Problem 8 IYPT 2027: 8. Upward-driven disc

Problem Description & Analysis

A circular disc is suspended from the point where it is sandwiched between two wheels, making it free to oscillate as a pendulum. One of the wheels is driven by a motor and thus exerts an upward driving force on the disc. Investigate the motion of the disc.

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Interactive Simulation (Python)

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فرکانس تحریک f (Hz) 30 Hz

دامنهٔ تحریک A (mm) 2 mm

ضریب بازگشت e 0.6

زمان شبیه‌سازی (s) 2 s

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Python Code Lab

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Python — Editable

# آزمایش دیسک بالا-رانده — ارتفاع پرش برحسب فرکانس تحریک
import numpy as np
import matplotlib.pyplot as plt

g = 9.81
A = 2e-3       # دامنهٔ لرزش صفحه (m)
e = 0.55       # ضریب بازگشت اندازه‌گیری شده

# --- بخش ۱: داده‌های شما ---
f_Hz  = np.array([10, 15, 20, 25, 30, 40, 50, 60, 80])
h_mm  = np.array([2.1, 6.4, 12.8, 22.5, 34.1, 60.3, 85.7, 115.2, 168.4])

# --- بخش ۲: نظریه: شبیه‌سازی سادهٔ Newton با restitution ---
def sim_height(f, A, e, T=1.0, dt=2e-5):
    w = 2*np.pi*f; y=0; v=0; peak=0
    for i in range(int(T/dt)):
        t = i*dt
        v -= g*dt; y += v*dt
        yp = A*np.sin(w*t); vp = A*w*np.cos(w*t)
        if y <= yp:
            y = yp; v = vp - e*(v-vp)
        peak = max(peak, y)
    return peak*1000    # mm

h_theory = np.array([sim_height(f, A, e) for f in f_Hz])

# --- بخش ۳: نمودار ---
fig, ax = plt.subplots(figsize=(8, 4.6))
ax.plot(f_Hz, h_theory, 'b-', lw=2, label=f'theory (A={A*1e3:.1f} mm, e={e})')
ax.plot(f_Hz, h_mm, 'ro', ms=9, label='measured')
ax.set_xlabel('drive frequency (Hz)'); ax.set_ylabel('max jump height (mm)')
ax.set_title('Upward-driven disc: h vs f')
ax.legend(); ax.grid(True, alpha=0.3)
plt.tight_layout()

# --- بخش ۴: نتایج ---
print(" f(Hz)   h_th(mm)  h_meas(mm)  Γ=Aω²/g")
for f, ht, hm in zip(f_Hz, h_theory, h_mm):
    Gam = A*(2*np.pi*f)**2/g
    print(f"  {f:3.0f}    {ht:6.2f}    {hm:6.2f}     {Gam:5.2f}")
rms = np.sqrt(np.mean((h_mm-h_theory)**2))
print(f"nRMS error = {rms:.2f} mm")

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Problem Description & Analysis

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