# On the vibrations of lumped parameter systems governed by differential-algebraic equations

## TL;DR

This paper investigates the vibrational behavior of lumped parameter systems modeled by differential-algebraic equations, highlighting numerical solution challenges through a system with a Bingham fluid dashpot.

## Contribution

It introduces an analysis of vibratory systems governed by differential-algebraic equations, emphasizing the complexities when constitutive relations are not explicitly definable.

## Key findings

- Illustrates issues with numerical solutions of DAE systems
- Uses a Bingham fluid dashpot example to demonstrate modeling challenges
- Highlights the need for specialized solution methods for such systems

## Abstract

In this paper, we consider the vibratory motions of lumped parameter systems wherein the components of the system cannot be described by constitutive expressions for the force in terms of appropriate kinematical quantities. Such physical systems reduce to a system of differential-algebraic equations, which invariably need to be solved numerically. To illustrate the issues with clarity, we consider a simple system in which the dashpot is assumed to contain a "Bingham" fluid for which one cannot describe the force in the dashpot as a function of the velocity. On the other hand, one can express the velocity as a function of the force.