Module 1 : Introduction

Lecture 1:Introduction to nonlinear mechanical systems

 


INTRODUCTION TO NONLINEAR MECHANICAL SYSTEMS

In this lecture the vibration of linear and nonlinear dynamical systems have been briefly discussed. Both inertia and energy based approaches have been introduced to derive the equation of motion. Review of linear single degree of freedom system free vibration is carried out.

Introduction 

Any motion that repeats itself after an interval of time is called vibration or oscillation. The swinging of a pendulum (Fig.1.1.1) and the motion of a plucked string are typical examples of vibration. The theory of vibration deals with the study of oscillatory motion of bodies and forces associated with them.

Elementary Parts of Vibrating system

  • • A means of storing potential energy (Spring or elasticity)

  • • A means of storing kinetic energy (Mass or inertia)

  • • A means by which energy is gradually lost (damper)

The forces acting on the systems are

  • • Disturbing forces
    •
  • Restoring force

  • • Inertia force
    •
  • Damping force

Fig. 1.1.1: Swinging of a Pendulum

Degree of Freedom : The minimum number of independent coordinates required to determine completely the position of all parts of a system at any instant of time defines the degree of freedom of the system.

System with a finite number of degrees of freedom are called discrete or lumped parameter system, and those with an infinite number of degrees of freedom are called continuous or distributed systems.

Classification of Vibration:

•  Free and forced

•  Damped and undamped

•  Linear and nonlinear

•  Deterministic and Random

Free vibration: If a system after initial disturbance is left to vibrate on its own, the ensuing vibration is called free vibration.

Forced Vibration: If the system is subjected to an external force (often a repeating type of force) the resulting vibration is known as forced vibration

Damped and undamped: If damping is present, then the resulting vibration is damped vibration and when damping is absent it is undamped vibration. The damped vibration can again be classified as under-damped, critically-damped and over-damped system depending on the damping ratio of the system.

Linear vibration: If all the basic components of a vibratory system – the spring the mass and the damper behave linearly, the resulting vibration is known as linear vibration. Principle of superposition is valid in this case.