In this talk, we consider how to create a real world problem with applications and formulate a differential equation system and contact conditions. Contact between bodies happens in every single day life. In this mathematical modeling, we employ two viscoelastic (Kelvin-Voigt type) objects: a linear Timoshenko beam and a nonlinear spring. The normal compliance condition and 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. When the top of the spring moves down to hit a rigid foundation, Signorini’s condition is applied. We prove the existence of solutions satisfying the differential equation system and all the conditions. Time discretizations and finite element methods are utilized to propose the fully discrete numerical schemes. Several groups of data are selected to present numerical simuylations.