MEAM 535

University of Pennsylvania 1

Kinematics

Velocity and Acceleration Analysis

MEAM 535

University of Pennsylvania 2

General Approach to Analyzing Multi-Body System

1. Points common to adjacent bodies

3. Pair of points on adjacent bodies whose relative motions can be easily described

2. Pair of points fixed to the same body

4. Write equations relating pairs of points either on same or adjacent bodies.

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Differentiation of vectors 2. Vector not fixed to B

A

B O

P r b1

b2 b3

r can be any vector

Recall

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Relationship between Velocities and Accelerations in A of Two Points fixed to B

  Points P and Q fixed to (in) B

  O is a point fixed in A

  Position vectors for P and Q in A are denoted by p and q

  Velocities for P and Q in A

  Accelerations for P and Q in A

B

r

a1

a3

a2

P

Q

O

A

p q

Notice consistency in leading superscripts!

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Velocities of P and Q (both fixed in A)

  Triangle law of vector addition for points P and Q

q = p + r

B

r

a1

a3

a2

P

Q

O

A

p q

  Differentiate both sides

  Substitute definitions of velocities

  Velocities for P and Q in A

0

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Angular Acceleration The angular acceleration of B in A, denoted by AαB, is defined as the first time-derivative in A of the angular velocity of B in A:

€

Aα B =A ddt

AωB( )

Notice consistency in leading superscripts!

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centripetal (normal) acceleration

tangential acceleration

Accelerations of P and Q

  Velocities for P and Q in A

B

r

a1

a3

a2

P

Q

O

A

p q

  Differentiate both sides

  Accelerations for P and Q in A

MEAM 535

University of Pennsylvania 9

Relationship between Velocities and Accelerations in A of Points described in B

B

r

a1

a3

a2

P

Q

O

A

p q

Q

Q

Q is moving in B

fixed in B

  Point P fixed to (in) B   Point Q moving in B (but easily

described in B)

MEAM 535

University of Pennsylvania 10 non zero

  Triangle law of vector addition for points P and Q

q = p + r   Differentiate both sides

  Substitute definitions of velocities

Velocities of P and Q

B

r

a1

a3

a2

P

Q

O

A

p q

  Velocities for P and Q in A

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Velocity and Acceleration of Q in B

  Point Q moving in B has position vector s in B

B

r

a1

a3

a2

P

Q

O

A

p q

Q

Q

Q is moving in B

fixed in B

s

  Velocity   Acceleration

Note: s may be 0, but v, a may not be!

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  Velocities for P and Q in A

  Differentiate in A

Acceleration of P and Q

B

r

a1

a3

a2

P

Q

O

A

p q

€

AaQ =A ddt

A vQ( ) =A ddt

A vP( ) +A ddt

B vQ( ) +A ddt

AωB × r( )

€

AaP{ }

€

B ddt

B vQ( ) +A ωB × BvQ

€

A ddt

AωB( ) × r + AωB ×B drdt

+ AωB × r

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Coriolis acceleration

centripetal (normal) acceleration

tangential acceleration

Acceleration of P and Q

B

r

a1

a3

a2

P

Q

O

A

p q Special case: r = 0

MEAM 535

University of Pennsylvania 15

Example Two bugs are on a rotating turntable. The one at P is stationary on the turntable while the one at Q is moving radially away from P at a uniform speed. (the rate of change of r, the magnitude of PQ, is constant).

Ω

r

b1 b2 θ

a1

a2

O

Q

P

B

Choose θ = 0.

Let both P and Q be instantaneously coincident, but Q is moving radially outward.

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Solution

b1 b2

a1 a2 A

C B

ω0

P P

G b2

c2

b3, c3

θ

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Example: Gyroscope

D

C

B

€

AωD = − ˙ φ sinθc1 + ˙ θ c2 + ˙ φ cosθ + ˙ ψ ( )c3

a3, b3

b2, c2

c3, d3

c1