Summary: 3D Moving Reference Frame Kinematics 1
!
PROBLEM: A person attached to a moving body (reference frame) is
v
Problem III-27
observing the motion of point B.
Given: A shaft rotates about a fixed vertical axis aty a constant rate of ! = 5 rad/sec,
!
Moving Reference Frame Kinematics Homework Problems
ME 274
B
! !
!
!
!
vB = v A + vB / A
+ ω × rB / A
rel
! !
! !
!
!
!
aB = a A + aB / A
+ α × rB / A + 2ω × v B / A
(
(
)
)rel
(
)rel
as shown below. A straight bar OA, having a length ofBL = 2 meters,
is pinned
aB
observer
to point O on the shaft,
with O being on the rotation
! axis of the shaft. At the
rB/ Aof 4 rad/sec from the
instant when " = 0, bar OA is being raised at a rate
horizontal plane. In addition, the rate at which the bar is
x being raised is
A
decreasing at a rate 3 rad / sec 2 .
!
! !
+ ω × ω A×setrofB xyz
OA withframe
its origin at O. A
/ Acoordinate axes is attached to barreference
(
where:
)
second set of coordinate axes, XYZ, arezfixed to ground. At the instant when "
= 0, the xyz and XYZ axes are aligned with each other.
!
!
Find:
For the instant when " = 0:
ω
•
and α are the angular velocity/acceleration of the a)observer
exceptions).
find the angular(no
velocity
and angular acceleration of bar OA.
!
!
b) find the acceleration of point A on the bar.
• ( v B / A ) and ( a B / A ) are the velocity/acceleration of B as seen by the observer.
rel
rel
All answers should be expressed in vector form. You may choose to write these vectors
terms of either their xyz or XYZ components.
• A is ANY point on the same reference frame as inthe
observer.
• Generally, you are free to choose your observer.
Y
QUESTION: How is this different from the 2D case? For
observer on arm OA:
!
ω = ΩĴ + θ"k̂
0
!
! dω "
!
" "
"
α=
= ΩĴ + ΩJˆ + θ""k̂ + θ"kˆ = Ω
Ĵ + θ""k̂ + θ" ω × k̂
dt
(
A
y
vertical shaft
L
"
bar
O
!
)
bearing
X
me 274 - cmk
x