A Model to Solve a System of SHM Equations

Following to article of “A Model to Solve a System of Nonlinear Equations by Using Microsoft Excel plus VBA” (http://emfps.blogspot.com/2018/02/a-model-to-track-location-of-particle.html), the purpose of this article is, to present another example about solving a system of nonlinear equations for Simple Harmonic Motion (SHM).

Suppose we have two oscillatory motions which are in two perpendicular directions. We can write a system of equations for them as follows:

Assume the angular frequency for both SHMs is the same and also there is the difference between initial phases equal 90 degree. In this case, we will have below system of equations:

This model is able to solve above system of two equations for given data of “x”, “y”, “Ax” and “Ay”.

 There are two methods for finding “ω”, “t” and “φ”. 

Method (1):

The convert of two equations to one equation and solving only one equation for three independent variables in accordance with the method mentioned in article of “Solving a Nonlinear Equation with Many Independent Variables by Using Microsoft Excel plus VBA” (https://emfps.blogspot.com/2016/10/can-we-solve-nonlinear-equation-with.html) as follows:

Below figure as well as shows you the components of this model:

Let me explain you about the components of above model as follows:

1. In right side on cells L6:O6, we have inputs including given data of “x”, “y”, “Ax” and “Ay”.

2. In left side on cells H6:J7, we have other inputs including lower and upper ranges for independent variables of “ω”, “t” and “φ” to reach the answers which are the solution for system of two nonlinear equations. Here, there are lower and upper ranges which are changed by click on cell A2 and also this change will again go back by click on cell B2 (Go & Back).

3. On cells H10:J24, we have outputs which are the answers to above system of two nonlinear equations.

4. On cell M7, we have Error which is the difference between equation of outputs (-y.Ax /x.Ay) and equation of inputs (Tan (ωt+φ))

5. On cells K10:K24, we have the solution of “x” by replacing the answers.

6. On cells M10:M24, we have the solution of “y” by replacing the answers.

7. On cells O10:O24, we have the difference between item 5 (“x”) and cell L6 which are the errors of our answers.

8. On cells P10:P24, we have the difference between item 6 (“y”) and cell M6 which are the errors of our answers.

You can see below clip as the examples for this model:

Method (2):

Using the method stated in article of “A Model to Solve a System of Nonlinear Equations by Using Microsoft Excel plus VBA” (http://emfps.blogspot.com/2018/02/a-model-to-track-location-of-particle.html

In this method, we directly solve a system of two nonlinear equations.  

Below figure as well as shows you the components of this model:

The explanations of the components are the same with method (1) except the error.

You can see below clip as the examples for this model:

All researchers and individual people, who are interested in having this model, don’t hesitate to send 

their request to below addresses:

Is Angular Frequency a Vector or Scalar Quantity?

Angular frequency is the changes of angular displacement to the changes of time:

Therefore, if we say that the angular frequency is a vector, then an angle should be also a vector. In all reference books in physics, you can find that only a very small angles can be considered as a vector. But I am willing to tell you that there are several another factors (except very small angles) which can be defined as impact factors such as Amplitude of an oscillatory motion in which high amplitude does not allow you to use the rules dominated on vectors to analyze an oscillatory system with combination of many harmonic motions.

The surprising news is: "The most important impact factors are not very small angles and amplitude but the direction of the motion. In some situations and conditions, you can even consider a very large angle as a vector where this reverses the concept defined in reference books in Physics."

If you do not truly apply the concept of vector for angular frequency, the result of your analysis and design for an oscillatory system with combination of many harmonic motions, will go in wrong way.

Workshop Series: The Big Data Science Analysis with Microsoft Excel plus VBA

"The Big Data Science Analysis with Microsoft Excel plus VBA"

 There are two big strategies for these workshop series as follows:

1. Solving the complicated problems by producing new methods and models

2.  Creating the value from the big data 

Here is PowerPoint movie about "The Big Data Science Analysis with Microsoft Excel plus VBA"

The Distances among The Particles in The Space (3)

You can preview the conjectures (1), (2), (3) and (4) on below links:

 http://www.emfps.org/2018/04/the-distances-among-particles-in-space.html

 http://www.emfps.org/2018/04/the-distances-among-particles-in-space-2.html

Conjecture (5): 

By using this theorem or conjecture, I am willing to show you that there is below figure among five points in the space:

Suppose we have point P1 (x, y, z) and an independent variable “t” where there is below functions between them:

If points Pm and Pn have below coordination:

Pm (f(x,y,z,t), f(x,y,z,t), f(x,y,z,t))

Pn (g(x,y,z,t), g(x,y,z,t), g(x,y,z,t))

Then, in according to conjecture (3), the distance of points Pm and Pn will be equal with the points P1, P2 and P3 stated in conjecture (3). (Please see conjecture (3) on above links).

