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- Homework Statement
- Obtain mass and stiffness matrices given equation of motion
- Relevant Equations
- [tex] A_{11e} \frac{d^2 u_1}{dx^2} + (A_{12e} + A_{66e}) \frac{d^2 v_1}{dx,dy} + A_{66e} \frac{d^2 u_1}{dy^2} + \frac{G_2}{h_2} \left( u_3 - u_1 - d \frac{dw}{dx} \right) = (\rho_p h_p + \rho_s h_s) \frac{d^2 u_1}{dt^2} [/tex]
Hello,
This is not homework but I am trying to replicate some results I found in a paper. In short, the situation is as follows. The following equation is given:
A_{11e} \frac{d^2 u_1}{dx^2} + (A_{12e} + A_{66e}) \frac{d^2 v_1}{dxdy} + A_{66e} \frac{d^2 u_1}{dy^2} + \frac{G_2}{h_2} \left( u_3 - u_1 - d \frac{dw}{dx} \right) = (\rho_p h_p + \rho_s h_s) \frac{d^2 u_1}{dt^2}
where u_1(x,y), v_1(x,y), u_3(x,y), and w(x,y) are given by:
u_1(x,y) = \sum_{m=0}^{\infty} \sum_{n=0}^{\infty} U_{1mn}(t) \frac{dX(x)}{dx} Y(y)
u_3(x,y) = \sum_{m=0}^{\infty} \sum_{n=0}^{\infty} U_{3mn}(t) \frac{dX(x)}{dx} Y(y)
v_1(x,y) = \sum_{m=0}^{\infty} \sum_{n=0}^{\infty} V_{1mn}(t) X(x) \frac{dY(y)}{dy}
w(x,y) = \sum_{m=0}^{\infty} \sum_{n=0}^{\infty} W_{mn}(t) X(x) Y(y)
with X(x) = \sin\left( \frac{m \pi x}{a} \right) and Y(y) = \sin\left( \frac{n \pi y}{b} \right).
All other quantities are constants.
As far as I understand, the author substitutes the proposed solutions and groups terms according to the temporal variable that accompanies them.
He then rewrites the equation as:
K_{11} U_{1mn} + K_{12} V_{1mn} + K_{13} U_{3mn} + K_{15} W_{mn} + M_{11} \ddot{U}_{1mn} = 0
where (according to him):
K_{11} = -A_{11e} \left( \frac{m \pi}{a} \right)^2 - A_{66e} \left( \frac{n \pi}{b} \right)^2 - \frac{G_2}{h_2}
K_{12} = -(A_{12e} + A_{66e}) \left( \frac{m \pi}{a} \right) \left( \frac{n \pi}{b} \right)
K_{13} = \frac{G_2}{h_2}
K_{15} = -\frac{G_2 d}{h_2} \left( \frac{m \pi}{a} \right)
M_{11} = -(\rho_p h_p + \rho_s h_s)
The problem is that he doesn’t explain how these elements K_{ij} and M_{ij} are obtained.
I assume he multiplied both sides by \frac{dX(x)}{dx} Y(y) and integrated over x \in [0,a] and y \in [0,b], as if applying orthogonality conditions.
However, when I do that, I only recover some of the terms:
K_{11} = -A_{11e} \left( \frac{m \pi}{a} \right)^2 - A_{66e} \left( \frac{n \pi}{b} \right)^2 - \frac{G_2}{h_2}
K_{12} = -(A_{12e} + A_{66e}) \left( \frac{n \pi}{b} \right)^2
K_{13} = \frac{G_2}{h_2}
K_{15} = -\frac{G_2 d}{h_2}
M_{11} = -(\rho_p h_p + \rho_s h_s)
I cross-checked my results with a Mathematica script, and they seem consistent.
Does anyone have an idea of what the author might be doing differently?
Thanks in advance.
This is not homework but I am trying to replicate some results I found in a paper. In short, the situation is as follows. The following equation is given:
A_{11e} \frac{d^2 u_1}{dx^2} + (A_{12e} + A_{66e}) \frac{d^2 v_1}{dxdy} + A_{66e} \frac{d^2 u_1}{dy^2} + \frac{G_2}{h_2} \left( u_3 - u_1 - d \frac{dw}{dx} \right) = (\rho_p h_p + \rho_s h_s) \frac{d^2 u_1}{dt^2}
where u_1(x,y), v_1(x,y), u_3(x,y), and w(x,y) are given by:
u_1(x,y) = \sum_{m=0}^{\infty} \sum_{n=0}^{\infty} U_{1mn}(t) \frac{dX(x)}{dx} Y(y)
u_3(x,y) = \sum_{m=0}^{\infty} \sum_{n=0}^{\infty} U_{3mn}(t) \frac{dX(x)}{dx} Y(y)
v_1(x,y) = \sum_{m=0}^{\infty} \sum_{n=0}^{\infty} V_{1mn}(t) X(x) \frac{dY(y)}{dy}
w(x,y) = \sum_{m=0}^{\infty} \sum_{n=0}^{\infty} W_{mn}(t) X(x) Y(y)
with X(x) = \sin\left( \frac{m \pi x}{a} \right) and Y(y) = \sin\left( \frac{n \pi y}{b} \right).
All other quantities are constants.
As far as I understand, the author substitutes the proposed solutions and groups terms according to the temporal variable that accompanies them.
He then rewrites the equation as:
K_{11} U_{1mn} + K_{12} V_{1mn} + K_{13} U_{3mn} + K_{15} W_{mn} + M_{11} \ddot{U}_{1mn} = 0
where (according to him):
K_{11} = -A_{11e} \left( \frac{m \pi}{a} \right)^2 - A_{66e} \left( \frac{n \pi}{b} \right)^2 - \frac{G_2}{h_2}
K_{12} = -(A_{12e} + A_{66e}) \left( \frac{m \pi}{a} \right) \left( \frac{n \pi}{b} \right)
K_{13} = \frac{G_2}{h_2}
K_{15} = -\frac{G_2 d}{h_2} \left( \frac{m \pi}{a} \right)
M_{11} = -(\rho_p h_p + \rho_s h_s)
The problem is that he doesn’t explain how these elements K_{ij} and M_{ij} are obtained.
I assume he multiplied both sides by \frac{dX(x)}{dx} Y(y) and integrated over x \in [0,a] and y \in [0,b], as if applying orthogonality conditions.
However, when I do that, I only recover some of the terms:
K_{11} = -A_{11e} \left( \frac{m \pi}{a} \right)^2 - A_{66e} \left( \frac{n \pi}{b} \right)^2 - \frac{G_2}{h_2}
K_{12} = -(A_{12e} + A_{66e}) \left( \frac{n \pi}{b} \right)^2
K_{13} = \frac{G_2}{h_2}
K_{15} = -\frac{G_2 d}{h_2}
M_{11} = -(\rho_p h_p + \rho_s h_s)
I cross-checked my results with a Mathematica script, and they seem consistent.
Does anyone have an idea of what the author might be doing differently?
Thanks in advance.