15 - St. Francis Xavier University



KINEMATICS OF RIGID BODIES IN PLANE AND 3-D MOTION

15.90 (Beer & Johnston)

[pic]

The disk shown has a constant angular velocity of 500 r/min counterclockwise. Keeping that rod BD is 250 mm long, determine the acceleration of collar D when (a) ( = 90(, (b) ( = 180(.

(a) ( = 90(

[pic]

Velocity Analysis

( B rotates about a fixed axis through A (

[pic]

( D translates ( BD is in general plane motion (B is chosen as reference point)

[pic]

i – components: 0.229 (BD = 0 ( (BD = 0

Acceleration Analysis

[pic]

[pic]

[pic]

i-components: 137.08 + 0.229(BD = 0 ( (BD = [pic] = -598.6 rad/s2

j-components: aD = 0.1(BD = -0.1 (-598.6) = 59.9 m/s2 ( [pic]

Vector Polygon

Not to scale!

b) ( = 180(

[pic]

Velocity Analysis

( [pic]

( [pic]

( [pic]

[pic]

i-components: 2.62 + 0.2 (BD = 0 ( (BD = [pic] = -13.1 rad/s

Acceleration Analysis

[pic]

[pic]

[pic]

i-components: 0 = 0.2(BD + 25.74 ( (BD = [pic] = -128.7 rad/s2

j-components: aD = 137.08 – 0.15(BD + 34.32 = 190.7 m/s2 ( [pic]

Vector Polygon

Not to scale!

Example: In the four-bar linkage shown, control link OA has a counterclockwise angular velocity (0 = 10 rad/s during a short interval of motion. When link CB passes the vertical position shown, point A has coordinates x = -60 mm and y = 80 mm. Determine, by means of vector algebra, the angular velocity of AB and BC.

• Link AO is in rotation about a fixed axis through 0

← [pic]

( Link CB is in rotation about a fixed axis through C

([pic]

( Link AB is in general plane motion ( [pic]

[pic]

[pic]

j-components: 0 = -600 + 240(AB

(AB = 600/240 = 2.5 rad/s [pic]

i-components: -180WBC = -800 - 100(AB

180(BC = 800 + 100(2.5)

(BC = 1050/180 = 5.83 rod/s [pic]

15.93

[pic]

AB rotates with a constant angular velocity of 60 r/min clockwise. Knowing that gear A does not rotate, determine the acceleration of the tooth of gear B which is in contact with gear A.

Velocity Analysis

( B rotates about a fixed axis through A

( [pic]

( Gear A does not rotate ( [pic]

( C is the instantaneous center of rotation of gear B

[pic]

Acceleration Analysis

[pic]

[pic] [pic]

Note: Gear B is in general plane motion; B is chosen as reference point.

Vector Polygon

Not to scale!

RATE OF CHANGE OF A VECTOR WITH RESPECT TO A ROTATINT FRAME OF REFERENCE

( XY frame is fixed

( x(y( frame rotates with angular velocity ( about he z-axis (i.e. perpendicular to plane of screen)

[pic]

[pic] not fixed since x(y( rotating.

[pic]

Evaluation of [pic] and [pic]

Introduce cross-product

[pic]

[pic]

[pic]

[pic]

Generalization

For any vector A

[pic]

Background

Vector A swings to A1 in time dt observer attached to frame x(y( (i.e. rotating frame) sees that [pic]consists of two components.

- A dB/dt due to rotation of A through d/B in x(y(.

- dA/dt due to change in magnitude of A.

Part of absolute rate of change is A not seen by rotating observer is [pic].

A( is magnitude of vector ((A.

Plan motion in a rotating frame

[pic]

Acceleration

[pic]

[pic]

[pic]

[pic]

[pic]

[pic]

normal or centripetal acceleration due to rotation of rotating frame

tangential acceleration due to angular acceleration of rotating frame

2((VAB – CORIOLIS ACCELERATION

[pic]

[pic]

[pic]

[pic]

[pic]

[pic]

[pic]

( Consider a rotating disk with a radial slot

( A small particle A is confined to slide in the slot

( Let ( = constant and Vrel = constant

( The velocity of A has two components:

x( (due to rotation of the disk)

vrel (due to motion of A in the slot)

( Consider the rate of change of the velocity of A:

- no change in magnitude of Vrel since Vrel = constant.

