Useful Equations in Planar Rigid-Body Dynamics Ken Youssefi Mechanical Engineering 1 Angular Motion nstant Angular Acceleration d Body Kinematics, relative velocity and acceleration eq Kinematics (Ch. 16) Constant linear Acceleration Linear Motion r = r{t} o o o s s a v v 2 2 2 t a v t v o o } { 2 2 1 } { t a t v s t s o o o dt d l v a
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Useful Equations in Planar Rigid-Body Dynamics Ken YoussefiMechanical Engineering 1 Angular Motion Constant Angular Acceleration 2D Rigid Body Kinematics,
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Mechanical Engineering 1
Useful Equations in Planar Rigid-Body Dynamics
Ken Youssefi
Angular Motion
Constant Angular Acceleration
2D Rigid Body Kinematics, relative velocity and acceleration equations
Kinematics (Ch. 16)
ooo ssavv 222
tavtv oo }{
221}{ tatvsts ooo
Constant linear Acceleration
Linear Motion
r = r{t} t
ttt
dt
dt
}{}{lim
0
vvva
Mechanical Engineering 2
Useful Equations in Planar Rigid-Body Dynamics
Ken Youssefi
Equations of Motion (Ch. 17)
Mass Moment of Inertia Parallel Axis Theorem Radius of Gyration
Translation F = ma
Rotation about the axis thru the center of gravity MG = IGα
Rotation about a fixed axis thru point O MO = IOα
MO = (MO)k
Mechanical Engineering 3
Work and Energy (Ch. 18)
Ken Youssefi
Total kinetic energy of a rigid body rotating and translating
Principle of Work and Energy
Work done by a force Work done by a moment
Conservation of Energy
Work done by gravity and spring
Due to gravity Due to linear springDue to torsional spring
Mechanical Engineering 4
Linear and Angular Momentum (Ch. 19)
Ken Youssefi
Principle of Linear Impulse and Momentum
Principle of Angular Impulse and Momentum
Non-Centroidal Rotation
Mechanical Engineering 5
Solving Dynamic Problems
Ken Youssefi
• If all forces are conservatives (negligible friction forces), the conservation of energy is easier to use than the principle of work and energy.
• Energy method tends to be more intuitive and easier to use.
• Momentum method is less intuitive, but sometime necessary.
• Newton’s second law (force and acceleration method) may be the most thorough method, but can sometimes be more difficult.
• Conservation of linear and angular impulse and momentum can be used if the external impulsive forces are zero (conservation of linear impulse and momentum, impact) or the moment of the forces are zero (conservation of angular momentum)
Mechanical Engineering 6
Solving Dynamic Problems
Ken Youssefi
The type of unknowns and given information could point to the best method to use.
Acceleration suggests that the equations of motion should be used, kinematic equations are used when there are no forces or moments involved.
Displacement or velocity (linear or angular) indicates that the work and energy method is easier to use.
Time suggests that the impulse and momentum method is useful
If springs (linear or torsional) exist, the work and energy is a useful method.
Mechanical Engineering 7
Example 1
Ken Youssefi
A homogeneous hemisphere of mass M is released from rest in the position shown. The moment of inertia of a hemisphere about its center of mass is (83/320)mR2.What is the angular velocity when the object’s flat surface is horizontal?
Conservation of energy
Mechanical Engineering 8
Example 2
Ken Youssefi
The angular velocity of a satellite can be altered by deploying small masses attached to very light cables. The initial angular velocity of the satellite is 1 = 4 rpm, and it is desired to slow it down to 2 = 1 rpm.Known information:
IA = 500 kg·m2 (satellite) mB = 2 kg (small weights)
What should be the extension length d to slow the satellite as required?
Mechanical Engineering 9
Example 2
Ken Youssefi
The only significant forces and moments in this problem are those between the two bodies, so conservation of momentum applies. The point about the deployed masses being small mean they have insignificant I, and the (r x mv) for the main body is zero because it spins about its center of mass (r = 0).
Mechanical Engineering 10
Example 3
Ken Youssefi
The left disk rolls at constant 2 rad/s clockwise. Determine the linear velocities of joints A and B, vA and vB. Also determine the angular velocities AB and BD .
A
B
C D
Relative velocity equation for points A and C
Mechanical Engineering 11
Example 3
Ken Youssefi
Relative velocity equation for points B and D
Relative velocity equation for points B and A
A
B
C D
Mechanical Engineering 12
Example 3
Ken Youssefi
Setting the i and j component equal:
Mechanical Engineering 13
Example 4
Ken Youssefi
The system shown is released from rest with the following conditions:
mA = 5 kg, mB = 10 kg, Ipulley = 0.2 kg·m2, R = 0.15 m
No moment is applied at the pivot. What is the velocity of mass B when it has fallen a distance h = 1 m?
The only force or moment that exists and does work is gravity, and it is a conservative force, so conservation of energy applies.
Mechanical Engineering 14
Example 4
Ken Youssefi
0
1
2
A B
C
Mechanical Engineering 15
Example 5
Ken Youssefi
If bar AB rotates at 10 rad/s, what is the rack velocity vR?
Relative velocity equations
Mechanical Engineering 16
Example 5
Ken Youssefi
Mechanical Engineering 17
Example 5
Ken Youssefi
i components
j components
Mechanical Engineering 18
Example 6
Ken Youssefi
Mechanical Engineering 19
Example 6
Ken Youssefi
Force and motion diagrams for the plate.
Relative acceleration equation for points G and A
KinematicsG
A
Mechanical Engineering 20
Example 6
Ken Youssefi
Mechanical Engineering 21
Example 7
Ken Youssefi
Force diagram
Motion diagram
Mechanical Engineering 22
Example 7
Ken Youssefi
G
Mechanical Engineering 23
Example 8
Ken Youssefi
vB = 21 AB
vC slider dir.
vC/B CB
= (vC/B)/CB
Mechanical Engineering 24
Example 8
Ken Youssefi
Mechanical Engineering 25
Example 9
Ken Youssefi
At the instant shown, the disk is rotating with an angular velocity of and has an angular acceleration of α. Determine the velocity and acceleration of cylinder B at this instant. Neglect the size of the pulley at C
Use position coordinate method
Determine the length s = AC in terms of the angle θ ( Law of Cosines)