Homework H6.N - Fa22

DISCUSSION

Four-step plan (for deriving EOM)

Step 1: FBD
Draw a free-body diagram of bar OA. Be sure have your spring force drawn correctly based on the coordinates shown.

Step 2: Kinetics - Newton/Euler
Write down the Euler equation for the bar.

Step 3: Kinematics
The horizontal displacement of end A of the bar is given by Lsinθ.

Step 4: Equation of motion
Linearize your EOM for small angles of θ through the use of cosθ ≈ 1 and sinθ ≈ θ.

Homework H6.K - Fa22

Low and high frequency excitations: ω < ωn and ω > ωn

• The top animation below shows the particular solution for the response corresponding to an excitation frequency that is less than the natural frequency. For this range of excitation frequencies, the response is in phase with the base motion.
• The lower animation is for the excitation frequency larger than the natural frequency. As expected, this response is 180° out of phase with the excitation.

Are these phase differences apparent to you as you view the animations?

Homework H6.L - Fa22

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 xB(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 < ωn, and the bottom animation has ω > ωn, where ωis 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: xP(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?

Homework H6.I - Fa22

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,ωn, the response is in phase with the excitation, and when the frequency is greater than ωn, 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: xP(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?

Homework H6.J - Fa22

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 - Fa22

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.H - Fa22

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?

Homework H6.E - Fa22

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.F - Fa22

Any questions??

DISCUSSION

Four-step plan (for deriving EOM)

Step 1: FBD
Draw a free-body diagram of block A. Be sure have your spring force and dashpot forces drawn correctly based on the coordinate x shown.

Step 2: Kinetics - Newton/Euler
Write down the Newton equation for the block.

Step 3: Kinematics
None needed here.

Step 4: Equation of motion

Homework H6.C - Fa22

CLARIFICATION:  Consider the bar to be parallel to the incline. This is made possible by having half-thickness of the bar being R, thus keeping the centerline of the bar parallel to the incline.

DISCUSSION

Four-step plan (for deriving EOM)

Step 1: FBD
Draw individual free-body diagrams for the two cylinders and for bar B. Do not forget the friction forces at the no-slip contact points when drawing your FBDs of the cylinders and bar. Define rotation coordinates β and ϕ for the upper and lower cylinders, respectively. Let's say that you choose both of these rotation coordinates to be positive in the CW sense. Be sure that the direction of the spring force on the bar is consistent with the definition of x.

Step 2: Kinetics - Newton/Euler
Write down the Newton/Euler equations for the two cylinders and the bar. Be sure to abide by the sign conventions defined for β, ϕ and x when writing down these equations.

Step 3: Kinematics
The kinematics that you need here are to relate β andϕ to x. Be reminded that the contact points of the cylinders with the fixed incline are the instant centers for the cylinders. As before, be sure to abide by the sign conventions defined for β, ϕ and x. For β and ϕ defined positive in the CW sense, we have = +3Rβ and = +2Rϕ .

Step 4: Equation of motion
Combine your equations from Steps 2 and 3 to end up with the differential equation of motion for the system in terms of x.