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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.

Can anything be said of the

[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 its

[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.

Can anything be said of the

**derivatives**of 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 its

**second**and**fourth derivatives**are 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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