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8.1 Raketa Asoslari — Nazariya

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Kirish​

Raketatexnika — kosmosga yo'l. Tsiolkovsky tenglamasidan boshlab, orbital mexanikagacha.

1. Raketa Prinsipi​

Nyuton III Qonuni​

Gaz orqaga — raketa oldinga.

Fraketa=−Fexhaust\mathbf{F}_{raketa} = -\mathbf{F}_{exhaust}

Thrust (Itarish Kuchi)​

F=m˙⋅ve+(pe−pa)⋅AeF = \dot{m} \cdot v_e + (p_e - p_a) \cdot A_e
  • m˙\dot{m} — massa oqimi (kg/s)
  • vev_e — exhaust tezligi
  • pep_e — nozzle exit bosimi
  • pap_a — atmosfera bosimi
  • AeA_e — exit yuza

Vakuumda: F=m˙⋅veF = \dot{m} \cdot v_e

2. Tsiolkovsky Tenglamasi​

Raketa asosiy tenglamasi:

Δv=veln⁡m0mf=Isp⋅g0ln⁡m0mf\Delta v = v_e \ln\frac{m_0}{m_f} = I_{sp} \cdot g_0 \ln\frac{m_0}{m_f}
  • m0m_0 — boshlang'ich massa
  • mfm_f — yakuniy massa (yonilg'isiz)
  • vev_e — effective exhaust velocity
  • IspI_{sp} — specific impulse (s)

Mass Ratio​

MR=m0mf=eΔv/veMR = \frac{m_0}{m_f} = e^{\Delta v / v_e}

Payload Fraction​

λ=mpayloadm0\lambda = \frac{m_{payload}}{m_0}

3. Specific Impulse (IspI_{sp})​

Yonilg'i samaradorligi o'lchovi:

Isp=Fm˙⋅g0=veg0I_{sp} = \frac{F}{\dot{m} \cdot g_0} = \frac{v_e}{g_0}
Yonilg'iIspI_{sp} (s)vev_e (m/s)
Solid250-2802450-2750
Kerosene/LOX300-3502940-3430
LH2/LOX420-4504120-4410
Ion engine1500-500014700-49000

4. Gravitatsiya Ta'siri​

Gravity Loss​

Δvgravity=g⋅tburn\Delta v_{gravity} = g \cdot t_{burn}

Tez yonish — kam yo'qotish.

Ko'tarish bilan​

Δvactual=veln⁡m0mf−g⋅tburn\Delta v_{actual} = v_e \ln\frac{m_0}{m_f} - g \cdot t_{burn}

Trajectory Optimization​

  • Gravity turn — samarali traektoriya
  • Vertikal boshlash, sekin gorizontalga

5. Ko'p Bosqichli Raketalar​

Nima uchun?​

Bir bosqichli raketa samarasiz — bo'sh tanklar tortib yuradi.

Staging Tenglamasi​

Δvtotal=∑i=1nve,iln⁡m0,imf,i\Delta v_{total} = \sum_{i=1}^{n} v_{e,i} \ln\frac{m_{0,i}}{m_{f,i}}

Optimal Staging​

Teng Δv\Delta v har bir bosqichda (taxminan).

6. Orbital Mexanika Asoslari​

Orbital Tezlik​

Aylanali orbit:

vorbit=GMr=μrv_{orbit} = \sqrt{\frac{GM}{r}} = \sqrt{\frac{\mu}{r}}

Yer yuzasida (LEO, 200 km):

vLEO≈7.8 km/sv_{LEO} \approx 7.8 \text{ km/s}

Qochish Tezligi​

vescape=2GMr=2⋅vorbitv_{escape} = \sqrt{\frac{2GM}{r}} = \sqrt{2} \cdot v_{orbit}

Yerdan: vescape≈11.2v_{escape} \approx 11.2 km/s

Delta-V Budget​

ManzilΔv\Delta v (km/s)
LEO~9.4
GTO~12.5
Moon~15
Mars~18

7. Aerodinamik Qarshilik​

Drag​

FD=12ρv2CDAF_D = \frac{1}{2} \rho v^2 C_D A

Max-Q​

Maksimal dinamik bosim — eng xavfli nuqta.

q=12ρv2q = \frac{1}{2} \rho v^2

8. Guidance, Navigation, Control (GNC)​

Guidance​

Traektoriyani hisoblash.

Hozirgi holatni aniqlash (IMU, GPS, star tracker).

Control​

Yo'nalishni sozlash:

  • Gimbal: Motor burchagini o'zgartirish
  • Fins: Aerodinamik
  • RCS: Reaction Control System

9. Stability​

Static Stability​

CP>CG (masofada)CP > CG \text{ (masofada)}
  • CP — Center of Pressure
  • CG — Center of Gravity

Active Control​

Gyroscope + computer → gimbal adjustment

10. Model Raketa Parametrlari​

Hobby rockets​

KlassImpulse (Ns)
A1.26-2.5
B2.5-5
C5-10
D10-20
E20-40
F40-80

Altitude Estimation​

hmax≈vburnout22gh_{max} \approx \frac{v_{burnout}^2}{2g}

(Drag e'tiborga olinmagan)

Xulosa​

TushunchaFormula
TsiolkovskyΔv=veln⁡(m0/mf)\Delta v = v_e \ln(m_0/m_f)
Specific ImpulseIsp=ve/gI_{sp} = v_e / g
Orbital velocityv=μ/rv = \sqrt{\mu/r}
Escape velocityvesc=2μ/rv_{esc} = \sqrt{2\mu/r}
ThrustF=m˙veF = \dot{m} v_e

Keyingi Qadam​

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