The approximate solution that satisfies the boundary conditions is:
The below equation should be minimized:
So we have this integral:
Initially, for no deflection, for w = 0, the value of λ from the above equation, is found as:
I can not understand, how λ is calculated.
This is the qustion, how λ is calculated?
Maybe! ,but it is related to mechanical structure. "W" is the displacement of structure. I have trouble in solving mathematical equation.
Hi,
I think, the approach, which has been shown in the document, is called the weighted residual method or the weak form of the strong form ( i.e. the partial differential equation (PDE) ). Generally, the weight (or W(x,y) which is multiplied in the left-side of the PDE as shown in the document) and the domain of integration, which means -a to +a, can be chosen in different ways or arbitrary, by regarding some limitations on the smoothness of the weight function. The most general approach is called Local Petrov-Galerkin method. The simplified, and maybe the most common, version is called Bubnov-Galerkin or Galerkin method. Noting that the function W(x,y) has been chosen the same for trial and test functions in this solution, the approach is the Galerkin method. As a result, you should apply the integration by part technique to this integral equation and extract the lambda value. In other words, the divergence theorem should be applied to this integral, as you may be familiar with. Just be careful when you define the boundary conditions for the boundary integral which stems from the divergence theorem. I hope this would be helpful.
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