Problems in Chapter 20 (3D Kinematics) kinematics ω α

Problems in Chapter 20 (3D Kinematics)
Ú How to analyze the kinematics problems
r
r
1. Determine the angular velocity ( ω ) and angular acceleration (α ) of a
body at a specific instance
or
r
r
Determine the velocity ( v ) and acceleration ( a ) of a point in the body at
a specific instance.
Select the appropriate XYZ (fixed) and/or xyz (moving) frames
- Coincident origins & axes at the instance of interest
2. In case of motion of one moving object (Fixed or Translational R.F.)
r r
r
r
- vB = v A + ω × rB / A
r r r
r r
r
r
a B = a A + α × rB / A + ω × (ω × rB / A )
3. In case of motions of several moving objects (Rotating R.F.)
r&
r
- xyz: Attached to a moving object then, determine Ω & Ω
of xyz
r r
r
r
r
- vB = v A + Ω × rB / A + (vB / A ) xyz
r r r
r r
r& r
r
r
r
aB = a A + Ω
× rB / A + Ω × (Ω × rB / A ) + 2Ω × (vB / A ) xyz + (a B / A ) xyz
Example 1. The conical spool rolls
on the plane without slipping. If the
axle has an angular velocity of ω1 =
3 rad/s and an angular acceleration of
α1 = 2 rad/s2 at the instant shown,
r
r
determine ω and α of the spool at
this instant.
Example 2. One end of the rigid
bar CD shown in figure slides
along the horizontal member AB,
and the other end slides along the
vertical member EF. If the collar
at C is moving towards B at a
speed of 3 m/s, determine the velocity of the collar at D and the angular
velocity of the bar at the instant shown. The bar is connected to the collar
at its end points by ball-and-socket joints.
Example 3. The pendulum shown in
figure consists of two rods; AB is pinsupported at A and swings only in the YZ plane, whereas a bearing at B allows
the attached rod BD to spin about rod
AB.
At a given instant, the rod have the
angular motions shown. Also a collar C,
located 0.2 m from B, has a velocity of 3 m/s and an acceleration of 2 m/s2
r
r
along the rod. Determine v and a of the collar at this instant.
Example 4. At the instant shown, rod
BD is rotating about the vertical axis with
an angular velocity ω BD = 7 rad/s and an
angular acceleration α BD = 4 rad/s2. Also
θ = 60o and link AC is rotating downward
such that θ& = 2 rad/s and θ&& = 3 rad/s2.
r
r
Determine v and a of point A on the link
at this instant.
Z, z,
z’
X, x,
x’
Y, y,
y’