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KINEMATICS OF RIGID BODIES IN PLANE AND 3-D MOTION
15.90 (Beer & Johnston)
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
Velocity Analysis
B rotates about a fixed axis through A
D translates BD is in general plane motion (B is chosen as reference point)
i – components: 0.229 BD = 0 BD = 0
Acceleration Analysis
i-components: 137.08 + 0.229BD = 0 BD = = -598.6 rad/s2
j-components: aD = 0.1BD = -0.1 (-598.6) = 59.9 m/s2
Vector Polygon
Not to scale!
b) = 180
Velocity Analysis
i-components: 2.62 + 0.2 BD = 0 BD = = -13.1 rad/s
Acceleration Analysis
i-components: 0 = 0.2BD + 25.74 BD = = -128.7 rad/s2
j-components: aD = 137.08 – 0.15BD + 34.32 = 190.7 m/s2
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
Link CB is in rotation about a fixed axis through C
Link AB is in general plane motion
j-components: 0 = -600 + 240AB
AB = 600/240 = 2.5 rad/s
i-components: -180WBC = -800 - 100AB
180BC = 800 + 100(2.5)
BC = 1050/180 = 5.83 rod/s
15.93
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
Gear A does not rotate
C is the instantaneous center of rotation of gear B
Acceleration Analysis
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
xy frame rotates with angular velocity about he z-axis (i.e. perpendicular to plane of screen)
not fixed since xy rotating.
Evaluation of and
Introduce cross-product
Generalization
For any vector A
Background
Vector A swings to A1 in time dt observer attached to frame xy (i.e. rotating frame) sees that consists of two components.
- A dB/dt due to rotation of A through d/B in xy.
- dA/dt due to change in magnitude of A.
Part of absolute rate of change is A not seen by rotating observer is .
A is magnitude of vector A.
Plan motion in a rotating frame
Acceleration
normal or centripetal acceleration due to rotation of rotating frame
tangential acceleration due to angular acceleration of rotating frame
2VAB – CORIOLIS ACCELERATION
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
- change in magnitude of x is dx
- change in direction of x is xd
Rates of change are:
are in the (+) y-direction
is in the (-) x-direction
Total rate of change of VA:
(normal) (Coriolis)
since Vrel = constant and slot has no curvature
since is constant
XY : Fixed Frame
xy : Rotating Frame
Recall for a fixed frame:
Now for a rotating frame:
XY : Fixed Frame
Xy : Rotating Frame
: normal acceleration of a point (P) fixed in the rotating frame
: tangential acceleration of a point (P) fixed in the rotating frame
: acceleration of point A in the rotating frame
: Coriolis acceleration brought about by the rotating () of the rotating frame and relative motion (Vrel) in the rotating frame
15.119The 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
Equateand
Coordinate transformation:
j-component: -1.1547 + 0.722 sin 30 + VP/BD cos 30 = 0
i-components: -VP/AE + 0.722 cos 30 - VP/BD sin 30 = 0
or
15.123At 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)
(b)
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.
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 .