Category Archives: Chapter 3 Homework

Homework H3.J - Fa22

Problem statement
Solution video

DISCUSSION THREAD

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Discussion and hints:

Your first decision on this problem is to choose your observer. Since an observer on the plate will have the simplest view of the motion of the insect, attaching the observer to the plate is recommended. Also, attach you xyz-axes to the plate.

Next write down the angular velocity and angular acceleration of the plate. Based on what we have been doing up to this point in Chapter 3, hopefully it is clear that the plate (and observer) has two components of angular velocity: Ω about the fixed X-axis and θ_dot about the moving z-axis. Take a time derivative of the angular velocity vector to find the angular acceleration of the plate (observer).

Following that, determine the motion of the insect as seen by the observer on the plate.

Use these results with the moving reference frame kinematics equation to determine the velocity and acceleration of the insect.


 

Homework H3.H - Fa22

Problem statement
Solution video

DISCUSSION THREAD

Any questions??


Discussion and hints:

It is recommended that you use an observer attached to the wheel. As we have discussed in class, your choice of observer directly affects four terms in the acceleration equation: ω and α  (how the observer moves), and the relative velocity and relative acceleration terms (what the observer sees). Note that the remainder of the discussion here is based on having the observer attached to the wheel.

The wheel shown above has TWO components of rotation:

  • a rotation rate of ω1 about a fixed axis (the "+" Y-axis), and,
  • a rotation rate of ω2 about a moving axis (the "+" z-axis)

(Be sure to make a clear distinction between the lower case and upper case symbols.)

Therefore, the angular velocity of the wheel is given by:

ω = ω1J + ω2 k

The angular acceleration vector α is simply the time derivative of the angular velocity vector ω : α = dω/dt. In taking this time derivative,

  • Recall that the J-axis is fixed. Since J is fixed, then dJ/dt = 0.
  • Recall that the k-axis is NOT fixed. Knowing that, how do you find dk/dt?

With the observer attached to the wheel, what motion does the observer see for points A and B? That is, what are (vA/O)rel and (aA/O)rel, and (vB/O)rel and (aB/O)rel?

NOTE: Pay particular attention to the motion of the reference point O. What path does O follow? And, based on that, how do you write down the acceleration vector of O, aO?


 

Homework H3.E - Fa22

Problem statement
Solution video

DISCUSSION THREAD

Ask your questions here. You can also answer questions of others. You can learn from helping others.


DISCUSSION and HINTS

Bar OA has two components of rotation:

  • One component of Ω about the fixed J-axis.
  • The second component of θ_dot about the moving k-axis.

Write out the angular velocity vector ω in terms of the two components described above.

Take a time derivative of ω to get the angular acceleration α of the bar. When taking this derivative, you will need to find the time derivative of the unit vector k. How do you do this? Read back over Section 3.2 of the lecture book. There you will see: k_dot = ω x k, where ω is the total angular velocity vector of bar OA that you found above.

Homework H3.C - Fa22

Problem statement
Solution video

DISCUSSION THREAD

Any questions??


Discussion:

Let's first take a look at the motion of point D. This motion of D is shown in the simulation results below.

The motion of D is circular, with the center of the path located at point A. This is expected since link AD is pinned to ground at point A.

Now, let's attach an observer to link BE. Keep in mind that this observer is unaware that they are moving. The motion that this observer sees is straight, with this straight path aligned with the slot cut into link BE. You can see this is the following animation shown from the perspective of the observer attached to link BE.


 

Homework H3.B - Fa22

Problem statement
Solution video

DISCUSSION THREAD

Any questions??


Discussion

Suppose that you decide to use the following velocity and acceleration equations:
vA = vO + (vA/O)rel + ω x rA/O
 aA = aO + (aA/O)rel + α x rA/O + 2ω x (vA/O)rel + ω x (ω x rA/O)

using an observer that is attached to link OB. With this choice, the angular velocity and acceleration of the observer are those of link OB:
ω  = ωOB k
α = α
OBk

Now to the question of what motion does the observer see for point A. The observer see A moving back and forth along the x-axis, where the x-axis is attached to OB:
(vA/O)rel = b_dot i
(aA/O)rel = b_ddot i

Note that A is traveling on a circular path of radius R at a constant speed vA. What does this say about the velocity and acceleration vectors for A and B,  vand aAat this instant? Watch the animation above to confirm your response.