26-mavzu. Tebranishlarni qo‘shish

1-masala: Bir xil yo'nalishli tebranishlarni qo'shish

x₁ = A₁cos ω(t + τ₁) va x₂ = A₂cos ω(t + τ₂) tenglamalar bilan ifodalanadigan tebranishlarni qo'shing.

Formulalar:
φ₁ = ωτ₁, φ₂ = ωτ₂
A = √(A₁² + A₂² + 2A₁A₂cos(φ₂ - φ₁))
φ = arctg((A₁sinφ₁ + A₂sinφ₂)/(A₁cosφ₁ + A₂cosφ₂))

A₁ (m):

A₂ (m):

τ₁ (s):

τ₂ (s):

ω (rad/s):

2-masala: Perpendikulyar tebranishlar va trayektoriya

x = A₁cos ωt va y = A₂cos (ω/2)t tenglamalar bilan berilgan nuqta trayektoriyasini toping.

Formula:
y = A₂√[(1 + x/A₁)/2]
cos(α/2) = √[(1 + cosα)/2] formulasidan foydalanish

A₁ (m):

A₂ (m):

ω (rad/s):

3-masala: Fazalar farqi bilan tebranishlar

x₁ = A sin(ωt + φ₁) va x₂ = A sin(ωt + φ₂) tebranishlarning yig'indisini toping.

Formulalar:
A_rez = 2A cos((φ₂ - φ₁)/2)
φ_rez = (φ₁ + φ₂)/2
x = A_rez sin(ωt + φ_rez)

A (amplituda, m):

φ₁ (rad):

φ₂ (rad):

ω (rad/s):

4-masala: Lissaju egrilari

x = A₁sin(ω₁t) va y = A₂sin(ω₂t) tenglamalar bilan berilgan nuqtaning trayektoriyasi.

Lissaju egrilari:
Chastotalar nisbati ω₁/ω₂ egri shaklini belgilaydi
1:1 - ellips, 1:2 - sakkizcha shakli, 2:3 - murakkab egri

A₁ (m):

A₂ (m):

ω₁ (rad/s):

ω₂ (rad/s):

5-masala: Fazalar interferensiyasi

Bir xil amplituda va chastotali, lekin turli fazali tebranishlar yig'indisi.

Maxsus holatlar:
Δφ = 0 → A_max = A₁ + A₂ (konstruktiv)
Δφ = π → A_min = |A₁ - A₂| (destruktiv)
Δφ = π/2 → A = √(A₁² + A₂²)

A₁ (m):

A₂ (m):

Δφ (rad):

ω (rad/s):

6-masala: Umumiy holda tebranishlar qo'shish

Turli amplituda, chastota va fazali tebranishlarning umumiy holda qo'shilishi.

Umumiy yondashuv:
x₁ = A₁cos(ω₁t + φ₁)
x₂ = A₂cos(ω₂t + φ₂)
x = x₁ + x₂ (raqamli hisoblash)

A₁ (m):

ω₁ (rad/s):

φ₁ (rad):

A₂ (m):

ω₂ (rad/s):

φ₂ (rad):

26-mavzu: Tebranishlarni qo'shish masalalari

1-masala: Bir xil yo'nalishli tebranishlarni qo'shish

2-masala: Perpendikulyar tebranishlar va trayektoriya

3-masala: Fazalar farqi bilan tebranishlar

4-masala: Lissaju egrilari

5-masala: Fazalar interferensiyasi

6-masala: Umumiy holda tebranishlar qo'shish