For the spring (K)-mass (M)-viscous damper (D) system shown moving horizontally

Question

For the spring (K)-mass (M)-viscous damper (D) system shown moving horizontally below (as usual, the cross-hatched base can move but is assumed to be so massive that the forces in the system do not influence it's motion): Write the differential equations describing the displacement, X, in terms of the force, F, Xb and the system parameters (K, M & D). Assume that the units are a consistent set (ie no conversion necessary) and that the system is initially at equilibrium (ie all IC's = 0). Derive an expression for the transfer function relating X to F in monic form (le write X(s)/F(s)). Derive an expression for the transfer function relating X to Xb in monic form. (ie write X(s)/Xb(s)). Write expression for the equivalent natural frequency, Win, damping ratio zeta, and damped natural frequency, Wd, for this system in terms of K, D & M. If K = 1000, D = 50, and M = 10, sketch the approx frequency response plots (gain & phase vs log freq) for X/F. Clearly label the plots and indicated where Win is on the freq axis on both of the curves. Write a time-domain expression for the steady-state motion of the mass, X(t), if F(t) = 1.5 Sin(10t)?

Explanation / Answer

1a)    Differential equation

=>    d2x/dt2 + (D/M)dx/dt + (K/M)x = F(t)

1b)     X(s)/F(s)   =   1/[s2 + (D/M)s + (K/M)]

1c)     X(s)/Xb(s)   =   [F(s) * (s-a)]/[s2 + (D/M)s + (K/M)]

1d)    wn = sqrt(K/M)

        damping ratio , t = D/[2*sqrt(M * K)]

      wd =   wn * sqrt(1 - 2t2)

1e)    Curve will be exponential increasing line which become constant after a time .

1f)     X(t) =   15 * cos(10t)

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