Mechanical Engineering Chapter 20 Problem The Inertia Matrix Rigid Body Terms Bodyxed Coordinate System With Its

Problem 20.104 The inertia matrix of a rigid body in

terms of a body-ﬁxed coordinate system with its origin

at the center of mass is

[I]=41−1

12 0

−10 6

kg-m2.

If the rigid body’s angular velocity is ω=10i−5j+

10k(rad/s), what is its angular momentum about its cen-

ter of mass?

Problem 20.105 What is the moment of inertia of the

rigid body in Problem 20.104 about the axis that passes

through the origin and the point (4, −4, 7) m?

Strategy: Determine the components of a unit vector

parallel to the axis, and use Eq. (20.43).

Problem 20.106 Determine the inertia matrix of the

0.6-slug thin plate in terms of the coordinate system

shown.

y

x

6 in 1.5 in

3 in

0.04 slug. The moments and products of inertia of the cut-out are

IC

xx =mCR2

C

IC

yy =mCR2

C

mC=2.656 ×10−3slug-ft2,

C

[I]0=



0.04 0 0

00.04 0



1.563 ×10−400

714

Problem 20.107 At t=0, the plate in Problem 20.106

has angular velocity ω=10i+10j(rad/s) and is sub-

jected to the force F=−10k(lb) acting at the point

(0, 6, 0) in. No other forces or couples act on the plate.

What are the components of its angular acceleration at

that instant?

Solution: The coordinates of the center of mass are (−0.01667, 0,

0) ft. The vector from the center of mass to the point of application of

the force is rF/G =0.01667i+0.5j(ft). The moment about the center

of mass of the plate is

ijk

Problem 20.108 The inertia matrix of a rigid body in

terms of a body-ﬁxed coordinate system with its origin

mass?

Solution: Use general motion, Eq. (20.19),

with α=dω

dt =0. The coordinate system is rotating with angular

velocity ω, from which ω=ω. Eq. (20.19) reduces to



Mφx



Problem 20.109 If the total moment about the center

of mass of the rigid body described in Problem 20.108

is zero, what are the components of its angular acceler-

ation?

716

Problem 20.110 The slender bar of length land

mass mis pinned to the L-shaped bar at O. The L-

shaped bar rotates about the vertical axis with a constant

angular velocity ω0. Determine the value of ω0necessary

for the bar to remain at a constant angle βrelative to

the vertical.

β

O

ω0

ω

b

l

Solution: Since the point Ois not ﬁxed, this is general motion, in

aG=−ω2

0+bsin β+L

2sin2βi

MG=rO/G ×A=rO/G ×(maG−W)

Carry out the matrix multiplication:



Mx

0



mω2

0bL

2cos β−mgL

2sin β+mω2

0L2

4sin βcos β

mL2

Problem 20.111 A slender bar of length land mass m

is rigidly attached to the center of a thin circular disk of

radius Rand mass m. The composite object undergoes

a motion in which the bar rotates in the horizontal plane

with constant angular velocity ω0about the center of

mass of the composite object and the disk rolls on the

ﬂoor. Show that ω0=2√g/R.

R

l

0

ω

0

ω

Solution: Measuring from the left end of the slender bar, the dis-

tance to the center of mass is

2m+mL

y, Z

3L

L

4

718

Problem 20.112* The thin plate of mass mspins

about a vertical axis with the plane of the plate

perpendicular to the ﬂoor. The corner of the plate at

Orests in an indentation, so that it remains at the same

point on the ﬂoor. The plate rotates with constant angular

velocity ω0and the angle βis constant.

(a) Show that the angular velocity ω0is related to the

angle βby

hω2

0

g=2 cos β−sin β

sin2β−2 sin βcos β−cos2β

(b) The equation you obtained in (a) indicates that

ω0=0 when 2 cos β−sin β=0. What is the

interpretation of this result?

2h

O

h

β

ω

0

Problem 20.113* In Problem 20.112, determine the

range of values of the angle βfor which the plate will

remain in the steady motion described.

Solution: From the solution to Problem 20.112

8=67.50◦.

f( ) vs

β

0 10

20 30 40 50 60 70 80 90

–2

–4

–5

1

2

3

4

5

720

c

currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.

Problem 20.114* Arm BC has a mass of 12 kg, and

its moments and products of inertia, in terms of the coor-

dinate system shown, are Ixx =0.03 kg-m2,Iyy =Izz =

4 kg-m2, and Ixy =Iyz =Ixz =0. At the instant shown,

arm AB is rotating in the horizontal plane with a con-

stant angular velocity of 1 rad/s in the counterclockwise

direction viewed from above. Relative to arm AB, arm

BC is rotating about the zaxis with a constant angu-

lar velocity of 2 rad/s. Determine the force and couple

exerted on arm BC at B.

2 rad/s

C

BA

x

y

1 rad/s

40°

700 mm

300 mm

Solution: In terms of the body-ﬁxed coordinate system shown, the

In terms of the angle θ, the angular velocity of the coordinate system is





MBx

MBy +0.3FBz

=



0.03 0 0

02.92 0





αx

αy



xx

The acceleration of Btoward Ais (1 rad/s)2(0.7m)=0.7 m/s2,so

FBz =m(0.771).

Problem 20.115 Suppose that you throw a football in a

z

(0.001 −0.003)cos 25◦=2.21 rev/s.

Problem 20.116 Sketch the body and space cones for

the motion of the football in Problem 20.115.

Solution: The angle βis given by tan β=Izz

x

X

cone

ψ

722

c

currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.

Problem 20.117 The mass of the homogeneous thin

plate is 1 kg. For a coordinate system with its origin at

O, determine the plate’s principal moments of inertia

and the directions of unit vectors parallel to the corre-

sponding principal axes.

x’

y’

z’

160

mm

160

mm

160

mm

O

400 mm

I(p)

xx =ρT bh3

yy =ρT hb3

The moments and products of inertia of the object : The moments and

products of inertia about Oare

Ixx =I(p)

xx −I(0)

xx =0.02316 kg-m2,

Iyy =I(p)

yy −I(0)

yy =0.04053 kg-m2,

Izz =I(p)

zz −I(0)

zz =0.06370 kg-m2,

Ixy =I(p)

xy −I(0)

xy =0.01691 kg-m2,

Ixz =Iyz =0.

f (I) vs I

.00002

724

Problem 20.118 The airplane’s principal moments of

inertia, in slug-ft2, are Ixx =8000, Iyy =48,000, and

Izz =50,000.

(a) The airplane begins in the reference position shown

and maneuvers into the orientation ψ=θ=φ=

45◦. Draw a sketch showing the plane’s orientation

relative to the XY Z system.

(b) If the airplane is in the orientation described in (a),

the rates of change of the Euler angles are ˙

ψ=0,

˙

θ=0.2 rad/s, and ˙

φ=0.2 rad/s, and the second

derivatives of the angles with respect to time are

zero, what are the components of the total moment

about the airplane’s center of mass?

Z, z

X, x

Y, y

Problem 20.119 What are the x,y, and zcomponents

of the angular acceleration of the airplane described in

Problem 20.118?

Problem 20.120 If the orientation of the airplane in

0.2 rad/s, and ˙

φ=0.1 rad/s, and the components of the

total moment about the center of mass of the plane

are Mx=400 ft-lb, My=1200 ft-lb, and Mz=

0, what are the x,y, and zcomponents of the airplane’s

angular acceleration?

726