Example C1: Forced Response of a 2DoF Oscillator with Instantaneous Nonlinearity

This example is designed to expose the core features of the MDOFUTILS module. As already noted in Example b2, much of the setup while doing these analyses are very repetitive and common. MDOFUTILS abstracts much of this so the user can focus on the dynamics.

The dynamical system that will be studied here is:

\[\underbrace{\begin{bmatrix} 1&0\\0&1 \end{bmatrix}}_{\mx{M}} \begin{bmatrix} \ddot{x_1}\\ \ddot{x_2} \end{bmatrix} + \left( 0.01 \mx{M} + 0.001 \mx{K}\right) \begin{bmatrix} \dot{x_1}\\ \dot{x_2} \end{bmatrix} + \underbrace{\begin{bmatrix} 2&-1\\-1&2 \end{bmatrix}}_{\mx{K}} \begin{bmatrix} x_1\\x_2 \end{bmatrix} + \begin{bmatrix} 0\\ \beta x_2^3 \end{bmatrix} = \begin{bmatrix} 1\\0 \end{bmatrix}\cos\Omega t.\]

The steps followed in this file are:

  1. First we specify the system under study in terms of its "MCK" matrices and the nonlinearities it contains.
  2. Next we setup the parameters for Harmonic Balance and setup the harmonic excitation vector.
  3. And that's it! We're ready to compute the forced responses and visualize the results.

Preamble: Load Packages

using GLMakie
using LinearAlgebra
using SparseArrays
using ForwardDiff
using NonlinearSolve
using DSP

using juliajim.HARMONIC
using juliajim.CONTINUATION
using juliajim.MDOFUTILS

System Setup

We specify the linear "MCK" matrices of the system here. Also check out the documentation for the MDOFGEN struct.

M = collect(1.0I(2));
K = [2. -1.;-1. 2.];
C = 0.01*M+0.001*K;

mdl = MDOFGEN(M, C, K);

Next we specify the nonlinearity using a function handle. This function handle must take 3 input arguments (time, displacement, velocity) and output 3 results (force, displacement derivative, velocity derivative). Although the nonlinearity is a simple spring here, this same routine also supports more complicated nonlinearities involving multiple degrees of freedom. In such cases, u,ud must be an (N\times d) matrix where (N) is the number of time samples and (d) is the number of nonlinear degrees-of-freedom. It should also return $d$ forces (for adjoint forcing).

Along with this, we also specify a "selector matrix" (\mx{L}) such that the nonlinear DoFs are (\mx{L}\vc{u}) when (\vc{u}) is the vector of all the DoFs.

# Nonlinearity
β = 0.1;
fnl = (t,u,ud)->return β.*u.^3, 3β.*u.^2, zeros(size(u));;
L = [0.0 1.0];

We "add" the nonlinearity to the MDOFGEN object (mdl here) using the ADDNL routine. The second argument here specifies that the nonlinearity can be evaluated "instantaneously", like a cubic spring (UNlike a Jenkins element, as in Example b2. ADDNL also supports non-self adjoint forcing - check its documentation for this. Under the hood, this creates an object of the NONLINEARITY struct and pushes it into the field NLTs (which is a vector of NONLINEARITY objects) of the MDOFGEN object. Each NONLINEARITY object contains a type, the force evaluation function and a DoF selection (and force distribution) matrix.

mdl = ADDNL(mdl, :Inst, fnl, L);

Setup HB

Next we setup the harmonics necessary and the number of time samples to use per period for the Alternating Frequency Time (AFT) step. We setup the excitation vector in the frequency domain by using the indices returned by HINDS - we set Fl to be non-zero at the first cosine harmonic index, which will correspond to the first harmonic of the first DoF.

h = 0:5;
N = 128;
t = (0:N-1)*2π/N;

Nhc = sum(all(h.==0, dims=2) + 2any(h.!=0, dims=2));

_, _, zinds, rinds, iinds = HINDS(mdl.Ndofs, h)
Fl = zeros(Nhc*mdl.Ndofs, 1);
Fl[rinds[1]] = 1.0;

Wst = 0.6;
E, dEdw = HARMONICSTIFFNESS(mdl.M, mdl.C, mdl.K, Wst, h);

Uw0 = [E\Fl; Wst];

Evaluate the Nonlinear Forces

The NLEVAL! routine is defined in MDOFUTILS. This uses the appropriate methodology to evaluate each nonlinearity that has been added to the given MDOFGEN model. The function is written in the in-place evaluation for efficiency. The routine directly just returns the nonlinear force harmonics (FNL here).

