# How Bridges Work 9, Exercise 1: resonance against flutter, on paper
# A deck mode has a frequency of 0.2 Hz. (a) Under a steady periodic push of
# 0.1 m/s2 at exactly that frequency, the peak amplitude is F / (2 zeta omega^2).
# Find it for damping ratios zeta = 0.005, 0.01 and 0.02.
# (b) Suppose the wind adds "negative damping" a = k U (illustrative), with k
# chosen so that a equals the structure's own damping 2 zeta omega at U = 15.6 m/s
# (35 mph) for zeta = 0.01. For wind speeds of 12, 15.6, 17.9 and 22 m/s
# find the net growth rate s = a / 2 - zeta omega (per second) and, when it is
# positive, the time for the motion to double (ln 2 / s).
import math

omega = 2 * math.pi * 0.2
for zeta in (0.005, 0.01, 0.02):
    print(zeta, round(0.1 / (2 * zeta * omega**2), 2))

zeta = 0.01
k = 2 * zeta * omega / 15.6
for u in (12, 15.6, 17.9, 22):
    s = k * u / 2 - zeta * omega
    doubling = round(math.log(2) / s, 1) if s > 1e-9 else None
    print(u, round(s, 5), doubling)

# Output when run:
# 0.005 6.33
# 0.01 3.17
# 0.02 1.58
# 12 -0.0029 None
# 15.6 0.0 None
# 17.9 0.00185 374.1
# 22 0.00516 134.4
#
# Part (a) lines: damping ratio, resonant amplitude (m). Halving the damping
# doubles the amplitude (6.33, 3.17, 1.58 m): resonance is limited by damping, and
# stays at a fixed size. Part (b) lines: wind speed, net growth rate (per second),
# doubling time (s). At 12 m/s the rate is negative (the motion dies away), at
# 15.6 m/s it is zero (the critical speed), and above it the motion grows with no
# limit in this model: doubling in 374 s at 17.9 m/s (40 mph) and in 134 s at
# 22 m/s. The difference is the point: resonance needs an outside rhythm and levels
# off, while flutter feeds itself from a steady wind and keeps growing until
# something gives. The linear model ignores the limits that real nonlinear
# effects set; the numbers are illustrations, not Tacoma's.