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VECTOR MECHANICS FOR ENGINEERS: DYNAMICSDYNAMICS
Tenth EditionTenth Edition
CHAPTER
© 2013 The McGraw-Hill Companies, Inc. All rights reserved.
18 Ferdinand P. BeerFerdinand P. Beer
E. Russell Johnston, Jr.E. Russell Johnston, Jr.
Phillip J. CornwellPhillip J. Cornwell
Lecture Notes:Lecture Notes:
Brian P. SelfBrian P. SelfCalifornia Polytechnic State UniversityCalifornia Polytechnic State University
Kinetics of Rigid Bodies in Three Dimensions
© 2013 The McGraw-Hill Companies, Inc. All rights reserved.
Vector Mechanics for Engineers: DynamicsVector Mechanics for Engineers: Dynamics
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Contents
18 - 2
Introduction
Rigid Body Angular Momentum in Three Dimensions
Principle of Impulse and Momentum
Kinetic Energy
Sample Problem 18.1
Sample Problem 18.2
Motion of a Rigid Body in Three Dimensions
Euler’s Equations of Motion and D’Alembert’s Principle
Motion About a Fixed Point or a Fixed Axis
Sample Problem 18.3
Motion of a Gyroscope. Eulerian Angles
Steady Precession of a Gyroscope
Motion of an Axisymmetrical Body Under No Force
© 2013 The McGraw-Hill Companies, Inc. All rights reserved.
Vector Mechanics for Engineers: DynamicsVector Mechanics for Engineers: Dynamics
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2 - 3
Three dimensional analyses are needed to determine the forces and moments on the gimbals of gyroscopes, the rotors of amusement park rides, and the housings of wind turbines.
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Vector Mechanics for Engineers: DynamicsVector Mechanics for Engineers: Dynamics
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Introduction
18 - 4
GG HM
amF
• The fundamental relations developed for the plane motion of rigid bodies may also be applied to the general motion of three dimensional bodies.
IHG • The relation which was used to determine the angular momentum of a rigid slab is not valid for general three dimensional bodies and motion.
• The current chapter is concerned with evaluation of the angular momentum and its rate of change for three dimensional motion and application to effective forces, the impulse-momentum and the work-energy principles.
© 2013 The McGraw-Hill Companies, Inc. All rights reserved.
Vector Mechanics for Engineers: DynamicsVector Mechanics for Engineers: Dynamics
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Rigid Body Angular Momentum in Three Dimensions
18 - 5
• Angular momentum of a body about its mass center,
n
iiii
n
iiiiG mrrmvrH
11Δ
• The x component of the angular momentum,
n
iiiiz
n
iiiiy
n
iiiix
n
iiixiziiyixi
n
iiyiiziix
mxzmyxmzy
mzxzxyy
mrzryH
111
22
1
1
ΔΔΔ
Δ
Δ
dmzxdmxydmzyH zyxx 22
zxzyxyxx III
zzyzyxzxz
zyzyyxyxy
IIIH
IIIH
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Vector Mechanics for Engineers: DynamicsVector Mechanics for Engineers: Dynamics
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Rigid Body Angular Momentum in Three Dimensions
18 - 6
zzyzyxzxz
zyzyyxyxy
zxzyxyxxx
IIIH
IIIH
IIIH
• Transformation of into is characterized by the inertia tensor for the body,
GH
zzyzx
yzyyx
xzxyx
III
III
III
• With respect to the principal axes of inertia,
z
y
x
I
I
I
00
00
00
zzzyyyxxx IHIHIH
• The angular momentum of a rigid body and its angular velocity have the same direction if, and only if, is directed along a principal axis of inertia.
GH
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Vector Mechanics for Engineers: DynamicsVector Mechanics for Engineers: Dynamics
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Rigid Body Angular Momentum in Three Dimensions
18 - 7
• The momenta of the particles of a rigid body can be reduced to:
vm
L
momentumlinear
GHG about momentumangular
zzyzyxzxz
zyzyyxyxy
zxzyxyxxx
IIIH
IIIH
IIIH
• The angular momentum about any other given point O is
GO HvmrH
© 2013 The McGraw-Hill Companies, Inc. All rights reserved.
