Lesson test: Driving, resonance and the frequency response

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Driving, resonance and the frequency response

1.[2p]
Why is the steady state the only part of the answer that matters after a while?

The complementary function decays whenever there is damping, leaving only the particular solution

Correct

The particular solution is always larger than the complementary function

The answer is: The complementary function decays whenever there is damping, leaving only the particular solution

The initial conditions apply only to the particular solution

The answer is: The complementary function decays whenever there is damping, leaving only the particular solution

The forcing eventually stops

The answer is: The complementary function decays whenever there is damping, leaving only the particular solution

2.[3p]
A lightly damped oscillator with $ζ = 0.02$ is driven at its natural frequency. By what factor is the static deflection amplified?

CorrectNot quite: 25

3.[3p]
For $ζ = 0.2$ , at what value of $ρ = ω / ω_{0}$ does the amplitude peak?

CorrectNot quite: 0.959

4.[2p]
What is the phase lag of the response exactly at $ρ = 1$ ?

90 degrees, whatever the damping ratio

Correct

0 degrees, since the drive and response are in step

The answer is: 90 degrees, whatever the damping ratio

180 degrees, since inertia dominates

The answer is: 90 degrees, whatever the damping ratio

It depends on the amplitude of the forcing

The answer is: 90 degrees, whatever the damping ratio

5.[2p]
The Tacoma Narrows bridge failed because the wind supplied a periodic force at the bridge's natural frequency.

True

The answer is: False

False

Correct

6.[3p]
An undamped oscillator with $m = 2$ kg and $ω_{0} = 5$ rad s⁻¹ is driven at resonance by a force of amplitude 10 N. Its response is $\frac{F_{0}}{2 m ω_{0}} t \sin ω_{0} t$ . What is the amplitude, in metres, after 20 s?

CorrectNot quite: 10

7.[2p]
Above what value of $ρ$ does a spring mount begin to isolate rather than amplify?

CorrectNot quite: 1.414

8.[3p]
Which statements about the frequency response are correct?

Select all that apply

Well below resonance the response equals the static deflection and is in phase with the drive

Correct

Well above resonance the amplitude falls as $1 / ρ^{2}$ and the response is in antiphase

Correct

For $ζ > 0.707$ the amplitude curve has no peak at all

Correct

Adding damping always reduces the motion transmitted through a mount, at any frequency

The answer is: Well below resonance the response equals the static deflection and is in phase with the drive, Well above resonance the amplitude falls as $1 / ρ^{2}$ and the response is in antiphase, For $ζ > 0.707$ the amplitude curve has no peak at all

9.[3p]
Match each driving regime to the property that controls the response.

Well below the natural frequency
Close to the natural frequency
Well above the natural frequency

the mass
the stiffness
the damping

Show the answer

Well below the natural frequency: the stiffness Close to the natural frequency: the damping Well above the natural frequency: the mass

1.[2p] Why is the steady state the only part of the answer that matters after a while?

2.[3p] A lightly damped oscillator with ζ=0.02 is driven at its natural frequency. By what factor is the static deflection amplified?

3.[3p] For ζ=0.2, at what value of ρ=ω/ω0 does the amplitude peak?

4.[2p] What is the phase lag of the response exactly at ρ=1?

5.[2p] The Tacoma Narrows bridge failed because the wind supplied a periodic force at the bridge's natural frequency.

6.[3p] An undamped oscillator with m=2 kg and ω0=5 rad s⁻¹ is driven at resonance by a force of amplitude 10 N. Its response is F02mω0tsinω0t. What is the amplitude, in metres, after 20 s?

7.[2p] Above what value of ρ does a spring mount begin to isolate rather than amplify?

8.[3p] Which statements about the frequency response are correct?

9.[3p] Match each driving regime to the property that controls the response.

1.[2p]
Why is the steady state the only part of the answer that matters after a while?

2.[3p]
A lightly damped oscillator with $ζ = 0.02$ is driven at its natural frequency. By what factor is the static deflection amplified?

3.[3p]
For $ζ = 0.2$ , at what value of $ρ = ω / ω_{0}$ does the amplitude peak?

4.[2p]
What is the phase lag of the response exactly at $ρ = 1$ ?

5.[2p]
The Tacoma Narrows bridge failed because the wind supplied a periodic force at the bridge's natural frequency.

6.[3p]
An undamped oscillator with $m = 2$ kg and $ω_{0} = 5$ rad s⁻¹ is driven at resonance by a force of amplitude 10 N. Its response is $\frac{F_{0}}{2 m ω_{0}} t \sin ω_{0} t$ . What is the amplitude, in metres, after 20 s?

7.[2p]
Above what value of $ρ$ does a spring mount begin to isolate rather than amplify?

8.[3p]
Which statements about the frequency response are correct?

9.[3p]
Match each driving regime to the property that controls the response.