Sunday, December 21, 2014

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


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.






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Friday, December 19, 2014

Oscillation tutorial - 2) Basic Idea - Simple Harmonic Oscillation


We continue to study Oscillation Basic Idea and this time - Simple Harmonic Oscillation. Basic but most difficult part for some, especially those who have not studied and understood differential equations and solution. If we pass this stage we will get Basic Idea more deeply and fully, which can apply to electromechanical (piezo devices) and electrical (LC resonance) oscillations.

 Simple harmonic motion,  Harmonic oscillator- from wiki 20-Dec-2014 
from Harmonic oscillator

 "
In classical mechanics, a harmonic oscillator is a system that, when displaced from its equilibrium position, experiences a restoring force, F, proportional to the displacement, x:
\vec F = -k \vec x \,
where k is a positive constant.
If F is the only force acting on the system, the system is called a simple harmonic oscillator, and it undergoes simple harmonic motion: sinusoidal oscillations about the equilibrium point, with a constant amplitude and a constant frequency (which does not depend on the amplitude).

 "

 from Simple harmonic motion

Introduction


Simple harmonic motion shown both in real space and phase space. The orbit is periodic. (Here the velocity and position axes have been reversed from the standard convention in order to align the two diagrams)
In the diagram a simple harmonic oscillator, comprising a weight attached to one end of a spring, is shown. The other end of the spring is connected to a rigid support such as a wall. If the system is left at rest at the equilibrium position then there is no net force acting on the mass. However, if the mass is displaced from the equilibrium position, a restoring elastic force which obeys Hooke's law is exerted by the spring.
Mathematically, the restoring force F is given by
\mathbf{F}=-k\mathbf{x},
where F is the restoring elastic force exerted by the spring (in SI units: N), k is the spring constant (N·m−1), and x is the displacement from the equilibrium position (in m).
For any simple harmonic oscillator:
  • When the system is displaced from its equilibrium position, a restoring force which resembles Hooke's law tends to restore the system to equilibrium.
Once the mass is displaced from its equilibrium position, it experiences a net restoring force. As a result, it accelerates and starts going back to the equilibrium position. When the mass moves closer to the equilibrium position, the restoring force decreases. At the equilibrium position, the net restoring force vanishes. However, at x = 0, the mass has momentum because of the impulse that the restoring force has imparted. Therefore, the mass continues past the equilibrium position, compressing the spring. A net restoring force then tends to slow it down, until its velocity reaches zero, whereby it will attempt to reach equilibrium position again.
As long as the system has no energy loss, the mass will continue to oscillate. Thus, simple harmonic motion is a type of periodic motion.

 "

 We have a very basic idea of  Simple Harmonic Oscillation but not precise as there is no equation(s) to prove it. And you may have some questions to which you want to have answers.

First of all, why 'harmonic'? Harmonic is usually used in the study of music or sound. Normal sound is a vibration of air to the ear(s) and usually in the form of sine (sinusoidal) vibration - vibration is a sine wave. Oscillation is a sine wave.

 Basic formula
\vec F = -k \vec x \,

"
Repeating the above wiki explanation

From Harmonic oscillator (wiki)

 In classical mechanics, a harmonic oscillator is a system that, when displaced from its equilibrium position, experiences a restoring force, F, proportional to the displacement, x:
\vec F = -k \vec x \,
where k is a positive constant.

 "

 F is a restoring force.

From Simple harmonic motion (wiki)

 " If the system is left at rest at the equilibrium position then there is no net force acting on the mass. However, if the mass is displaced from the equilibrium position, a restoring elastic force which obeys Hooke's law is exerted by the spring.
Mathematically, the restoring force F is given by
\mathbf{F}=-k\mathbf{x},
where F is the restoring elastic force exerted by the spring (in SI units: N), k is the spring constant (N·m−1), and x is the displacement from the equilibrium position (in m).

 "

 F is a restoring force (in SI units: N) derived from the spring (spring's elasticity) and proportional (or counter proportional as the minus sine) to the displacement from the equilibrium position, x. That is

When x is at the equilibrium position, F = 0 as x = 0. Once the movement started the mass passes at the equilibrium point at the maximum speed (velocity) as the restoring Force (against the movement and therefore a minus sign) is zero.

 When x is at the most extended position, F = max value as x = max value. but please note the minus sign.
In somewhere in the middle between x = 0 and x =max value F changes with x proportionally but please note again the minus sign.

 As we cannot see this restoring force (F), we must use our imagination. And the point is that the the direction of the restoring force (F) is the opposite to the direction of the motion of the displacement, x and the word <restoring> suggests that some energy is stored and used and re-stored. 'Force' is not stored and restored.

 The units are simple. Force (F) - Newton while  k is N·m−1 as x is in meter.

The next is a differential equation. An equation relates A and B or A = B.

 From From Harmonic oscillator (wiki)

 "

Simple harmonic oscillator

Main article: Simple harmonic motion

Simple harmonic motion
A simple harmonic oscillator is an oscillator that is neither driven nor damped. It consists of a mass m, which experiences a single force, F, which pulls the mass in the direction of the point x=0 and depends only on the mass's position x and a constant k. Balance of forces (Newton's second law) for the system is
F = m a = m \frac{\mathrm{d}^2x}{\mathrm{d}t^2} = -k x.
Solving this differential equation, we find that the motion is described by the function
x(t) = A\cos\left( \omega t+\phi\right),
where
\omega = \sqrt{\frac{k}{m}} = \frac{2\pi}{T}.
The motion is periodic, repeating itself in a sinusoidal fashion with constant amplitude, A. In addition to its amplitude, the motion of a simple harmonic oscillator is characterized by its period T, the time for a single oscillation or its frequency f = 1⁄T, the number of cycles per unit time. The position at a given time t also depends on the phase, φ, which determines the starting point on the sine wave. The period and frequency are determined by the size of the mass m and the force constant k, while the amplitude and phase are determined by the starting position and velocity.
The velocity and acceleration of a simple harmonic oscillator oscillate with the same frequency as the position but with shifted phases. The velocity is maximum for zero displacement, while the acceleration is in the opposite direction as the displacement.
The potential energy stored in a simple harmonic oscillator at position x is
U = \frac{1}{2}kx^2.
" 