Vector Mechanics for Engineers: DynamicsVector Mechanics for Engineers: Dynamics
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Rigid Body Angular Momentum in Three Dimensions
18 - 8
• The angular momentum of a body constrained to rotate about a fixed point may be calculated from
GO HvmrH
zzyzyxzxz
zyzyyxyxy
zxzyxyxxx
IIIH
IIIH
IIIH
• Or, the angular momentum may be computed directly from the moments and products of inertia with respect to the Oxyz frame.
n
iiii
n
iiiO
mrr
mvrH
1
1
Δ
Δ
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Vector Mechanics for Engineers: DynamicsVector Mechanics for Engineers: Dynamics
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Principle of Impulse and Momentum
18 - 9
• The principle of impulse and momentum can be applied directly to the three-dimensional motion of a rigid body,
Syst Momenta1 + Syst Ext Imp1-2 = Syst Momenta2
• The free-body diagram equation is used to develop component and moment equations.
• For bodies rotating about a fixed point, eliminate the impulse of the reactions at O by writing equation for moments of momenta and impulses about O.
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Vector Mechanics for Engineers: DynamicsVector Mechanics for Engineers: Dynamics
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Concept Question
2 - 10
At the instant shown, the disk shown rotates with an angular velocity 2 with respect to arm ABC, which rotates around the y axis as shown (1). Determine the directions of the angular momentum of the disc about point A (choose all that are correct).
+x +y +z
-x -y -z
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Vector Mechanics for Engineers: DynamicsVector Mechanics for Engineers: Dynamics
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Concept Question
2 - 11
A homogeneous disk of mass m and radius r is mounted on the vertical shaft AB as shown Determine the directions of the angular momentum of the disc about the mass center G (choose all that are correct).
+x +y +z
-x -y -z
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Vector Mechanics for Engineers: DynamicsVector Mechanics for Engineers: Dynamics
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Kinetic Energy
18 - 12
• Kinetic energy of particles forming rigid body,
)22
2(
Δ
Δ
222212
21
1
2212
21
1
2212
21
xzzxzyyz
yxxyzzyyxx
n
iii
n
iii
II
IIIIvm
mrvm
vmvmT
• If the axes correspond instantaneously with the principle axes,
)( 222212
21
zzyyxx IIIvmT
• With these results, the principles of work and energy and conservation of energy may be applied to the three-dimensional motion of a rigid body.
© 2013 The McGraw-Hill Companies, Inc. All rights reserved.
Vector Mechanics for Engineers: DynamicsVector Mechanics for Engineers: Dynamics
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Kinetic Energy
18 - 13
• Kinetic energy of a rigid body with a fixed point,
)22
2( 22221
xzzxzyyz
yxxyzzyyxx
II
IIIIT
• If the axes Oxyz correspond instantaneously with the principle axes Ox’y’z’,
)( 22221
zzyyxx IIIT
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Vector Mechanics for Engineers: DynamicsVector Mechanics for Engineers: Dynamics
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Concept Question
2 - 14
22
1
2 zI
At the instant shown, the disk shown rotates with an angular velocity 2 with respect to arm ABC, which rotates around the y axis as shown (1). What terms will contribute to the kinetic energy of the disk (choose all that are correct)?
21
1
2 yI 21
1
2 xI
2 21
1
2mL
21
1
2 zI
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Vector Mechanics for Engineers: DynamicsVector Mechanics for Engineers: Dynamics
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Sample Problem 18.1
18 - 15
Rectangular plate of mass m that is suspended from two wires is hit at D in a direction perpendicular to the plate.
Immediately after the impact, determine a) the velocity of the mass center G, and b) the angular velocity of the plate.
SOLUTION:
• Apply the principle of impulse and momentum. Since the initial momenta is zero, the system of impulses must be equivalent to the final system of momenta.
