2.1 Kinematika — Nazariya Hafta: 3 | Masalalar: 30 | Qiyinlik: ⭐⭐
Kirish
Kinematika — harakat haqidagi fan, kuchlarni hisobga olmagan holda. Robot, dron, raketa — hammasi harakat qiladi, va biz avval harakatni tasvirlashni o'rganamiz.
1. To'g'ri Chiziqli Harakat
Asosiy tushunchalar
Joylashuv (pozitsiya): x ( t ) x(t) x ( t )
Siljish: Δ x = x 2 − x 1 \Delta x = x_2 - x_1 Δ x = x 2 − x 1
Masofa: s s s
Tezlik:
v = d x d t = x ˙ v = \frac{dx}{dt} = \dot{x} v = d t d x = x ˙ Tezlanish:
a = d v d t = d 2 x d t 2 = x ¨ a = \frac{dv}{dt} = \frac{d^2x}{dt^2} = \ddot{x} a = d t d v = d t 2 d 2 x = x ¨ O'rtacha va lahzali
O'rtacha tezlik:
v ˉ = Δ x Δ t \bar{v} = \frac{\Delta x}{\Delta t} v ˉ = Δ t Δ x Lahzali tezlik:
v = lim Δ t → 0 Δ x Δ t = d x d t v = \lim_{\Delta t \to 0} \frac{\Delta x}{\Delta t} = \frac{dx}{dt} v = Δ t → 0 lim Δ t Δ x = d t d x 2. Tekis Tezlanuvchan Harakat
Agar a = const a = \text{const} a = const
v = v 0 + a t v = v_0 + at v = v 0 + a t x = x 0 + v 0 t + 1 2 a t 2 x = x_0 + v_0 t + \frac{1}{2}at^2 x = x 0 + v 0 t + 2 1 a t 2 v 2 = v 0 2 + 2 a ( x − x 0 ) v^2 = v_0^2 + 2a(x - x_0) v 2 = v 0 2 + 2 a ( x − x 0 ) x = x 0 + v 0 + v 2 t x = x_0 + \frac{v_0 + v}{2}t x = x 0 + 2 v 0 + v t Bu to'rt formula — kinematikaning asosi!
Erkin tushish
Yerda: g ≈ 9.81 g \approx 9.81 g ≈ 9.81
y = y 0 + v 0 t − 1 2 g t 2 y = y_0 + v_0 t - \frac{1}{2}gt^2 y = y 0 + v 0 t − 2 1 g t 2 v = v 0 − g t v = v_0 - gt v = v 0 − g t 3. Vektor Kinematika (2D/3D)
Pozitsiya vektori
r ⃗ ( t ) = x ( t ) i ^ + y ( t ) j ^ + z ( t ) k ^ \vec{r}(t) = x(t)\hat{i} + y(t)\hat{j} + z(t)\hat{k} r ( t ) = x ( t ) i ^ + y ( t ) j ^ + z ( t ) k ^ Tezlik vektori
v ⃗ ( t ) = d r ⃗ d t = x ˙ i ^ + y ˙ j ^ + z ˙ k ^ \vec{v}(t) = \frac{d\vec{r}}{dt} = \dot{x}\hat{i} + \dot{y}\hat{j} + \dot{z}\hat{k} v ( t ) = d t d r = x ˙ i ^ + y ˙ j ^ + z ˙ k ^ Tezlik kattaligi:
∣ v ⃗ ∣ = x ˙ 2 + y ˙ 2 + z ˙ 2 |\vec{v}| = \sqrt{\dot{x}^2 + \dot{y}^2 + \dot{z}^2} ∣ v ∣ = x ˙ 2 + y ˙ 2 + z ˙ 2 Tezlanish vektori
a ⃗ ( t ) = d v ⃗ d t = x ¨ i ^ + y ¨ j ^ + z ¨ k ^ \vec{a}(t) = \frac{d\vec{v}}{dt} = \ddot{x}\hat{i} + \ddot{y}\hat{j} + \ddot{z}\hat{k} a ( t ) = d t d v = x ¨ i ^ + y ¨ j ^ + z ¨ k ^ 4. Proektil Harakati
Gravitatsiya maydonida otilgan jism:
Boshlang'ich shartlar
Boshlang'ich tezlik: v 0 v_0 v 0
Otish burchagi: θ \theta θ
Boshlang'ich pozitsiya: ( x 0 , y 0 ) (x_0, y_0) ( x 0 , y 0 )
Komponentlar
v 0 x = v 0 cos θ , v 0 y = v 0 sin θ v_{0x} = v_0 \cos\theta, \quad v_{0y} = v_0 \sin\theta v 0 x = v 0 cos θ , v 0 y = v 0 sin θ Harakat tenglamalari
Gorizontal (tezlanishsiz):
x ( t ) = x 0 + v 0 cos θ ⋅ t x(t) = x_0 + v_0 \cos\theta \cdot t x ( t ) = x 0 + v 0 cos θ ⋅ t Vertikal (gravitatsiya bilan):
y ( t ) = y 0 + v 0 sin θ ⋅ t − 1 2 g t 2 y(t) = y_0 + v_0 \sin\theta \cdot t - \frac{1}{2}gt^2 y ( t ) = y 0 + v 0 sin θ ⋅ t − 2 1 g t 2 Traektoriya
t t t
y = x tan θ − g x 2 2 v 0 2 cos 2 θ y = x\tan\theta - \frac{gx^2}{2v_0^2\cos^2\theta} y = x tan θ − 2 v 0 2 cos 2 θ g x 2 Bu parabola tenglamasi.
