Mechanical Vibration/Lagrange form Applied

Contents

Why Use Lagrange Form?Edit

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

GivensEdit

$k=1000$$k=1000$

$m=10kg$$m=10kg$

Lagrangian Form \& Energy EquationsEdit

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

workEdit

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