GRE Physics Companion: Small Oscillations and the Normal-Mode Eigenvalue Problem

GRE Physics Companion: Small Oscillations and the Normal-Mode Eigenvalue Problem

For small conservative oscillations,

(1)

The normal modes satisfy

(2)

1 High-value GRE facts

1. At equilibrium, the first derivatives of the potential vanish.
2. The stiffness matrix is the potential-energy Hessian evaluated at equilibrium.
3. The mass matrix need not be diagonal in generalized coordinates.
4. small oscillations obey Mη + Kη = 0.
5. The generalized eigenvalue is ω², not ω.
6. Distinct modes satisfy a_r^T Ma _s = 0.
7. mass-normalized modes satisfy Φ^T MΦ = 1.
8. The Rayleigh quotient estimates squared frequencies.
9. A zero eigenvalue gives a neutral mode.
10. A negative eigenvalue indicates linear instability.

Part I: Original GRE-style problems

Problem 1: equilibrium expansion

At a mechanical equilibrium,

1. all second derivatives of U vanish
2. all first derivatives of U vanish
3. the kinetic energy vanishes for all time
4. the mass matrix is the identity
5. every Normal frequency is zero

Problem 2: stiffness matrix

The small-oscillation stiffness matrix is

1. K_ij = ∂U∕∂q_i
2. K_ij = ∂²U∕∂q _i∂q_j at equilibrium
3. K_ij = M_ij⁻¹
4. K_ij = q_iq_j
5. always diagonal

Problem 3: generalized eigenproblem

The mode equation is

1. Ka = ωMa
2. Ka = ω²Ma
3. Ma = ω²Ka only
4. Ka = 0 for every mode
5. MKa = 0

Problem 4: mass orthogonality

For two distinct-frequency modes of a conservative symmetric system,

1. a_r^T a _s = 1
2. a_r^T Ma _s = 0
3. a_r^T Ka _s = 1
4. a_r = a_s
5. ω_r = ω_s

Problem 5: Rayleigh quotient

The Rayleigh quotient is

1. (y^T My)∕(y^T Ky)
2. (y^T Ky)∕(y^T My)
3. det M∕ det K
4. y^T y
5. tr K

Problem 6: zero mode

If

(3)

the modal equation of motion is

1. Q + ω₀²Q = 0
2. Q = 0
3. Q = 0 necessarily
4. Q = 0 necessarily
5. Q − σ²Q = 0

Problem 7: unstable eigenvalue

If a generalized eigenvalue is negative,

1. the mode has a real stable oscillation frequency
2. the mode is a zero mode
3. the linearized motion contains exponential growth and decay
4. the mass matrix is singular
5. the mode disappears

Problem 8: double-pendulum frequencies

For the equal-mass equal-length small-angle double pendulum,

(4)

are

1. (g∕ℓ)(1 ± 1)
2. (g∕ℓ)(2 ±)
3. (g∕ℓ)(2 ± 2)
4. (g∕ℓ)(1 ±)
5. both g∕ℓ

Problem 9: lower double-pendulum mode shape

A mode vector for the lower-frequency equal-mass double-pendulum mode is proportional to

1. 
2. 
3. 
4. 
5. 

Problem 10: numerical low frequency

For

(5)

the lower double-pendulum normal frequency is closest to

1. 1.00 rad/s
2. 2.40 rad/s
3. 3.13 rad/s
4. 5.79 rad/s
5. 9.81 rad/s

Problem 11: mass normalization

If the columns of Φ are mass-normalized modes, then

1. Φ^T MΦ = 1
2. Φ^T KΦ = 1 always
3. M = 1
4. K = 0
5. det Φ = 0

Problem 12: modal projection

For mass-normalized modes, the initial modal displacement is

1. Q₀ = Φ^T η ₀
2. Q₀ = Φ^T Mη ₀
3. Q₀ = Kη₀
4. Q₀ = M⁻¹η ₀
5. Q₀ = 0

Part II: Complete worked solutions

Solution 1

At equilibrium,

(6)

This eliminates the linear term in the potential expansion. Answer: (B).

Solution 2

The local quadratic potential is determined by the Hessian:

(7)

Answer: (B).

Solution 3

Substituting

(8)

into the small-oscillation equations gives

(9)

Answer: (B).

Solution 4

For distinct eigenfrequencies and symmetric

(10)

the modes satisfy

(11)

Answer: (B).

Solution 5

The Rayleigh quotient is

(12)

For an exact mode it equals

(13)

Answer: (B).

Solution 6

With

(14)

the modal equation becomes

(15)

Answer: (B).

Solution 7

Write

(16)

Then

(17)

whose solutions contain

(18)

and

(19)

Answer: (C).

Solution 8

The equal-mass equal-length double pendulum gives

(20)

Answer: (B).

Solution 9

The lower-frequency eigenvector has same-sign angular components:

(21)

Answer: (C).

Solution 10

ω−	=	(22)
	≈ 2.40 rad∕s.	(23)

Answer: (B).

Solution 11

Mass normalization gives

(24)

Thus

(25)

Answer: (A).

Solution 12

From

(26)

multiply by

(27)

Therefore

(28)

Answer: (B).

2 GRE checklist

1. Confirm that the expansion point is actually an equilibrium.
2. Form both M and K; do not assume either is diagonal.
3. Remember that the eigenvalue is ω².
4. Use the mass inner product for modal orthogonality.
5. Mass normalize before applying the simplest projection formulas.
6. Use the Rayleigh quotient as a frequency-squared estimate.
7. Treat zero modes separately from oscillatory modes.
8. Interpret negative eigenvalues as local instability.

References

References