• Assume that the supporting cables remain taut such that the vertical velocity and the rotation about an axis normal to the plate is zero.
• Principle of impulse and momentum yields to two equations for linear momentum and two equations for angular momentum.
• Solve for the two horizontal components of the linear and angular velocity vectors.
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Sample Problem 18.1
18 - 16
SOLUTION:• Apply the principle of impulse and momentum. Since the initial momenta is zero,
the system of impulses must be equivalent to the final system of momenta.
• Assume that the supporting cables remain taut such that the vertical velocity and the rotation about an axis normal to the plate is zero.
kvivv zx
ji yx
Since the x, y, and z axes are principal axes of inertia,
jmaimbjIiIH yxyyxxG
21212
121
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Vector Mechanics for Engineers: DynamicsVector Mechanics for Engineers: Dynamics
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Sample Problem 18.1
18 - 17
• Principle of impulse and momentum yields two equations for linear momentum and two equations for angular momentum.
• Solve for the two horizontal components of the linear and angular velocity vectors.
xmv0
0xvzvmtF Δ
mtFvz Δ
kmtFv
Δ
x
x
mb
HtbF
2121
21 Δ
mbtFx Δ6
y
y
ma
HtaF
2121
21 Δ
matFy Δ6
jbiamab
tF Δ6
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Vector Mechanics for Engineers: DynamicsVector Mechanics for Engineers: Dynamics
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Sample Problem 18.1
18 - 18
kmtFv
Δ
jbiamab
tF Δ6
jmaimbH yxG
21212
121
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Sample Problem 18.2
18 - 19
A homogeneous disk of mass m is mounted on an axle OG of negligible mass. The disk rotates counter-clockwise at the rate 1 about OG.
Determine: a) the angular velocity of the disk, b) its angular momentum about O, c) its kinetic energy, and d) the vector and couple at G equivalent to the momenta of the particles of the disk.
SOLUTION:
• The disk rotates about the vertical axis through O as well as about OG. Combine the rotation components for the angular velocity of the disk.
• Compute the angular momentum of the disk using principle axes of inertia and noting that O is a fixed point.
• The kinetic energy is computed from the angular velocity and moments of inertia.
• The vector and couple at G are also computed from the angular velocity and moments of inertia.
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Vector Mechanics for Engineers: DynamicsVector Mechanics for Engineers: Dynamics
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Sample Problem 18.2
18 - 20
SOLUTION:
• The disk rotates about the vertical axis through O as well as about OG. Combine the rotation components for the angular velocity of the disk.
ji
21
Noting that the velocity at C is zero,
Lr
krL
jriLji
rv CC
12
12
210
0
jLri
11
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Vector Mechanics for Engineers: DynamicsVector Mechanics for Engineers: Dynamics
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Sample Problem 18.2
18 - 21
• Compute the angular momentum of the disk using principle axes of inertia and noting that O is a fixed point.
jLri
11
kIjIiIH zzyyxxO
002412
12
412
12
21
mrmLIH
LrmrmLIH
mrIH
zzz
yyy
xxx
jLrrLmimrHO
12
412
12
21
• The kinetic energy is computed from the angular velocity and moments of inertia.
2
12
4122
12
21
22221
LrrLmmr
IIIT zzyyxx
212
22
81 6
L
rmrT
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Vector Mechanics for Engineers: DynamicsVector Mechanics for Engineers: Dynamics
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Sample Problem 18.2
18 - 22
jLri
11
• The vector and couple at G are also computed from the angular velocity and moments of inertia.
kmrvm
1
jLrmrimr
kIjIiIH zzyyxxG
241
12
21
j
L
rimrHG
212
21
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Motion of a Rigid Body in Three Dimensions
18 - 23
GHM
amF
• Angular momentum and its rate of change are taken with respect to centroidal axes GX’Y’Z’ of fixed orientation.
• Convenient to use body fixed axes Gxyz where moments and products of inertia are not time dependent.