FNL = zeros(mdl.Ndofs*Nhc);
dFNLdU = zeros(mdl.Ndofs*Nhc, mdl.Ndofs*Nhc);
dFNLdw = zeros(mdl.Ndofs*Nhc);
NLEVAL!(Uw0, mdl, h, N; FNL=FNL, dFNLdU=dFNLdU, dFNLdw=dFNLdw);
FNL
22-element Vector{Float64}:
  0.0
  0.0
  0.0
  0.0155453316569695
  0.0
  0.00019691813922932257
  0.0
  0.0
  0.0
  0.0
  ⋮
  0.00019687601555559817
  0.0
  0.0
  0.0
  0.0
  0.0
  2.598369207793659e-20
  0.0
 -4.239758677462978e-19

Test HB Residue

HBRESFUN! implements the Harmonic Balance forced response residue. This can be used in the same way that the RESFUN! functions were used in Example b1 and Example b2 for forced response evaluation.

R = zeros(mdl.Ndofs*Nhc);
dRdU = zeros(mdl.Ndofs*Nhc, mdl.Ndofs*Nhc);
dRdw = zeros(mdl.Ndofs*Nhc);
HBRESFUN!(Uw0, mdl, Fl, h, N; R=R, dRdU=dRdU, dRdw=dRdw)

fun = NonlinearFunction((r,u,p)->HBRESFUN!([u;p], mdl, Fl, h, N; R=r),
    jac=(J,u,p)->HBRESFUN!([u;p], mdl, Fl, h, N; dRdU=J),
    paramjac=(Jp,u,p)->HBRESFUN!([u;p], mdl, Fl, h, N; dRdw=Jp));

prob = NonlinearProblem(fun, Uw0[1:end-1], Uw0[end]);
sol = solve(prob, show_trace=Val(true));

Algorithm: NewtonRaphson(
    descent = NewtonDescent(),
    autodiff = AutoForwardDiff(),
    vjp_autodiff = AutoFiniteDiff(
        fdtype = Val{:forward}(),
        fdjtype = Val{:forward}(),
        fdhtype = Val{:hcentral}(),
        dir = true
    ),
    jvp_autodiff = AutoForwardDiff(),
    concrete_jac = Val{false}()
)

----    	-------------       	-----------
Iter    	f(u) inf-norm       	Step 2-norm
----    	-------------       	-----------
0       	1.55453317e-02      	0.00000000e+00
1       	2.50631608e-05      	2.32525851e-02
2       	4.84989670e-10      	8.06504063e-05
3       	1.05818132e-16      	9.82129830e-10
Final   	1.05818132e-16
----------------------

Continuation

HBRESFUN! can also be used in tandem with CONTINUATE for getting the full forced response.

Om0 = 0.1;
Om1 = 3;

dOm = 0.01; cpars = (parm=:arclength, nmax=2000, Dsc=:auto, minDsc=1e-5); # 1e-5

dOm = 0.1;
cpars = (parm=:arclength, nmax=2000, Dsc=:none);  # TODO: Needs tuning!!

sols, its, dss, xis, Dsc = CONTINUATE(Uw0[1:end-1], fun, [Om0, Om1], dOm; cpars...);

uh = zeros(Complex, maximum(h)+1, length(sols), 2);
for i in 1:2
    uh[h.+1, :, i] = hcat([[s.up[zinds[i:2:end]];
                            s.up[rinds[i:2:end]]+1im*s.up[iinds[i:2:end]]]
                          for s in sols]...);
end
Oms = [s.up[end] for s in sols];

Algorithm: NewtonRaphson(
    descent = NewtonDescent(),
    autodiff = AutoForwardDiff(),
    vjp_autodiff = AutoFiniteDiff(
        fdtype = Val{:forward}(),
        fdjtype = Val{:forward}(),
        fdhtype = Val{:hcentral}(),
        dir = true
    ),
    jvp_autodiff = AutoForwardDiff(),
    concrete_jac = Val{false}()
)

