Small oscillations - introductionbased on FW-21

We will consider small-amplitude oscillations of mechanical systems about static equilibrium, e.g. coupled pendulums:

applications: vibrations of molecules, crystals,...

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Consider a system described by a set of n independent generalized coordinates, with time-independent potential and no time-varying constraints:

Static equilibrium:

equations of motion:

all generalized forces have to vanish:

if the potential has a minimum, the equilibrium is stable

Stability:

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Consider a small displacement from equilibrium:

Kinetic energy:

evaluated at the equilibrium - constant matrix!

Potential energy:

evaluated at the equilibrium - constant matrix!

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Lagrangian:

constant real symmetric matrices

Equations of motion:

linear in small displacements and derivatives

quadratic in small displacements and derivatives

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Normal modes - pendulumbased on FW-22

Small displacement problem for a system described by 1 generalized coordinate:

Introduce complex coordinate:

We seek a solution of the form:

eigenvalue equation:

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General solution:

General solution of the original problem:

we wrote the general solution using only positive eigenvalue

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Normal modes - general - Part 1based on FW-22

In general, we need to solve a set of n linear homogeneous coupled differential equations with constant coefficients. It is convenient to introduce complex parameters:

We will look first for normal modes:

all the coordinate oscillate with the same frequency

n linear homogeneous coupled algebraic equations

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Coupled pendulums - Part 1based on FW-23

Small-amplitude oscillations coupled pendulums:

Lagrangian:

length of the spring - natural length

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Langrange’s equations:

both pendulums oscillate with the same frequency

Solving for normal modes:

in matrix notation

this is the general solution (we will prove it later)

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Linear equations - math review:Consider a set of n linear inhomogeneous equations (real coefficients):

Solution:

If then

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Consider a set of n linear homogeneous equations:

Solution:

If then

with all

= 0!

only trivial solution

If

at least one equation is linearly dependent, and can be discarded. Then, assuming the n-th component of x is non-zero, we can divide all remaining equations by it...

then

and obtain a set of n-1 inhomogeneous equationsgo to previous page for n-1

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Back to coupled pendulums:

has a non-trivial solution only if:

two solutions for normal-mode frequencies:

free pendulum

higher frequency

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Corresponding normal-mode eigenvectors:

must be linearly dependent

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Normal modes - general - Part 2based on FW-22

Our set of linear homogeneous equations has a nontrivial solution only if:

this leads to an n-th order polynomial, which has n roots:

all the roots are real

given non-trivial solution

differs only by interchange of dummy summation indices

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positive-definite, since m is the mass matrix

Stability:

unstable

stable

imaginary frequency leads to runaway solutions

Form of the general solution:

positive-definite if the potential is a minimum at equilibrium

n-1 ratios of components are real, there can be only an overall phase

one component can be chosen arbitrarily115

Form of the general solution (continued):

s-th eigenvalue: t-th eigenvalue:

For non-degenerate eigenvalues:we can normalize the eigenvectors according to

and write the general solution as:

for degenerate eigenvalues we can use Gram-Schmidt orthogonalization procedure to get

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