Ask your questions here. Or, answer questions of others here. Either way, you can learn.

DISCUSSION and HINTS

In this problem, the excitation does not come from a prescribed force, but, instead, it arises from a prescribed displacement on one body in the system. The support B here is given a prescribed motion of x_B(t) = b sinωt, where ω is the frequency of excitation. Our goal is to solve for the particular solution of the response. Shown below is an animation of this forced response for a given value of excitation frequency? Can you tell from watching the motion of the system if the excitation frequency is less than or greater than the natural frequency of the system?

Recall the following four-step plan outline in the lecture book and discussed in lecture:

Step 1: FBDs
Draw a free body diagram (FBD) of the disk. You will need to temporarily define a coordinate representing the rotation of the disk. For example, you could use the angle θ as the angle of rotation for the disk, measured positive in the clockwise direction. Take care in getting the directions correct on the spring forces acting on the disk. Also, be sure to include the friction force acting on the disk by the cart B. Note that you do NOT need an FBD of B since you already know its motion.

Step 2: Kinetics (Newton/Euler)
Write down both Newton’s 2nd law and Euler’s equation for the disk. Note that with C being the no-slip point on the disk: ΣM_C ≠ I_C α.  Why is that?

Step 3: Kinematics
Here you need to relate x_ddot to θ_ddot. How do you do this? Please note that C is an accelerating point on the disk.

Step 4: EOM
Combine your results from Step 2 and Step 3 to arrive at your EOM.

Once you have determined the EOM for the system, identify the natural frequency from the EOM. From the EOM we know that the particular solution of the EOM is given by: x_P(t) = A*sin(ωt)+ B*sin(ωt). How do you find the forced response coefficients A and B? And, how do you determine the phase of this forced response?

Any questions?

Discussion and hints:

The derivation of the dynamical equation of motion (EOM) for a system is a straight-forward application of what we have learned from Chapter 5 in using the Newton-Euler equations. The goal in deriving the EOM is to end up with a single differential equation in terms of a single dependent variable that describes the motion of the system. Here in this problem, we want our EOM to be in terms of x(t), and then later in terms of z(t).

Recall the following four-step plan outline in the lecture book and discussed in lecture:

Step 1: FBDs
Draw an FBD of the block. Take care in expressing the force on the block due to the spring. The force in the spring depends in magnitude on the relative displacements of the block (using the x-coordinate) and the ground, with the direction of the spring force found by the thought process covered in lecture.

Step 2: Kinetics (Newton/Euler)
Use Newton’s second law on your FBD found in Step 1.

Step 3: Kinematics
Do you need any additional kinematics here?

Step 4: EOM
Your single differential equation of motion for the system in terms of the x-coordinate was found in Step 2. Next, we need to convert this EOM to being in terms of the z-coordinate. To this end, use the definition of z = x – x_B. From this, we see that x = z + x_B and x_ddot = z_ddot + x_B_ddot. Substitute these into your x-coordinate  EOM above so that it is now in terms of z and its second time derivative. Note that the left-hand side of the EOM is unchanged except it is now in terms of z. The right-hand sides of the EOMs are different.

For this problem, you need to determine the particular solution for each EOM corresponding to the excitation. Make a sketch of the amplitude of response in each case as a function of Omega. Do you see a consistency between these two plots?

Any questions?

Ask your questions here. Or, answer questions of others here. Either way, you can learn.

DISCUSSION and HINTS

In this problem, the excitation does not come from a prescribed force, but, instead, it arises from a prescribed displacement on one body in the system. The support B here is given a prescribed motion of x_B(t) = b cosωt, where ω is the frequency of excitation. Our goal is to solve for the particular solution of the response. Shown below are animations of this forced response corresponding to two different frequencies: the top animation has ω < ω_n, and the bottom animation has ω > ω_n, where ω_n is the natural frequency of the system. Can you see the difference between these two simulation in terms of the the phase of the response? Study both the time histories and the animation of motion.

Recall the following four-step plan outline in the lecture book and discussed in lecture:

Step 1: FBDs
Draw a free body diagram (FBD) of block A. Take care in getting the directions correct on the spring forces acting on A. Note that you do NOT need an FBD of B since you already know its motion.

Step 2: Kinetics (Newton/Euler)
Use Newton’s 2nd law to arrive at the EOM.

Step 3: Kinematics
None needed here.

Step 4: EOM
The EOM was found back in Step 2.

Once you have determined the EOM for the system, identify the natural frequency from the EOM. From the EOM we know that the particular solution of the EOM is given by: x_P(t) = A*sin(ωt)+ B*sin(ωt). How do you find the forced response coefficients A and B?

