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E1C02

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102

Chapter 2 1 N-m/rad

T(t)

Modeling in the Frequency Domain θ 2(t)

1 kg-m2 1 N-m-s /rad 1 N-m-s /rad

1 N-m-s /rad

35. Find the transfer function, GðsÞ ¼ u4 ðsÞ=TðsÞ, for the rotational system shown in Figure P2.21. [Section: 2.7] θ 4(t)

T(t) θ 1(t)

26 N-m-s/rad

N1 = 26 N4 = 120

θ 3(t)

θ 2(t)

FIGURE P2.17

N3 = 23

N2 = 110

32. For the rotational mechanical system with gears shown in Figure P2.18, find the transfer function, GðsÞ ¼ u3 ðsÞ=TðsÞ. The gears have inertia and bearing friction as shown. [Section: 2.7]

2 N-m/rad

FIGURE P2.21

36. For the rotational system shown in Figure P2.22, find the transfer function, GðsÞ ¼ uL ðsÞ=TðsÞ. [Section: 2.7]

T(t) N1 J1, D1 N2

N3

2 N-m-s /rad 3 N-m /rad

J3, D3 θ (t) 3

J2, D2

N2 = 33 T(t)

N4 J4, D4

N3 = 50

1 kg-m2

J5, D5

θ L(t)

N4 = 10

N1 = 11

0.04 N-m-s /rad

FIGURE P2.18

Apago PDF Enhancer

33. For the rotational system shown in Figure P2.19, find the transfer function, GðsÞ ¼ u2 ðsÞ=TðsÞ. [Section: 2.7]

FIGURE P2.22

T(t)

N1 = 4 J1 = 2 kg-m2 D = 1 N-m-s /rad 1 θ 2(t) N3 = 4 N2 = 12 J2 = 1 kg-m2 D2 = 2 N-m-s/rad K = 64 N-m /rad N4 = 16 J3 = 16 kg-m2 D3 = 32 N-m-s /rad

37. For the rotational system shown in Figure P2.23, write the equations of motion from which the transfer function, GðsÞ ¼ u1 ðsÞ=TðsÞ, can be found. [Section: 2.7]

FIGURE P2.19

34. Find the transfer function, GðsÞ ¼ u2 ðsÞ=TðsÞ, for the rotational mechanical system shown in Figure P2.20. [Section: 2.7] 1000 N-m-s/rad 200 kg-m2

N3 = 25

3 N-m/rad N2 = 50

N1

Ja

J1 N2

K

D

N3

J2

J3 N4

T(t) 3 kg-m2

T(t) θ 1(t)

J4

N1 = 5 θ 2(t)

250 N-m/rad

JL DL

FIGURE P2.23

200 kg-m2

FIGURE P2.20

38. Given the rotational system shown in Figure P2.24, find the transfer function, GðsÞ ¼ u6 ðsÞ=u1 ðsÞ. [Section: 2.7]

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103

Problems θ1(t)

T(t)

N1 J1, D

Radius = r

J2

K1

Ideal D3 gear 1:1

J1

N2

N3

J2, D

J3

K1 T(t)

D

N4

J5

J4, D

J3

θ 6(t) M

J6

x(t)

D

K2

K2

fv

FIGURE P2.24

39. In the system shown in Figure P2.25, the inertia, J, of radius, r, is constrained to move only about the stationary axis A. A viscous damping force of translational value f v exists between the bodies J and M. If an external force, f(t), is applied to the mass, find the transfer function, GðsÞ ¼ uðsÞ=FðsÞ. [Sections: 2.5; 2.6]

FIGURE P2.27

42. For the motor, load, and torque-speed curve shown in Figure P2.28, find the transfer function, GðsÞ ¼ uL ðsÞ=Ea ðsÞ. [Section: 2.8] Ra +

θ (t) fv

J r

–

A

N1 = 50

J1 = 5 kg-m2

ea(t)

N2 = 150

D1 = 8 N-m-s/rad

θ L(t)

J2 = 18 kg-m2

D2 = 36 N-m-s/rad

K fv

M

f(t)

FIGURE P2.25

T (N-m)

