User:Ben pcc/Tex

Am using this page to render TEX for whatever reason (likely school).

(old stuff removed August 2007)

$$Bo = \frac{\rho g R^2}{\sigma}$$

=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:

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

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

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

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

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

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