Lesson test: Arbitrary forcing

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Arbitrary forcing

1.[3p]
In variation of parameters, why is the condition $u_{1}' y_{1} + u_{2}' y_{2} = 0$ imposed?

It is a free choice that keeps second derivatives of the unknowns out of the calculation

Correct

It is required for $y_{p}$ to satisfy the initial conditions

The answer is: It is a free choice that keeps second derivatives of the unknowns out of the calculation

It forces the Wronskian to be one

The answer is: It is a free choice that keeps second derivatives of the unknowns out of the calculation

Without it the particular solution would not be unique

The answer is: It is a free choice that keeps second derivatives of the unknowns out of the calculation

2.[2p]
What is the Wronskian of $y_{1} = \cos t$ and $y_{2} = \sin t$ ?

CorrectNot quite: 1

3.[2p]
What does the impulse response $h (t)$ of a system describe?

The motion following a very short blow of unit impulse, starting from rest

Correct

The steady state under sinusoidal driving

The answer is: The motion following a very short blow of unit impulse, starting from rest

The general solution of the homogeneous equation

The answer is: The motion following a very short blow of unit impulse, starting from rest

The response to a suddenly applied constant force

The answer is: The motion following a very short blow of unit impulse, starting from rest

4.[3p]
A second order system with $ζ = 0.1$ is given a sudden constant force. By what percentage does its response overshoot the final value?

CorrectNot quite: 72.9

5.[3p]
A system has $ω_{0} = 10$ rad s⁻¹ and $ζ = 0.3$ . At what time, in seconds, does its step response first peak?

CorrectNot quite: 0.329

6.[1p]
The convolution formula $y_{p} = \int_{0}^{t} h (t - s) g (s) d s$ applies to any equation, linear or not.

True

The answer is: False

False

Correct

7.[3p]
Which statements about a linear time invariant system are correct?

Select all that apply

The step response is the integral of the impulse response

Correct

The frequency response and the impulse response carry the same information

Correct

The response to a periodic forcing follows by treating each Fourier component separately

Correct

A system whose mass changes with time still has a well-defined impulse response

The answer is: The step response is the integral of the impulse response, The frequency response and the impulse response carry the same information, The response to a periodic forcing follows by treating each Fourier component separately

8.[3p]
A square wave at frequency $f$ drives a lightly damped system whose natural frequency is $5 f$ . What dominates the output?

The fifth harmonic, because its gain near resonance outweighs its small share of the input

Correct

The fundamental, because it is the largest component of the input

The answer is: The fifth harmonic, because its gain near resonance outweighs its small share of the input

The seventh harmonic, because it is closest to the high frequency limit

The answer is: The fifth harmonic, because its gain near resonance outweighs its small share of the input

Nothing, since a square wave contains no sinusoids

The answer is: The fifth harmonic, because its gain near resonance outweighs its small share of the input

9.[3p]
Match each description of a system to its name.

Response to a tap
Response to a suddenly applied constant force
Amplitude and phase against driving frequency
Integral of forcing against a kernel of elapsed time

the step response
the impulse response
the frequency response
a convolution

Show the answer

Response to a tap: the impulse response Response to a suddenly applied constant force: the step response Amplitude and phase against driving frequency: the frequency response Integral of forcing against a kernel of elapsed time: a convolution

1.[3p] In variation of parameters, why is the condition u1'y1+u2'y2=0 imposed?

2.[2p] What is the Wronskian of y1=cost and y2=sint?

3.[2p] What does the impulse response h(t) of a system describe?

4.[3p] A second order system with ζ=0.1 is given a sudden constant force. By what percentage does its response overshoot the final value?

5.[3p] A system has ω0=10 rad s⁻¹ and ζ=0.3. At what time, in seconds, does its step response first peak?

6.[1p] The convolution formula yp=∫0th(t-s)g(s)ds applies to any equation, linear or not.

7.[3p] Which statements about a linear time invariant system are correct?

8.[3p] A square wave at frequency f drives a lightly damped system whose natural frequency is 5f. What dominates the output?

9.[3p] Match each description of a system to its name.

1.[3p]
In variation of parameters, why is the condition $u_{1}' y_{1} + u_{2}' y_{2} = 0$ imposed?

2.[2p]
What is the Wronskian of $y_{1} = \cos t$ and $y_{2} = \sin t$ ?

3.[2p]
What does the impulse response $h (t)$ of a system describe?

4.[3p]
A second order system with $ζ = 0.1$ is given a sudden constant force. By what percentage does its response overshoot the final value?

5.[3p]
A system has $ω_{0} = 10$ rad s⁻¹ and $ζ = 0.3$ . At what time, in seconds, does its step response first peak?

6.[1p]
The convolution formula $y_{p} = \int_{0}^{t} h (t - s) g (s) d s$ applies to any equation, linear or not.

7.[3p]
Which statements about a linear time invariant system are correct?

8.[3p]
A square wave at frequency $f$ drives a lightly damped system whose natural frequency is $5 f$ . What dominates the output?

9.[3p]
Match each description of a system to its name.