Small Oscillations and the Normal-Mode Eigenvalue Problem

Small Oscillations and the Normal-Mode Eigenvalue Problem

Many mechanical systems are nonlinear when written exactly.

Near a stable equilibrium, however, their motion often simplifies dramatically.

To second order in small displacements, a broad class of conservative systems takes the form

(1)

Here

• η is the displacement from equilibrium in generalized coordinates,
• M is the mass matrix evaluated at equilibrium,
• K is the stiffness matrix, equal to the Hessian of the potential at equilibrium.

The associated normal modes satisfy

(2)

This is the generalized normal-mode eigenvalue problem.

Its eigenvalues

(3)

determine the small-oscillation frequencies, while its eigenvectors determine the collective mode shapes.

M06-04 introduced this structure using coupled springs.

The present article derives it from the local mechanics of a general conservative system and develops the mathematical properties that make modal analysis work.

1 Equilibrium and displacement coordinates

Let the system have generalized coordinates

(4)

Suppose

(5)

is an equilibrium configuration.

Define the small displacement vector

(6)

At equilibrium,

(7)

for every independent coordinate.

This removes the linear term from the local expansion of the potential energy.

2 Quadratic expansion of the potential

Expand

(8)

in a Taylor series:

U	= U0 + ∑ i0ηi	(9)
	+ ∑ i,j0ηiηj + .	(10)

The first derivatives vanish at equilibrium.

Therefore, to quadratic order,

(11)

where

(12)

The matrix

(13)

is the Hessian of the potential at equilibrium.

For a smooth potential,

(14)

3 Stable equilibrium and the stiffness matrix

For a strict stable equilibrium, the quadratic change in potential must be positive for every nonzero sufficiently small displacement:

(15)

Thus, in the nondegenerate stable case,

(16)

If

(17)

is only positive semidefinite, zero-curvature directions can occur.

These directions lead to zero modes.

If the quadratic form has a negative direction, the equilibrium is linearly unstable.

4 Quadratic kinetic energy

For a broad class of mechanical systems,

(18)

Near equilibrium,

(19)

The velocities are already first order in the small motion.

Therefore replacing

(20)

by its equilibrium value introduces only higher-order corrections.

To quadratic order,

(21)

where

(22)

For independent physical degrees of freedom,

(23)

and normally

(24)

5 Why coordinate-dependent inertia can be frozen at equilibrium

Suppose

(25)

The correction

(26)

is first order in

(27)

But

(28)

contains one factor of displacement and two factors of velocity.

It is third order in the small-motion bookkeeping.

Therefore it is omitted in the quadratic small-oscillation theory.

6 Quadratic Lagrangian

Ignoring the irrelevant constant

(29)

the quadratic Lagrangian is

(30)

The Euler-Lagrange equations give

(31)

This is the universal linear system for conservative small oscillations.

7 Normal-mode ansatz

Seek a collective motion

(32)

Then

(33)

Substitution yields

(34)

For a nonzero mode vector,

(35)

8 Why the eigenvalues are frequencies squared

The generalized eigenvalue equation is

(36)

Comparison with the mode equation identifies

(37)

Thus a positive eigenvalue gives an oscillatory angular frequency

(38)

A zero eigenvalue gives a zero mode.

A negative eigenvalue corresponds to an imaginary

(39)

and therefore exponential behavior rather than stable oscillation.

9 Rayleigh quotient

Multiply

(40)

on the left by

(41)

Then

(42)

Therefore any exact mode satisfies

(43)

This ratio motivates the Rayleigh quotient

(44)

The normal modes are stationary directions of this quotient.

For a positive-definite conservative system, the lowest mode minimizes the Rayleigh quotient.

10 Reality and positivity of stable frequencies

If

(45)

is positive definite and

(46)

is positive definite, then for every nonzero

(47)

(48)

and

(49)

Hence

(50)

The mass-weighting transformation below also shows that the eigenvalues are real.

11 Mass-weighted coordinates

Because

(51)

is positive definite, it has a positive square root

(52)

Define

(53)

Then

(54)

The equation of motion becomes

(55)

where

(56)

Since

(57)

and

(58)

are symmetric,

(59)

Thus the generalized eigenvalue problem becomes an ordinary symmetric eigenvalue problem.

