Simple pendulum simulation using LabVIEW

 

A simple harmonic oscillator is a linear oscillator which is governed by the following linear characteristics equation:

 A pendulum is a non-linear system due to presence of sine term in its characteristics equation.  It exhibits harmonic motion only for small angle oscillations.

 

 

 

 For small values of θ, sinθ = θ. Thus the non-linear equation becomes

Let us put the above equations in ordinary differential equation form.

 

 We can assume any initial condition to solve the above ODEs. Let’s assume the following:

 

 

The position of Pendulum bob can be found out using simple trigonometric equations:

We will use above sets of equation to write our code and simulate the behavior of a simple harmonic oscillator and a non-linear pendulum.

Symbols used:

m= mass of the pendulum (neglecting mass of the hanging string)

L= Length of the string

b = damping coefficient

g = acceleration due to gravity = 9.81 m/s²

θ = angle subtended by the string with vertical position

 

LabVIEW implementation

Small initial angle; x1(0) = 25⁰

The response of both linear (simple harmonic oscillator) and non-linear (Pendulum) system overlaps and identical. It holds true because for small θ, sinθ = θ. Thus equation (5) and (6) become same.

 

For large value of initial angle x1(0), the response of Linear and Non-linear pendulum varies as they have different angular displacements.

 

 

The “Damping enable” button allows us to enable or disable damping in our simulation.

 

Block Diagram of code