Apago PDF Enhancer

E1C02 11/03/2010 11:29:55 Page 102

1 N-m-s/rad1 N-m-s/rad

1 N-m/rad

1 N-m-s/rad

θ2(t)T(t)

1 kg-m2

FIGURE P2.17

32. For the rotational mechanical system with gearsshown in Figure P2.18, find the transfer function,GðsÞ ¼ u3ðsÞ=TðsÞ. The gears have inertia and bear-ing friction as shown. [Section: 2.7]

N2

N4

θ 3(t)

N3

N1

T(t)

J2, D2

J4, D4

J1, D1

J3, D3

J5, D5

FIGURE P2.18

33. For the rotational system shown in Figure P2.19, findthe transfer function, GðsÞ ¼ u2ðsÞ=TðsÞ. [Section: 2.7]

J1 = 2 kg-m2

J2 = 1 kg-m2

J3 = 16 kg-m2

K = 64 N-m/rad

N1 = 4T(t)

D1 = 1 N-m-s/rad

N2 = 12

D2 = 2 N-m-s/rad

N3 = 4

N4 = 16

θ 2(t)

D3 = 32 N-m-s/rad

FIGURE P2.19

34. Find the transfer function, GðsÞ ¼ u2ðsÞ=TðsÞ, forthe rotational mechanical system shown in FigureP2.20. [Section: 2.7]

N1 = 5

N3 = 25

N2 = 50

θ

1000 N-m-s/rad

250 N-m/rad

200 kg-m2

3 kg-m2

2(t)

T(t)200 kg-m2

3 N-m/rad

FIGURE P2.20

35. Find the transfer function, GðsÞ ¼ u4ðsÞ=TðsÞ, forthe rotational system shown in Figure P2.21.[Section: 2.7]

θ3(t)θ 2(t)

N3 = 23N2 = 110

2 N-m/rad

N1 = 26 N4 = 120

26 N-m-s/radT(t) θ1(t) θ4(t)

FIGURE P2.21

36. For the rotational system shown in Figure P2.22,find the transfer function, GðsÞ ¼ uLðsÞ=TðsÞ. [Sec-tion: 2.7]

3 N-m/rad

0.04 N-m-s/rad

2 N-m-s/rad

θL(t)T(t)1 kg-m2N2 = 33

N1 = 11 N4 = 10

N3 = 50

FIGURE P2.22

37. For the rotational system shown inFigure P2.23, write the equations ofmotion from which the transfer func-tion, GðsÞ ¼ u1ðsÞ=TðsÞ, can be found.[Section: 2.7]

K

N1

N4

DLJ4

J1 DN3

J3

N2

J2

Ja

θT(t) 1(t)

JL

FIGURE P2.23

38. Given the rotational system shown in Figure P2.24,find the transfer function, GðsÞ ¼ u6ðsÞ=u1ðsÞ.[Section: 2.7]

102 Chapter 2 Modeling in the Frequency Domain

Apago PDF Enhancer

E1C02 11/03/2010 11:29:55 Page 103

θ1(t)N1

J1, DN2

N3

J3

K1

N4

K2

D T(t) θ6(t)

D

J2, D

J4, DJ5 J6

FIGURE P2.24

39. In the system shown in Figure P2.25, the inertia, J, ofradius, r, is constrained to move only about the station-ary axis A. A viscous damping force of translationalvalue f v exists between the bodies J and M. If anexternal force, f(t), is applied to the mass, find thetransfer function, GðsÞ ¼ uðsÞ=FðsÞ. [Sections: 2.5; 2.6]

Kfv

f(t)Mfv

θr

JA

M

(t)

FIGURE P2.25

40. For the combined translational and rotational sys-tem shown in Figure P2.26, find the transfer func-tion, GðsÞ ¼ XðsÞ=TðsÞ. [Sections: 2.5; 2.6; 2.7]

D2 = 1 N-m-s/rad

3 kg-m2N1 = 10

N4 = 60N2 = 20

Radius = 2 m1 N-m-s/rad J = 3 kg-m2

Idealgear 1:1

3 N/m2 N-s/m

T(t)

x(t)

N3 = 30

2 kg

FIGURE P2.26

41. Given the combined translational androtational system shown in FigureP2.27, find the transfer function,GðsÞ ¼ XðsÞ=TðsÞ. [Sections: 2.5; 2.6]

