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

• 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?

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Question: Is the excitation frequency ω less than or larger than the natural frequency ω_n for the parameters used in the animation below?

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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?

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Discussion and hints

The above shows the results of simulations on this problem for two frequencies of excitation, ω. The upper plot where ω is less than the natural frequency, ω_n, and the lower plot where ω is greater than ω_n. In each plot, the RED curve is the forcing F(t) and the BLUE curve is the particular solution of the EOM, x_P(t). As can be seen from these simulation results:

• The response for ω < ω_n has the steady-state response of x_P(t) moving in phase with the forcing F(t). That is, during the time that the block is displaced to the right, the force is acting to the right. Conversely, during the time that the block is displaced to the left, the force is acting to the left.
• The response for ω > ω_n has the steady-state response of x_P(t) moving 180° out of phase with the forcing F(t). That is, during the time that the block is displaced to the right, the force is acting to the left, and during the time that the block is displaced to the left, the force is acting to the right.

In your analysis for this problem, you will be determining the natural frequency ω_n of the system, along with the steady-state response. Be sure to check your answer in the end: does it demonstrate the phase relations seen in the above simulation results?

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The above simulation results are for a set of generic parameters that  may be different from those assigned this semester.

Question: Can you tell from the animation if the excitation frequency ω is smaller than or larger than the natural frequency ω_n of the system?

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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.

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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.)

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Discussion and hints

Discussion and hints

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