PROOF OF MODE ORTHOGONALITY OF STRUCTURES VIA SYMPLECTIC INNER PRODUCT

This article introduces an alternative method based on the symplectic inner product to establish the mode orthogonality relations for linear vibrations of continuous structures. Based on symplectic elasticity, the definition of symplectic inner product is introduced, which then is applied to vibration problems of Euler-Bernoulli beams and Timoshenk beams. It is shown that the mode orthogonality relations can be established in a unified matter for the two types of beams subject to complex boundary conditions, illustrating that the new method boasts greater generality and demonstrates certain advantages when handling complex problems. Its incorporation enriches the scope of teaching structural dynamics or vibrations, as well as helps students make connections across different areas of knowledge, thus enhancing their learning effectiveness.