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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