Analysis and Design Open Oscillatory Systems with Forced Harmonic Motion (1)

Consider a child, who is playing with a swing. During the period of the time, he learns to apply the optimum force to the swing in order to minimize efforts and maximize the amplitude of the swing. How? The answer is that driving force should be applied periodically and should be timed to coincide closely with the natural motion of the swing.In other words, a driven oscillator responds most strongly when driven by a periodically varying force, the frequency of which is closely matched to the frequency with which the system would freely oscillate if left to it. This frequency is called the natural frequencyof the oscillator.

The purpose of this article is to utilize some methodologies such as sensitivity analysis and Monte Carlo simulation model to analyse and design open systems which have the damped harmonic motion and are also forced by external oscillatory forces. A case of “Is There Any Mechanical Oscillatory System Where Maximum Velocity of Resonance Will Increase More Than Speed of the Light?” has been analysed by using of the methodology stated in article of “EMFPS: How Can We Get the Power Set of a Set by Using of Excel?” posted on link: http://emfps.blogspot.com/2012/08/emfps-how-can-we-get-power-set-of-set.html.

Introduction

There are three types of oscillatory motions as follows:

1. Mechanical waves:These involve motions that are governed by Newton’s laws and can exist only within a material medium such as air, water, rock, etc. Common examples are: sound waves, seismic waves, etc.

2. Matter (or material) waves:All microscopic particles such as electrons, protons, neutrons, atoms etc. have a wave associated with them governed by Schrödinger's equation.

3. Electromagnetic waves:These waves involve propagating disturbances in the electric and magnetic field governed by Maxwell’s equations. They do not require a material medium in which to propagate but they travel through vacuum. Common examples are: radio waves of all types, visible, infra-red, and ultra-violet light, x-rays, and gamma rays. All electromagnetic waves propagate in vacuum with the same speed of the light(c = 300,000 km/s).

First of all, I am willing to start the analysis and design of a mechanical system which is harmonically moving and it has been referred to Mechanical waves (Item 1). Before that, let me tell you a summary of damped and forced SHM.

Damped Harmonic Motion:

We know that a SHM can infinitely continue its motion, if there is not any friction force. In this case, a mass connected to a spring will have oscillatory motion forever. But the amplitude of SHM usually decreases and is closed to zero due tofriction force. We say that is a Damped Harmonic Motion (DHM). The damped force depends on the velocity of the particle and it can be calculated from formula: - b(dx/dt) where “b” is a positive constant number. The equation of the motion is obtained by using of Newton’s laws(F = ma) as follows:

Reference: K. R. Symon, Mechanics. Third edition, Addison – Wesley Publishing Company, 1971, Section 2.9.

Forced Harmonic Motion (FHM):

But if an external oscillatory force is affecting on an open system with DHM, we can analyze the equation of motion in accordance with below formula:

 Reference: K. R. Symon, Mechanics. Third edition, Addison – Wesley Publishing Company, 1971, Section 10.2.

In this case, when the frequency of external force reaches to natural frequency of our system, we will have the resonance.

Regarding to above equations, we can see that the most important parameters for analysis and designing of an open system are as follows:

Fm = External force (N)

k = Restoring constant of system (N/m)

m = mass of system (kg)

b = Damped force constant of system (kg/s)

ω'' = Angular velocity of external force (rad/s)

Methodologies

I used from three methods in which each one is assigned to one type of the oscillatory motions as follows:

- For mechanical waves, I consider to utilize the method mentioned in article of “EMFPS: How Can We Get the Power Set of a Set by Using of Excel?” posted on link: http://emfps.blogspot.com/2012/08/emfps-how-can-we-get-power-set-of-set.html.As an example, I will analyze a case by using of this method where the result will be the options for designing.

- For matter (or material) waves, I will use fromMonte Carlo simulation method stated in my previous articles such as “Application of Pascal’s Triangular plus Monte Carlo Analysis to Find the Least Squares Fitting for a Limited Area” posted on link: http://emfps.blogspot.com/2012/05/application-of-pascals-triangular-plus_23.html.As an example, I will examine the oscillatory motion of a free neutron to find out its coordination in related with the time.

