Problem 5 IYPT 2027: 5. Y-shaped pendulum

Problem Description & Analysis

Construct a Bowditch pendulum (a pendulum whose strings form a “Y” shape). Investigate how the Lissajous patterns generated by the bob depend on the relevant parameters.

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پاندول Y

الگوی لیساژو

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طول شاخهٔ بالا L1 (m) 0.4 m

طول شاخه‌های پایین L2 (m) 0.3 m

جرم گلولهٔ انتهایی m (kg) 0.1 kg

نصف زاویهٔ باز شدن α (°) 30 °

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Python — Editable

# آزمایش پاندول Y — دو فرکانس نرمال برحسب زاویهٔ باز شدن
import numpy as np
import matplotlib.pyplot as plt

g  = 9.81
L1 = 0.40   # طول شاخهٔ بالا (m)
L2 = 0.30   # طول شاخه‌های پایین (m)

# --- بخش ۱: داده‌های شما ---
alpha_deg = np.array([15, 30, 45, 60, 75])
# فرکانس مود درون‌صفحه (Hz) — از شمردن ۱۰ نوسان
f1_meas = np.array([0.771, 0.795, 0.827, 0.878, 0.945])
# فرکانس مود عمود بر صفحه (Hz)
f2_meas = np.array([0.802, 0.845, 0.925, 1.05, 1.24])

# --- بخش ۲: نظریه ---
def mode_freqs(a):
    L_sym  = L1 + L2*np.cos(a)
    L_asym = L1 + (L2*np.sin(a))**2 / max(L2*np.cos(a), 1e-6)
    return 1/(2*np.pi)*np.sqrt(g/L_sym), 1/(2*np.pi)*np.sqrt(g/L_asym)

a_th   = np.radians(np.linspace(5, 80, 200))
f1_th  = np.array([mode_freqs(a)[0] for a in a_th])
f2_th  = np.array([mode_freqs(a)[1] for a in a_th])

# --- بخش ۳: نمودار ---
fig, ax = plt.subplots(figsize=(8, 4.8))
ax.plot(np.degrees(a_th), f1_th, 'b-', lw=2, label='f1 theory')
ax.plot(np.degrees(a_th), f2_th, 'r-', lw=2, label='f2 theory')
ax.plot(alpha_deg, f1_meas, 'bo', ms=9, label='f1 measured')
ax.plot(alpha_deg, f2_meas, 'rs', ms=9, label='f2 measured')
ax.set_xlabel('α (deg)'); ax.set_ylabel('frequency (Hz)')
ax.set_title('Y-pendulum normal modes')
ax.legend(); ax.grid(True, alpha=0.3)
plt.tight_layout()

# --- بخش ۴: نتایج ---
print(" α    f1_th  f1_meas   f2_th  f2_meas")
for a, m1, m2 in zip(alpha_deg, f1_meas, f2_meas):
    t1, t2 = mode_freqs(np.radians(a))
    print(f"{a:3.0f}°  {t1:.3f}   {m1:.3f}   {t2:.3f}   {m2:.3f}")
err1 = np.sqrt(np.mean((f1_meas - np.array([mode_freqs(np.radians(a))[0] for a in alpha_deg]))**2))
err2 = np.sqrt(np.mean((f2_meas - np.array([mode_freqs(np.radians(a))[1] for a in alpha_deg]))**2))
print(f"nRMS error mode 1 = {err1*1000:.1f} mHz")
print(f"RMS error mode 2 = {err2*1000:.1f} mHz")

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Problem Description & Analysis

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