Seminar Report on

Applications of

Differential Equations in

Simple Harmonic

Motion

Presented by-Pratheek Manjunath

2nd Semester, EEE RVCE, Bangalore

INTRODUCTION

The Linear Differential Equation with constant coefficients find their most important applications in the study of electrical, mechanical and other linear systems, especially oscillatory systems.What is Simple Harmonic Motion?When the acceleration of a particle is proportional to tis displacement from a fixed point and is always directed towards it, then the motion is said to be Simple Harmonic.

Case 1: Spring-Mass Systems Consider a mass m is attached to the free end of a mass-less spring.

x=0 is the mean position; B depicts compression and C, elongation by a distance ‘x’. Let the maximum displacement be ‘A’-Amplitude

The spring force is given by:F(x)= -kx where ‘k’ is the Spring constant.

By force balance, -kx=ma m d 2 x + kx = 0 dt2

d 2 x + µ2x=0 where µ2

= k dt2 m

(D2+ µ2)x=0

Solution of the differential equation is x=c1cos µt + c2sin µt

Velocity ‘v’= d x = µ(-c1sin µt+c2cos µt) dt at t=0, x=a => c1=A at t=0, v=0 =>c2=0

Therefore x =A cos µt From the above equation of motion, we can find the following:

Time period of oscillation T=2 π = 2π√(m ) µ k

frequency = T-1 = 1 √(k) 2π√(m)

Maximum velocity = Aµ Maximum acceleration= Aµ2