(a) Torsional spring-mass system. (b) Spring element.

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1 m v s T s v a (a) T a (b) T a FIGURE 2.1 (a) Torsional spring-mass system. (b) Spring element.

2 by ky Wall friction, b Mass M k y M y r(t) Force r(t) (a) (b) FIGURE 2.2 (a) Spring-mass-damper system. (b) Free-body diagram.

3 Voltage v(t) e a 2t 0 Time 2(p/b 2 ) FIGURE 2.4 Typical voltage response for underdamped RLC circuit.

4 f Mass M f y 2 Nonlinear spring y Spring force f 0 df dy y y 0 Equilibrium (operating point) y 0 y (a) (b) FIGURE 2.5 (a) A mass sitting on a nonlinear spring. (b) The spring force versus y.

5 T u Length L p p 2 p 2 p u Mass M (a) (b) FIGURE 2.6 Pendulum oscillator.

6 jv s pole zero FIGURE 2.7 An s-plane pole and zero plot.

7 s 1 jv jv n 1 z 2 u cos 1 z v n 2zv n zv n 0 s s 2 jv n 1 z 2 FIGURE 2.9 An s-plane plot of the poles and zeros of Y(s).

8 z 1 z increasing z 0 v n jv jv n z 1 z 1 z 1 0 s FIGURE 2.10 The locus of roots as z varies with vn constant.

9 y(t) y 0 Overdamped case 0 Time Underdamped case e zv nt envelope FIGURE 2.12 Response of the spring-mass-damper system.

10 Friction b 2 Friction b 1 M 2 k Velocity v 2 (t) Current r(t) v 1 (t) R 1 v 2 (t) C 1 R 2 C 2 L M 1 Velocity v 1 (t) Force r(t) (a) (b) FIGURE 2.16 (a) Two-mass mechanical system. (b) Two-node electric circuit analog C1 = M1, C2 = M2, L = 1/k, R1 = 1/b1, R2 = 1/b2.

11 Armature R a L a Rotor windings Brush Stator winding N Shaft i a V f R f i f (t) Field L f i a v, u Load Inertia J Friction b Inertia load Angle u S Bearings Brush Commutator (a) (b) FIGURE 2.17 A dc motor (a) wiring diagram and (b) sketch.

12 FIGURE 2.18 A pancake dc motor with a flat-wound armature and a permanent magnet rotor. These motors are capable of providing high torque with a low rotor inertia. A typical mechanical time constant is in the range of 15 ms. (Courtesy of Mavilor Motors.)

13 Disturbance T d (s) Field Load Speed I T 1 f (s) m (s) T L (s) 1 v(s) V f (s) K 1 m R f L f s Js b s Position u(s) Output FIGURE 2.19 Block diagram model of field-controlled dc motor.

14 Dorf/Bishop Disturbance T d (s) V a (s) Armature K m R a L a s T m (s) T L (s) 1 Js b Speed v(s) 1 s Position u(s) Back emf K b FIGURE 2.20 Armature-controlled dc motor.

15 V f (s) R f L f I a J, b I f u, v Table 2.5 Transfer Functions of Dynamic Elements and Networks 5. dc motor, field-controlled, rotational actuator

16 V a (s) R a I a L a V b I f u, v J, b Table 2.5 Transfer Functions of Dynamic Elements and Networks 6. dc motor, armature-controlled, rotational actuator

17 Shaft u(s), v(s) V 2 (s) Table 2.5 Transfer Functions of Dynamic Elements and Networks 13. Tachometer, velocity sensor

18 V f (s) K G(s) m s(js b)(l f s R f ) Output u(s) FIGURE 2.22 Block diagram of dc motor.

19 R 1 (s) R 2 (s) Inputs System Outputs Y 1 (s) Y 2 (s) FIGURE 2.23 General block representation of two-input, two-output system.

20 R 1 (s) G 11 (s) Y 1 (s) R 2 (s) G 12 (s) G 21 (s) G 22 (s) Y 2 (s) FIGURE 2.24 Block diagram of interconnected system.

