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MEAM 535

University of Pennsylvania 1

Position, Velocity and Angular Velocity Vectors

MEAM 535


Comment about Notation

MEAM 535


Position, Velocity and Acceleration Vectors p is a position vector of P in A

!  Emanates from a point fixed to A !  Ends up at P

AvP is the velocity of P in A

AaP is the acceleration of P in A

O

p

P

A

!

AaP =A d A vP( )

dt

O'

p'

Velocity and acceleration vectors are independent of choice of origin

MEAM 535


Velocity of P in A is independent of choice of “origin” in A!

p is a position vector of P in A !  Emanates from a point fixed to A !  Ends up at P

AvP is the velocity of P in A O

p

P

A O'

p' r

MEAM 535


Example 1

A robotic arm is a system of rigid bodies (reference frames) B, C, and D. A is the inertial or the laboratory reference frame that is considered fixed.

1. Are the following equal?

2. If motor (joint) rates are given, calculate

Q

3. Find the velocity of Q in A

a1

a2 a3

MEAM 535


Angular Velocity (Kane) The angular velocity of B in A, denoted by A!B, is defined as:

!  Defined in terms of a reference triad attached to B

!  Independent of reference triad attached to A

!  Generalizes to three dimensions !  Yields simple results for

derivatives of vectors

Example

MEAM 535


Differentiation of vectors 1. Vector fixed to B

r = r1 b1 + r2 b2 + r3 b3

A

B P Q r

b1

b2 b3 Composition and Projection rule

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Differentiation of vectors (cont’d)

Conclusion: For any unit vector fixed in B,

A

B P Q r

b1

b2 b3

Key Result (any vector fixed in B)

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Velocity of P (attached to B) in A when A and B have a common point O

A

B

O

P p=r

b1

b2 b3

Choose p to be a position vector of P in A

O is fixed in A and in B

MEAM 535


Example 2

b1

b3 c3

c1

p

!

B vP =B dpdt

= ?

1. B rotates in A and C rotates in B.

2. Assuming B is fixed to A

!

A dpdt

= ?

A

MEAM 535


Alternative Formulation: Angular Velocity B rotates as seen by an observer attached to A

Consider the position vector, p, in A, of a point P fixed to B.

!  What is the velocity of P in A in terms of components with respect to the SRT ai?

!

A vP[ ]A

=ddtp[ ]A

!

ddtp[ ]A =

ddt

A RB p[ ]B( )A

B

O

P p

b1

b2 b3

a1

a2 a3

O is fixed in A and in B

MEAM 535


Angular Velocity and Rotation Matrix

!

ARB[ ] =

b1 " a1 b2 " a1 b3 " a1b1 " a2 b2 " a2 b3 " a2b1 " a3 b2 " a3 b3 " a3

#

$

% % %

&

'

( ( (

!

ARB[ ]T A ˙ R B[ ] =

b1 " a1 b2 " a1 b3 " a1

b1 " a2 b2 " a2 b3 " a2

b1 " a3 b2 " a3 b3 " a3

#

$

% % %

&

'

( ( (

T ˙ b 1 " a1˙ b 2 " a1

˙ b 3 " a1

˙ b 1 " a2˙ b 2 " a2

˙ b 3 " a2

˙ b 1 " a3˙ b 2 " a3

˙ b 3 " a3

#

$

% % %

&

'

( ( (

!

ARB[ ]T A ˙ R B[ ] =

b1 " ˙ b 1 b1 " ˙ b 2 b1 " ˙ b 3b2 " ˙ b 1 b2 " ˙ b 2 b2 " ˙ b 3b3 " ˙ b 1 b3 " ˙ b 2 b3 " ˙ b 3

#

$

% % %

&

'

( ( (

!

ARB[ ]T A ˙ R B[ ] =

0 "b2 # ˙ b 1 b1 # ˙ b 3b2 # ˙ b 1 0 "b3 # ˙ b 2"b1 # ˙ b 3 b3 # ˙ b 2 0

$

%

& & &

'

(

) ) )

Compare

!

˙ b i =Adbi

dt

The angular velocity vector can be obtained by differentiating the rotation matrix and premultiplying!

Define ( ) .

Kane

MEAM 535


Recall every 3"1 vector a has a 3"3 skew symmetric matrix counterpart A or a

For any vector b a" b = A b

a

A

^

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Angular velocity components in A and B A - Inertial B - Body fixed frame

!

ddtp[ ]A

=ddt

A RB p[ ]B( )= A ˙ R B p[ ]B

Pre multiply by BRA to get components of AvP in B

!

A RB[ ]T d

dtp[ ]A

= A RB[ ]T A ˙ R B p[ ]B

Components of AvP in B

Components of A!B in B

Components of position vector of P in A in B

= x!Components of AvP in A

!

