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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