-For electromagnetic waves,I will utilize from Sensitivity Analysis and as an example, I will analyze a case of energy carried by Gamma ray.

1. A Case of Mechanical Waves

Case: Is There Any Mechanical Oscillatory System Where Maximum Velocity of Resonance Will Increase More Than Speed of the Light?”

Assume we are designing an open system under force harmonic motion. What are the parameters of designing? According to above mentioned, they are as follows:

Fm = External force (N)

k = Restoring constant of system (N/m)

m = mass of system (kg)

b = Damped force constant of system (kg/s)

ω'' = Angular velocity of external force (rad/s)

We are willing to know if there is any mechanical system with FHM  in which maximum velocity of this system will go up more than 3E+8 m/s. What is the range for parameters of designing?

I used from the method stated in article of “EMFPS: How Can We Get the Power Set of a Set by Using of Excel?” posted on link: http://emfps.blogspot.com/2012/08/emfps-how-can-we-get-power-set-of-set.html.

I would like to remind you that we applied VB code written by Myrna Larson where the method of designing is step by step as follows:

- I know that the velocity of our system is the function of the above parameters (independent variables): V = f (Fm, k, m, b, ω’’) and we need to have Vm> 3E+8 m/s

- I consider a random domain for all five parameters for instance: 0.1 <(Fm, k, m, b, ω’’)< 1

- I start my calculation by using of Myrna Larson’s VB code and excel spreadsheet program.I have to analyse only 30240 column forcalculations simultaneously (=Permut(10,5))becuse my PC has not necessary instruments to analyse big data.

- I change the domain for all five parameters: 0.001 <(Fm, k, m, b, ω’’)< 100

- I continue to change the domain where I reach: 0.000001 ≤ (Fm, k, m, b, ω’’) ≤ 1000

In this domain, I found 17 types of the parameters where maximum velocity of our system is equal to 1E+9 m/s > c = 3E+8 m/s. It means that we can have 17 types of design for our system to reach maximum velocity more than speed of the light. All parameters for designing have been arranged in below Table:

As we can see, the most crucial thing is that our system will reach to maximum velocity more than speed of the light, if external oscillatory force goes up more than 1KN and damped force constant decrease less than 1E-6 kg/s. In fact, the boundary conditions are:

Fm ≥ 1KN               and                  b ≤ 1E-6 kg/s

2. A Case of Matter (or material) waves

Case:How Can We Find the Coordination of Free Neutrons in the Space of Entropy? 

The neutron is electrically neutral as its name implies. Because the neutron has no charge, it was difficult to detect with early experimental apparatus and techniques. Today, neutrons are easily detected with devices such as plastic scintillators.Neutrons are elementary particles with mass mN= 1.67 × 10−27 kg.

Free neutrons are unstable. They undergo beta-decay whereits half-life is approximately between 614 to 885.7 ± 0.8 s. Neutrons emitted in nuclear reactions can be slowed down by collisions with matter. They are referred to as thermal neutrons after they come into thermal equilibrium with the environment. The average kinetic energy of a thermal neutron is approximately 0.04 eV. This moderated (thermal) neutrons move about 8 times the speed of sound. Typical wavelength (λ)values for thermal neutrons(also callednon-relativistic neutronscold) are between 0.1 and 1 nm. Their properties are described in the framework of material wave mechanics. Therefore, we can easily calculate de Broglie wavelength of these neutrons. But can de Broglie wavelength help us to solve this case? How?

As I stated, the analysis of an oscillatory neutroncan be done by Schrödinger's equation. The general figure of this equation is as follows:

To solve above equation for boundary conditions, we need to apply a strong method. Can Monte Carlo Simulation method help us to analyse this case?

For using of Monte Carlo simulation model, I firstly choose the probability distribution inferred from Binomial and Bayesian method to obtain a framework referred to entropy of these neutrons…..

Note:  “All spreadsheets and calculation notes are available. The people, who are interested in having my spreadsheets of this method as a template for further practice, do not hesitate to ask me by sending an email to: soleimani_gh@hotmail.com or call me on my cellphone: +989109250225. Please be informed these spreadsheets are not free of charge.”

To be Continued……………