Problem 13 IYPT 2027: 13. Vortex pendulum

Problem Description & Analysis

A pendulum with a vertically oriented cylindrical bob is suspended above a channel of flowing water. When the bob is partially submerged, the flow may cause the pendulum to oscillate in a direction perpendicular to the flow. Explain this phenomenon and determine how it depends on relevant parameters.

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قطر استوانه D (mm) 30 mm

فرکانس طبیعی fn (Hz) 1.2 Hz

عدد استروهال St 0.2

ضریب کیفیت Q 40

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Python — Editable

# آزمایش پاندول گردابی — دامنه برحسب سرعت جریان
import numpy as np
import matplotlib.pyplot as plt

D  = 0.030       # قطر استوانه (m)
fn = 1.2         # فرکانس طبیعی پاندول (Hz)
St = 0.20

# --- بخش ۱: داده‌های شما ---
# سرعت جریان اندازه‌گیری‌شده (m/s)
U   = np.array([0.05, 0.10, 0.14, 0.16, 0.18, 0.20, 0.22, 0.25, 0.30, 0.40])
# دامنهٔ نوسان اندازه‌گیری‌شده (mm)
amp = np.array([0.4,  1.6,  4.2,  8.5, 14.1, 11.7, 7.3,  3.8,  1.9,  1.0])

# --- بخش ۲: نظریه ---
U_lock = fn*D/St
print(f"Predicted lock-in speed = fn·D/St = {U_lock:.3f} m/s")

fs = St*U/D
Q  = 40
r  = fs/fn
amp_th = 1.0/np.sqrt((1-r**2)**2 + (r/Q)**2)
# normalize
amp_th = amp_th / amp_th.max() * amp.max()

# --- بخش ۳: نمودار ---
fig, ax = plt.subplots(figsize=(8, 4.6))
ax.plot(U, amp,     'ro-', ms=9, label='measured')
ax.plot(U, amp_th,  'b--', lw=2, label='SHO model (Q=40)')
ax.axvline(U_lock, color='g', ls=':', label=f'U_lock={U_lock:.3f} m/s')
ax.set_xlabel('flow speed U (m/s)')
ax.set_ylabel('oscillation amplitude (mm)')
ax.set_title('Vortex pendulum — amplitude vs flow speed')
ax.legend(); ax.grid(True, alpha=0.3)
plt.tight_layout()

# --- بخش ۴: نتایج ---
i_peak = np.argmax(amp)
U_peak = U[i_peak]
print(f"Measured peak at U = {U_peak:.3f} m/s")
print(f"Deviation from prediction = {(U_peak-U_lock)/U_lock*100:+.1f}%")
St_fit = fn*D/U_peak
print(f"Fitted Strouhal number St = {St_fit:.3f} (literature ~0.20)")

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