----    	-------------       	-----------
Iter    	f(u) inf-norm       	Step 2-norm
----    	-------------       	-----------
0       	3.39647780e-01      	NaN
1       	7.19434350e-03      	3.88965697e-01
2       	2.25630863e-06      	5.63043188e-03
3       	3.10513334e-13      	2.03363086e-06
Final   	3.10513334e-13
----------------------
2. 0.20 with step 0.0980 (0.0990) converged in 2 iterations.
3. 0.29 with step 0.0948 (0.0990) converged in 2 iterations.
4. 0.37 with step 0.0890 (0.0990) converged in 3 iterations.
5. 0.44 with step 0.0659 (0.0808) converged in 2 iterations.
6. 0.49 with step 0.0594 (0.0808) converged in 2 iterations.
7. 0.54 with step 0.0523 (0.0808) converged in 2 iterations.
8. 0.49 with step 0.0452 (0.0808) converged in 39 iterations.
9. 0.52 with step 0.0262 (0.0404) converged in 2 iterations.
10. 0.54 with step 0.0244 (0.0404) converged in 2 iterations.
11. 0.56 with step 0.0226 (0.0404) converged in 2 iterations.
12. 0.57 with step 0.0185 (0.0404) converged in 6 iterations.
13. 0.58 with step 0.0026 (0.0233) converged in 2 iterations.
14. 0.58 with step 0.0013 (0.0233) converged in 2 iterations.
15. 0.58 with step 0.0009 (0.0233) converged in 2 iterations.
16. 0.58 with step 0.0006 (0.0233) converged in 2 iterations.
17. 0.58 with step 0.0005 (0.0233) converged in 2 iterations.
18. 0.58 with step 0.0004 (0.0233) converged in 2 iterations.
19. 0.58 with step 0.0004 (0.0233) converged in 2 iterations.
20. 0.58 with step 0.0004 (0.0233) converged in 2 iterations.
21. 0.58 with step 0.0004 (0.0233) converged in 2 iterations.
22. 0.58 with step 0.0004 (0.0233) converged in 2 iterations.
23. 0.58 with step 0.0005 (0.0233) converged in 2 iterations.
24. 0.58 with step 0.0007 (0.0233) converged in 2 iterations.
25. 0.58 with step 0.0010 (0.0233) converged in 2 iterations.
26. 0.59 with step 0.0018 (0.0233) converged in 2 iterations.
27. 0.59 with step 0.0039 (0.0233) converged in 3 iterations.
28. 0.60 with step 0.0073 (0.0198) converged in 2 iterations.
29. 0.61 with step 0.0087 (0.0198) converged in 2 iterations.
30. 0.62 with step 0.0086 (0.0198) converged in 1 iterations.
31. 0.63 with step 0.0119 (0.0280) converged in 2 iterations.
32. 0.64 with step 0.0113 (0.0280) converged in 1 iterations.
33. 0.66 with step 0.0153 (0.0396) converged in 2 iterations.
34. 0.67 with step 0.0143 (0.0396) converged in 1 iterations.
35. 0.69 with step 0.0189 (0.0560) converged in 2 iterations.
36. 0.70 with step 0.0173 (0.0560) converged in 1 iterations.
37. 0.73 with step 0.0224 (0.0792) converged in 2 iterations.
38. 0.74 with step 0.0199 (0.0792) converged in 2 iterations.
39. 0.76 with step 0.0179 (0.0792) converged in 1 iterations.
40. 0.78 with step 0.0229 (0.1120) converged in 2 iterations.
41. 0.80 with step 0.0202 (0.1120) converged in 2 iterations.
42. 0.82 with step 0.0179 (0.1120) converged in 2 iterations.
43. 0.83 with step 0.0162 (0.1120) converged in 1 iterations.
44. 0.85 with step 0.0208 (0.1584) converged in 2 iterations.
45. 0.87 with step 0.0185 (0.1584) converged in 2 iterations.
46. 0.89 with step 0.0167 (0.1584) converged in 2 iterations.
47. 0.90 with step 0.0153 (0.1584) converged in 2 iterations.
48. 0.92 with step 0.0141 (0.1584) converged in 1 iterations.
