# Mechanical Vibration/Lagrange form Applied

## Why Use Lagrange Form?

The largest benefit of using the Lagrange form is the deriving the equation of motion is easier for complex systems.

To start out we will start out applying the Lagrange formulation on a spring and mass system defined within the givens. We will

1. Determine the equation of motion
2. Plot the equation of motion

## Givens

$k=1000$

$m=10kg$

## Lagrangian Form \& Energy Equations

$\frac{d}{dt}(\frac{dT}{\dot{x}})-\frac{dT}{dx} +\frac{dU}{dx}=0$
where T equals the kinetic energy of the system
and
U equals the potential energy in the system.
$T=1/2*k*x^2$
$U=1/2*m*g*h$ or
$U=1/2*m*g*-\delta x$

## work

The derivative of dT with respect to x is: $\frac{dT}{dx}=k*x$