Amass is moving at a constant velocity v0 when thruct force f0 is in equilibrium

Question

Amass is moving at a constant velocity v0 when thruct force f0 is in equilibrium with non damping force .

A mass is moving at a constant velocity,vomega when thrust force, f0x is in equilibrium with nonlinear damping force: falpha = fb(valpha) = b . Valpha3 where b is a known constant. Considering small deviation around the equilibrium, where small additional force results in small velocity change(i.e.f = falpha + deltaf causing v = valpha + deltav), find a transfer function G(s) = deltaV (s) /Delta F(s) by linearizing the nonlinear damping force near v alpha. Find the time response, deltav(t),for a step input force, deltaf=Au(t), where A[N] is a "small" force, m=10[kg], b=1/3 [N/(m/s)3], and valpha=2[m/s]. Using Matlab, plot the nonlineardamping force fb vs v. In the same figure, plot the linearized damping force. Also generate step response plot and bode plot. Repeat (b) and (c) with valpha=4 [m/s].

Explanation / Answer

a)

      ?V/?F = 3*b*Vo^2

b) for Vo = 2 m/s
       ?V/?F = 3*(1/3)*2^2 = 4

       a = ?F(t) / m = A*u(t) / 10

       ?V(t) = Vo + a*t = 2 + A*u(t)*t

   for Vo = 4 m/s

?V/?F = 3*(1/3)*4^4 = 16

       a = ?F(t) / m = A*u(t) / 10

       ?V(t) = Vo + a*t = 4 + A*u(t)*t

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