0주: 환영(Welcome)
W0.1 강좌 예고(Course Trailer)
W0.2~5 강좌 개요(Course Overview)
W0.6~7/1강 미분 방정식 소개 및 연습문제
(Introduction to Differential Equations & Practice Quiz)
1주: 1차 미분 방정식(Frst-order Differential Equation)
W1.1/2강 수치해석:오일러 방법(Euler Method)
W1.2/3강 1차 미분 방정식 세우기(Separable First-order equations)
W1.3/4강 1차 미분 방정식 세우기 예제(Separable First-order equations:Example)
W1.4 연습문제(Practice Quiz:Separable First-order ODEs)
W1.5/5강 선형 1차 미분 방정식(Linear First-order Equations)
W1.6/6강 선형 1차 미분 방정식:예제(Linear First-order Equations:Example)
W1.7 연습문제(Practice Quiz:Linear First-Order ODE)
W1.8/7강 선형 1차 미방 응용:예금(Application: Compound interest)
W1.9/8강 선형 1차 미방 응용:종말속도(Application: Terminal Velocity)
W1.10/9강 선형 1차 미방 응용:RC회로(Application: RC Circuit)
W1.11 연습문제(Practice Quiz: Applications)
W1.12 평가문제(Week-1 Graded Quiz)
2주: 2차 미분 방정식(Second-order Differential equations)
W2.1/10강 고차 미분 방정식에 대한 오일러 방법(Euler method for higher-order odes)
W2.2/11강 중첩의 원리(Principle of Superposition)
W2.3/12강 론스키 행렬식(The Wronskian)
W2.4/보강1 복소수(complex numbers)
W2.5/13강 일원 2차 상미분 방정식(Homogeneous 2nd-order ODEs)
W2.6/14강 예제1: 2개의 실근(Case1: Distinct Real Roots)
W2.7/15강 예제2A: 켤레 복소수 근 A(Case 2: Complex-Conjugate Roots (Part A))
W2.8/16강 예제2B: 켤레 복소수 근 B(Case 2: Complex-Conjugate Roots (Part B))
W2.9/17강 예제3A: 중근 A(Case 3: Repeated Roots (Part A))
W2.10/18강 예제3B: 중근 B(Case 3: Repeated Roots (Part B))
W2.11 연습문제(Practice quiz: Homogeneous 2nd-order ODEs)
W2.12/19강 이원 2차 상미분 방정식(Inhomogeneous 2nd-order ODEs)
W2.13/20강 지수함수꼴 이원항(Inhomogeneous term:Exponential func.)
W2.14/21강 삼각함수꼴 이원항(A)(Inhomogeneous term:sine or cosine-Part A)
W2.15/22강 삼각함수꼴 이원항(B)(Inhomogeneous term:sine or cosine-Part B)
W2.16/23강 다항함수꼴 이원항(Inhomogeneous term:Polynomials)
W2.17/24강 공진현상(Resonance)
W2.18 연습문제(Practice quiz: Inhomogeneous 2nd-order ODEs)
W2.19/보강2 차원해석(Dimensional Analysis: Nondimensionalization) /동영상/영문자막
Non-Dimensionalization: Making an equation dimensionless
For an example, I'll use this equation which governs the oscillation of a pendulum.
Looking at this equation there're a lot of parameters, m, l, c, g, f_0 and ω.
- the mass of the pendulum,
- the length of the rod connecting the mass to the pivot point,
- the time (the second & first derivative of the angle with time)
- friction co-efficient c,
- gravity, the gravitational acceleration.
Then, an external force has
- an amplitude f_0
- a frequency ω, cosine function of
You have six parameters. But those parameters have dimensions. If you do nondimensionalize this equation, it's possible to reduce those six parameters into a smaller set.
When you do nondimensionalization, then you can explore the solutions to this equation with a fewer number of parameters.
Also, another very important point maybe that the nondimensional equation may be equivalent to other physical systems provided you also nondimensionalize those equations associated with the other physical systems.
How do you nondimensionalize equation?
The fundamental units are, mass(m), length(l), time(t).
What are the units of the different terms?
That happens to have units of force. A very fundamental understanding of physical equations is that a physical equation is only valid if every term in that equation has the same units.
The units across an equation always have to be the same, otherwise it's not a valid physical equation. We have this damping parameter c in second term. Let's figure out what are the units of c. In order to match the terms in this equation, we need a mass divided by a time. So, the units of c is mass per unit time.
Similarly, g, that's the gravitational acceleration. We know what the units have to be. If we match the terms, then it would be a length per unit time squared.
Then, F_0 and ω. The units of F_0 is force. The units of ω has to make the argument of
cosine dimensionless. So, the units of omega has to be t to the minus one.
Now that we're familiar with what the units of everything is in this equation, we can try to make a dimensionless form of this equation. Let's divide through by ml.
How do we make this equation dimensionless?
- The dependent variable is θ, which is already dimensionless in radians.
- The independent variable is t, which is the only variable that we need to make dimensionless.
We have to choose how to measure time.
The square root of (g/l) would be the natural frequency of a simple harmonic oscillator.
Frequency ω has units of one over t. Define ω_0 to be the square root of (g/l), which also has units of one over t. We can then use ω_0 to make time non-dimensional.
We can define the time 𝜏, which will be our dimensionless time.
We need to replace the derivatives of dimensionless time.
Replace dimensional differential equation by a dimensionless equation.
Divide through by (ω_0)^2 and we end up then with the dimensionless differential equation.
Groupings of dimensional parameters,
We have our α, r, β.
What have we done?
We started with an equation that had six parameters: m, l, c, g, F_0, and ω. By non-dimensionalizing the equation, we end up with a final equation that has only three parameters: α, r, β.
The reason we were able to go from six to three is because there are three fundamental units in this problem: mass, length, and time.
So, six minus three gives us three. That has a funny name. It's called the Buckingham Pi theorem. It's interesting to learn about that, but actually, in practice, engineers don't really need to use this theorem.
What they need to do is know how to nondimensionalize an equation. Believe me, if you're doing a numerical solution, you don't want to deal with six parameters when you can deal with only three.
Let me review. In this video, I'm trying to show you how, if you have a physical problem, where everything has units, you can redefine your variables to be dimensionless so that you can reduce that physical problem with so many parameters to a lot fewer parameters, and all of the parameters are dimensionless.
[참고] '마구잡이 수학 1장(Street-Fighting Mathematics)
1장 차원/Ch1. Dimension)
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