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TRANSPORTATION
FIGURE 8l Damped vibrator.
the position Y and the velocity df/dt arc both positive. At this particular instant, the following forces are acting on the block: 1. The force exerted by the spring (toward the left) of KY where K is a positive constant, called Hooke s constant. 2. The viscous friction force (acting to the left) of C dY/dt, where C is a positive constant called the damping coefficient. 3. The external force F(t) (acting toward the right). Newton s law of motion, which states that the sum of all forces acting on the mass is equal to the rate of change of momentum (mass X acceleration), takes the form W  d2Y = KY  Cg + F(t) (8.1) gc dt2 Rearrangement gives W d2Y + C$+ + KY = F(t) (8.2) gc dt2 where W = mass of block, lb, gc = 32.2(lb,)(ft)/(lbf)(sec2) C = viscous damping coefficient, lbf/(ft/sec) K = Hooke s constant, lbf/ft F(t) = driving force, a function of time, lbf Dividing Eq. (8.2) by K gives  W d2Y +cd iy = F(t) K dt g,K dt2 K For convenience, this is written as ,d2Y 7 @ + 2{ % + Y = X(t) where W 72 = gcK (8.5) (8.4) (8.3) LINEAR
OPENLOOP
SYSTEMS
257 = 4 F(t) X(t) = 7 Solving for T and 5 from Eqs. (8.5) and (8.6) gives set dimensionless
(8.6) (8.7) (8.9) By definition, both T and f must be positive. The reason for introducing T and f in the particular form shown in Eq. (8.4) will become clear when we discuss the solution of Eq. (8.4) for particular forcing functions X(t). Equation (8.4) is written in a standard form that is widely used in control theory. Notice that, because of superposition, X(t) can be considered as a forcing function because it is proportional to the force F(t). If the block is motionless (dYldt = 0) and located at its rest position (Y = 0) before the forcing function is applied, the Laplace transform of Eq. (8.4) becomes T*S*Y(S) + 25TSY(S) + Y(S) = x(S) (8.10) From this, the transfer function follows: 1 Y(s)  = (8.11) T*S* + 267s + 1 X(s) The transfer function given by Eq. (8.11) is written in standard form, and we shall show later that other physical systems can be represented by a transfer function having the denominator T*S* + 2573 + 1. All such systems are defined as secondorder. Note that it requires two parameters, T and 5, to characterize the dynamics of a secondorder system in contrast to only one parameter for a firstorder system. For the time being, the variables and parameters of Eq. (8.11) can be interpreted in terms of the damped vibrator. We shall now discuss the response of a secondorder system to some of the common forcing functions, namely, step, impulse, and sinusoidal.