49. 0.93 with step 0.0186 (0.2240) converged in 2 iterations.
50. 0.95 with step 0.0170 (0.2240) converged in 2 iterations.
51. 0.96 with step 0.0157 (0.2240) converged in 2 iterations.
52. 0.98 with step 0.0147 (0.2240) converged in 2 iterations.
53. 0.99 with step 0.0138 (0.2240) converged in 2 iterations.
54. 1.00 with step 0.0129 (0.2240) converged in 2 iterations.
55. 1.02 with step 0.0122 (0.2240) converged in 2 iterations.
56. 1.03 with step 0.0116 (0.2240) converged in 1 iterations.
57. 1.04 with step 0.0155 (0.3168) converged in 2 iterations.
58. 1.06 with step 0.0144 (0.3168) converged in 2 iterations.
59. 1.07 with step 0.0134 (0.3168) converged in 2 iterations.
60. 1.08 with step 0.0125 (0.3168) converged in 2 iterations.
61. 1.09 with step 0.0117 (0.3168) converged in 2 iterations.
62. 1.10 with step 0.0109 (0.3168) converged in 2 iterations.
63. 1.11 with step 0.0102 (0.3168) converged in 2 iterations.
64. 1.12 with step 0.0096 (0.3168) converged in 2 iterations.
65. 1.13 with step 0.0090 (0.3168) converged in 2 iterations.
66. 1.14 with step 0.0084 (0.3168) converged in 2 iterations.
67. 1.15 with step 0.0079 (0.3168) converged in 2 iterations.
68. 1.15 with step 0.0075 (0.3168) converged in 2 iterations.
69. 1.16 with step 0.0071 (0.3168) converged in 2 iterations.
70. 1.17 with step 0.0067 (0.3168) converged in 2 iterations.
71. 1.17 with step 0.0063 (0.3168) converged in 2 iterations.
72. 1.18 with step 0.0060 (0.3168) converged in 2 iterations.
73. 1.19 with step 0.0057 (0.3168) converged in 2 iterations.
74. 1.19 with step 0.0054 (0.3168) converged in 2 iterations.
75. 1.20 with step 0.0051 (0.3168) converged in 1 iterations.
76. 1.20 with step 0.0069 (0.4480) converged in 2 iterations.
77. 1.21 with step 0.0064 (0.4480) converged in 2 iterations.
78. 1.21 with step 0.0060 (0.4480) converged in 2 iterations.
79. 1.22 with step 0.0056 (0.4480) converged in 2 iterations.
80. 1.22 with step 0.0053 (0.4480) converged in 2 iterations.
81. 1.23 with step 0.0050 (0.4480) converged in 2 iterations.
82. 1.23 with step 0.0047 (0.4480) converged in 2 iterations.
83. 1.24 with step 0.0045 (0.4480) converged in 2 iterations.
84. 1.24 with step 0.0042 (0.4480) converged in 2 iterations.
85. 1.25 with step 0.0040 (0.4480) converged in 2 iterations.
86. 1.25 with step 0.0038 (0.4480) converged in 2 iterations.
87. 1.25 with step 0.0036 (0.4480) converged in 2 iterations.
88. 1.26 with step 0.0034 (0.4480) converged in 2 iterations.
89. 1.26 with step 0.0033 (0.4480) converged in 2 iterations.
90. 1.26 with step 0.0031 (0.4480) converged in 2 iterations.
91. 1.27 with step 0.0030 (0.4480) converged in 2 iterations.
92. 1.27 with step 0.0029 (0.4480) converged in 2 iterations.
93. 1.27 with step 0.0027 (0.4480) converged in 2 iterations.
94. 1.27 with step 0.0026 (0.4480) converged in 2 iterations.
95. 1.28 with step 0.0025 (0.4480) converged in 2 iterations.
96. 1.28 with step 0.0024 (0.4480) converged in 2 iterations.
97. 1.28 with step 0.0023 (0.4480) converged in 2 iterations.
98. 1.28 with step 0.0022 (0.4480) converged in 2 iterations.
99. 1.29 with step 0.0022 (0.4480) converged in 2 iterations.
100. 1.29 with step 0.0021 (0.4480) converged in 2 iterations.
101. 1.29 with step 0.0020 (0.4480) converged in 2 iterations.
