We know that modes of vibration of an Euler-Bernoulli beam are given by eigenfunctions, with the natural frequency of each mode being given by its eigenvalue. Thus these modes are all mutually orthogonal.(adsbygoogle = window.adsbygoogle || []).push({});

Can anything be said of thederivativesof these eigenfunctions? For example, I have the general solution

[tex]

w(x,t)=\sum_{n=1}^{\infty}q_{n}(t)\Big[ \left(C_{1}cosh(s_{1}x)+C_{2}sinh(s_{1}x)+C_{3}cos(s_{2}x)+C_{4}sin(s_{2}x)\right)\Big]

[/tex]

where [itex] q_{n}(t) [/itex] is the dynamic component of the solution, [itex]s_{1,2}[/itex] are known constants and [itex] C_{1-4} [/itex] are known coefficients.

Can we say that itssecondandfourth derivativesare also orthogonal, such that

[tex]

\int_{0}^{L} \left(\frac{\partial^{2}w_{n}(x,t)}{\partial x^{2}}\right) \left(\frac{\partial^{2}w_{m}(x,t)}{\partial x^{2}}\right) dx = 0 [/tex]

and

[tex]

\int_{0}^{L} \left(\frac{\partial^{4}w_{n}(x,t)}{\partial x^{4}}\right) \left(\frac{\partial^{4}w_{m}(x,t)}{\partial x^{4}}\right) dx = 0 \text{?} [/tex]

And what about the combination of the two:

[tex]\int_{0}^{L} \left(\frac{\partial^{2}w_{n}(x,t)}{\partial x^{2}}\right) \left(\frac{\partial^{2}w_{n}(x,t)}{\partial x^{2}}\right) dx = 0 \text{?} [/tex]

Thanks in advance for the help, it would be much appreciated.

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# I Are the derivatives of eigenfunctions orthogonal?

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