From Simple harmonic motion (wiki)

 "

Dynamics of simple harmonic motion

For one-dimensional simple harmonic motion, the equation of motion, which is a second-order linear ordinary differential equation with constant coefficients, could be obtained by means of Newton's second law and Hooke's law.
F_{net} = m\frac{\mathrm{d}^2 x}{\mathrm{d}t^2} = -kx,
where m is the inertial mass of the oscillating body, x is its displacement from the equilibrium (or mean) position, and k is the spring constant.

 Therefore,

\frac{\mathrm{d}^2 x}{\mathrm{d}t^2} = -\left(\frac{k}{m}\right)x,

 Solving the differential equation above, a solution which is a sinusoidal function is obtained.
x(t) = c_1\cos\left(\omega t\right) + c_2\sin\left(\omega t\right) = A\cos\left(\omega t - \varphi\right),
where
\omega = \sqrt{\frac{k}{m}},
A = \sqrt{{c_1}^2 + {c_2}^2},
\tan \varphi = \left(\frac{c_2}{c_1}\right),
In the solution, c1 and c2 are two constants determined by the initial conditions, and the origin is set to be the equilibrium position.[A] Each of these constants carries a physical meaning of the motion: A is the amplitude (maximum displacement from the equilibrium position), ω = 2πf is the angular frequency, and φ is the phase.[B]
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}}.
These equations demonstrate that the simple harmonic motion is isochronous (the period and frequency are independent of the amplitude and the initial phase of the motion).

 "

 Newton's second law
F = m a = m \frac{\mathrm{d}^2x}{\mathrm{d}t^2} = -k x.
 or (the same thing)

\mathbf{F} = m\,\frac{\mathrm{d}\mathbf{v}}{\mathrm{d}t} = m\mathbf{a},
Hook's law (as shown above)
\mathbf{F}=-k\mathbf{x},
From Simple harmonic motion (wiki)

 "

 \frac{\mathrm{d}^2 x}{\mathrm{d}t^2} = -\left(\frac{k}{m}\right)x,

 "
This is fine. The next is the the key part as we are talking about 'harmonic' oscillation or sine (sinusoidal) wave and difficult part for those who have not studied and understood differential equations and solution as from the above simple equation tone trigonometric equation comes up.

 From From Harmonic oscillator (wiki)

 "
F = m a = m \frac{\mathrm{d}^2x}{\mathrm{d}t^2} = -k x.
Solving this differential equation, we find that the motion is described by the function
x(t) = A\cos\left( \omega t+\phi\right),
where
\omega = \sqrt{\frac{k}{m}} = \frac{2\pi}{T}.
"

 Here we need to know how to solve the above differential equation. Without understanding this (how to solve a differential equation) we cannot get a Basic Idea of Oscillation.

 From wiki <Examples of differential equations> (20-Dec-2014)

 "

Second-order linear ordinary differential equations

A simple example

Suppose a mass is attached to a spring which exerts an attractive force on the mass proportional to the extension/compression of the spring. For now, we may ignore any other forces (gravity, friction, etc.). We shall write the extension of the spring at a time t as x(t). Now, using Newton's second law we can write (using convenient units):
m\frac{d^2x}{dt^2} +kx=0,
where m is the mass and k is the spring constant that represents a measure of spring stiffness. For simplicities sake, let us take m=k as an example.

 If we look for solutions that have the form Ce^{\lambda t}, where C is a constant, we discover the relationship \lambda^2+1=0, and thus \lambda must be one of the complex numbers i or -i. Thus, using Euler's theorem we can say that the solution must be of the form:
x(t) = A \cos t + B \sin t
To determine the unknown constants A and B, we need initial conditions, i.e. equalities that specify the state of the system at a given time (usually t = 0).
For example, if we suppose at t = 0 the extension is a unit distance (x = 1), and the particle is not moving (dx/dt = 0). We have
x(0) = A \cos 0 + B \sin 0 = A = 1, \,
and so A = 1.
x'(0) = -A \sin 0 + B \cos 0 = B = 0, \,
and so B = 0.

 Therefore x(t) = cos t. This is an example of simple harmonic motion.

 "

 This example is the exact one. And the conclusion is nice but under the special condition of (for simplicity) of <m = k>. And we are still at <without understanding this (how to solve a differential equation we cannot get a Basic Idea of Oscillation>. The problem is in the middle.

 1) Where does  Ce^{\lambda t} come from ?

 2) Where do we discover the relationship \lambda^2+1=0  ?

 3) What is Euler's theorem ?

Either

Solving the differential equation above, a solution which is a sinusoidal function is obtained.
x(t) = c_1\cos\left(\omega t\right) + c_2\sin\left(\omega t\right) = A\cos\left(\omega t - \varphi\right),
or

Solving this differential equation, we find that the motion is described by the function
x(t) = A\cos\left( \omega t+\phi\right),
we need some knowledge of solving a differential equation. Solving the following equation 

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

  m\frac{d^2x}{dt^2} +kx=0,


is a big jump as a trigonometric equation comes up. This is an important supporting knowledge to understand Oscillation more deeply. Once having gained the some key knowledge of a solving differential equations our understanding will jump as well.


 --  continue --


sptt



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