Saturday, November 29, 2014

Oscillation tutorial - 1) Basic Idea


Oscillation is not easy to understand. I studied the basic for sales reason 15 years ago and I thought I understood to a certain extent but now I forgot almost completely what I studied probably because the sales was not successful and surely because my understanding was very shallow. I decided to try to study it again and want to share my idea with those who want to get some idea of Oscillation and fix what we have learned to our brains for sales reason or what ever. Since Oscillation is a very important function in every field or in the world or even in the universe not to mention Electronics it is worth studying it. As a tutorial I will try to show some idea about Oscillation I acquire during the process of study step by step from very basic to some practical ones. Let's start now.

Very Basic

Wiki definition (as of 29-Nov-2014)  - to the current writing some improvement is required, which indicates that Oscillation is not easy to explain to general public.

"
Oscillation is  the relative variation, typically in time, of some measure about a central value (often a point of equilibrium) or between two or more different states. Familiar examples include a swinging pendulum and alternating current power. The term vibration is sometimes used narrowly to mean a mechanical oscillation but is sometimes as a synonym of "oscillation".

"

"alternating current power" is strange as current and power are different things and simply "alternating current" is fine. This wiki article does not elaborate a swinging pendulum and alternating current power. Instead it explains Simple harmonic oscillator by using a spring-mass system.We will look at a swinging pendulum for a while as it was widely used in the past and still used in some antique clocks. Swinging pendulums have been largely replaced by so called "quartz" for time keeping. The function is the same either a swinging pendulum or "quartz". Mechanical watches use some other mechanical systems to get Oscillation - special springs for power and Oscillation. "Quartz" is a sub-topic of this post as this is electrical Oscillator.

Pendulum (wiki 13-Dec-2014)

"

Main article: Pendulum (mathematics)

The period of swing of a simple gravity pendulum depends on its length, the local strength of gravity, and to a small extent on the maximum angle that the pendulum swings away from vertical, θ0, called the amplitude.[8] It is independent of the mass of the bob. If the amplitude is limited to small swings,[Note 1] the period T of a simple pendulum, the time taken for a complete cycle, is:[9]
T \approx 2\pi \sqrt\frac{L}{g} \qquad \qquad \qquad \theta_0 \ll 1 \qquad (1)\,
where L is the length of the pendulum and g is the local acceleration of gravity.
For small swings the period of swing is approximately the same for different size swings: that is, the period is independent of amplitude. This property, called isochronism, is the reason pendulums are so useful for timekeeping.[10] Successive swings of the pendulum, even if changing in amplitude, take the same amount of time.

"

g = acceleration of gravity = 9.80665 m/s2 and L is in m (meter)
So the the unit in the square root is s2. So T is in s (second). The unit is OK.
And although L is under the square root  T is considered to be proportional with L or the Longer the length of a pendulum longer the period. We can imagine this easily from your experience.

Equilibrium

Equilibrium is interesting in that

When the bob stays (does not move) the staying location is at Equilibrium while once the bob starts to move back and forth or up and down (or oscillates) the velocity is the fastest at Equilibrium, i.e. the least <to want to stay> point.

But why 2π ?   Where does it come from ?

This comes from cycle or frequency, which is a reciprocal of T (period). The period in this case is the time required for one cycle while frequency is the number of cycles (or how many cycles) per unit time (usually second). 

T (period) = 1 / frequency (number /sec) = sec / number

 Keeping these basic things in mind we see the following equation.

 From wiki "Mechanical resonance" (16-Dec-2014)

"

Description

The natural frequency of a simple mechanical system consisting of a weight suspended by a spring is:
f = {1\over 2 \pi} \sqrt {k\over m}
where m is the mass and k is the spring constant.
A swing set is a simple example of a resonant system with which most people have practical experience. It is a form of pendulum. If the system is excited (pushed) with a period between pushes equal to the inverse of the pendulum's natural frequency, the swing will swing higher and higher, but if excited at a different frequency, it will be difficult to move. The resonance frequency of a pendulum, the only frequency at which it will vibrate, is given approximately, for small displacements, by the equation:[1]
f = {1\over 2 \pi} \sqrt {g\over L}
where g is the acceleration due to gravity (about 9.8 m/s2 near the surface of Earth), and L is the length from the pivot point to the center of mass.(An elliptic integral yields a description for any displacement). Note that, in this approximation, the frequency does not depend on mass.
Mechanical resonators work by transferring energy repeatedly from kinetic to potential form and back again. In the pendulum, for example, all the energy is stored as gravitational energy (a form of potential energy) when the bob is instantaneously motionless at the top of its swing. This energy is proportional to both the mass of the bob and its height above the lowest point. As the bob descends and picks up speed, its potential energy is gradually converted to kinetic energy (energy of movement), which is proportional to the bob's mass and to the square of its speed. When the bob is at the bottom of its travel, it has maximum kinetic energy and minimum potential energy. The same process then happens in reverse as the bob climbs towards the top of its swing.

" Look at the following pendulum frequency equation.
f = {1\over 2 \pi} \sqrt {g\over L}
where  f  is frequency and T (period) = 1 / frequency

1/T = 

f = {1\over 2 \pi} \sqrt {g\over L}

then we can get


T \approx 2\pi \sqrt\frac{L}{g} \qquad \qquad \qquad \theta_0 \ll 1 \qquad (1)\,

actually the same thing.

But from where these equations come from ? We must think about the force and energy (mechanical energy)
The above explanation is only statement and no equations of force and energy. (we may go to these equations later). But the point to understand oscillation is <Mechanical resonators work by transferring energy repeatedly from kinetic to potential form and back again.> and this relates with piezoelectric (mechanical-electrical) oscillation and electrical oscillation (LC resonant circuit).

As electronic components we have Ceramic 'Resonators' and we seldom heard Crystal or Crystal Quartz Resonators. Due to some reasons (which I do not know) we seldom heard Crystal Quartz Resonators instead the suppliers use <Crystal Quartz Units> and separately <Crystal Quartz Oscillators>. We could say that resonance is required to make oscillation continuously but in a well control manner - a little push at an exact timing (cycle). And generally the resonance frequency is equal to the oscillation frequency. So as a component a resonator (a device or a system) is a helping component for oscillator.

Back to the force and energy issue. See the next post.

sptt


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