Author Archives: baitingz

The Spring Model Continued – Baiting

ΦΘΙΝΟΥΣΕΣ ΤΑΛΑΝΤΩΣΕΙΣ(6)

In the past two weeks, I worked on Second Order Differential Equations with constant coefficients and learned more about the Spring-Damper Model. In this blog, I will provide a brief recap of the basic knowledge, and then provide further analysis of the same model. If you are interested in my last blog, please visit here.

Recap of Critical Definition from my Last Blog:

1. The standard form of homogeneous second order differential equations is:

WeChat Screenshot_20181019091555

  • In which y is a function of tt is the independent variable, and A, B are arbitrary constants. While analyzing the spring-damper model, the independent variable t represents time.

2. The characteristic equation is:

WeChat Screenshot_20181019092258

  • To solve a homogeneous second order differential equation with constant coefficient, set

20181103113629

  • Solve for r, then use WeChat Screenshot_20181019091131.png to find y.

3. The spring-damper model is in the form of

WeChat Screenshot_20181021171514.png

  • m is the mass of the block, b is the damping constant, and k is the spring constant.
  • If you wonder how we get this equation, please check the link above and read my last blog.

 

Oscillation:

After reviewing the basic knowledge, we are now ready to analyze the damping of the spring-damper model! In this case, we will first transform

WeChat Screenshot_20181021171514.png

into

20181103131113.png

Remember that mass never equal to 0, so these two equations are equivalent.

And then the characteristic equation becomes

WeChat Screenshot_20181103131532.png

We first need to analyze how the roots (r) behave in the characteristics equation. We can use the same analysis for quadratic equation:

WeChat Screenshot_20181103132049.png

In this blog, we will look at “underdamping” in detail. If you are interested in other two types of damping, please first try to analyze them yourself then visit here.

First, I want to look at the implications of b^2<4mk in general. This may mean three things:

  1. The damping constant b is small, which means the damper is weak.
  2. The spring constant k is large, which means the spring is strong.
  3. The mass is large, which means the block is heavy.

Imagine either one of the three cases, then you will find that as long as the mass(m) is not too big to break the spring, the block will oscillate and goes back to its equilibrium position. Let’s now try to prove this assumption mathematically!

First, since we haveWeChat Screenshot_20181103132049.png, we can set

WeChat Screenshot_20181103141617

So that

WeChat Screenshot_20181103142027.png

Plug it back into x and take the real part, we have

WeChat Screenshot_20181103145701.png

Notice that since the value in cos and sin are the same, we can use the sinusoidal identities and transform the equation into

WeChat Screenshot_20181103145723.png

In which

WeChat Screenshot_20181104193319

Following this equation, we would find:

  1. When t (time) approaches infinite, (position) approaches 0. Which means it goes back to its equilibrium position. This agrees with our assumption.
  2. The position of the block versus time should somehow follow the cosine function, which means it has a period and oscillates up and down. This agrees with our assumption.

If assign arbitrary values to b, m, and and plot the function, we would find something look like:

WeChat Screenshot_20181103151009.png

Graph from MIT Open Course Ware

And this is how underdamping looks like!! And conventionally, you can imagine the damper as gravity, and thus the motion would look like this:

Image result for spring damper gif

In this blog, we looked at how a block oscillate with a spring and a damper when the damper is weak. We also analyzed and found the math agrees with our common sense in real life. In fact, many of our observations have an origin in physics or mathematics. As long as we use deliberately use math as a tool, we will be able to explain them!

Starting now, I will be still working on the online course and building my knowledge. But I will also start to refer more to other models not covered in the course material. Hope you will find them interesting and meaningful!

References

Mattuck, A., Miller, H., Orloff, J., & Lewis, J. (2011, Fall). 18.03SC Differential Equations. Retrieved November 5, 2018, from MIT Open Course Ware website: https://ocw.mit.edu/courses/mathematics/18-03sc-differential-equations-fall-2011/index.htm

Zhu, B. (2018, October 21). Spring Model – Baiting [Blog post]. Retrieved from Independent Seminar Blog: https://independentseminarblog.com/2018/10/21/spring-model-baiting/

(Images)

ΠΕΛΛΗΣ, Σ. (2012, October 20). ΦΘΙΝΟΥΣΕΣ ΤΑΛΑΝΤΩΣΕΙΣ [Illustration]. Retrieved from http://physiclessons.blogspot.com/2012/10/blog-post.html#.W9-SM5NKjOg

Pasami. (2018, January 12). File:Spring-mass under-damped.gif [Illustration]. Retrieved from https://commons.wikimedia.org/wiki/File:Spring-mass_under-damped.gif

Euler’s Formula – Baiting

As mentioned in my last post, I continued to develop my algebraic skills for solving differential equations during the past two weeks. In this blog, I want to introduce Euler’s Formula, the most impressive piece of math work I have ever seen. In specific, I will dive deep into the mathematical proof of this formula and explain its broad application.