21 X 1 G 1 (s) X 2 X 3 X 1 X 3 G 2 (s) G 1 G 2 or X 1 G 2 G 1 X 3 Table 2.6 Block Diagram Transformations 1.Combining blocks in cascade

22 X 1 G X 3 X 1 G X 3 X 2 G X 2 Table 2.6 Block Diagram Transformations 2. Moving a summing point behind a block

23 X 1 G X 2 X 1 G X 2 X 2 X 2 G Table 2.6 Block Diagram Transformations 3. Moving a pickoff point ahead of a block

24 X 1 G X 2 X 1 G X 2 X 1 X 1 1 G Table 2.6 Block Diagram Transformations 4. Moving a pickoff point behind a block

25 X 1 G X 3 X 1 G X 3 X 2 1 G X 2 Table 2.6 Block Diagram Transformations 5. Moving a summing point ahead of a block

26 X 1 X 2 G X 1 G 1 GH X 2 H Table 2.6 Block Diagram Transformations 6. Eliminating a feedback loop

27 H 2 R(s) G 1 G 2 G 3 G 4 Y(s) H 1 H 3 FIGURE 2.26 Multiple-loop feedback control system.

28 H 2 G 4 R G 1 G 2 G 3 G 4 Y(s) H 1 H 3 (a) H 2 G 4 R G 1 G 2 G 3 G 4 1 G 3 G 4 H 1 Y(s) H 3 (b) R G 1 G 2 G 3 G 4 1 G 3 G 4 H 1 G 2 G 3 H 2 H 3 Y(s) R(s) G 1 G 2 G 3 G 4 1 G 3 G 4 H 1 G 2 G 3 H 2 G 1 G 2 G 3 G 4 H 3 Y(s) (c) (d) FIGURE 2.27 Block diagram reduction of the system of Fig

29 V f (s) G(s) u(s) FIGURE 2.28 Signal-flow graph of the dc motor.

30 R 1 (s) R 2 (s) G 11 (s) G 12 (s) G 21 (s) G 22 (s) Y 1 (s) Y 2 (s) FIGURE 2.29 Signal-flow graph of interconnected system.

31 a 11 1 r 1 x 1 a 21 a 12 1 r 2 x 2 a 22 FIGURE 2.30 Signal-flow graph of two algebraic equations.

32 H 2 H 3 L 1 L 2 R(s) G 1 G 2 G 3 G 4 Y(s) G 5 G 6 G 7 G 8 L 3 L 4 H 6 H 7 FIGURE 2.31 Two-path interacting system.

33 V a (s) K G m 1 (s) (R a L a s) T d (s) 1 1 G 1 (s) T m (s) 1 1 T L (s) 1 G 2 (s) (Js b) G 2 (s) 1 s u(s) K b FIGURE 2.32 The signal-flow graph of the armature-controlled dc motor.

34 G 7 G 8 R(s) 1 G 1 G 2 G 3 G 4 G 5 G 6 Y(s) H 4 H 1 H 2 H 3 FIGURE 2.33 Multiple-loop system.

35 v d (t) Desired velocity Power amplifier Armaturecontrolled motor v d (t) Vehicle velocity Sensor (a) R 2 T d (s) v t R 1 v 1 v in R 3 v2 2e 3v 1 v v 2 K m R a L a s 1 Js b v(s) Speed R4 K b v t K t (b) v in v d (s) T d (s) G 3 (s) G 1 (s) G 2 (s) (s 1) 2s 0.5 v(s) 0.1 (c) v d (s) G 1 G 2 v(s) (d) FIGURE 2.35 Speed control of an electric traction motor.

36 >>y0=0.15; >>wn=sqrt(2); >>zeta=1/(2*sqrt(2)); >>t=[0:0.1:10]; >>unforced unforced.m %Compute Unforced Response to an Initial Condition % c=(y0/sqrt(1-zeta^2)); y=c*exp(-zeta*wn*t).*sin(wn*sqrt(1-zeta^2)*t+acos(zeta)); % bu=c*exp(-zeta*wn*t);bl=-bu; % plot(t,y,t,bu,'--',t,bl,'--'), grid xlabel('time (sec)'), ylabel('y(t) (meters)') legend(['\omega_n=',num2str(wn),' \zeta=',num2str(zeta)]) v n z y(0)/1 z 2 e zv nt envelope FIGURE 2.40 Script to analyze the spring-mass-damper.

37 v n , z y(t) (meters) y(0) 1 z 2 ezv nt y(0) 1 z 2 ezv nt Time (sec) FIGURE 2.41 Spring-mass-damper unforced response.

38 >>p=[ ]; >>r=roots(p) r = i i >>p=poly(r) p = p(s) s 3 3s 2 4 Calculate roots of p(s) 0. Reassemble polynomial from roots FIGURE 2.42 Entering the polynomial p(s) = s3 1 3s2 1 4 and calculating its roots.

39 >>p=[3 2 1]; q=[1 4]; >>n=conv(p,q) n= >>value=polyval(n,-5) value = Multiply p and q. n(s) 3s 3 14s 2 9s 4 Evaluate n(s) at s FIGURE 2.43 Using conv and polyval to multiply and evaluate the polynomials (3s2 1 2s 1 1)(s 1 4).

