A Spring–Beam System with Signorini’S Condition and the Normal Compliance Condition

. This paper provides mathematical and numerical analyses for a dynamic frictionless contact problem in which both of Signorini’s condition and the normal compliance condition are used. The contact problem is considered by employing two viscoelastic (Kelvin-Voigt type) objects: a linear Timoshenko beam and a nonlinear spring. In addition, a transmission condition is imposed on one end of the beam and the top of the spring so that they can touch and vibrate together. We prove the existence of solutions satisfying all the conditions. Time discretizations and finite element methods are utilized to propose the fully discrete numerical schemes. We select several groups of data to present and discuss numerical simulations.