102. 1.29 with step 0.0019 (0.4480) converged in 2 iterations.
103. 1.29 with step 0.0019 (0.4480) converged in 2 iterations.
104. 1.30 with step 0.0018 (0.4480) converged in 2 iterations.
105. 1.30 with step 0.0017 (0.4480) converged in 2 iterations.
106. 1.30 with step 0.0017 (0.4480) converged in 2 iterations.
107. 1.30 with step 0.0016 (0.4480) converged in 2 iterations.
108. 1.30 with step 0.0016 (0.4480) converged in 2 iterations.
109. 1.30 with step 0.0015 (0.4480) converged in 2 iterations.
110. 1.30 with step 0.0015 (0.4480) converged in 1 iterations.
111. 1.31 with step 0.0016 (0.4949) converged in 2 iterations.
112. 1.31 with step 0.0015 (0.4949) converged in 2 iterations.
113. 1.31 with step 0.0015 (0.4949) converged in 2 iterations.
114. 1.31 with step 0.0014 (0.4949) converged in 2 iterations.
115. 1.31 with step 0.0014 (0.4949) converged in 2 iterations.
116. 1.31 with step 0.0014 (0.4949) converged in 2 iterations.
117. 1.31 with step 0.0013 (0.4949) converged in 2 iterations.
118. 1.32 with step 0.0013 (0.4949) converged in 2 iterations.
119. 1.32 with step 0.0012 (0.4949) converged in 2 iterations.
120. 1.32 with step 0.0012 (0.4949) converged in 2 iterations.
121. 1.32 with step 0.0012 (0.4949) converged in 2 iterations.
122. 1.32 with step 0.0011 (0.4949) converged in 2 iterations.
123. 1.32 with step 0.0011 (0.4949) converged in 2 iterations.
124. 1.32 with step 0.0011 (0.4949) converged in 2 iterations.
125. 1.32 with step 0.0010 (0.4949) converged in 2 iterations.
126. 1.33 with step 0.0010 (0.4949) converged in 2 iterations.
127. 1.33 with step 0.0010 (0.4949) converged in 2 iterations.
128. 1.33 with step 0.0010 (0.4949) converged in 2 iterations.
129. 1.33 with step 0.0009 (0.4949) converged in 2 iterations.
130. 1.33 with step 0.0009 (0.4949) converged in 2 iterations.
131. 1.33 with step 0.0009 (0.4949) converged in 2 iterations.
132. 1.33 with step 0.0009 (0.4949) converged in 2 iterations.
133. 1.33 with step 0.0009 (0.4949) converged in 2 iterations.
134. 1.33 with step 0.0008 (0.4949) converged in 2 iterations.
135. 1.33 with step 0.0008 (0.4949) converged in 2 iterations.
136. 1.33 with step 0.0008 (0.4949) converged in 2 iterations.
137. 1.33 with step 0.0008 (0.4949) converged in 2 iterations.
138. 1.34 with step 0.0008 (0.4949) converged in 2 iterations.
139. 1.34 with step 0.0007 (0.4949) converged in 2 iterations.
140. 1.34 with step 0.0007 (0.4949) converged in 2 iterations.
141. 1.34 with step 0.0007 (0.4949) converged in 2 iterations.
142. 1.34 with step 0.0007 (0.4949) converged in 2 iterations.
143. 1.34 with step 0.0007 (0.4949) converged in 2 iterations.
144. 1.34 with step 0.0007 (0.4949) converged in 2 iterations.
145. 1.34 with step 0.0006 (0.4949) converged in 2 iterations.
146. 1.34 with step 0.0006 (0.4949) converged in 2 iterations.
147. 1.34 with step 0.0006 (0.4949) converged in 2 iterations.
148. 1.34 with step 0.0006 (0.4949) converged in 2 iterations.
149. 1.34 with step 0.0006 (0.4949) converged in 2 iterations.
150. 1.34 with step 0.0006 (0.4949) converged in 2 iterations.
151. 1.34 with step 0.0006 (0.4949) converged in 2 iterations.
152. 1.34 with step 0.0006 (0.4949) converged in 2 iterations.
153. 1.34 with step 0.0005 (0.4949) converged in 2 iterations.
