Thin oscillating bending bar with free ends


Solution of u´´´´ = freq² u with boundary condition u´´´(1)=u´´´(-1)= u´´(1)=u´´(-1)=0. Here freq is proportional to the frequency of oscillation.

Thin oscillating disc with free boundary

Three views of the same disc at a given moment. Black is below the neutral plane, white is above it.

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Solution of  Δ Δ u  = freq² u  with as boundary conditions that  Δ u  and its gradient vanish on the edge of the disc. Here  Δ  is the Laplacian.

Thin oscillating square plate with free boundary


Picture for square plate obtained from solutions for the thin bar by the method of Wheatstone.

Many more pictures.

Wilberd van der Kallen