2.4 Tebranishlar va To'lqinlar — Nazariya

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Kirish

Tebranishlar — vaqt bo'yicha takrorlanuvchi harakat. Robotika va dronlarda:

• 🎯 PID tuning — so'nuvchi tebranishlar
• 🚁 Dron barqarorligi — rezonansdan qochish
• 🔧 Sensor signallari — filtrlar va chastota tahlili

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1. Oddiy Garmonik Harakat (SHM)

Ta'rif

Tiklash kuchi pozitsiyaga proporsional:

$$F = -kx$$

Harakat tenglamasi

$$m\frac{d^2x}{dt^2} = -kx$$ $$\frac{d^2x}{dt^2} + \omega_0^2 x = 0$$

Bu yerda $\omega_0 = \sqrt{\frac{k}{m}}$ — tabiiy burchak chastota.

Yechim

$$x(t) = A\cos(\omega_0 t + \phi)$$

• $A$ — amplituda
• $\omega_0$ — burchak chastota (rad/s)
• $\phi$ — boshlang'ich faza

Tezlik va tezlanish

$$v(t) = -A\omega_0\sin(\omega_0 t + \phi)$$ $$a(t) = -A\omega_0^2\cos(\omega_0 t + \phi) = -\omega_0^2 x$$

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2. Davriy Kattaliklar

Davr (Period)

$$T = \frac{2\pi}{\omega_0} = 2\pi\sqrt{\frac{m}{k}}$$

Chastota

$$f = \frac{1}{T} = \frac{\omega_0}{2\pi}$$

Maxsus tizimlar

Prujina-massa:

$$T = 2\pi\sqrt{\frac{m}{k}}$$

Oddiy mayatnik:

$$T = 2\pi\sqrt{\frac{L}{g}}$$

Fizik mayatnik:

$$T = 2\pi\sqrt{\frac{I}{mgh}}$$

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3. Energiya

Kinetik va potensial

$$KE = \frac{1}{2}mv^2 = \frac{1}{2}m\omega_0^2 A^2\sin^2(\omega_0 t + \phi)$$ $$PE = \frac{1}{2}kx^2 = \frac{1}{2}kA^2\cos^2(\omega_0 t + \phi)$$

Umumiy energiya

$$E = KE + PE = \frac{1}{2}kA^2 = \frac{1}{2}m\omega_0^2 A^2 = \text{const}$$

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4. So'nuvchi Tebranishlar

Ishqalanish yoki qarshilik mavjud bo'lganda:

Harakat tenglamasi

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

$c$ — damping koeffitsienti

Damping nisbati

$$\zeta = \frac{c}{2\sqrt{km}} = \frac{c}{2m\omega_0}$$

Uch holat

1. Underdamped ($\zeta < 1$):

$$x(t) = Ae^{-\zeta\omega_0 t}\cos(\omega_d t + \phi)$$

Bu yerda $\omega_d = \omega_0\sqrt{1-\zeta^2}$ — damped chastota.

2. Critically damped ($\zeta = 1$):

$$x(t) = (A + Bt)e^{-\omega_0 t}$$

Eng tez muvozanatga qaytish, tebranishsiz.

3. Overdamped ($\zeta > 1$):

$$x(t) = Ae^{-\alpha_1 t} + Be^{-\alpha_2 t}$$

Sekin qaytish, tebranishsiz.

Kontrolda

PID tuning — ko'pincha $\zeta \approx 0.7$ maqsad qilinadi (tez va minimal overshoot).

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5. Majburiy Tebranishlar

Tashqi kuch bilan haydalganda:

$$m\frac{d^2x}{dt^2} + c\frac{dx}{dt} + kx = F_0\cos(\omega t)$$

Steady-state yechim

$$x(t) = X\cos(\omega t - \phi)$$

Amplituda:

$$X = \frac{F_0/k}{\sqrt{(1-r^2)^2 + (2\zeta r)^2}}$$

Bu yerda $r = \omega/\omega_0$ — chastota nisbati.

Faza kechikishi:

$$\phi = \arctan\frac{2\zeta r}{1-r^2}$$

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6. Rezonans

$\omega \approx \omega_0$ da amplituda keskin ortadi.

Rezonans chastotasi

$$\omega_r = \omega_0\sqrt{1-2\zeta^2}$$

($\zeta < 0.707$ bo'lganda mavjud)

Sifat omili (Q-factor)

$$Q = \frac{1}{2\zeta}$$

Yuqori Q = tor rezonans, sekin so'nish.

Muhandislikda

Rezonansdan qochish kerak! Binolar, ko'priklar, dronlar — hammasi rezonansda buzilishi mumkin.

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7. Bog'langan Tebranishlar

Ikki yoki undan ko'p tebranuvchi tizim:

Ikki mayatnik

$$m_1\ddot{x}_1 + k_1 x_1 + k_c(x_1-x_2) = 0$$ $$m_2\ddot{x}_2 + k_2 x_2 + k_c(x_2-x_1) = 0$$

Normal modlar

Tizim tabiiy chastotalari — xos qiymatlardan topiladi.

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8. Robotikada Qo'llanilish

PID va Tebranishlar

Yomon tuning qilingan PID — tebranadi:

• $K_p$ yuqori → underdamped, tebranadi
• $K_d$ past → sekin so'nadi
• $K_i$ yuqori → barqarorsiz bo'lishi mumkin

Dron Vibratsiyasi

Propeller muvozanatsizligi → vibratsiya → sensor xatolari

Filtr qo'llaniladi (low-pass):

$$y_n = \alpha x_n + (1-\alpha)y_{n-1}$$

Suspension tizimlari

Robot chassis vibratsiyasini kamaytirish — damper dizayni.

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9. To'lqinlar (Qisqacha)

To'lqin tenglamasi

$$\frac{\partial^2 y}{\partial t^2} = v^2\frac{\partial^2 y}{\partial x^2}$$

Asosiy kattaliklar

• To'lqin uzunligi: $\lambda$
• Chastota: $f$
• Tezlik: $v = \lambda f$

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Xulosa

Tushuncha	Formula	Ahamiyati
SHM	$x = A\cos(\omega t)$	Asosiy model
Davr	$T = 2\pi\sqrt{m/k}$	Vaqt xarakteristikasi
Damping	$\zeta = c/(2\sqrt{km})$	So'nish darajasi
Rezonans	$\omega \approx \omega_0$	Xavfli holat
Q-factor	$Q = 1/(2\zeta)$	Sifat o'lchovi

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