The Mathematical Theories of Contact Mechanics (MTCM) are reaching a mature stage, with increasing attention being given to computational and

numerical methods. This study contributes to MTCM through the development of a mathematical model of a mass-spring–damper system incorporating

friction and additional effects such as adhesion. A brief review of previous work is provided. The simple but fundamental model exemplifies

the application of abstract theorems, such as those concerning differential set-valued inclusions. Dynamic friction and frictionless contact problems

inherently produce nonsmooth solutions. As a result, regularization techniques are required to construct effective numerical schemes. Some of the

numerical results validate the theoretical findings.