All posts by CMK

Homework H2.J - Sp24

Problem statement
Solution video


DISCUSSION THREAD
For this problem, please use a CLOCKWISE rotation for link AB (that is, use what the problem figure indicates).

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.


Any questions?? Please ask/answer questions regarding this homework problem through the "Leave a Comment" link above.

Homework H2.G - Sp24

Problem statement
Solution video


DISCUSSION THREAD

Discussion

Consider the motion of the system shown below,

along with a freeze-frame of this motion that follows:

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 vB which points straight down, with the same speed vA as the horizontal guide A.
  • D has a velocity vD 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 vand vD. Note that point B is much closer to the IC than point D. What does this say about the relative sizes of vD and vB? 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.

Any questions?? Please ask/answer questions regarding this homework problem through the "Leave a Comment" link above.

Homework H2.H - Sp24

Problem statement
Solution video


DISCUSSION THREAD

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 is 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?


Any questions?? Please ask/answer questions regarding this homework problem through the "Leave a Comment" link above.

Homework H2.E - Sp24

Problem statement
Solution video


DISCUSSION THREAD

DISCUSSION

Note that with the disk not slipping on the outer cable, and with section BC of that cable being at rest, the velocity of point C on the disk has zero velocity. This means that the disk rolls without slipping on the cable, which is the same is it were rolling without slipping on a fixed vertical wall.

With the above in mind, consider the following:

  • Let M represent the point on the left side of the pulley from which the outer cable leaves contact with the disk. Note that M is moving vertically upward with a speed of vA.
  • Write a rigid body kinematics equation relating the motion of points C and M: vM = vC + ω x rM/C . That equation will produce an expression for ω, the angular speed of the disk.
  • Then, you can use vN = vC + ω x rN/C and  vH = vC + ω x rH/C to find the velocities of point N and H on the disk. Since the inner cable does not slip on the disk, then vN = vD and vH = vQ.
  • Also,  vE = vC + ω x rE/C.

Any questions?? Please ask/answer questions regarding this homework problem through the "Leave a Comment" link above.

Homework H2.F - Sp24

Problem statement
Solution video


DISCUSSION THREAD

DISCUSSION
For the velocity analysis of this mechanism, it is recommended that you write down three vector velocity equations: 1) relating the motion of O and C; 2) the motion of B and C; and, 3) relating the motion of A and B. Solve the resulting scalar equations from these vector equations for the angular velocities of AB and of the disk.

For the acceleration analysis, repeat the above process, but now use the three acceleration equations. Solve the resulting scalar equations from these vector equations for the angular accelerations of AB and of the disk. Please note that the acceleration of the no-slip contact point is NOT zero. Please read this page from the lecture book for an explanation as to why C can have a non-zero component of acceleration in the direction normal to the surface on which it rolls..

For example, the velocity and acceleration equations for link AB are given by:
vB = vA + ωAB x rB/A
aB = aA + αAB x rB/A - ωAB2*rB/A


 

Homework H2.C - Sp24

Problem statement
Solution video



DISCUSSION THREAD

We encourage you to interact with your colleagues here in conversations about this homework problem.


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:

vB = vA + ω x rB/A
aB = aA + α x rB/A - ω2*rB/A

where vB = vBj, aB = aB*j and aA = aA*(cos(θ)*+ sin(θ)*j)Each of the two vector equations above represents two scalar equations, providing us with the necessary equations to solve for vB, ω, aB 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?


 

Homework H2.D - Sp24

Problem statement
Solution video



DISCUSSION THREAD

 

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.


Any questions?? Please ask/answer questions regarding this homework problem through the "Leave a Comment" link above.

Homework H2.A - Sp24

Problem statement
Solution video

DISCUSSION THREAD

Discussion

Note that the plate rotates about point O. Therefore, O is the center of the circular paths of points A and B. From the animation above, we see that the velocities of A and B are tangent to their circular paths, as expected. The accelerations of A and B are NOT perpendicular to the paths of A and B since the speeds of A and B are increasing in time (and consequently, A and B each have positive tangential components of acceleration).

Initially, the acceleration for these two points is nearly aligned with velocity, since the speeds are small and therefore the centripetal components of acceleration are small. Near the end of the first revolution of the plate, the speeds have increased to the point where the centripetal components of acceleration dominate, and acceleration is nearly perpendicular to the path.

Solution hints
For Part a) of this problem, it is recommended that you use the rigid body kinematics equations using point O as the reference point, since the velocity and acceleration of O are zero. That is, you should use v_B = v_O + Ω x r_B/O and a_B = a_O + Ω_dot x r_B/O - Ω^2*r_B/O. Repeat the process for finding the velocity and acceleration of A.

For Part b) of this problem, it is recommended that you use the rigid body kinematics equations with point A first. This will give you the equations needed to find Ω_dot. Then, use the rigid body kinematics equations to find the acceleration of B.


Any questions?? Please ask/answer questions regarding this homework problem through the "Leave a Comment" link above.