40 >> num1=[10];den1=[1 2 5]; >> sys1=tf(num1,den1) Transfer function: Transfer function object G(s) num den s^2 + 2 s + 5 G 1 (s) sys = tf(num,den) >> num2=[1];den2=[1 1]; >> sys2=tf(num2,den2) Transfer function: s + 1 >> sys=sys1+sys2 Transfer function: s^ s s^3 + 3 s^2 + 7 s + 5 G 2 (s) G 1 (s) G 2 (s) FIGURE 2.44 (a) The tf function. (b) Using the tf function to create transfer function objects and adding them using the 1 operator.

41 Poles >> sys=tf([1 10],[1 2 1]) Transfer function: s s^2 + 2 s + 1 sys p=pole(sys) z=zero(sys) Transfer funtion object >> p=pole(sys) p= -1-1 The system poles. Zeros >> z=zero(sys) z= -10 The system zeros. (a) (b) FIGURE 2.45 (a) The pole and zero functions. (b) Using the pole and zero functions to compute the pole and zero locations of a linear system.

42 P: pole locations in column vector Z: zero locations in column vector num G(s) sys den [P,Z]=pzmap(sys) FIGURE 2.46 The pzmap function.

43 2 Pole-Zero Map Zeros Imag. Axis Poles Real Axis FIGURE 2.47 Pole zero map for G(s)/H(s).

44 >>numg=[6 0 1]; deng=[ ];sysg=tf(numg,deng); >>z=zero(sysg) z = i i Compute poles and zeros of G(s). >>p=pole(sysg) p = i i Expand H(s). >>n1=[1 1]; n2=[1 2]; d1=[1 2*i]; d2=[1-2*i]; d3=[1 3]; >>numh=conv(n1,n2); denh=conv(d1,conv(d2,d3)); >>sysh=tf(numh/denh) Transfer function: s^2 + 3 s + 2 s^3 + 3 s^2 + 4 s + 12 >>sys=sysg/sysh Transfer function: 6 s^ s^ s^ s^2 + 4 s +12 s^5 + 6 s^ s^ s^2 + 9 s + 2 >>pzmap(sys) H(s) G(s) sys H(s) Pole-zero map. FIGURE 2.48 Transfer function example for G(s) and H(s).

45 U(s) System 1 G 1 (s) System 2 G 2 (s) Y(s) (a) T(s) Y(s) sys U(s) G 1 (s) sys1 G 2 (s) sys2 [sys]=series(sys1,sys2) (b) FIGURE 2.50 (a) Block diagram. (b) The series function.

46 R(s) G c (s) s 1 s 2 U(s) 1 G(s) 500 s 2 Y(s) (a) >>numg=[1]; deng=[ ]; sysg=tf(numg,deng); >>numh=[1 1]; denh=[1 2]; sysh=tf(numh,denh); >>sys=series(sysg,sysh); >>sys Transfer function: s s^ s^2 G c (s)g(s) (b) FIGURE 2.51 Application of the series function.

47 U(s) System 1 G 1 (s) System 2 G 2 (s) Y(s) (a) T(s) Y(s) sys U(s) G 1 (s) sys1 G 2 (s) sys2 [sys]=parallel(sys1,sys2) (b) FIGURE 2.52 (a) Block diagram. (b) The parallel function.

48 R(s) E a (s) Controller G c (s) U(s) Process G(s) Y(s) FIGURE 2.53 A basic control system with unity feedback.

49 R(s) System 1 G c (s)g(s) Y(s) (a) T(s) Y(s) R(s) sys G c (s)g(s) sys1 1 - positive feedback 1 - negative feedback (default) [sys]=feedback(sys1,[1],sign) (b) FIGURE 2.54 (a) Block diagram. (b) The feedback function with unity feedback.

50 R(s) System 1 G(s) Y(s) System 2 H(s) (a) T(s) Y(s) sys R(s) G(s) sys1 H(s) sys2 1 - pos. feedback 1 - neg. feedback (default) [sys]=feedback(sys1,sys2,sign) (b) FIGURE 2.55 (a) Block diagram. (b) The feedback function.

51 R(s) E a (s) G c (s) s 1 s 2 U(s) 1 G(s) 500 s 2 Y(s) (a) >>numg=[1]; deng=[ ]; sys1=tf(numg,deng); >>numc=[1 1]; denc=[1 2]; sys2=tf(numc,denc); >>sys3=series(sys1,sys2); >>sys=feedback(sys3,[1]) Transfer function: s s^ s^2 + s + 1 Y(s) R(s) G c (s)g(s) 1 G c (s)g(s) (b) FIGURE 2.56 (a) Block diagram. (b) Application of the feedback function.

