math

scripts/string_model_de.py

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+
+import matplotlib.pyplot as plt
+import numpy as np
+import math
+
+# https://people.cs.uchicago.edu/~ridg/stabil/pianostring.pdf
+# a differential equations model for a piano string
+
+# eq 1: governing wave equation
+# d2y/dt2 = c^2*d2y/dx2 - stiffness*c^2*L^2*d4y/dx4 - 2*b_1*dy/dt + 2*b_3*d3y/dt3 + f(x, x_0, t)
+# where y = string's transverse displacement, b_1, b_3 = damping coefficients, f = force density
+# c = sqrt(T/mu), transverse wave velocity , T = string tension, mu = string linear mass density)
+
+# eq 2: string stiffness
+# stiffness = K^2*(E*S/(T*L^2))
+# where K = string's radius of gyration (r/2 for a circular string), E = string's Young's modulus,
+# S = cross sectional string area, T = string tension, L = string length
+
+# eq 3: decay
+# sigma = 1/tau = b_1 + b_3*omega^2
+# where sigma = decay rate, tau = decay time, omega = angular frequency
+
+# eq 4: excitation
+# f(x, x_0, t) = f_H(t) * g(x, x_0)
+# where f_H(t) = hammer force, g(x, x_0) = hammer dimensional effect
+
+# eq 5: hammer time history
+# read the paper if you care, useful only for derivation
+
+# eq 6: power law
+# F_H(t) = K*|eta(t) - y(x_0, t)|^p
+# where eta(t) is the transverse displacement of the hammer head
+# and p = stiffness nonlinear exponent
+
+# eq 7: hammer displacement
+# M_H*d2eta/dt2 = -F_H(t)
+# where M_H is the mass of the hamemr head
+
+# eq 8: boundary conditions
+# y(0, t) = y(L, t) = 0, fixed ends dont move
+# d2y/dx2(0, t) = d2y/dx2(L, t) = 0, displacement along the string approaching the ends is continuous
+
+# discrete time implementation
+
+# eq 9: continuous to discrete
+# y(x, t) -> y(x_i, t_n) -> y(i, n)
+# divide the string into i segments and iterate over n timesteps
+# where x_i = delta_x * i and t_n = delta_t * n
+
+# eq 10: recurrence derivation
+# y(i, n+1) = a_1*y(i, n) + a_2*y(i, n-1) + a_3*[y(i+1, n) + y(i-1, n)] + a_4*[y(i+2,n) + y(i-2,n)]
+#               + a_5*[y(i+1, n-1) + y(i-1, n-1) + y(i, n-2)]
+#               + [delta_t^2 * N*F_H(n) * g(i, i_0)]/M_S
+# where a_1 through a_5 are defined as follows:
+# a_1 = [2 - 2*r^2 + b_1/delta_t - 6*stiffness*N^2*r^2]/D
+# a_2 = [-1 + b_1*delta_t + 2*b_3/delta_t]/D
+# a_3 = [r^2*(1 + 4*stiffness*N^2)]/D
+# a_4 = [b_3/delta_t - stiffness*N^2*r^2]/D
+# a_5 = [-b_3/delta_t]/D
+# where D = 1 + b_1*delta_t + 2*b_3/delta_t
+# and r = c*delta_t/delta_x
+
+# eq 11: stability condition
+# N_max = sqrt{[-1 + sqrt(1+16*stiffness*gamma^2)]/(8*stiffness)}
+# where
+# eq 12: idk what gamma represents
+# gamma = f_e/(2*f_1), f_e = sampling frequency and f_1 = fundamental frequency
+# or
+# eq 13: if neglecting stiffness
+# N_max = gamma
+
+# eq 14: rest condition
+# y(i, 0) = 0
+
+# eq 15: discrete hammer displacement
+# at t = delta_t (n = 1)
+# eta(1) = V_H_0 * delta_t
+
+# eq 16: discrete truncated taylor series
+# y(i, 1) = [y(i + 1, 0) + y(i - 1, 0)]/2
+
+# eq 17: hammer force exertion
+# F_H(1) = K*|eta(1) - y(i_0, 1)|^p
+
+# eq 18: string displacement iteration
+# y(i, 2) = y(i-1, 1)] + y(i + 1, 1) - y(i, 0) + [delta_t^2 * N*F_H(1) * g(i, i_0)]/M_S
+
+# eq 19: hammer displacement iteration
+# eta(2) = 2*eta(1) - eta(0) - [delta_t^2 * F_H(1)]/M_H
+
+# eq 20: hammer force iteration
+# F_H(2) = K*|eta(2) - y(i_0, 2)|^p
+
+# eq 21: hammer rest condition
+# eta(n + 1) < y(i_0, n + 1)
+
+# eq 22: spacial boundary conditions
+# y(0, n) = y(N, n) = 0
+
+# eq 23: temporal boundary conditions
+# y(-1, n) = -y(1, n)
+# y(N + 1, n) = -y(N - 1, n)