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CHAPTER
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HIGHER-ORDER SYSTEMS: SECOND-ORDER AND TRANSPORTATION LAG
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This section introduces a basic system called a second-order system or a quadratic lug. A second-order transfer function will be developed by considering a classical example from mechanics. This is the damped vibrator, which is shown in Fig. 8.1. A block of mass Wresting on a horizontal, frictionless table is attached to a linear spring. A viscous damper (dashpot) is also attached to the block. Assume that the system is free to oscillate horizontally under the influence of a forcing function F(t). The origin of the coordinate system is taken as the right edge of the block when the spring is in the relaxed or unstretched condition. At time zero, the block is assumed to be at rest at this origin. * Positive directions for force and displacement are indicated by the arrows in Fig. 8.1. Consider the block at some instant when it is to the right of Y = 0 and when it is moving toward the right (positive direction). Under these conditions,
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*In effect, this assumption makes the displacement variable Y(z) a deviation variable. Also, the assumption that the block is initially at rest permits derivation of the second-order transfer function i n i t s standard form. An initial velocity has the same effect as a forcing function. Hence, this assumption is in no way restrictive.
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SYSTEMS:
SECOND-ORDER
TRANSPORTATION
FIGURE 8-l 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
OPEN-LOOP
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 second-order. Note that it requires two parameters, T and 5, to characterize the dynamics of a second-order 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 second-order system to some of the common forcing functions, namely, step, impulse, and sinusoidal.
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