GRE Physics Companion: Driven Harmonic Motion and Resonance

GRE Physics Companion: Driven Harmonic Motion and Resonance

For

(1)

the steady-state displacement amplitude is

(2)

The phase lag is

(3)

1 High-value GRE facts

1. The steady-state response oscillates at the driving frequency.
2. Initial conditions determine the transient, not the long-time steady-state amplitude.
3. X = F₀∕.
4. δ = π∕2 at ω = ω₀.
5. displacement resonance occurs at ω_r = when γ < ω₀∕.
6. velocity amplitude and average absorbed Power peak at ω₀.
7. The half-power bandwidth is Δω = 2γ = b∕m.
8. Q = ω₀∕(2γ) = mω₀∕b.
9. Low-frequency displacement tends to F₀∕k.
10. Exact undamped Resonance produces amplitude growing linearly with time.

Part I: Original GRE-style problems

Problem 1: steady-state frequency

A damped oscillator is driven sinusoidally at angular frequency ω. After transients disappear, the displacement oscillates at

1. ω₀ only
2. ω_d only
3. ω
4. ω + ω₀
5. zero frequency

Problem 2: amplitude

The steady-state amplitude is

1. F₀∕(k − mω²)
2. F₀∕
3. F₀∕(bω) for all ω
4. F₀∕k for all ω
5. F₀mω²

Problem 3: phase at natural frequency

At

(4)

the displacement lags the force by

1. 0
2. π∕4
3. π∕2
4. 3π∕2
5. 2π

Problem 4: displacement resonance

If damping is weak enough for a nonzero displacement-response maximum,

(5)

equals

1. 
2. 
3. 
4. 2γ
5. ω₀ + γ

Problem 5: resonance condition

A nonzero-frequency displacement peak exists if

1. γ > ω₀
2. γ = ω₀
3. γ < ω₀∕
4. γ > ω₀∕
5. b = 0 only

Problem 6: power resonance

The average absorbed power is maximum at

1. ω = 0
2. ω = ω_r
3. ω = ω_d
4. ω = ω₀
5. ω = 2ω₀

Problem 7: bandwidth

For viscous damping, the half-power bandwidth is

1. γ
2. 2γ
3. ω₀
4. ω₀∕γ
5. γ²∕ω ₀

Problem 8: quality factor

If

(6)

then

(7)

is

1. 5
2. 10
3. 20
4. 40
5. 400

Problem 9: low-frequency response

As

(8)

the displacement amplitude approaches

1. zero
2. F₀∕k
3. F₀∕b
4. F₀∕m
5. infinity

Problem 10: high-frequency response

For

(9)

the displacement amplitude scales approximately as

1. ω²
2. ω
3. 1∕ω
4. 1∕ω²
5. a constant

Problem 11: exact undamped resonance

For

(10)

a resonant particular solution is proportional to

1. cos ω₀t
2. e^−ω₀t
3. t sin ω₀t
4. t²
5. sin 2ω₀t

Problem 12: impedance

At

(11)

the mechanical impedance

(12)

becomes

1. 0
2. ib
3. b
4. mω₀
5. k∕ω₀

Part II: Complete worked solutions

Solution 1

A linear time-invariant system driven at one sinusoidal frequency responds in steady state at that same frequency. The natural frequency appears in the amplitude and phase, not as a different long-time response frequency. Answer: (C).

Solution 2

The spring-inertia term and damping term combine in quadrature:

(13)

Answer: (B).

Solution 3

At

(14)

(15)

Thus the phase denominator is purely quadrature and

(16)

Answer: (C).

Solution 4

Maximizing displacement response gives

(17)

Answer: (C).

Solution 5

The quantity under the square root must be positive:

(18)

Therefore

(19)

Answer: (C).

Solution 6

For viscous damping the average absorbed power peaks exactly at

(20)

Answer: (D).

Solution 7

The exact half-power bandwidth is

(21)

Answer: (B).

Solution 8

(22)

Answer: (B).

Solution 9

At very low frequency the response is quasistatic:

(23)

Answer: (B).

Solution 10

At high frequency inertia dominates:

(24)

Thus

(25)

Answer: (D).

Solution 11

At exact undamped resonance the ordinary sinusoidal trial duplicates a homogeneous solution. Multiplying by

(26)

gives

(27)

Answer: (C).

Solution 12

Since

(28)

(29)

Therefore

(30)

Answer: (C).

2 GRE checklist

1. Separate transient and steady-state behavior before solving.
2. Remember that the steady-state frequency is the drive frequency.
3. Use atan2 for the response phase.
4. Distinguish ω_d, ω_r, and ω₀.
5. Remember that displacement resonance and power resonance are not identical with finite damping.
6. Use Δω = 2γ and Q = ω₀∕Δω as fast resonance checks.
7. At exact undamped resonance, use the secular t sin ω₀t solution.
8. At ω₀, the mechanical impedance is purely dissipative.

References

References