)/√((ω_0^2-ω_d^2 )^2+(γω_d )^2 )
and
tan⁡δ=(γω_d)/(ω_0^2-ω_d^2 ).
The symbols have the usual meanings.
A sketch of the variation of amplitude versus driving frequency is shown in Figure 1:

Figure 1: Variation of amplitude with driving frequency
Explain why the amplitude is F_0/k when the driving frequency is zero and the type of forcing the system is subjected to.
Show that the frequency at which the amplitude attains a maximum is given by
ω_m=√(ω_0^2-γ^2/2)
and that the maximum amplitude is given by
A=A_0 Q/[1-1/(4Q^2 )]^(1/2)
where Q is the quality factor and A_0=F_0/k

Answer 1

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