Problem statementSolution video |

**DISCUSSION THREAD**

Any questions?

Problem statementSolution video |

**DISCUSSION THREAD**

**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 < ω _{n}* . 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 > ω _{n}* . 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 *f**our-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

* 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?

Problem statementSolution video |

**DISCUSSION THREAD**

* Question*: Is the excitation frequency

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

Problem statementSolution video |

**DISCUSSION THREAD**

**Low frequency: ω < ω _{n}**

As expected, the response is in-phase with the base motion. Can you see this?

**High frequency: ω > ω _{n}**

As expected, the response is 180° out-of-phase with the base motion. Can you see this?

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

Problem statementSolution video |

**DISCUSSION THREAD**

**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.**

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

Problem statementSolution video |

**DISCUSSION THREAD**

* 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 < ω

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 *f**our-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

* 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?

Problem statementSolution video |

**DISCUSSION THREAD**

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

**DISCUSSION and HINTS
**

Recall the following *f**our-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 }*[

Problem statementSolution video |

**DISCUSSION THREAD**

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 *f**our-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

**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 }*[

Problem statementSolution video |

**DISCUSSION THREAD**

**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 *f**our-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?

Problem statementSolution video |

**DISCUSSION THREAD**

*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.

*x*(t).

Recall the following *f**our-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?