• Transformation of into is independent of the system of coordinate axes.
GH
• Define rate of change of change of with respect to the rotating frame,
GH
kHjHiHH zyxGxyzG
Then,
GGxyzGG HHH
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Euler’s Eqs of Motion & D’Alembert’s Principle
18 - 24
GGxyzGG HHM
• With and Gxyz chosen to correspond to the principal axes of inertia,
yxyxzzz
xzxzyyy
zyzyxxx
IIIM
IIIM
IIIM
Euler’s Equations:
• System of external forces and effective forces are equivalent for general three dimensional motion.
• System of external forces are equivalent to the vector and couple, . and GHam
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Group Problem Solving
18 - 25
Two L-shaped arms, each weighing 4 lb, are welded at the third points of the 2-ft shaft AB. Knowing that shaft AB rotates at the constant rate = 240 rpm, determine (a) the angular momentum of the body about B, and b) its kinetic energy.
SOLUTION:
• The part rotates about the axis AB. Determine the mass moment of inertia matrix for the part.
• Compute the angular momentum of the disk using the moments inertia.
• Compute the kinetic energy from the angular velocity and moments of inertia.
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Group Problem Solving
2 - 26
244 lb, 0.12422 lb s /ft ,
32.28 in. 0.66667 ft
W m
a
(2 )(240)8 rad/s, 0, 0
60z x y
Given:
Find: HB, T
There is only rotation about the z-axis, what relationship(s) can you use?
( )B x xzH I ( )B y yzH I ( )B z zH I
Split the part into four different segments, then determine Ixz, Iyz, and Iz. What is the mass of each segment?
•Mass of each of the four segments 20.06211 lb s /ft2
mm
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Group Problem Solving
2 - 27
Fill in the table below to help you determine Ixz and Iyz. 1
2
3 4
Ixz Iyz
1
2
3
4
( )( )m a a
( 0.5 )( )m a a
( )(2 )m a a
a=
(0.5 )(2 )m a a
(0.5 )( )m a a
(0)( )m a
(0)(2 )m a
(0.5 )(2 )m a a
22 22 1
1 4 2
1
12z zI I m a m a a
21.5m a 21.5m a
Determine Iz by using the parallel axis theorem – do parts 1 and 4 first
Parts 2 and 3 are also equal to one another
22 3
1
3z zI I m a
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Group Problem Solving
2 - 28
Calculate Hx
2
2
3( )
23
(0.06211)(0.66667) (8 )21.04067 lb s ft
B x xzH I m a
2
2
3( )
23
(0.06211)(0.66667) (8 )21.04067 lb s ft
B y yzH I m a
2
2
10( )
310
(0.06211)(0.66667) (8 )3
2.3126 lb s ft
B z zH I m a
(1.041 lb ft s) (1.041 lb ft s) (2.31 lb ft s)B H i j k
Calculate Hy
Calculate Hz
Total Vector
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Group Problem Solving
2 - 29
2 2 2 21 1 1 1
2 2 2 2x x y y z z xy x y xz x z yz y z zT I I I I I I I
2 2 2 21 10 1 10(0.06211)(0.66667) (8 )
2 3 2 3T m a
29.1 ft lbT
Calculate the kinetic energy T
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2 - 30
Retracting the landing gear while the wheels are still spinning can result in unforeseen moments being applied to the gear.
When turning a motorcycle, you must “steer” in the opposite direction as you lean into the turn.
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Motion About a Fixed Point or a Fixed Axis
18 - 31
• For a rigid body rotation around a fixed point,
OOxyzO
OO
HH
HM
• For a rigid body rotation around a fixed axis, zzyzyxzx IHIHIH
2
iIjIkIjIiI
kIjIiIk
kIjIiI
HHM
yzxzzyzxz
zyzxz
zyzxz
OOxyzOO
zz
xzyzy
yzxzx
IM
IIM
IIM
2
2
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Rotation About a Fixed Axis
18 - 32
zz
xzyzy
yzxzx
IM
IIM
IIM
2
2• For a rigid body rotation around a fixed axis,
• If symmetrical with respect to the xy plane,
zzyx IMMM 00
• If not symmetrical, the sum of external moments will not be zero, even if = 0,
022 zxzyyzx MIMIM
• A rotating shaft requires both static and dynamic balancing to avoid excessive vibration and bearing reactions.