ARB[ ]T A ˙ R B[ ] =

0 "b2 # ˙ b 1 b1 # ˙ b 3b2 # ˙ b 1 0 "b3 # ˙ b 2"b1 # ˙ b 3 b3 # ˙ b 2 0

$

%

& & &

'

(

) ) )

MEAM 535


Differentiation of vectors 2. Vector not fixed to B

A

B O

P r b1

b2 b3

r can be any vector

MEAM 535


Example 3 Two bugs are on a rotating turntable with angular speed !. The one at P is stationary on the turntable while the one at Q is moving radially away from P at a uniform speed.

Let OP be r0, a constant. Let the magnitude of PQ be r. r and ! are variables. Let the rate of change of r be v. !#

r

b1 b2 $#

a1

a2

O

Q

P

B

Find the velocity of both bugs in A when $ = 0.

MEAM 535


Simple Angular Velocity

Angular velocity of B in A !  is along a2 as seen in A !  is along b2 as seen in B

A rigid body B has a simple angular velocity in A, when there exists a unit vector k whose orientation (as seen) in both A and in B is constant (independent of time).

a1 a2

a3

In each frame, the angular velocity has a constant direction (magnitude may change)

MEAM 535


Simple Angular Velocity

D!A, A!B and C!B are simple angular velocities.

However, D!C is not a simple angular velocity. The motion of C relative to D is such that there is no vector fixed in D that also remains fixed in C.

How about D!C ?

MEAM 535


Addition Theorem for Angular Velocities Let A, B, and C be three rigid bodies. The addition theorem for angular velocities states:

Proof Let r be fixed to C.

!

A drdt

=B drdt

+ A"B # r

=C drdt

+ B"C # r + A"B # r

=C drdt

+ B"C +A "B( ) # r

= B"C +A "B( ) # r

A

B

C rAnd,

MEAM 535


Angular velocities can be found by adding up “simple” angular velocities

D!A, A!B and C!B are simple angular velocities.

D!C is not a simple angular velocity. The motion of C relative to D is such that there is no vector fixed in D that also remains fixed in C.

But,

MEAM 535


Example 4

A disk of radius r is mounted on an axle OG. The disk rotates counter-clockwise at the constant rate of !1 about OG.

Determine the angular velocity of the disk in an inertial frame if the disk rolls on the plane.

Determine the velocity of the point P in an inertial frame.

P A

B

C

Definition: The two points of contact have the same velocity.

MEAM 535


Example 4 (continued)

b1 b2

a1 a2 A

C B

!0

P P

G b2

c2

b1, c1

$#

MEAM 535


Example 5 The rolling (and sliding) disk on a horizontal plane

A circular disk C of radius R is in contact with a horizontal plane (not shown in the figure) at the point P. The point P is attached to the disk. The plane is the x-y plane. It is rigidly attached to the earth. The standard reference triad bi is chosen so that b1 is along the direction of progression of the disk (parallel to the tangent to the disk at P), b2 is parallel to the plane of the disk, and b3 is normal to the disk. Note that this triad is not fixed to the disk. Call the earth-fixed reference frame A and choose the standard reference triad ax, ay, and az in an obvious fashion along the x, y, and z axes shown in the figure.

MEAM 535


Reference Triads

!  Rotate triad A about z through q1 followed by rotation about x by 90 deg to get E

!  Rotate triad E about -x through q2 to get B

!  Rotate triad B about z through q3 to get C (not shown)

x

y

z

P

q2

q3 q1

q5

q4

C

b1

b2

b3

C*

Locus of the point of contact Q on the plane A

e1

e2

e3

a1

a2

a3

Imagine E to be a virtual body that is attached to Q

Imagine B to be a virtual body that is attached to C*

MEAM 535


Transformations Find the transformation

!  ai in terms of bi

x

y

z

P

q2

q3 q1

q5

q4

C

b1

b2

b3

C*

e1

e2

e3

a1

a2

a3

MEAM 535


Transformations Two coordinate transformations

!  ai in terms of ei

!  ei in terms of bi

x

y

z

P

q2

q3 q1

q5

q4

C

b1

b2

b3

C*

e1

e2

e3

a1

a2

a3

!

BRE

!

ERA

MEAM 535


Angular Velocity: Components A!C = u1 b1 + u2 b2 + u3 b3

!  ui are the components of the angular velocity of the disk with respect to the reference triad B

A!C = ux ax + uy ay + uz az !  u" are the components of the angular

velocity of the disk with respect to the reference triad A

MEAM 535


Notation Revisited…

!  Points P fixed to (in) B and Q

!  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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