Does your result here agree with the expected relationship between excitation frequency and response with regard to the “phase” of the solution, and with the animations above?

Ask your questions here. Or, answer questions of others here. Either way, you can learn.

DISCUSSION and HINTS

Recall the following four-step plan outline in the lecture book and discussed in lecture:

Step 1: FBDs
Draw a free body diagram (FBD) of block A.

Step 2: Kinetics (Newton/Euler)
Use Newton’s 2nd law to write down the dynamical equation for the system in terms of the coordinate x. Be careful with the directions of your spring forces, and that these are consistent with the coordinate x defined. here.

Step 3: Kinematics
None needed here.

Step 4: EOM
Your EOM was found at Step 2.

Once you have determined the EOM for the system, identify the natural frequency from the EOM. From the EOM we know that the particular solution of the EOM is given by: x_P(t) = A*sin(ωt)+ B*cos(ωt). How do you find the forced response coefficients A and B?

Ask your questions here. Or, answer questions of others here. Either way, you can learn.

DISCUSSION and HINTS

The particular solution for the equation of motion for this system is shown below for two values of the frequency of excitation, ω. As we have seen in lecture, and as shown below, when the excitation frequency is less than the natural frequency,ω_n, the response is in phase with the excitation, and when the frequency is greater than ω_n, the response is 180° out of phase with the excitation. Can you see this in the animations below?

Recall the following four-step plan outline in the lecture book and discussed in lecture:

Step 1: FBDs
Draw a free body diagram (FBD) of the disk. It is important to temporarily define a translation coordinate that describes the position of the center of the disk, O. Let’s call that variable x, and have it defined as being positive to the right.

Step 2: Kinetics (Newton/Euler)
Use Euler’s equation about the no-slip contact point C.

Step 3: Kinematics
Relate x_ddot to θ_ddot through kinematics.

Step 4: EOM
Combine your equations from Steps 2 and 3 to arrive at your EOM in terms of θ.

Once you have determined the EOM for the system, identify the natural frequency from the EOM. From the EOM we know that the particular solution of the EOM is given by: theta_P(t) = A*sin(ωt)+ B*cos(ωt). How do you find the forced response coefficients A and B?

Does your result here agree with the expected relationship between excitation frequency and response with regard to the “phase” of the solution?

Ask your questions here. Or, answer questions of others here. Either way, you can learn.

DISCUSSION and HINTS

Recall the following four-step plan outline in the lecture book and discussed in lecture:

Step 1: FBDs
Draw individual free body diagrams (FBDs) of the wheel and block A.   Don’t forget the equal-and-opposite pair of reaction forces acting between the wheel and the block atO. It is important to temporarily define a coordinate that describes the rotation of the wheel. Let’s call this θ,  and define it to be positive in the clockwise sense.

Step 2: Kinetics (Newton/Euler)
Using your FBDs from above, write down the Euler equation for the disk (about the no-slip point C), and the Newton equation for the block. Combine these two equations into a single equation through the elimination of the reaction forces at O for the equations.

Step 3: Kinematics
You need to relate the angular acceleration of the disk and the acceleration of the block. How is this done? What are the results?

Step 4: EOM
From your equations in Steps 2 and 3, derive the equation of motion (EOM) of the system in terms of x.

Once you have determined the EOM for the system, identify the natural frequency, damping ratio and the damped natural frequency from the EOM. Also, from the EOM we know that the response of the system in terms of x is given by: x(t) = e^-ζω_nt [C*cos(ω_dt)+ S*sin(ω_dt)]. How do you find the response coefficients C and S?

Ask your questions here. Or, answer questions of others here. Either way, you can learn.

DISCUSSION and HINTS

After block B strikes and sticks to block A, the two blocks move together as a single rigid body. In deriving your equation of motion here, focus on a system with a single block (of mass 3m) connected to ground with a spring and dashpot.

Recall the following four-step plan outline in the lecture book and discussed in lecture:

Step 1: FBDs
Draw a free body diagram (FBD) of A+B.

Step 2: Kinetics (Newton/Euler)
Use Newton’s 2nd law to write down the dynamical equation for the system in terms of the coordinate x.

Step 3: Kinematics
None needed here.

Step 4: EOM
Your EOM was found at Step 2.

Once you have determined the EOM for the system, identify the natural frequency, damping ratio and the damped natural frequency from the EOM. Also, from the EOM we know that the response of the system in terms of x is given by: x(t) = e^-ζωnt [C*cos(ω_dt)+ S*sin(ω_dt)]. How do you find the response coefficients C and S? What will you use for the initial conditions of the problem? (Consider linear momentum of A+B during impact.)