154. 1.35 with step 0.0005 (0.4949) converged in 2 iterations.
155. 1.35 with step 0.0005 (0.4949) converged in 2 iterations.
156. 1.35 with step 0.0005 (0.4949) converged in 2 iterations.
157. 1.35 with step 0.0005 (0.4949) converged in 2 iterations.
158. 1.35 with step 0.0005 (0.4949) converged in 2 iterations.
159. 1.35 with step 0.0005 (0.4949) converged in 2 iterations.
160. 1.35 with step 0.0005 (0.4949) converged in 2 iterations.
161. 1.35 with step 0.0005 (0.4949) converged in 2 iterations.
162. 1.35 with step 0.0005 (0.4949) converged in 2 iterations.
163. 1.35 with step 0.0004 (0.4949) converged in 2 iterations.
164. 1.35 with step 0.0004 (0.4949) converged in 2 iterations.
165. 1.35 with step 0.0004 (0.4949) converged in 2 iterations.
166. 1.35 with step 0.0004 (0.4949) converged in 2 iterations.
167. 1.35 with step 0.0004 (0.4949) converged in 2 iterations.
168. 1.35 with step 0.0004 (0.4949) converged in 2 iterations.
169. 1.35 with step 0.0004 (0.4949) converged in 2 iterations.
170. 1.35 with step 0.0004 (0.4949) converged in 2 iterations.
171. 1.35 with step 0.0004 (0.4949) converged in 2 iterations.
172. 1.35 with step 0.0004 (0.4949) converged in 2 iterations.
173. 1.35 with step 0.0004 (0.4949) converged in 2 iterations.
174. 1.35 with step 0.0004 (0.4949) converged in 2 iterations.
175. 1.35 with step 0.0004 (0.4949) converged in 2 iterations.
176. 1.35 with step 0.0004 (0.4949) converged in 2 iterations.
177. 1.36 with step 0.0003 (0.4949) converged in 2 iterations.
178. 1.36 with step 0.0003 (0.4949) converged in 2 iterations.
179. 1.36 with step 0.0003 (0.4949) converged in 2 iterations.
180. 1.36 with step 0.0003 (0.4949) converged in 2 iterations.
181. 1.36 with step 0.0003 (0.4949) converged in 2 iterations.
182. 1.36 with step 0.0003 (0.4949) converged in 2 iterations.
183. 1.36 with step 0.0003 (0.4949) converged in 2 iterations.
184. 1.36 with step 0.0003 (0.4949) converged in 2 iterations.
185. 1.36 with step 0.0003 (0.4949) converged in 2 iterations.
186. 1.36 with step 0.0003 (0.4949) converged in 2 iterations.
187. 1.36 with step 0.0003 (0.4949) converged in 2 iterations.
188. 1.36 with step 0.0003 (0.4949) converged in 2 iterations.
189. 1.36 with step 0.0003 (0.4949) converged in 2 iterations.
190. 1.36 with step 0.0003 (0.4949) converged in 2 iterations.
191. 1.36 with step 0.0003 (0.4949) converged in 2 iterations.
192. 1.36 with step 0.0003 (0.4949) converged in 2 iterations.
193. 1.36 with step 0.0003 (0.4949) converged in 2 iterations.
194. 1.36 with step 0.0003 (0.4949) converged in 2 iterations.
195. 1.36 with step 0.0002 (0.4949) converged in 2 iterations.
196. 1.36 with step 0.0002 (0.4949) converged in 2 iterations.
197. 1.36 with step 0.0002 (0.4949) converged in 2 iterations.
198. 1.36 with step 0.0002 (0.4949) converged in 2 iterations.
199. 1.36 with step 0.0002 (0.4949) converged in 2 iterations.
200. 1.36 with step 0.0002 (0.4949) converged in 2 iterations.
201. 1.36 with step 0.0002 (0.4949) converged in 2 iterations.
202. 1.36 with step 0.0002 (0.4949) converged in 2 iterations.
203. 1.36 with step 0.0002 (0.4949) converged in 2 iterations.
204. 1.36 with step 0.0002 (0.4949) converged in 2 iterations.
205. 1.36 with step 0.0002 (0.4949) converged in 2 iterations.
