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How can I interpret the normal modes of this mechanical system?

$\def \b {\mathbf}$ The EOM's are: $$ \b M\, \begin{bmatrix} \ddot{u}_1 \\ \ddot{u}_2 \\ 0 \\ \end{bmatrix}+\b Q\, \begin{bmatrix} {u}_1 \\ {u}_2 \\ \alpha \\ \end{bmatrix}= \...
Eli's user avatar
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2 votes

How can I interpret the normal modes of this mechanical system?

Ok, I'm not doing the math beyond counting degrees-of-freedom (DoF), but here's my thought process: $3 x 3$ matrix: 3 dimensions should have 3 eigenvalues for 3 DoF. But you have 2 masses, so it's a 2 ...
JEB's user avatar
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3 votes

How can I interpret the normal modes of this mechanical system?

In this example all the linkage does is convert a displacement of $u_3$ at the top to a displacement of $-\frac ba u_3$ at the bottom instantaneously so you should not expect there to be any ...
Farcher's user avatar
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5 votes

How can I interpret the normal modes of this mechanical system?

The system you wrote is a system of DAEs (differential-algebraic equations) since the third equation is a algebraic equation representing an algebraic constraints. As the rod is massless, its "...
basics's user avatar
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5 votes

How can I interpret the normal modes of this mechanical system?

I have not attempted to do the algebra, but I presume that you have found that the coeffeicient of $(\omega^2)^3$ in your characteristic equation is zero. To understand what this means consider the ...
mike stone's user avatar
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6 votes

Why potential energy is not considered in the internal energy of diatomic molecules?

You're right - there is an additional degree of freedom from the potential energy of the bond! In fact, there is also another kinetic degree of freedom from the relative motion of atoms as they ...
Carmeister's user avatar
1 vote

Why potential energy is not considered in the internal energy of diatomic molecules?

There is intramolecular potential energy and intermolecular potential energy. The first applies to the chemical and nuclear bonds of the molecule/atom. The second applies to the potential energy ...
Bob D's user avatar
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9 votes

Why potential energy is not considered in the internal energy of diatomic molecules?

Note that potential energy is defined on a relative scale, and so we are free to chose the state which we call the zero potential energy state. Since the diatomic molecules are considered rigid, their ...
CompassBearer's user avatar
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Is it possible there can be a non-Fourier model of string vibration? Is there an exact solution?

Assuming linearity, the response of the system can be written as the superposition of its modes. How many modes does a continuous medium have? An infinite number, in general. How many modes have non-...
basics's user avatar
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2 votes

Is it possible there can be a non-Fourier model of string vibration? Is there an exact solution?

A vibrating string can support a very large number of vibrational modes simultaneously. This is because waves on strings superimpose linearly. Note here that an electric guitar string struck hard with ...
niels nielsen's user avatar
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Understanding Loop Formation in a Plucked String

One simple way to get some intuition is to think of it like this: After plucking, the string would like to relax into a sinusoidal wave which is zero at the edges and has a maximum amplitude (antinode)...
Codename 47's user avatar
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