Muhim kattaliklar
Maksimal balandlik:
h m a x = v 0 2 sin 2 θ 2 g h_{max} = \frac{v_0^2 \sin^2\theta}{2g} h ma x = 2 g v 0 2 sin 2 θ Uchish masofasi (range):
R = v 0 2 sin 2 θ g R = \frac{v_0^2 \sin 2\theta}{g} R = g v 0 2 sin 2 θ Maksimal masofa θ = 45 ° \theta = 45° θ = 45°
Uchish vaqti:
T = 2 v 0 sin θ g T = \frac{2v_0 \sin\theta}{g} T = g 2 v 0 sin θ 5. Egri Chiziqli Harakat
Tangensial va Normal Tezlanish
Tezlanishni ikki komponentga ajratamiz:
Tangensial tezlanish a t a_t a t
a t = d ∣ v ⃗ ∣ d t a_t = \frac{d|\vec{v}|}{dt} a t = d t d ∣ v ∣ Normal (markazga intilma) tezlanish a n a_n a n
a n = v 2 r a_n = \frac{v^2}{r} a n = r v 2 Bu yerda r r r
Umumiy tezlanish:
∣ a ⃗ ∣ = a t 2 + a n 2 |\vec{a}| = \sqrt{a_t^2 + a_n^2} ∣ a ∣ = a t 2 + a n 2 6. Aylanma Harakat
Burchak o'zgaruvchilari
Burchak pozitsiya: θ \theta θ
Burchak tezlik:
ω = d θ d t \omega = \frac{d\theta}{dt} ω = d t d θ Burchak tezlanish:
α = d ω d t \alpha = \frac{d\omega}{dt} α = d t d ω Chiziqli va burchak o'zgaruvchilar
s = r θ s = r\theta s = r θ v = r ω v = r\omega v = r ω a t = r α a_t = r\alpha a t = r α a n = r ω 2 = v 2 r a_n = r\omega^2 = \frac{v^2}{r} a n = r ω 2 = r v 2 Tekis aylanish (α = const \alpha = \text{const} α = const
ω = ω 0 + α t \omega = \omega_0 + \alpha t ω = ω 0 + α t θ = θ 0 + ω 0 t + 1 2 α t 2 \theta = \theta_0 + \omega_0 t + \frac{1}{2}\alpha t^2 θ = θ 0 + ω 0 t + 2 1 α t 2 ω 2 = ω 0 2 + 2 α ( θ − θ 0 ) \omega^2 = \omega_0^2 + 2\alpha(\theta - \theta_0) ω 2 = ω 0 2 + 2 α ( θ − θ 0 ) Davriy harakat
Davr (period): T = 2 π ω T = \frac{2\pi}{\omega} T = ω 2 π
Chastota: f = 1 T = ω 2 π f = \frac{1}{T} = \frac{\omega}{2\pi} f = T 1 = 2 π ω
Aylanish tezligi: n n n
ω = 2 π f = 2 π n 60 (rpm dan rad/s ga) \omega = 2\pi f = \frac{2\pi n}{60} \text{ (rpm dan rad/s ga)} ω = 2 π f = 60 2 π n (rpm dan rad/s ga) 7. Nisbiy Harakat
Ikki sanoq sistemasi
A dan B ning tezligi:
v ⃗ B / A = v ⃗ B − v ⃗ A \vec{v}_{B/A} = \vec{v}_B - \vec{v}_A v B / A = v B − v A Bog'lanish
r ⃗ B / A = r ⃗ B − r ⃗ A \vec{r}_{B/A} = \vec{r}_B - \vec{r}_A r B / A = r B − r A v ⃗ B / A = d r ⃗ B / A d t \vec{v}_{B/A} = \frac{d\vec{r}_{B/A}}{dt} v B / A = d t d r B / A a ⃗ B / A = d v ⃗ B / A d t \vec{a}_{B/A} = \frac{d\vec{v}_{B/A}}{dt} a B / A = d t d v B / A Dron va maqsad orasidagi nisbiy harakat — tracking algoritmlari uchun.
8. Robotikada Qo'llanilish
Differensial Drive Robot
Ikki g'ildirakli robot kinematikasi:
v = v R + v L 2 v = \frac{v_R + v_L}{2} v = 2 v R + v L ω = v R − v L L \omega = \frac{v_R - v_L}{L} ω = L v R − v L Bu yerda L L L
Dron Traektoriyasi
Waypoint navigatsiya:
r ⃗ ( t ) = r ⃗ 0 + v ⃗ t + 1 2 a ⃗ t 2 \vec{r}(t) = \vec{r}_0 + \vec{v}t + \frac{1}{2}\vec{a}t^2 r ( t ) = r 0 + v t + 2 1 a t 2 Raketa Uchishi
Ko'tarilish bosqichida:
h ( t ) = 1 2 F − m g m t 2 = 1 2 a n e t t 2 h(t) = \frac{1}{2}\frac{F - mg}{m}t^2 = \frac{1}{2}a_{net}t^2 h ( t ) = 2 1 m F − m g t 2 = 2 1 a n e t t 2 Xulosa
Tushuncha Formula Birlik Siljish Δ x = x 2 − x 1 \Delta x = x_2 - x_1 Δ x = x 2 − x 1 m Tezlik v = d x / d t v = dx/dt v = d x / d t m/s Tezlanish a = d v / d t a = dv/dt a = d v / d t m/s² Burchak tezlik ω = d θ / d t \omega = d\theta/dt ω = d θ / d t rad/s Markazga intilma a n = v 2 / r a_n = v^2/r a n = v 2 / r m/s²
Keyingi Qadam
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