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Discussion

The motion of this four-bar mechanism is quite complicated. However, we can make some sense out of the velocities of points B and D by using instant centers (ICs). For a given position of the mechanism, can you visualize the location of the IC for link BD? And from this location, do the directions for the velocities for points B and D make sense?

HINTS:
You are asked to perform the complete kinematical analysis for the mechanism for the position shown below.

Velocity
For the velocity analysis, you are encouraged to use the IC center approach. Where is the IC for link BD located? What does the location of this IC say about the angular velocity of link BD? Use that result to determine the angular velocity of link DE.

Acceleration
This is a standard kinematics of a mechanism problem. For this, consider the following steps:

1. Write down the acceleration of point B using link AB.
2. Write down the acceleration expression for D using link DE.
3. Write down the acceleration expression for B using link DB and your result for the acceleration of D from Step 2.
4. Equate the expressions for the acceleration of B in Steps 1 and 3 above. From this, solve for the angular accelerations of links BD and DE.

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Discussion

Points B and D on the disk roll without slipping on the horizontal guide A and the wedge E, respectively. Because of this:

• B has a velocity v_B which points straight down, with the same speed v_A as the horizontal guide A.
• D has a velocity v_D which points directly to the left with the same speed as the wedge E.

The instant center (IC) of the disk (call it point C) therefore exists at the intersection of the perpendiculars to v_B and v_D. Note that point B is much closer to the IC than point D. What does this say about the relative sizes of v_D and v_B? Is this consistent with the speeds of D and B shown in the above animation?

HINTS:

• Use the above discussion points to locate the IC for the disk.
• Determine the angular speed of the disk based on the speed of B and the distance from B to C.
• The speed of the wedge E is found by using the angular speed from above and the distance from C to D.

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Discussion

The animation above shows the motion of the mechanism over a range of input angles of link OA. For a given position, envision the location of the instant centers (ICs) for links AB and CD. Do the directions and magnitudes for the velocities of points B, C and D agree with the location of these ICs?

Shown below if a freeze-frame of the mechanism motion at the position for which you are asked to do analysis. From this figure, where are the two ICs for links CD and BC? In particular, how does the position of the IC for AB relate to the relative sizes of the speeds of points A, C and B? What is the angular velocity of link AB at this position?

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Discussion

As you plan your solution, consider that for the velocity part of the problem, you will need to use a velocity equation for bar AB and two velocity equations for the disk:

• Write the velocity of point B referenced to point A on bar AB.
• Write the velocity of point O referenced to the contact point C on the disk. Note that since the disk does not slip, the velocity of point C is zero.
• Write the velocity of point B referenced to point O on the disk.

These equations together will provide you with the equations that you need for the angular velocities of the bar and disk.

Repeat this same process above for accelerations. Please note that the acceleration of C on the disk is NOT zero - instead, you know that its acceleration is strictly in the y-direction.

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Discussion and hints:

The solution for the velocity and acceleration of end B is a straight-forward application of the rigid body velocity and acceleration equations for member AB:

v_B = v_A + omega x r_B/A
a_B = a_A + α x r_B/A - ω²*r_B/A

where v_B = v_B* j_hat, a_B = a_B*j_hat and a_A = a_A*(cos(θ)*i_hat + sin(θ)*j_hat). Each of the two vector equations above represents two scalar equations, providing us with the necessary equations to solve for v_B, ω, a_B and α. All of the observations made above can be predicted by the above kinematics equations. Instant centers (later on in the course) can prove useful in providing explanations.

For the inclination angle used in the above simulation, we see that point B moves DOWNWARD along the vertical wall as A moves up along the incline. As B moves onto the same horizontal plane as A, the acceleration of B becomes very large (although A continues to move with a constant speed). Can you provide a physical explanation for this?

If we now consider a steeper inclination angle for A, as used above, we see that end B initially moves UPWARD along the wall; however, at some point B reverses its direction and begins to move DOWNWARD along the wall. Can you provide a physical explanation for this? Note also that the acceleration of B becomes very large as B moves onto the same horizontal plane as A, as it was for the initial value of inclination angle.

What is the value of the incline angle theta that defines the boundary between the types of initial motions for bar AB shown in the above two simulations? For the numerical value of the angle theta provided in the problem statement, which of the two simulations above agree with your results?

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Animation of motion

Note that as link BD passes through the horizontal orientation, the velocity of B (as well as the angular velocity of AB) instantaneously goes to zero, yet the acceleration of B is not zero. In a couple days when we cover instant centers of zero velocity we will be able to predict when point B has zero velocity.

Solution plan

1. Note that since D is traveling on a straight line with constant speed, the acceleration of D is zero. Therefore, we can write: a_B = a_D + alpha_BD x r_B/D - omega_BD^2*r_B/D = alpha_BD x r_B/D - omega_BD^2*r_B/D.
2. Note that since A is a pin joint, the acceleration of A is zero. Therefore, we can write: a_B = a_A + alpha_AB x r_B/A - omega_AB^2*r_B/A = alpha_AB x r_B/A - omega_AB^2*r_B/A.
3. Equate the expressions of a_B  from 1. and 2. above. This gives you two SCALAR equations in order to solve for two unknowns, alpha_AB and alpha_BD.

Note that you need to solve the velocity problem first. Follow the above procedure to do so. Here. the velocity of D is known as v_D = -v_D*j_hat.

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