Timoshenko beam theory

Blatantly copied from http://ccrma.stanford.edu/~bilbao/master/node163.html Math isn't rendering right there.

$\rho A{\frac {\partial ^{2}w}{\partial t^{2}}}={\frac {\partial }{\partial x}}\left(A\kappa G({\frac {\partial w}{\partial x}}-\psi )\right)$

$\rho I{\frac {\partial ^{2}\psi }{\partial t^{2}}}={\frac {\partial }{\partial x}}\left(EI{\frac {\partial \psi }{\partial x}}\right)+A\kappa G\left({\frac {\partial w}{\partial x}}-\psi \right)$

$\rho A{\frac {\partial v}{\partial t}}={\frac {\partial q}{\partial x}}$

${\frac {1}{A\kappa G}}{\frac {\partial q}{\partial t}}={\frac {\partial v}{\partial x}}-\omega$

$\rho I{\frac {\partial \omega }{\partial t}}={\frac {\partial m}{\partial x}}+q$

${\frac {1}{EI}}{\frac {\partial m}{\partial t}}={\frac {\partial \omega }{\partial x}}$

$v\triangleq {\frac {\partial w}{\partial t}}\qquad \omega \triangleq {\frac {\partial \psi }{\partial t}}\qquad m\triangleq EI{\frac {\partial \psi }{\partial x}}\qquad q\triangleq A\kappa G\left({\frac {\partial w}{\partial x}}-\psi \right)$

w is transverse displacement, ψ is angular displacement, "κ is a constant which depends on the geometry of the beam" <- what the hell?

G is shear modulus. ρ is (volume) density, so A is area. All variables are assumed functions of x.

"The Euler-Bernoulli system (5.4) is recovered in the limit as AκG [increases with no limit] and ρI [approaches 0] [131]."

For a square, κ = 5/6 (called "Timoshenko's coefficient), reference given to Shock and Vibration Handbook.

For derivation, refer to:

Wave Motion in Elastic Solids.

Mechanical Vibrations, 2nd Ed.

Vibrations in Mechanical Systems.

Theory of Vibrations with Applications.

Principles of Vibration.

Kavrayskiy VII, according to D. Goldberg:

$x={\frac {3\lambda }{2\pi }}{\sqrt {{\frac {\pi ^{2}}{3}}-\phi ^{2}}}$

$y=k\phi \,$

You get to pick k, it's usually taken to be something like 1.2.

Addendum: no you don't. k = 1, period.

Some high(er) order finite differences:

${\frac {\partial ^{2}f}{\partial x^{2}}}={\frac {11f(x+4h)-56f(x+3h)+114f(x+2h)-104f(x+h)+35f(x)}{12h^{2}}}+O(h^{3})$

${\frac {\partial ^{2}f}{\partial x^{2}}}={\frac {-f(x+3h)+4f(x+2h)+6f(x+h)-20f(x)+11f(x-h)}{12h^{2}}}+O(h^{3})$

${\frac {\partial ^{2}f}{\partial x^{2}}}={\frac {-f(x+2h)+16f(x+h)-30f(x)+16f(x-h)-f(x-2h)}{12h^{2}}}+O(h^{3})$

${\frac {\partial f}{\partial x}}={\frac {-3f(x+4h)+16f(x+3h)-36f(x+2h)+48f(x+h)-25f(x)}{12h}}+O(h^{4})$

${\frac {\partial f}{\partial x}}={\frac {f(x+3h)-6f(x+2h)+18f(x+h)-10f(x)-3f(x-h)}{12h}}+O(h^{4})$

${\frac {\partial f}{\partial x}}={\frac {-f(x+2h)+8f(x+h)-8f(x-h)+f(x-2h)}{12h}}+O(h^{4})$