Frahm’s vibration absorber and Den Hartog’s tuning (1909)

Hermann Frahm (patent filed 1909) stopped ships’ hulls and machines from vibrating at their running speed by attaching a small spring-mass system tuned to that speed. At the tuned frequency the absorber swings so that its spring force cancels the applied force, and the machine stands still. The cure works at one frequency only: the system now has two resonances, one on each side. Jacob Pieter Den Hartog and John Ormondroyd (1928) added damping to the absorber. Every response curve then passes through two fixed points, whatever the damping, and the best design tunes the absorber to make them equal and puts the peaks there (tuned_mass_damper()). This is the tuned mass damper of tall buildings.

import matplotlib.pyplot as plt
import numpy as np

from engineeringkit.mechanical_engineering import tuned_mass_damper

Frahm’s undamped absorber

Tuned to the primary frequency (\(f = 1\)) with mass ratio \(\mu\), the primary mass stops at \(r = 1\). The two new resonances are at \(r^2 = 1 + \mu/2 \pm \sqrt{\mu + \mu^2/4}\).

m, k, mu = 500.0, 2e6, 0.1
frahm = tuned_mass_damper(m, k, mu, frequency_ratio=1.0, damping_ratio=0.0)
split = np.sqrt(1 + mu / 2 + np.array([-1, 1]) * np.sqrt(mu + mu**2 / 4))
print(f"amplification at r = 1: {frahm.amplification(1.0):.2e}; new resonances at r = {np.round(split, 4).tolist()}")
print(f"absorber stroke at r = 1: {abs(frahm.response(1.0)[1]):.1f} x static deflection")
assert frahm.amplification(1.0) < 1e-12
assert np.all(frahm.amplification(split) > 1e9)

amplification at r = 1: 0.00e+00; new resonances at r = [0.8543, 1.1705]
absorber stroke at r = 1: 10.0 x static deflection

Den Hartog’s fixed points

With \(f = 1/(1 + \mu)\) the fixed points are at \(r^2 = (1 \pm \sqrt{\mu/(2 + \mu)})/(1 + \mu)\) and both have height \(\sqrt{1 + 2/\mu}\). The optimum damping puts the peaks on them.

fixed = np.sqrt((1 + np.array([-1, 1]) * np.sqrt(mu / (2 + mu))) / (1 + mu))
r = np.linspace(0.6, 1.4, 4001)
fig, axes = plt.subplots(1, 2, figsize=(10, 3.8))
for zeta in (0.02, 0.1, 0.4):
    design = tuned_mass_damper(m, k, mu, damping_ratio=zeta)
    assert np.allclose(design.amplification(fixed), np.sqrt(1 + 2 / mu), rtol=1e-9)
    axes[0].semilogy(r, design.amplification(r), label=f"zeta = {zeta}")
optimum = tuned_mass_damper(m, k, mu)
peak = optimum.amplification(r).max()
print(f"optimum: f = {optimum.frequency_ratio:.4f}, zeta = {optimum.damping_ratio:.4f}")
print(f"peak {peak:.3f}, fixed-point height sqrt(1 + 2/mu) = {np.sqrt(1 + 2 / mu):.3f}")
assert np.isclose(peak, np.sqrt(1 + 2 / mu), rtol=0.01)
axes[0].semilogy(r, optimum.amplification(r), "k", lw=2, label="Den Hartog optimum")
axes[0].plot(fixed, optimum.amplification(fixed), "ko")
axes[0].set(xlabel="forcing / primary frequency", ylabel="amplification", title=f"mu = {mu}, f = 1/(1 + mu)")
axes[0].legend(fontsize=8)
optimum.plot(ax=axes[1])
axes[1].set(title="With and without the optimum absorber")
fig.tight_layout()

plt.show()

optimum: f = 0.9091, zeta = 0.1679
peak 4.590, fixed-point height sqrt(1 + 2/mu) = 4.583