- change in direction of Vrel is [pic]

- change in magnitude of x( is (dx

- change in direction of x( is x(d(

Rates of change are: [pic]

[pic] are in the (+) y-direction

[pic] is in the (-) x-direction

( Total rate of change of VA: [pic]

(normal) (Coriolis)

[pic]since Vrel = constant and slot has no curvature

[pic] since ( is constant

XY : Fixed Frame

xy : Rotating Frame

( Recall for a fixed frame: [pic]

( Now for a rotating frame: [pic]

[pic]

XY : Fixed Frame

Xy : Rotating Frame

[pic]: normal acceleration of a point (P) fixed in the rotating frame

[pic]: tangential acceleration of a point (P) fixed in the rotating frame

[pic]: acceleration of point A in the rotating frame

[pic]: Coriolis acceleration brought about by the rotating (() of the rotating frame and relative motion (Vrel) in the rotating frame

15.119 The motion of pin P is guided by slots cut in rods AE and BD. Knowing that the rods rotate with the constant angular velocity (A = 4 rad/s ↓ and (B = 5 rad/s ↓, determine the velocity of pin P for the position shown.

( Pin P moves in BD and AE both of which rotate ( relative motion in a rotating frame

( [pic]

[pic]

[pic] (

[pic]

[pic]

[pic] (

Equate(and ( ( [pic]

Coordinate transformation:

[pic]

[pic]

← [pic]

[pic]

j-component: -1.1547 + 0.722 sin 30( + VP/BD cos 30( = 0

[pic]

i-components: -VP/AE + 0.722 cos 30( - VP/BD sin 30( = 0

[pic]

[pic]

or

[pic]

15.123 At the instant shown the length of the boom is being decreased at the constant rate of 150 mm/s and the boom is being lowered at the constant rate of 0.075 rad/s. Knowing that ( = 30(, determine (a) the velocity, (b) the acceleration of point B.

( There is relative motion of B in the rotating x-y frame

(a) [pic]

[pic]

[pic]

(b) [pic]

[pic]

[pic]

The vertical shaft and attached clevis rotate about the z-axis at the constant rate ( = 4 rad/s. Simultaneously, the shaft B revolves about its axis OA at the constant rate (0 = 3 rad/s, and the angle ( is decreasing at the constant rate of (/4 rad/s. Determine the angular velocity ( and the magnitude of the angular acceleration ( of shaft B when ( = 30(. The x-y-z axes are attached to the clevis and rotate with it.

[pic]

[pic]

[pic]

[pic]

[pic]

[pic]

[pic]

1. The circular plate and rod are rigidly connected and rotate about the ball-and-socket joint ( ) with an angular velocity ( = ( i + ( j + ( k. Knowing that VA = -(540 mm/s)i + 350 mm/s)j + (r4)2k and (ij = 4 rad/s. Determine (a) the angular velocity of the assembly, (b) the velocity of point B.

2. A disk of radius r rotates at a constant rate (2 with respect to the are ( ), which itself rotates at a constant rate (1 about the Y axis. Determine (a) the angular velocity and angular acceleration of the disk, (b) the velocity and acceleration of point A on the rim of the disk.

3. The bent rod ABC rotates at a constant rate (1. Knowing that the collar D moves downward along the rod at a constant relative speed u, determine for the position shown (a) the velocity of D, (b) the acceleration of D.

4. A disk of radius r spins at the constant rate (2 about an axle held by a fork-ended horizontal rod which rotates at the constant rate (1. Determine the acceleration of point I( for an arbitrary value of the angle (.

-----------------------

0.2299(BDi

0.1(BDj

aD/Bt

aB

aD

aD

0.15(BDj

aD/Bt

25.74 i

aD/Bn

aB

35.32i

0.2(BDi

A

B

j

i

0.15m

0.075m

aC/An

aC/At

aB/An

VB/At

aB/At

aC/Bn

aC

aB/An

IA/B

IB

IA

X

X1

Y1

B

A

Y

(

d(

d(

di = j d(

di = (1) d(

j

i

dj/dt = -id(/dt

di/dt = jd(/dt

i

k

j

Ad(

X11

Y1

Y

X1

X

(

|dA|XY

|dA|x1y1

dA=d(|A|)

(

(

Y11

Ad(

d(A

d(

(

X

x1

y1

Y

A

B

30o

30o

i*

j

j*

i

Vp

-0.167i

-1.155j

-0.722i*

Vp

-0.916j*

-0.15i

-0.45j

VB

aB

0.023j

-0.034i

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