自动控制理论第二章习题

E2.4A laser printer uses a laser beam to print copy rapidly for a computer. The laser is positioned by a control input,r(t),so that we haveY(s)=5(s+100)s2+60s+500R(s)This input r(t)represents the desired position of the laser beam.(a)If r(t) is a unit step input,find the output y(t).(b)What is thefinal value of y(t)?E2.14Obtain the differential equations in terms of and for the circuit in Figure E2.14.Figure E2.14Electric circuit.E2.15The position control system for a spacecraft platform is governed by the following equation.d2p dt2+2dpdt+4p=θv1=r−pdθdt=0.6v2v2=7v1The variable involved are as follows:r(t)=desired platform position;p(t)=desired platform position; v1(t)=amplifier input voltage;v2(t)=amplifier output voltage;θ(t)=motor shaft position1Sketch a signal-flow diagram or a block diagram of the system,identify-ing the component parts and their transmittances;then determine the system transfer function P(s)/R(s)E2.18The transfer function of a system isY(s) R(s)=10(s+2)s2+8s+15Determine y(t)when r(t)is a unit step input.E2.19Determine the transfer function V0(s)/V(s)of the operational amplifier circuit shown in Figure E2.19.Assume an ideal operational amplifier.Deter-mine the transfer function when R1=R2=100kΩC1=10µf and C2=5µfE2.20A high-precision positioning slide is shown in Figure E2.20.Determine the transfer function X p(s)/X in(s)when the drive shaft friction is b d=0.7,the drive shaft spring constant is k d=2,m c=1,and the sliding friction is b s=0.8.2E2.21The rotational velocityωof the satellite shown in Figure E2.21is adjusted by changing the length of the beam L.The transfer function between ω(s)and the incremental change in beam length∆L(s)isω(s)∆L(s)=2.5(s+2)(s+5)(s+1)2FIGURE E2.21E2.25Determine the transfer function X2(s)/F(s)for the system shown in Figure E2.25.Both masses slide on a frictionless surface.And k=1N/M.3FIGURE E2.25P2.1An electric circuit is shown in Figure P2.1.Obtain a set of simultaneous integrodifferential equations representing the network.FIGURE P2.1Electric circuitP2.2A dynamic vibration absorber is shown in Figure P2.2.This system is representative of many situations involving the vibration of machines containing unbalanced components.The parameters M2and k12may be chosen so that the main mass M1does not vibrate in the steady state when F(t)=a sinω0t.Obtain the differential equations describing the system.FIGURE P2.2Vibration absorber.P2.3A coupled spring-mass system is shown in Figure P2.3.The masses and springs are assumed to be equal.Obtain the differential equations describing the system.4FIGURE P2.3Two-mass system.P2.8Abridged-T network is often used in AC control systems as afilter network[8].The circuit of one bridged-T network is shown in Figure P2.8. Show that the transfer function of the network isV0(s) V in(s)=1+2R1Cs+R1R2C2s2 1+(2R1+R2)Cs+R1R2C2s2Sketch the pole-zero diagram when R1=0.5,R2=1,and C=0.5.FIGURE P2.8Bridged-T network.P2.17A mechanical system is shown in Figure P2.17,which is subjected to a known displacement with respect to the reference.(a)Determine the two independent equations of motions.(b)Obtain the equations of motion in terms of the Laplace transform,assuming that the initial conditions are zero.(c) Sketch a signal-flow graph representing the system of equations.(d)Obtain the relationship T13(s)between X1(s)andX3(s)by using Mason’s signal-flow gain5pare the work necessary to obtain T13(s)by matrix methods to that using Mason’s signal-flow gain formula.FIGURE P2.17Mechanical system.P2.28An LC ladder network is shown in Figure P2.18.One may write the equations describing the network as follows:I1=(V1−V a)Y1V a=(I1−I a)Z2I a=(V a−V2)Y3V2=I a Z4Construct aflow graph from the equations and determine the transfer function V2(s)/V1(s).FIGURE P2.18LC ladder network.6P2.20The source follower amplifier provides lower output impedance and essentially unity gain.The circuit diagram is shown in Figure P2.20(a),and the small signal model is shown in Figure P2.20(b).This circuit uses a FET and provides a gain of approximately unity.Assume that R2>>R1for biasing purposes and that R g>>R2.(a)Solve for the amplifier gain.(b)Solve for the gain when g m=2000µohms and R s=10Kohms where R s=R1+R2.(c)Sketch a block diagram that represents the circuit equations.FIGURE P2.20The source follower or common drain amplifier using an FET.7P2.22Figure P2.22shows two pendulums suspended from frictionless pivotsand connected at their midpoints by a spring[1].Assume that each pendulumcan be represented by a mass M at the end of a mass less bar of length L.Also assume that the displacement is small and linear approximations can be usedfor sinθand cosθ.The spring located in the middle of the bars is unstretched whenθ1=θ2.The input force is represented by f(t),which influences the left-handed bar only.(a)Obtain the equations of motion,and sketch a block diagram from them.(b)Determine the transfer function T(s)=θ1(s)/F(s).(c)Sketchthe location of the poles and zeros of T(s)on the s-plate.FIGURE P2.22The bars are each of length L and spring is located at L/2.P2.29We desire to balance a rolling ball on a tilting beam as shown in FigureP2.29.We will assume the motor input current controls the torque with negli-gible friction.Assume the beam may be balanced near the horizontal(φ=0); therefore,we have a small deviation ofφ.Find the transfer function X(s)/I(s),and draw a block diagram illustrating the transfer function showingφ(s),X(s)and I(s).FIGURE P2.29Tilting beam and ball.8P2.35The suspension system for one wheel of an old-fashioned pickup truck is illustrated in Figure P2.35.The mass of the vehicle is m1and the mass of the wheel is m2.The suspension spring has a spring constant k1,and the tire has a spring constant k2.The damping constant of the shock absorber is b. Obtain the transfer function Y1(s)/X(s),which represents the vehicle response to bumps in the road.FIGURE P2.35Pickup truck suspension.9。