differential-algebraic equations; existence and uniqueness of solutions; mechanical oscillators
The classical framework for studying the equations governing the motion of lumped parameter systems presumes one can provide expressions for the forces in terms of kinematical quantities for the individual constituents. This is not possible for a very large class of problems where one can only provide implicit relations between the forces and the kinematical quantities. In certain special cases, one can provide non-invertible expressions for a kinematical quantity in terms of the force, which then reduces the problem to a system of differential-algebraic equations. We study such a system of differential-algebraic equations, describing the motions of the mass-spring-dashpot oscillator. Assuming a monotone relationship between the displacement, velocity and the respective forces, we prove global existence and uniqueness of solutions. We also analyze the behavior of some simple particular models.
 Filippov, A. F.: Klassische Lösungen von Differentialgleichungen mit einer mehrdeutigen rechten Seite (Classical solutions of differential equations with multi-valued right-hand side)
. SIAM J. Control 5 (1967), 609-621 English. DOI 10.1137/0305040
| MR 0220995
 Meirovitch, L.: Elements of Vibration Analysis, 2nd ed. McGraw-Hill New York (1986).