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In the animation of the simulation shown below, the RED vectors shown are the forces of reaction acting on particles A and B (such as the force on each particle by member AB, and the normal forces of reaction by the floor and wall).

Since the motion of P is being described here in terms of polar variables of r and θ, it is recommended that you use a polar description for your forces and acceleration.

Use the Four-Step solution plan outlined in the lecture book:

Step 1 - FBD: Draw a free body diagram of C. NOTE: The arm rotates about a vertical axis, meaning that the arm moves in a horizontal plane; that is, the gravitational force acts perpendicular to the plane of the paper.

Step 2 - Kinetics (Newton): Resolve the forces in your FBD into their polar components. Sum forces in the r-direction and set equal m*a_r. Sum forces in the θ-direction and set equal to m*a_θ

Step 3 - Kinematics: Use the polar kinematics descriptions of a_r = r_ddot - r*θ_dot^2 and a_θ = r*θ_ddot + 2*r_dot*θ_dot.

Step 4 - Solve. When solving for the normal force, N, acting on C take note of the sign on your answer. What does this sign mean in terms of answering Part (c)?

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Any questions??

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 your 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.

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 (v_A/O)_rel and (a_A/O)_rel, and (v_B/O)_rel and (a_B/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, a_O?

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.

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DISCUSSION and HINTS

The disk has two components of rotation:

• One component of ω₁ about the fixed J-axis.
• The second component of ω₂ about the moving k-axis.

Part (a)
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.

Part (b)
Here you will use the MRF kinematics equations of:
v_P = v_O +(v_P/O)_rel + ω x r_P/O
a_P = a_O + (a_P/O)_rel + α x r_P/O + 2ω x (v_P/O)_rel + ω x (ω x r_P/O)

For these equations, employ an observer on the disk. The angular velocity and angular acceleration of the observer are the same as the ω and α of the disk found in Part (a). What are the relative velocity and relative acceleration terms ((v_P/O)_rel and (a_P/O)_rel ) in these equations? These represent the velocity and acceleration of P as seen by our observer. As our observer moves with the disk and with P being on the disk, what motion does the observer see for P?

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.