CHAPTER

HIGHER-ORDERSYSTEMS:SECOND-ORDERAND TRANSPORTATIONLAG

SECOND-ORDER SYSTEM

This section introduces a basic system called a second-order system or a quadraticlug. A second-order transfer function will be developed by considering a classi-cal example from mechanics. This is the damped vibrator, which is shown inFig. 8.1.

A block of mass Wresting on a horizontal, frictionless table is attached to alinear spring. A viscous damper (dashpot) is also attached to the block. Assumethat the system is free to oscillate horizontally under the influence of a forcingfunction F(t). The origin of the coordinate system is taken as the right edge ofthe 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 anddisplacement are indicated by the arrows in Fig. 8.1.

Consider the block at some instant when it is to the right of Y = 0 andwhen it is moving toward the right (positive direction). Under these conditions,

*In effect, this assumption makes the displacement variable a deviation variable. Also, theassumption that the block is initially at rest permits derivation of the second-order transfer functionin i t s form. An ini t ial veloci ty has the same effect as a forcing funct ion. Hence , thisassumption is in no way restrictive.

90

HIGHER-ORDER SYSTEMS: SECOND-ORDER AND TRANSPORTATION LAG 91

Ú×Ù ËÎÛDamped vibrator.

the position Y and the velocity 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 positiveconstant, called Hooke’s constant.

2. The viscous friction force (acting to the left) of where C is a positiveconstant called the damping coefficient.

3. The external force (acting toward the right).

Newton’s law of motion, which states that the sum of all forces acting onthe mass is equal to the rate of change of momentum (mass acceleration), takesthe form

W- - = -KY + F(t)

Rearrangement givesW- - + + KY = F(t)

where W = mass of block, lb, =

C = viscous damping coefficient, K = Hooke’s constant,

F(t) = driving force, a function of time, Dividing Eq. (8.2) by K gives

W- -K dt

For convenience, is written as

+ + Y = X(t)

where

(8.4)

W

92 LINEAR OPEN-LOOP SYSTEMS

=

X(t) =

dimensionless

(8.7)

Solving for and from Eqs. (8.5) and (8.6) gives

(8.9)

By definition, both and must be positive. The reason for introducing and in the particular form shown in Eq. (8.4) will become clear when we discuss thesolution of Eq. (8.4) for particular forcing functions X(t).

Equation (8.4) is written in a standard form that is widely used in controltheory. Notice that, because of superposition, X(t) can be considered as a forcingfunction because it is proportional to the force F(t).

If the block is motionless = 0) and located at its rest position(Y = 0) before the forcing function is applied, the transform of Eq.(8.4 ) becomes

+ + Y ( S ) = x ( S ) (8.10)From this, the transfer function follows:

1+ +

(8.11)

The transfer function given by Eq. (8.11) is written in standard form, andwe shall show later that other physical systems can be represented by a transferfunction having the denominator + + 1. All such systems are definedas second-order. Note that it requires two parameters, and to characterize thedynamics of a second-order system in contrast to only one parameter for a order system. For the time being, the variables and parameters of Eq. (8.11) canbe interpreted in terms of the damped vibrator. We shall now discuss the responseof a second-order system to some of the common forcing functions, namely, step,impulse, and sinusoidal.

Step ResponseIf the forcing function is a unit-step function, we have

X(s) =

In terms of the damped vibrator shown in Fig. 8.1 this is equivalent to suddenlyapplying a force of magnitude K directed toward the right at time = 0. Thisfollows from the fact that X is defined by the relationship X(t) = .