100 Apago PDF Enhancer 50 V 150

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

T(t) 3 kg-m2 N2 = 20

N1 = 10 N4 = 60

J = 3 kg-m2 1 N-m-s/rad Radius = 2 m Ideal gear 1:1

N3 = 30

D2 = 1 N-m-s/rad

2 kg x(t) 2 N-s/m

3 N/m

FIGURE P2.26

ω (rad/s)

FIGURE P2.28

43. The motor whose torque-speed characteristics are shown in Figure P2.29 drives the load shown in the diagram. Some of the gears have inertia. Find the transfer function, GðsÞ ¼ u2 ðsÞ=Ea ðsÞ. [Section: 2.8] + N1 = 10 ea(t) Motor – J1 = 1 kg-m2 N3 = 10 N2 = 20 J3 = 2 kg-m 2 J2 = 2 kg-m 2 θ2 (t) N4 = 20

D = 32 N-m-s/rad J4 = 16 kg-m 2

T(N-m) 5

41. Given the combined translational and rotational system shown in Figure P2.27, find the transfer function, GðsÞ ¼ XðsÞ=TðsÞ. [Sections: 2.5; 2.6]

5V 600 π

FIGURE P2.29

RPM

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Chapter 2

Modeling in the Frequency Domain

44. A dc motor develops 55 N-m of torque at a speed of 600 rad/s when 12 volts are applied. It stalls out at this voltage with 100 N-m of torque. If the inertia and damping of the armature are 7 kg-m2 and 3 N-m-s/rad, respectively, find the transfer function, GðsÞ ¼ uL ðsÞ=Ea ðsÞ, of this motor if it drives an inertia load of 105 kg-m2 through a gear train, as shown in Figure P2.30. [Section: 2.8] + ea(t) –

48. Find the series and parallel analogs for the rotational 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 the system linear? [Section: 2.10] 50. Consider the differential equation d2 x dx þ 3 þ 2x ¼ f ðxÞ dt2 dt

θ m(t) Motor

N1 = 12

N2 = 25

where f(x) is the input and is a function of the output, x. If f ðxÞ ¼ sinx, linearize the differential equation for small excursions. [Section: 2.10] a. x ¼ 0

N3 = 25

θ L(t) Load

N4 = 72

FIGURE P2.30

45. In this chapter, we derived the transfer function of a dc motor relating the angular displacement output to the armature voltage input. Often we want to control the output torque rather than the displacement. Derive the transfer function of the motor that relates output torque to input armature voltage. [Section: 2.8]

b. x ¼ p 51. Consider the differential equation d3 x d2 x dx þ 10 þ 31 þ 30x ¼ f ðxÞ 3 2 dt dt dt where f(x) is the input and is a function of the output, x. If f ðxÞ ¼ ex, linearize the differential equation for x near 0. [Section: 2.10] 52. Many systems are piecewise linear. That is, over a

large range of variable values, the system can be Apago PDF Enhancer

46. Find the transfer function, GðsÞ ¼ XðsÞ=Ea ðsÞ, for the system shown in Figure P2.31. [Sections: 2.5–2.8] + ea(t) Motor –

N1 = 10 J = 1 kg-m2

D = 1 N-m-s/rad

N2 = 20 Ideal gear 1:1

Radius = 2 m

For the motor: Ja = 1 kg-m2 Da = 1 N-m-s/rad Ra = 1Ω Kb = 1 V-s/rad Kt = 1 N-m/A

described linearly. A system with amplifier saturation is one such example. Given the differential equation d2 x dx þ 17 þ 50x ¼ f ðxÞ 2 dt dt assume that f(x) is as shown in Figure P2.32. Write the differential equation for each of the following ranges of x: [Section: 2.10] a. 1 < x < 3 b. 3 < x < 3 c. 3 < x < 1

M = 1 kg

f(x)

x(t)

6

fv = 1 N-s/m

FIGURE P2.31

47. Find the series and parallel analogs for the translational mechanical system shown in Figure 2.20 in the text. [Section: 2.9]

–3

3

–6

FIGURE P2.32

x