It means:

d(Pm,P1)= d(Pm,P2) = d(Pm,P3) = d(Pn,P1) = d(Pn,P2) = d(Pn,P3) =R

Example (1):

Suppose we have below data:

P1 (-11, 3, 31)

t = 129

The results will be as follows:

Example (2):

Suppose we have below data:

P1 (-3.57, 9.4, -1.84)

t = 32.83

The results will be as follows:

The Mapping A System of five Particle for Given Gravity Potential Energy

Suppose we have five particles Pm, Pn, P1, P2 and P3 with the distances among them in infinity which have the same masses of “m”. If an external force brings all these particles in new location just like above figure, how can we map the location of these particles for a constant gravity potential 

energy?

The method of analysis is just like the steps stated in previous article 

(http://www.emfps.org/2018/04/the-distances-among-particles-in-space-2.html)

The only difference is to solve a system of three nonlinear equations instead of a system of two nonlinear equations in previous articles as follows:

Example:

To simplify above system of equations, I consider “t” as a constant number.

Assume we have:

r = 0.24 m

t = 3

Thus, we should solve below system of three equations:

I found 190 models that some of them are as follows:

For testing of these models, I use model (1) in above table and we can see the results as follows:

The Distances among The Particles in The Space (2)

You can preview the conjectures (1) and (2) and (3) on below link:

  http://www.emfps.org/2018/04/the-distances-among-particles-in-space.html

Conjecture (4):

The distances between the point P0 (a, a, a) and the points mentioned in conjecture (3) are always the same where “a” is a member of real number.

Example:

Suppose we have point P1 with below coordination:

P1 (-3, 15, 34)

According to conjecture (3), the points P2 and P3 will be:

P2 (34, -3, 15)

P3 (15, 34, -3)

The distances among the points P1 and P2 and P3 are equal to 45.32108.

Assume we have point P0 with below coordination:

P0 (56, 56, 56)

The distances between the point P0 (56, 56, 56) and the points P1 and P2 and P3 is the same and 

equal to 75.13987.

d (P0, P1 ) = 75.13987

d (P0, P2) = 75.13987

d (P0, P3 ) = 75.13987

The Mapping A System of Four Particle for Given Gravity Potential Energy

Suppose we have four particles P0, P1, P2 and P3 with the distances among them in infinity   which have the same masses of “m”. If an external force brings all these particles in new location just like 

below figure, how can we map the location of these particles for a constant gravity potential energy?

  As I told you, suppose that mass of all particles is the same. Therefore, for our analysis, we have three independent variables of “m”, “r” and “R”.

Suppose the constant gravity potential energy is equal -1.008E-08 J

U (r) = -1.008E-08 J

  As I stated in my previous article, to find the coordination and mapping the particles, we should take two steps:

Step (1):

To find the mass of particles accompanied by the distances of “r” and “R”, we should apply the method mentioned in article of “Solving a Nonlinear Equation with Many Independent Variables” (http://www.emfps.org/2016/10/can-we-solve-nonlinear-equation-with.html)

I found 8 answers for three independent variables of the particles:

For example, I choose below three variables from above table:

m	r	R	T_G_P_E
1	0.02	2.8	-1.007949E-08
3	0.21	1.2	-1.007949E-08
3	0.24	0.7	-1.007949E-08

And by using these three set, I start step 2.

Step (2):

Since we should get the coordination of particles by using the distances “r’ and “R”, therefore, we have to solve a system of two nonlinear equations as follows:

By using the model presented in article of “A Model to Track the Location of a Particle in the Space” (http://www.emfps.org/2018/02/a-model-to-track-location-of-particle.html), we can easily find the coordination of the particles.

For instance, if we have:

 m = 1kg

r = 0.02 m

R = 2.8 m  

I found 444 models that some of them are as follows:

For testing of these models, I use model (1) in above table and we can see the results as follows:

If we have:

 m = 3kg

r = 0.21 m

R = 1.2 m  

I found 276 models that some of them are as follows:

For testing of these models, I use model (1) in above table and we can see the results as follows:

If we have:

 m = 3kg

r = 0.24 m

R = 0.7 m  

I found 276 models that some of them are as follows:

For testing of these models, I use model (1) in above table and we can see the results as follows:

Conclusion

Suppose you are squeezing a sponge by your hand and assume that the work done by your hand will stay the constant. How can you say that you are controlling the potential energy throughout the sponge when you deform the sponge and change the location of particles of the sponge?

In physics, the people usually use the vector fields, gradient vectors, curl and so on. But the problem is, to encounter with the chaos systems in the nature in which you are not able to find a real function for your subject. In this case, the people usually use the methods of the boundary conditions.

I think that the method mentioned in above article can help us to solve the problems which are defined as the chaos systems.