52 R(s) E a (s) 1 G(s) 500 s 2 Y(s) H(s) s 1 s 2 (a) >>numg=[1]; deng=[ ]; sys1=tf(numg,deng); >>numh=[1 1]; denh=[1 2]; sys2=tf(numh,denh); >>sys=feedback(sys1,sys2); >>sys Transfer function: s s^ s^2 + s + 1 Y(s) R(s) G(s) 1 G(s)H(s) (b) FIGURE 2.58 Application of the feedback function: (a) block diagram, (b) MATLAB script.

53 >>ng1=[1]; dg1=[1 10]; sysg1=tf(ng1,dg1); >>ng2=[1]; dg2=[1 1]; sysg2=tf(ng2,dg2); >>ng3=[1 0 1]; dg3=[1 4 4]; sysg3=tf(ng3,dg3); >>ng4=[1 1]; dg4=[1 6]; sysg4=tf(ng4,dg4); >>nh1=[1 1]; dh1=[1 2]; sysh1=tf(nh1,dh1); >>nh2=[2]; dh2=[1]; sysh2=tf(nh2,dh2); >>nh3=[1]; dh3=[1]; sysh3=tf(nh3,dh3); >>sys1=sys2/sys4; >>sys2=series(sysg3,sysg4); >>sys3=feedback(sys2,sysh1,+1); >>sys4=series(sysg2,sys3); >>sys5=feedback(sys4,sys1); >>sys6=series(sysg1,sys5); >>sys=feedback(sys6,[1]); Step 1 Step 2 Step 3 Step 4 Step 5 Transfer function: s^5 + 4 s^4 + 6 s^3 + 6 s^2 + 5 s s^ s^ s^ s^ s^ s FIGURE 2.59 Multiple-loop block reduction.

54 No common factors T(s) sys Possible common factors G(s) sys1 sys=minreal(sys1) FIGURE 2.60 The minreal function.

55 >>num=[ ]; den=[ ]; >>sys1=tf(num,den); >>sys=minreal(sys1); Cancel common factors. Transfer function: s^ s^ s^ s s^ s^ s^ s^ s FIGURE 2.61 Application of the minreal function.

56 >>num1=[10]; den1=[1 1]; sys1=tf(num1,den1); >>num2=[1]; den2=[2 0.5]; sys2=tf(num2,den2); >>num3=[540]; den3=[1]; sys3=tf(num3,den3); >>num3=[0.1]; den4=[1]; sys4=tf(num4,den4); >>sys5=series(sys1,sys2); >>sys6=feedback(sys5,sys4); >>sys7=series(sys3,sys6); >>sys=feedback(sys7,[1]) Eliminate inner loop. Compute closed-loop transfer function. Transfer function: s^ s v(s) v d (s) FIGURE 2.62 Electric traction motor block reduction.

57 u(t) Step input System G(s) Output y(t) t t (a) y(t) output response at t T simulation time G(s) sys t T: user-supplied time vector or t T final : simulation final time (optional) [y,t]=step(sys,t) (b) FIGURE 2.63 The step function.

58 Wheel velocity Time (sec) (a) mresp.m % This script computes the step % response of the Traction Motor % Wheel Velocity % num=[5400]; den=[ ]; sys=tf(num,den); t=[0:0.005:3]; [y,t]=step(sys,t); plot(t,y),grid xlabel('time (sec)') ylabel('wheel velocity') (b) FIGURE 2.64 (a) Traction motor wheel velocity step response. (b) Matlab script.

59 Motor Arm Flexure Head FIGURE 2.65 Head mount for reader, showing flexure.

60 Desired head position Error Control device Amplifier Input voltage Sensor Actuator and read arm dc motor and arm Actual head position Read head and index track on disk (a) R(s) Amplifier E(s) V(s) K a Sensor H(s) = 1 Motor and arm G(s) G(s) = K m s(js+b)(ls+r) Y(s) (b) FIGURE 2.66 Block diagram model of disk drive read system.

61 R(s) K a G(s) Y(s) FIGURE 2.67 Block diagram of closed-loop system.

62 y(t ) (rad) Time (sec) FIGURE 2.68 The system response of the system shown in Fig for R(s) s

63 F Force of material placed in truck bed F x 1 m v Truck vehicle mass b 1 k 1 Shock absorber Tire x 2 m t b 2 k 2 Tire (a) (b) FIGURE P2.47 Truck support model.

(a) Torsional spring-mass system. (b) Spring element.

(a) Torsional spring-mass system. (b) Spring element. m v s T s v a (a) T a (b) T a FIGURE 2.1 (a) Torsional spring-mass system. (b) Spring element. by ky Wall friction, b Mass M k y M y r(t) Force r(t) (a) (b) FIGURE 2.2 (a) Spring-mass-damper system. (b)

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