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## Main Question or Discussion Point

The standard equation for the damped angular frequency of a normal damped mass-spring system is ω[itex]_{d}[/itex] = [itex]\sqrt{\frac{k}{m}-\frac{b^{2}}{4m^{2}}}[/itex]. Let p=[itex]\frac{b}{2m}[/itex], we have ω[itex]_{d}[/itex] =[itex]\sqrt{ω_{0}^{2}-p^{2}}[/itex]

Now consider that damped mass-spring system being driven by a periodic force with the driving frequency ω[itex]_{r}[/itex]. So the question is: Which value of ω[itex]_{r}[/itex] gives the biggest amplitude of the mass-spring system? (i.e. what is the damped resonant frequency of the system?)

I originally assume that for the system to resonate at its biggest amplitudes, then ω[itex]_{r}[/itex]=ω[itex]_{d}[/itex] =[itex]\sqrt{ω_{0}^{2}-p^{2}}[/itex]. However, this is not correct; and in fact the driving frequency is supposed to be ω[itex]_{r}[/itex]=[itex]\sqrt{ω_{0}^{2}-2p^{2}}[/itex]. I get this information from a video lecture from MIT open course (watch the last 1 minute of the video: )

So my question is that why is ω[itex]_{r}[/itex]=[itex]\sqrt{ω_{0}^{2}-2p^{2}}[/itex]?

Thanks in advance!!!

Now consider that damped mass-spring system being driven by a periodic force with the driving frequency ω[itex]_{r}[/itex]. So the question is: Which value of ω[itex]_{r}[/itex] gives the biggest amplitude of the mass-spring system? (i.e. what is the damped resonant frequency of the system?)

I originally assume that for the system to resonate at its biggest amplitudes, then ω[itex]_{r}[/itex]=ω[itex]_{d}[/itex] =[itex]\sqrt{ω_{0}^{2}-p^{2}}[/itex]. However, this is not correct; and in fact the driving frequency is supposed to be ω[itex]_{r}[/itex]=[itex]\sqrt{ω_{0}^{2}-2p^{2}}[/itex]. I get this information from a video lecture from MIT open course (watch the last 1 minute of the video: )

So my question is that why is ω[itex]_{r}[/itex]=[itex]\sqrt{ω_{0}^{2}-2p^{2}}[/itex]?

Thanks in advance!!!

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