D3

J2Radius = r

K1 Idealgear 1:1

K2

M

fv

J1 J3

T(t)

x(t)

FIGURE P2.27

42. For the motor, load, and torque-speed curve shownin Figure P2.28, find the transfer function,GðsÞ ¼ uLðsÞ=EaðsÞ. [Section: 2.8]

J2 = 18 kg-m2

N1 = 50

N2 = 150

Ra

ea(t)θL(t)

+

D1 = 8 N-m-s/rad

D2 = 36 N-m-s/rad

T (N-m)

100

50 V

150 (rad/s)ω

θJ1 = 5 kg-m2

–

FIGURE P2.28

43. The motor whose torque-speed characteristics areshown in Figure P2.29 drives the load shown in thediagram. Some of the gears have inertia. Find thetransfer function, GðsÞ ¼ u2ðsÞ=EaðsÞ. [Section: 2.8]

J = 2 kg-m2 2

N1 = 10

N2 = 20 N3 = 10

θ2

J 16 kg-m42

D = 32 N-m-s/rad

T(N-m)

5

RPM

5 V

600π

J 1 kg-m1 = 2

J 2 kg-m32=

=

(t)N4 = 20

ea(t) Motor+

–

FIGURE P2.29

Problems 103

Apago PDF Enhancer

E1C02 11/03/2010 11:29:56 Page 104

44. A dc motor develops 55 N-m of torque at a speedof 600 rad/s when 12 volts are applied. It stalls outat this voltage with 100 N-m of torque. If theinertia and damping of the armature are 7 kg-m2

and 3 N-m-s/rad, respectively, find the transferfunction, GðsÞ ¼ uLðsÞ=EaðsÞ, of this motor if itdrives an inertia load of 105 kg-m2 through a geartrain, as shown in Figure P2.30. [Section: 2.8]

N1 = 12

N2 = 25 N3 = 25

θL

θm

Load

(t)

(t)

N4 = 72

ea(t) Motor+

–

FIGURE P2.30

45. In this chapter, we derived thetransfer function of a dc motorrelating the angular displace-ment output to the armaturevoltage input. Often we want to control the out-put torque rather than the displacement. Derivethe transfer function of the motor that relatesoutput torque to input armature voltage.[Section: 2.8]

46. Find the transfer function, GðsÞ ¼ XðsÞ=EaðsÞ, forthe system shown in Figure P2.31. [Sections: 2.5–2.8]

N2 = 20

Radius = 2 m

D = 1 N-m-s/radN1 = 10ea(t)J = 1 kg-m2

Idealgear 1:1

fv = 1 N-s/m

x(t)Ja = 1 kg-m2

Da = 1 N-m-s/rad

Ra = 1

Kb = 1 V-s/rad

Kt = 1 N-m/A

Ω

M = 1 kg

Motor+

For the motor:

–

FIGURE P2.31

47. Find the series and parallel analogs for the transla-tional mechanical system shown in Figure 2.20 in thetext. [Section: 2.9]

48. Find the series and parallel analogs for the rota-tional mechanical systems shown in Figure P2.16(b)in the problems. [Section: 2.9]

49. A system’s output, c, is related to the system’s input,r, by the straight-line relationship, c ¼ 5r þ 7. Is thesystem linear? [Section: 2.10]

50. Consider the differential equation

d2x

dt2þ 3

dx

dtþ 2x ¼ f ðxÞ

where f(x) is the input and is a function of theoutput, x. If f ðxÞ ¼ sinx, linearize the differentialequation for small excursions. [Section: 2.10]

a. x ¼ 0

b. x ¼ p

51. Consider the differential equation

d3x

dt3þ 10

d2x

dt2þ 31

dx

dtþ 30x ¼ f ðxÞ

where f(x) is the input and is a function of theoutput, x. If f ðxÞ ¼ e�x, linearize the differentialequation for x near 0. [Section: 2.10]

52. Many systems are piecewise linear. That is, over alarge range of variable values, the system can bedescribed linearly. A system with amplifier satura-tion is one such example. Given the differentialequation

d2x

dt2þ 17

dx

dtþ 50x ¼ f ðxÞ

assume that f(x) is as shown in Figure P2.32. Writethe differential equation for each of the followingranges of x: [Section: 2.10]

a. �1 < x < �3

b. �3 < x < 3

c. 3 < x < 1

–3 3

–6

6

x

f(x)

FIGURE P2.32

104 Chapter 2 Modeling in the Frequency Domain