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)

Mechanical Engineering 26

Example 9

Ken Youssefi

Mechanical Engineering 27

Example 10

Ken Youssefi

Mechanical Engineering 28

Example 10

Ken Youssefi