A priori estimates for the general dynamic Euler–Bernoulli beam equation: Supported and cantilever beams

This work is a further development of weak solution theory for the general Euler–Bernoulli beam equation ρ(x)u_tt+μ(x)u_t+r(x)u_{xx xx}−(T_r(x)u_x)_x=F(x,t) defined in the finite dimension domain Ω_T≔(0,l)×(0,T)⊂R², based on the energy method. Here r(x)=EI(x), E>0 is the elasticity modulus and I(x)>0 is the moment of inertia of the cross-section, ρ(x)>0 is the mass density of the beam, μ(x)>0 is the damping coefficient and T_r(x)≥0 is the traction force along the beam. Two benchmark initial boundary value problems with mixed boundary conditions, corresponding to supported and cantilever beams, are analyzed. For the weak and regular weak solutions of these problems a priori estimates are derived under the minimal conditions. These estimates in particular imply the uniqueness of the solutions of both problems.