0 0
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Concept Question
2 - 33
The device at the right rotates about the x axis with a non-constant angular velocity. Which of the following is true (choose one)?
a) You can use Euler’s equations for the provided x, y, z axes
b) The only non-zero moment will be about the x axis
c) At the instant shown, there will be a non-zero y-axis bearing force
d) The mass moment of inertia Ixx is zero
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Sample Problem 18.3
18 - 34
Rod AB with weight W = 40 lb is pinned at A to a vertical axle which rotates with constant angular velocity = 15 rad/s. The rod position is maintained by a horizontal wire BC.
Determine the tension in the wire and the reaction at A.
• Expressing that the system of external forces is equivalent to the system of effective forces, write vector expressions for the sum of moments about A and the summation of forces.
• Solve for the wire tension and the reactions at A.
SOLUTION:
• Evaluate the system of effective forces by reducing them to a vector attached at G and couple
am
.GH
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Sample Problem 18.3
18 - 35
SOLUTION:
• Evaluate the system of effective forces by reducing them to a vector attached at G and coupleam
.GH
I
ILIraa n
2
2212
sft450
cos
Ig
am
lb55945040
kIjIiIH zzyyxxG
0sincos
0 2212
21
zyx
zyx mLIImLI
imLHG
cos2121
kkmL
imLji
HHH GGxyzGG
ftlb645cossin
cossincos0
22121
2121
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Sample Problem 18.3
18 - 36
• Expressing that the system of external forces is equivalent to the system of effective forces, write vector expressions for the sum of moments about A and the summation of forces.
effAA MM
KKT
KIJJIITJ
64519348093.6
64555946.340293.6
lb384T
effFF
IJIKAJAIA ZYX
55940384
JIA
lb 40lb 175
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Group Problem Solving
18 - 37
A four-bladed airplane propeller has a mass of 160 kg and a radius of gyration of 800 mm. Knowing that the propeller rotates at 1600 rpm as the airplane is traveling in a circular path of 600-m radius at 540 km/h, determine the magnitude of the couple exerted by the propeller on its shaft due to the rotation of the airplane
• Determine the mass moment of inertia for the propeller
• Calculate the angular momentum of the propeller about its CG
SOLUTION:
• Determine the overall angular velocity of the aircraft propeller
• Calculate the time rate of change of the angular momentum of the propeller about its CG
• Calculate the moment that must be applied to the propeller and the resulting moment that is applied on the shaft
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Group Problem Solving
2 - 38
Establish axes for the rotations of the aircraft propeller
Determine the x-component of its angular velocity
540 km/h 150 m/sv
2 rad1600 rpm
60 s
167.55 rad/s
x
150 m/s0.25 rad/s
600 myv
•
Determine the y-component of its angular velocity
Determine the mass moment of inertia
2 2 2(160 kg)(0.8 m) 102.4 kg mxI mk
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Group Problem Solving
2 - 39
Angular momentum about G equation:
Determine the couple that is exerted on the shaft by the propeller
Determine the moment that must be applied to the propeller shaft
G x x y yI I H i j
( ) 0 ( )G G Gxyz G y x x y yI I H H Ω H j i j
2(102.4 kg m )(167.55 rad/s)(0.25 rad/s)G x x yI H k k
(4289 N m) = (4.29 kN m)G H k k
Time rate of change of angular momentum about G :
(4.29 kN m)G M H k
(4.29 kN m)on shaft M M k
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Vector Mechanics for Engineers: DynamicsVector Mechanics for Engineers: Dynamics
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Concept Question
2 - 40
The airplane propeller has a constant angular velocity +x. The plane begins to pitch up at a constant angular velocity x
y
zThe propeller has zero angular acceleration
TRUE FALSE
Which direction will the pilot have to steer to counteract the moment applied to the propeller shaft?
a) +x b) +y c) +z
d) -y e) -z
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2 - 41
Gyroscopes are used in the navigation system of the Hubble telescope, and can also be used as sensors (such as in the Segway PT). Modern gyroscopes can also be MEMS (Micro Electro-Mechanical System) devices, or based on fiber optic technology.