12 Orthogonality of distinct modes

Let

(60)

and

(61)

satisfy

(62)

and

(63)

Multiply the first equation by

(64)

and the second by

(65)

Using symmetry,

(66)

Therefore

(67)

For distinct eigenfrequencies,

(68)

Distinct modes are orthogonal in the mass inner product.

13 Stiffness orthogonality

For distinct mass-orthogonal modes,

(69)

Therefore

(70)

when

(71)

and the frequencies are distinct.

Thus normal modes diagonalize both quadratic forms.

14 Mass normalization

The scale of an eigenvector is arbitrary.

A particularly useful convention is

(72)

Then

(73)

and

(74)

15 Modal matrix and simultaneous diagonalization

Collect the mass-normalized mode vectors into

(75)

Then

(76)

and

(77)

where

(78)

Set

(79)

Then

(80)

The coupled small-oscillation system becomes a collection of independent harmonic oscillators.

16 Quadratic energy in normal coordinates

The quadratic energy is

(81)

Use

(82)

For mass-normalized modes,

(83)

Each normal coordinate carries the energy of an independent SHO.

17 Projecting arbitrary initial conditions

Suppose

(84)

and

(85)

Since

(86)

multiply by

(87)

Using mass normalization,

(88)

Similarly,

(89)

For every positive-frequency mode,

(90)

18 Degenerate normal frequencies

Two or more linearly independent modes can share the same frequency.

This is a degeneracy.

If

(91)

the earlier orthogonality proof does not force

(92)

However, within the degenerate eigenspace, one can choose a mass-orthonormal basis.

Thus degeneracy means the mode basis is not unique.

Any suitable linear combination within the degenerate eigenspace is also a mode with the same frequency.

19 Zero modes

If

(93)

then

(94)

The potential has no quadratic restoring curvature in that direction.

The modal equation becomes

(95)

Therefore

(96)

A zero mode is not an oscillator.

It represents a neutral direction, often associated with a continuous symmetry.

20 Unstable modes

Suppose an eigenvalue is negative:

(97)

Then the modal equation is

(98)

Its solution is

(99)

The growing exponential signals linear instability.

Thus the spectrum of the quadratic problem classifies local behavior:

• ω² > 0: stable oscillatory mode,
• ω² = 0: neutral zero mode,
• ω² < 0: unstable mode.

21 Example 1: equal-mass double pendulum

Consider a planar double pendulum with:

• two equal point masses m,
• two massless rods of equal length ℓ,
• small angular displacements 𝜃₁ and 𝜃₂ from the downward vertical.

The exact kinetic energy is

T	= mℓ22𝜃12 + 𝜃 22	(100)
	+ 2𝜃1𝜃2 cos(𝜃1 − 𝜃2).	(101)

For small oscillations,

(102)

Thus

(103)

The gravitational potential is

(104)

Using

(105)

the quadratic part is

(106)

apart from an additive constant.

22 Double-pendulum mass and stiffness matrices

Define

(107)

Then

(108)

and

(109)

This example is important because

(110)

is not diagonal.

Coupling can arise through kinetic energy as well as potential energy.

23 Double-pendulum eigenvalue equation

The mode condition is

(111)

Define the dimensionless quantity

(112)

Then

(113)

Thus

(114)

Therefore

(115)

The roots are

(116)

Hence

(117)

24 Double-pendulum mode shapes

For the lower frequency,

(118)

The eigenvector equation gives

(119)

The two angles have the same sign.

For the higher frequency,

(120)

and

(121)

The two angles have opposite signs.

25 Example 2: numerical double-pendulum frequencies

Let

(122)

Then

ω−	=	(123)
	≈ 2.40 rad∕s,	(124)

and

ω+	=	(125)
	≈ 5.79 rad∕s.	(126)

The corresponding periods are approximately

(127)

and

(128)

26 Example 3: Rayleigh estimate of the lowest double-pendulum frequency

Choose the simple trial direction

(129)

The Rayleigh quotient is

(130)

For the double pendulum,

(131)

while

(132)

Therefore

(133)

The trial estimate is

(134)

The exact lowest squared frequency is

(135)

The trial value is

(136)

which is already close.

This illustrates why the Rayleigh quotient is useful for estimating low modes of systems whose exact eigenproblem is difficult.

