Identification of unknown spatial load distributions in a vibrating Euler–Bernoulli beam from limited measured data

Two types of inverse source problems of identifying asynchronously distributed spatial loads governed by the Euler–Bernoulli beam equation ρ ( x ) w tt + μ ( x ) w t + ( EI ( x ) w xx ) xx − T r u xx = ∑ m = 1 M g m ( t ) f m ( x ) , ( x , t ) ∈ Ω T ≔ ( 0 , l ) × ( 0 , T ) , with hinged–clamped ends ( w ( 0 , t ) = w xx ( 0 , t ) = 0 , w ( l , t ) = w x