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Chapter 2

Q. 2.16

Transfer Function—One Equation of Motion

PROBLEM: Find the transfer function, X(s)/F(s), for the system of Figure 2.15(a).

Step-by-Step

Verified Solution

Begin the solution by drawing the free-body diagram shown in Figure 2.16(a). Place on the mass all forces felt by the mass. We assume the mass is traveling toward the
right. Thus, only the applied force points to the right; all other forces impede the motion and act to oppose it. Hence, the spring, viscous damper, and the force due to acceleration point to the left.

We now write the differential equation of motion using Newton’s law to sum to zero all of the forces shown on the mass in Figure 2.16(a):

M\frac{d^{2}x(t) }{dt^{2} } +ƒ_{\nu } \frac{dx(t)}{dt}+Kx(t) =ƒ(t)

Taking the Laplace transform, assuming zero initial conditions

Ms^{2} X(s)+ƒ_{\nu }X(s) +KX(s)=F(s)

or

(Ms^{2} +ƒ_{\nu }s+K)X(s)=F(s)

Solving for the transfer function yields

G(s)=\frac{X(s)}{F(s)} =\frac{1}{Ms^{2}+ƒ_{\nu }s+K }

which is represented in Figure 2.15(b).