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Motion of a Gyroscope. Eulerian Angles
18 - 42
• A gyroscope consists of a rotor with its mass center fixed in space but which can spin freely about its geometric axis and assume any orientation.
• From a reference position with gimbals and a reference diameter of the rotor aligned, the gyroscope may be brought to any orientation through a succession of three steps:
a) rotation of outer gimbal through j about AA’,
b) rotation of inner gimbal through q about
c) rotation of the rotor through y about CC’.
• and are called the Eulerian Angles and
spin of rate
nutation of rate
precession of rate
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Motion of a Gyroscope. Eulerian Angles
18 - 43
• The angular velocity of the gyroscope,
kjK
kji
jiK
cossin
cossinwith
• Equation of motion,
OOxyzOO HHM
jK
kIjIiIHO
cossin
cos
cossincossin
coscos2sin2
dt
dIM
IIM
IIM
z
y
x
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Steady Precession of a Gyroscope
18 - 44
Steady precession,
constant are ,,
ki
kIiIH
ki
zO
z
cossin
sin
sin
jII
HM
z
OO
sincos
Couple is applied about an axis perpendicular to the precession and spin axes
When the precession and spin axis are at a right angle,
jIM O
90
Gyroscope will precess about an axis perpendicular to both the spin axis and couple axis.
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Motion of an Axisymmetrical Body Under No Force
18 - 45
• Consider motion about its mass center of an axisymmetrical body under no force but its own weight, e.g., projectiles, satellites, and space craft.
constant 0 GG HH
• Define the Z axis to be aligned with and z in a rotating axes system along the axis of symmetry. The x axis is chosen to lie in the Zz plane.
GH
xGx IHH sinI
HGx
sin
yy IH 0 0y
zGz IHH cosI
HGz
cos
• = constant and body is in steady precession.
• Note:
tantanI
I
z
x
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Motion of an Axisymmetrical Body Under No Force
18 - 46
Two cases of motion of an axisymmetrical body which under no force which involve no precession:
• Body set to spin about its axis of symmetry,
aligned are and
0
G
xx
H
H
and body keeps spinning about its axis of symmetry.
• Body is set to spin about its transverse axis,
aligned are and
0
G
zz
H
H
and body keeps spinning about the given transverse axis.
© 2013 The McGraw-Hill Companies, Inc. All rights reserved.
Vector Mechanics for Engineers: DynamicsVector Mechanics for Engineers: Dynamics
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Motion of an Axisymmetrical Body Under No Force
18 - 47
The motion of a body about a fixed point (or its mass center) can be represented by the motion of a body cone rolling on a space cone. In the case of steady precession the two cones are circular.
• I < I’. Case of an elongated body. and the vector lies inside the angle ZGz. The space cone and body cone are tangent externally; the spin and precession are both counterclockwise from the positive z axis. The precession is said to be direct.
• I > I’. Case of a flattened body. and the vector lies outside the angle ZGz. The space cone is inside the body cone; the spin and precession have opposite senses. The precession is said to be retrograde.
© 2013 The McGraw-Hill Companies, Inc. All rights reserved.
Vector Mechanics for Engineers: DynamicsVector Mechanics for Engineers: Dynamics
Te
nth
Ed
ition
Concept Question
2 - 48
The rotor of the gyroscope shown to the right rotates with a constant angular velocity. If you hang a weight on the gimbal as shown, what will happen?
a) It will precess around to the right
b) Nothing – it will be in equilibrium
c) It will precess around to the left
d) It will tilt down so the rotor is horizontal