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

**has been analysed by using of the methodology stated in article of***“Is There Any Mechanical Oscillatory System Where Maximum Velocity of Resonance Will Increase More Than Speed of the Light?”***posted on link: http://emfps.blogspot.com/2012/08/emfps-how-can-we-get-power-set-of-set.html***“EMFPS: How Can We Get the Power Set of a Set by Using of Excel?”*__.__**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

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

**posted on link: http://emfps.blogspot.com/2012/05/application-of-pascals-triangular-plus_23.html***“Application of Pascal’s Triangular plus Monte Carlo Analysis to Find the Least Squares Fitting for a Limited Area”*__.__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

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

__F____m____≥ 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

*m**N*= 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 called*non-relativistic neutronscold*) are between 0.1 and 1 nm. Their properties are described in the framework**. Therefore, we can easily calculate***of material wave mechanics***of these neutrons. But can de Broglie wavelength help us to solve this case? How?***de Broglie wavelength*
As I stated,
the analysis of an oscillatory neutroncan be done by

**. The general figure of this equation is as follows:***Schrödinger's equation*
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

**and***Binomial***method to obtain a framework referred to entropy of these neutrons…..***Bayesian*

*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……………