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.

For this problem, you are asked to determine the amount of damping (i.e., the value of c) for which the system is CRITICALLY damped (ζ = 1). The animation below shows the response of such a critically damped system. Not that with this value for the damping ratio ζ, the oscillations are damped out, with the response asymptotically approaching the steady-state static equilibrium state.

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 θ(t).

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

Step 1: FBDs
Draw individual FBDs of the drum and the bar. Define a translation coordinate, x, for the bar.

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

Step 3: Kinematics
Use the no-slip condition between the drum and the bar to relate x to θ.

Step 4: EOM
Combine your Newton/Euler equations along with your kinematics to arrive at a single differential equation in terms of the dependent variable θ.

You will then need to find the static rotation of the disk from your EOM. Also, put the EOM in "standard form" in order to find the undamped natural frequency ω_n and the damping ratio ζ of terms of the given parameters for the system. Critical damping corresponds to ζ = 1.

Any questions?

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.)

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

Discussion and hints

Discussion and hints

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

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

DISCUSSION and HINTS
As the system moves, there is no slipping between the contact point C on the drum and block A.  This represents a constraint between the drum rotation and the block translation. We will deal with this in Step 3 of the derivation of the equation of motion below.

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 drum and the disk. Be sure to include the equal-and-opposite contact forces on both the drum and the block. It is important to temporarily define a coordinate that describes the motion of the block. Let's call that variable "x", and define it to be positive to the right. With this definition, the spring forces on the left and right side of the block are kx and 2kx, respectively, with both forces pointing to the left.

Step 2: Kinetics (Newton/Euler)
Using your FBDs from above, write down the Euler equation for the drum, and the Newton equation for the block. Combine these two equations through the elimination of the drum-to-block contact force.

Step 3: Kinematics
You need to relate the angular acceleration of the drum to the acceleration of the block. How is this done? What are the results? Also, how do you relate the stretch/compression x in the two springs in terms of θ?

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

Once you have determined the EOM for the system, identify the natural frequency from the EOM. Also from the EOM, we know that the general form of the response is: θ(t) = C*cos(ω_nt)+ S*sin(ω_nt). How do you find the response coefficients C and S?

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).

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

Step 1: FBDs
Draw a FBD of the disk. Define a rotation coordinate for the disk.

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

Step 3: Kinematics
Use the no-slip condition between the disk and the ramp to relate x to the rotational coordinate that you chose above.

Step 4: EOM
Combine your Newton/Euler equations along with your kinematics to arrive at a single differential equation in terms of the dependent variable x.

Any questions?

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

Discussion and hints

The four-step plan:

1. FBD: It is recommended that you draw individual free body diagrams of each disk and of bar AB. Note that the mass of AB is negligible and that forces are applied at only two points (i.e., a two-force member). What does this say about the direction of the forces on the disks at A and B?
2. Kinetics: Recommended that you use an Euler equation about the contact point of each disk since those are no-slip points for whose accelerations are directed toward the center of mass of each disk. Note that both of these two equations will involve the load carried by AB. You can eliminate that load from these equations, resulting in a single equation.
3. Kinematics: Note that rigid bar AB insures that the velocities of points A and B are the same. How can this be used to relate the angular accelerations of the two disks? Can you see this kinematic relationship between the angular accelerations in the animation above? Please note that the angular accelerations of the two disks are NOT the same, although their centers have the same accelerations.
4. EOM: Combine your results from Steps 2 and 3 to produce a single differential equation of motion (EOM) in terms of the x-coordinate.

Any questions? Ask/answer questions in the discussion thread below.

DISCUSSION

The animation below is for the case of c = 0 (undamped).

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 θ(t).

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

Step 1: FBDs
Draw individual free body diagrams for the drum and the block. Choose a translational coordinate (say x, defined as being positive to the left).  Be sure the get the correct direction for the spring and dashpot forces on the block. Also, take care in drawing the friction force on the drum as being equal and opposite to the friction force on the block.

Step 2: Kinetics (Newton/Euler)
Write down the Newton/Euler equations for the drum and block based on your FBDs above. Be sure to be in consistent in your sign conventions for forces/translation and moments/rotation.

Step 3: Kinematics
The contact point of the drum on the block (call it point A) is a no-slip point; that is, the horizontal component of acceleration of A is equal to the acceleration of the block.

Step 4: EOM
Combine your Newton/Euler equations along with your kinematics to arrive at a single differential equation in terms of the dependent variable θ.

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

DISCUSSION
As always, we should follow the four-step plan:

STEP 1: FBD
Draw an FBD of the bar. Since the support at B is smooth, the reaction on the bar at B will be perpendicular to the bar.

STEP 2: Kinetics
You should write down the two Newton equations, and the Euler equation. As always, take care in choosing your reference point for Euler's equation. Since we have no fixed points for the bar, you should choose the center of mass G as your reference point.

STEP 3: Kinematics
Note that the path of the center of mass G is tangent to the surface of the bar (that is, the bar can move only along the support, not into the support). With the bar being released from rest, the acceleration of G is tangent to the path. Specifically, the acceleration of G is directed along the direction of the bar. Write down the rigid body kinematics equation that relates the accelerations of A and G. This vector equation represents two scalar equations (x- and y-components).

STEP 4: Solve
At this point, you will have three kinetics equations and two kinematics equations, for a total of five equations. You will have five unknowns. Solve!

Any questions?