# Homework H6.M - Sp23

 Problem statement Solution video Any questions?

# Homework H6.N - Sp23

 Problem statement Solution video Discussion and hints:

Shown below is the particular (steady-state) solution of the EOM for this system with the frequency of excitation less than the natural frequency: ω < ωn . Note that the response in blue is completely in phase with the base motion in red. This is expected based on what we have see in the lecture book and in lecture. Now we look at the response with the frequency of excitation being greater than the natural frequency: ω > ωn . Note that the response in blue is 180° out of phase with the base motion in red. This is also expected based on what we have see in the lecture book and in lecture. 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).

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

Step 1: FBDs
Draw individual FBDs of the disk and the block. Define a rotational coordinate θ for the disk. Take care in expressing the force on the block due to the spring connecting the block to the base B. The force in that spring depends in magnitude on the relative displacements of the block and the base, with the direction of the spring force found by the thought process covered in lecture.

Step 2: Kinetics (Newton/Euler)
Write down the Newton/Euler equations for the disk and the block.

Step 3: Kinematics
Use the no-slip condition at both the top and bottom locations on the disk to relate x and θ.

Step 4: EOM
Combine your Newton/Euler equations and your kinematics equations to arrive at the single differential equation of motion for the system.

For this problem, you need to determine the particular solution for the EOM corresponding to the excitation.

Any questions?

# Homework H6.K - Sp23

 Problem statement Solution video Question: Is the excitation frequency ω less than or larger than the natural frequency ωn for the parameters used in the animation below? # Homework H6.L - Sp23

 Problem statement Solution video Low frequency: ω < ωn
As expected, the response is in-phase with the base motion. Can you see this? High frequency: ω > ωn
As expected, the response is 180° out-of-phase with the base motion. Can you see this? # Homework H6.I - Sp23

 Problem statement Solution video

NOTE: Please use the mass values provided in the FIGURE, not in the text of the problem. That is, A has a mass of 2m, and B has a mass of m. # Homework H6.J - Sp23

 Problem statement Solution video Discussion and hints:
Shown below are animations of the steady-state response of the system for two values of excitation frequency: the first with Ω < ωn, and the second with Ω > ωn. In comparing these two time histories, focus on the phase between the input M(t) and the steady-state response x(t). Do you see the difference?  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).

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

Step 1: FBDs
Draw individual FBDs of the disk and the block. Define a rotational coordinate θ for the disk.

Step 2: Kinetics (Newton/Euler)
Write down the Newton/Euler equations for the disk and the block.

Step 3: Kinematics
Use the no-slip condition at C to relate x and θ.

Step 4: EOM
Combine your Newton/Euler equations and your kinematics equations to arrive at the single differential equation of motion for the system.

For this problem, you need to determine the particular solution for the EOM corresponding to the excitation.

Any questions?

# Homework H6.G - Sp23

 Problem statement Solution video 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-ζωnt [C*cos(ωdt)+ S*sin(ωdt)]. How do you find the response coefficients C and S?

# Homework H6.H - Sp23

 Problem statement Solution video 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(ωdt)+ S*sin(ωdt)]. 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.)

# Homework H6.E - Sp23

 Problem statement Solution video Discussion and hints:

Shown  below is an animation of the results of a simulation of the motion corresponding to an UNDERDAMPED system. The response is oscillatory, however, the amplitude of the response decays away at an exponential rate. 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).

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

Step 1: FBDs
Draw an FBD of the block.

Step 2: Kinetics (Newton/Euler)
Write down the Newton/Euler equations for the block.

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

Step 4: EOM
Your EOM should come directly from what you do in Step 2.

Determine the static deformation of the block from your FBD. Also, put your EOM in "standard form." From this, identify the undamped natural frequency, the damping ratio and the damped natural frequency in terms of the system parameters of m, c and k.

Any questions?

# Homework H6.F - Sp23

 Problem statement Solution video

NOTE - Please use the following parameter values:  m = 2 kg, c = 16 kg/s and k = 800 N/m. Discussion and hints:

Shown  below is an animation of the results of a simulation of the motion corresponding to an UNDERDAMPED system. The response is oscillatory, however, the amplitude of the response decays away at an exponential rate. One of your tasks in solving this problem is to determine if the system is underdamped, critically damped or overdamped. 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).

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

Step 1: FBDs
Draw an FBD of the block.

Step 2: Kinetics (Newton/Euler)
Write down the Newton/Euler equations for the block.

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

Step 4: EOM
Your EOM should come directly from what you do in Step 2.

For this problem, you need to:

• Put your EOM in "standard form." From this, identify the undamped natural frequency, the damping ratio and the damped natural frequency in terms of the system parameters of m, c and k.
• If the system is underdamped (ζ < 1), then the solution can be written in terms of the decaying oscillation response derived in the lecture book and in class.
• You need to enforce the initial conditions on the problem (x(0) and dx/dt(0)) to determine the two response coefficients.

Any questions?