#C10 · Calculus & Analysis · Lv.4View in 3D Mind-Map

Differential Equations & Harmonic Oscillator

Newton, Euler & d'Alembert · 1743 CE

m\frac{d^2x}{dt^2} + c\frac{dx}{dt} + kx = 0,\quad \omega_0 = \sqrt{\frac{k}{m}}

x(t) = A e^{-\zeta\omega_0 t}\cos(\omega_d t + \phi),\quad \zeta = \frac{c}{2\sqrt{mk}}

Links position, velocity, and acceleration into a dynamic equation solved by complex exponentials—driving game spring physics, suspension engineering, and AI Diffusion ODEs.

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Mass m

Damping c

Stiffness k

\omega_0 = \sqrt{\frac{4}{1}} = 2.00, \; \zeta = 0.15 \implies x(t) = e^{-0.30t}\cos(1.98t)

고유 각진동수 ω₀ = √(k/m)

2.000 rad/s

감쇠비 ζ = c / (2√mk)

0.150 (부족감쇠 진동)

진동 주기 T = 2π/ω_d

3.18 s

Prerequisite Path (17 STEP)

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Click steps in order to follow the 3D neon path from basics to target.

STEP 1#G01Pythagorean Theorem

a^2 + b^2 = c^2 \implies c = \sqrt{a^2 + b^2}

STEP 2#G03Pi, Radian Measure & Circular Sector

l = r\theta

STEP 3#A01Linear Equation & Balance Axiom

ax + b = 0 \implies x = -\frac{b}{a}

STEP 4#A02Polynomial Expansion & Factoring

(a+b)^2 = a^2 + 2ab + b^2

STEP 5#A03Laws of Exponents & Radicals

a^m \cdot a^n = a^{m+n}

STEP 6#A06Logarithm Laws & Change of Base

\log_a(xy) = \log_a x + \log_a y

STEP 7#A07Arithmetic & Geometric Series Sum

S_n = \frac{a(1 - r^n)}{1 - r}

STEP 8#A08Binomial Theorem & Pascal's Triangle

(a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k

STEP 9#C01Limits, Continuity & Euler's Number e

e = \lim_{n \to \infty} \left(1 + \frac{1}{n}\right)^n \approx 2.718281828

STEP 10#G04Unit Circle Trigonometry & Identities

\cos^2\theta + \sin^2\theta = 1

STEP 11#A04Quadratic Formula & Discriminant

x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

STEP 12#C02Derivative Definition & Instantaneous Rate

f'(x) = \frac{df}{dx} = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}

STEP 13#C04Fundamental Theorem of Calculus (FTC)

\int_a^b f(x)\

STEP 14#A09Complex Numbers & Polar Plane

z = a + bi = r(\cos\theta + i\sin\theta)

STEP 15#C06Taylor & Maclaurin Series Expansion

f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!}(x-a)^n

STEP 16#C07Euler's Formula & Euler's Identity

e^{i\theta} = \cos\theta + i\sin\theta

STEP 17#C10Differential Equations & Harmonic Oscillator

m\frac{d^2x}{dt^2} + c\frac{dx}{dt} + kx = 0

Geometric & Intuitive Insight

해 x(t) = e^(λt) 를 대입하면 미분방정식이 이차방정식 mλ² + cλ + k = 0 (특성방정식)으로 바뀝니다! 판별식 D < 0 이면 복소수 근이 나와 오일러 공식에 의해 진동하면서 서서히 멈추는 파동이 됩니다.

Step-by-Step Derivation

Step 1.1단계: 뉴턴 제2법칙 F = ma 에 복원력 -kx 와 마찰 감쇠력 -c(dx/dt) 를 대입합니다.
Step 2.2단계: x(t) = e^(λt) 를 대입하면 특성방정식 mλ² + cλ + k = 0 을 얻습니다.
Step 3.3단계: 근의 공식과 오일러 공식을 적용하면 감쇠비 ζ에 따라 부족감쇠(진동)·임계감쇠·과감쇠 해가 도출됩니다.

AI & Coding Real-World Use

Generative AI Diffusion ODE solvers, React-Spring UI physics animations, and vehicle suspension control

#Diffusion AI ODE#Spring Physics UI#Control Engineering

const zeta = c / (2 * Math.sqrt(m * k)); const w0 = Math.sqrt(k / m);

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