Asnetcomp Tutorials: Oscillation tutorial - 3) Basic Idea

Now we must solve the following differential equation

F = m a = m \frac{\mathrm{d}^2x}{\mathrm{d}t^2} = -k x.

or (the same thing but the right hand side is zero)

m\frac{d^2x}{dt^2} +kx=0,

As the solution is already given we must understand from where the solution comes from. The solution (which is a trigonometric function) is

From From Harmonic oscillator (wiki)

x(t) = A\cos\left( \omega t+\phi\right),

where

\omega = \sqrt{\frac{k}{m}} = \frac{2\pi}{T}.

or

From Simple harmonic motion (wiki)

Using the techniques of differential calculus, the velocity and acceleration as a function of time can be found:

v(t) = \frac{\mathrm{d} x}{\mathrm{d} t} = - A\omega \sin(\omega t-\varphi),

Speed:

{\omega} \sqrt {A^2 - x^2}

Maximum speed

= wA

(at equilibrium point)

a(t) = \frac{\mathrm{d}^2 x}{\mathrm{d}t^2} = - A \omega^2 \cos( \omega t-\varphi).

Maximum acceleration =

A\omega^2

(at extreme points)
Acceleration can also be expressed as a function of displacement:

a(x) = -\omega^2 x.\!

Then since ω = 2πf,

f = \frac{1}{2\pi}\sqrt{\frac{k}{m}},

and, since T = 1/f where T is the time period,

T = 2\pi \sqrt{\frac{m}{k}}.

"

x(t) is the position of the mass attached to a spring and moves up and down very periodically. The movement has velocity and acceleration which are the 1st and the 2nd derivative of x(t) or the 1st and the 2nd differentiation of x(t) with time. And acceleration is the first derivative (differentiation). x(t) is a function with time and so are velocity and acceleration.

Mathematics sometimes show a tricky face and is sometimes a matter of <to believe or not to believe>. Many huge mathematics buildings are built upon the matters of <to believe or not to believe> and definitions, which are also often matters of <to believe or not to believe>. There are several ways to solve differential equation. Integration is the reverse mathematics operation to the differentiation. So integration is one major way of solving a differential equation as the reverse operation reverses the differential equation to a non-differential equation from which we can draw a solution of the given differential equation.

Asnetcomp Tutorials

Sunday, December 21, 2014

Oscillation tutorial - 3) Basic Idea - Solving a differential equation

No comments:

Post a Comment