A driven harmonic oscillator has the equation write an expression for the motion

Question

A driven harmonic oscillator has the equation write an expression for the motion x(t) which is valid at late times (after the initial conditions are forgotten and is responding only to the driving force). Let omega o = 4rad/s, omega = 4. 2rads/s, and beta = 1/s. By how many radians does the motion lag (or lead) the driving force? What is the resonant frequency of this oscillator (using values given in part b)? Compute the quality factor Q. Draw the appropriate curve from figure 3-16(b) and plot the point ( omega , Delta ) as well as ( omega o, phi /2),

Explanation / Answer

tough Q for a senior engineering student the key is finding the damping constant from the maximum A at resonace omega zero then assume the driver goes to zero and the oscillator loses its max energy exponentially a calculus problem or if you only have the results in algebra, a carful analaysis of the terms of the equation without a damping factor the Amplitute would increase without limit at resonance until the system was no longer the origianal SHM, it other words till it "blew up B. The most intuitive dimension for a damper is F/v or units of N/(m/s) or N-s/m. kg = N-s^2/m so kg/s (as given) = N-s/m. So we can believe kg/s. Units of sqrt(km) (as given) = sqrt(N/m*kg) = sqrt(N/m*N-s^2/m) = N-s/m, QED C. I prefer to do amplitude analysis in terms of the damping ratio zeta. I assume omega(d) is the driving frequency which in this case = sqrt(k/m) = the natural frequency omega, and likewise B = 0.2omega. Zeta = B*omega/(2k) = 0.2omega^2/(2k). Maximum amplitude occurs when omegad = omega (a given condition) and is equal to (Fmax/k)/(2zeta) = Fmax/(0.2omega^2)

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