Euler’s Formula consists of a simple line:

0011.png

In this equation, e represents the irrational constant 2.71828…, or more specifically WeChat Screenshot_20181009230402 , and i represents WeChat Screenshot_20181009230659.png. With an irrational number and an imaginary number involved, Euler’s Formula is not easy to visualize in the regular coordinate system. In order to make this blog more comprehensive to everyone, I will offer an algebraic proof, so that we don’t have to learn the complex Cartesian and Polar coordinate system from the beginning.

To prove Euler’s Formula, we have to prove it satisfies these three fundamental properties:

  1. WeChat Screenshot_20181007153513.png(Law of exponential)
  2. WeChat Screenshot_20181007153856 (Law and definition of constant e in calculus)
  3. Taylor’s series of the left should equal to Taylor’s series on the right as the number of terms approaches infinite.

So let’s start from the first one! Remember we are now assuming Euler’s Formula is correct and testing if it [actually] follows all of the properties.

WeChat Screenshot_20181007160416.png

After proving the Euler’s Formula follows exponential identity, we can move on to proving the identity of e.

The concept of constant e was first brought up by Euler himself,  and is sometimes referred to as “Euler’s Number.” The core definition of e is the rate of change always equal to its self. If you want to know more about this mysterious number, please visit here. Due to its ingenious definition, e is used in multiple areas including charging/discharging a capacitor or calculating quantities related to half-life.

Now here is a simple and straightforward proof for Part II.

WeChat Screenshot_20181007161023

In the last part, we are proving the polynomials of e^iθ equals the polynomials of cos(x)+i*sin(x) from Taylor’s series. If you remember some relevant knowledge from Calculus II, this would also be straightforward.

Now, we have proved the Euler’s Formula. And I will briefly and broadly cover its applications. If you are interested in knowing more about any of these topics, please tell me and I will discuss them in the future!

Trigonometry, Fourier transformation, Taylor’s series, …. the span of the impact of Euler’s Formula goes on and on. With Euler’s Formula, we can easily model rotations in complex coordinate system or even explain the spiral movements of the starts; with Euler’s Formula, the trigonometry relationships become easier than ever; and most importantly, with Euler’s Formula, we can, for the first time in history, cross the wall between transcendental numbers and algebraic numbers.

Euler’s Formula is a piece of math theorem, but also a piece of art. Like Mona Lisa’s Smile by Leonardo da Vinci, Euler’s Formula contains much more than it seems. When rewritten as e^(iπ)+1=0, the simple line involves plus/minus, multiplication/division, exponential, trigonometry, complex number, the concept of zero, and transcendental numbers. Even within itself, there is a sea of knowledge waiting for me to explore.

To me, Euler’s Formula is just a beginning. On my journey of learning, more fascinating math topics will be closely studied and researched. What I will not forget, however, is the excitement after seeing this beautiful math work. Like a line of a poem that eulogizes our life, Euler’s Formula thoroughly depicts the beauty of math.

 

 

References

Fourier Transforms [Illustration]. (2010, Fall). Retrieved from https://slideplayer.com/slide/10994153/

Arthur Mattuck, Haynes Miller, Jeremy Orloff, and John Lewis. 18.03SC Differential Equations. Fall 2011. Massachusetts Institute of Technology: MIT OpenCourseWare, https://ocw.mit.edu. License: Creative Commons BY-NC-SA.

 

Images:

Leonhard Euler [Photograph]. (n.d.). Retrieved from https://www.usna.edu/Users/math/meh/euler.html

O’Connor, J. J., & Robertson, E. F. (2001, September). The number e. Retrieved October 8, 2018, from MacTutor History of Mathematics archive website: http://www-history.mcs.st-and.ac.uk/HistTopics/e.html

Bank Account Model – Baiting

In the past two weeks, I continued my learning on First Order Linear Differential Equations. In this blog, I will focus on the Bank Account Model. If you find this blog is interesting and would like to learn more from my primary source, please go here.

Bank Account Model:

When you are putting money into your bank account, what would you care about the most? While I guess the answer for me and many will be the interest. So in this model, we will look at the logarithm behind the (theoretical) interest system. Continue reading

Introduction to Differential Equations (DE), Geometric Method, Isoclines, and Euler’s Method – Baiting

newton-leipnikhttp://www.idea.wsu.edu/

 

In the past week, I studied Differential Equations through an MIT Online Open Course, which can be found here:

https://ocw.mit.edu/courses/mathematics/18-03sc-differential-equations-fall-2011/index.htm

In this blog, I will introduce Differential Equations as well as some of the methods of solving or visualizing them. I will start from the place where most students left out since Calculus II to make it more comprehensible.

 

Differential Equations:

Differential Equations, also known as DE, means “an equation involving derivatives of a function of functions” (dictionary.com). Differential Equations have a broad application in subjects like Physics, Engineering, and Biology, which will be discussed in depth in my future blogs. The first several blogs, however, will be focusing on the math behind differential equations including how to solve and visualize the formulas.

One simple example of DE is wechat-screenshot_201809091742161, in which x is the independent variable and y is the dependent variable. Notice that taking integral is not a way to find a general solution of y; instead, we must employ a method called “Separation of Variables”. Continue reading