On wave-like differential equations in general Hilbert space with application to Euler–Bernoulli bending vibrations of a beam

Abstract

Wave-like differential equations occur in many engineering applications. Here the engineering setup is embedded into the framework of functional analysis of modern mathematical physics. After an overview, the L2–Hilbert space approach to free Euler–Bernoulli bending vibrations of a beam in one spatial dimension is investigated. We analyze in detail the corresponding positive, selfadjoint differential operators of 4-th order associated to the boundary conditions in statics. A comparison with free string wave swinging is outlined.

Keywords

• Wave-like differential equations in Hilbert space
• Sobolev spaces
• Positive selfadjoint differential operators of 4-th order
• Friedrichs extension
• Euler–Bernoulli (partial) differential equation
• Boundary conditions