27 Coordinate changes and physical invariance

Let

(137)

for an invertible constant matrix

(138)

Then

(139)

and

(140)

transform by congruence:

(141)

and

(142)

The component form changes, but the physical Normal frequencies do not.

Thus normal frequencies are properties of the local mechanical system, not artifacts of a particular generalized-coordinate basis.

28 Degeneracy and symmetry

Degenerate frequencies frequently reflect symmetry.

For example, an isotropic two-dimensional oscillator has equal stiffness in two independent directions.

Its two mode frequencies are equal.

Within that two-dimensional eigenspace, many different orthogonal pairs can be chosen as mode shapes.

Symmetry therefore often explains both degeneracy and freedom in the choice of mode basis.

29 Beyond the quadratic approximation

The small-oscillation theory neglects cubic and higher terms such as

(143)

At larger amplitude these terms can produce:

• amplitude-dependent frequencies,
• coupling between normal modes,
• internal Resonances,
• nonlinear energy transfer,
• chaotic behavior in sufficiently nonlinear systems.

Normal modes are exact for the quadratic linearized system and approximate for the original nonlinear system near equilibrium.

30 Why the theory is so general

The same generalized eigenvalue structure appears in:

• molecular vibrations,
• structural vibration analysis,
• lattice dynamics,
• acoustic cavities,
• elastic bodies,
• finite-element models,
• electrical small-signal systems,
• wave equations after spatial discretization.

The normal-mode eigenproblem is therefore one of the central mathematical structures linking classical mechanics to much broader physics.

31 Common mistakes

1. Expanding about a configuration that is not an equilibrium.
2. Keeping the linear term in the potential even though the expansion point is an equilibrium.
3. Treating the mass matrix as automatically diagonal in generalized coordinates.
4. Forgetting that coordinate dependence of M contributes only beyond quadratic order in the standard small-oscillation expansion.
5. Solving det K = 0 instead of det(K − ω²M) = 0.
6. Interpreting the generalized eigenvalue itself as ω rather than ω².
7. Using ordinary Euclidean orthogonality when the mass matrix is not proportional to the identity.
8. Forgetting that eigenvectors have arbitrary overall scale.
9. Forgetting to mass normalize before using Q = Φ^T Mη.
10. Applying the positive-frequency SHO formula to a zero mode.
11. Calling a negative eigenvalue an oscillation frequency instead of recognizing an instability.
12. Assuming degenerate mode vectors are unique.
13. Confusing a diagonal stiffness matrix with a fully decoupled problem when the mass matrix is not diagonal.
14. Assuming normal modes exchange energy in the exact conservative quadratic theory.
15. Extending the quadratic approximation to amplitudes where higher-order terms are no longer negligible.

32 Practice exercises

1. Starting from a Taylor expansion of U, derive the quadratic stiffness matrix.
2. Explain why the first derivatives of the potential vanish at equilibrium.
3. Show why coordinate dependence of the mass matrix can be neglected to quadratic order.
4. Derive the small-oscillation equations from the quadratic Lagrangian.
5. Derive the generalized normal-mode eigenvalue problem.
6. Prove mass orthogonality of distinct modes.
7. Show that distinct modes are also stiffness orthogonal.
8. Derive the mass-weighted symmetric matrix D.
9. Derive the modal projection formulas for initial displacement and velocity.
10. Derive the modal energy as a sum of independent oscillator energies.
11. Explain the difference among positive, zero, and negative eigenvalues.
12. Derive the double-pendulum mass and stiffness matrices.
13. Solve the double-pendulum characteristic equation and mode shapes.
14. Reproduce the numerical double-pendulum frequencies for ℓ = 1 m.
15. Use the Rayleigh quotient with a trial vector of your choice and compare the estimate with an exact normal frequency.

33 Summary

Near a stable equilibrium, a conservative mechanical system has the quadratic Lagrangian

(144)

The equations of motion are

(145)

Normal modes satisfy

(146)

Distinct modes obey

(147)

With mass-normalized modes,

(148)

and

(149)

The Rayleigh quotient is

(150)

Positive eigenvalues give stable oscillations, zero eigenvalues give neutral modes, and negative eigenvalues indicate linear instability.

The next step is to apply this finite-dimensional mode theory to extended discrete systems, where a chain of oscillators develops a dispersion relation and begins to behave like a wave-bearing medium.

References

References