206. 1.36 with step 0.0002 (0.4949) converged in 2 iterations.
207. 1.36 with step 0.0002 (0.4949) converged in 2 iterations.
208. 1.36 with step 0.0002 (0.4949) converged in 2 iterations.
209. 1.36 with step 0.0002 (0.4949) converged in 2 iterations.
210. 1.36 with step 0.0002 (0.4949) converged in 2 iterations.
211. 1.36 with step 0.0002 (0.4949) converged in 2 iterations.
212. 1.36 with step 0.0002 (0.4949) converged in 2 iterations.
213. 1.36 with step 0.0002 (0.4949) converged in 2 iterations.
214. 1.36 with step 0.0002 (0.4949) converged in 2 iterations.
215. 1.36 with step 0.0002 (0.4949) converged in 2 iterations.
216. 1.36 with step 0.0002 (0.4949) converged in 2 iterations.
217. 1.36 with step 0.0002 (0.4949) converged in 2 iterations.
218. 1.36 with step 0.0002 (0.4949) converged in 2 iterations.
219. 1.36 with step 0.0002 (0.4949) converged in 2 iterations.
220. 1.36 with step 0.0001 (0.4949) converged in 2 iterations.
221. 1.37 with step 0.0001 (0.4949) converged in 2 iterations.
222. 1.37 with step 0.0001 (0.4949) converged in 2 iterations.
223. 1.37 with step 0.0001 (0.4949) converged in 2 iterations.
224. 1.37 with step 0.0001 (0.4949) converged in 2 iterations.
225. 1.37 with step 0.0001 (0.4949) converged in 2 iterations.
226. 1.37 with step 0.0001 (0.4949) converged in 2 iterations.
227. 1.37 with step 0.0001 (0.4949) converged in 2 iterations.
228. 1.37 with step 0.0001 (0.4949) converged in 2 iterations.
229. 1.37 with step 0.0001 (0.4949) converged in 2 iterations.
230. 1.37 with step 0.0001 (0.4949) converged in 2 iterations.
231. 1.37 with step 0.0001 (0.4949) converged in 2 iterations.
232. 1.37 with step 0.0001 (0.4949) converged in 2 iterations.
233. 1.37 with step 0.0001 (0.4949) converged in 2 iterations.
234. 1.37 with step 0.0001 (0.4949) converged in 2 iterations.
235. 1.37 with step 0.0001 (0.4949) converged in 2 iterations.
236. 1.37 with step 0.0001 (0.4949) converged in 2 iterations.
237. 1.37 with step 0.0001 (0.4949) converged in 2 iterations.
238. 1.37 with step 0.0001 (0.4949) converged in 2 iterations.
239. 1.37 with step 0.0001 (0.4949) converged in 2 iterations.
240. 1.37 with step 0.0001 (0.4949) converged in 2 iterations.
241. 1.37 with step 0.0001 (0.4949) converged in 2 iterations.
242. 1.37 with step 0.0001 (0.4949) converged in 2 iterations.
243. 1.37 with step 0.0001 (0.4949) converged in 2 iterations.
244. 1.37 with step 0.0001 (0.4949) converged in 2 iterations.
245. 1.37 with step 0.0001 (0.4949) converged in 2 iterations.
246. 1.37 with step 0.0001 (0.4949) converged in 2 iterations.
247. 1.37 with step 0.0001 (0.4949) converged in 2 iterations.
248. 1.37 with step 0.0001 (0.4949) converged in 2 iterations.
249. 1.37 with step 0.0001 (0.4949) converged in 2 iterations.
250. 1.37 with step 0.0001 (0.4949) converged in 2 iterations.
251. 1.37 with step 0.0001 (0.4949) converged in 2 iterations.
252. 1.37 with step 0.0001 (0.4949) converged in 2 iterations.
253. 1.37 with step 0.0000 (0.4949) converged in 2 iterations.
254. 1.37 with step 0.0000 (0.4949) converged in 2 iterations.
255. 1.37 with step 0.0000 (0.4949) converged in 2 iterations.
256. 1.37 with step 0.0000 (0.4949) converged in 2 iterations.
257. 1.37 with step 0.0000 (0.4949) converged in 2 iterations.
258. 1.37 with step 0.0000 (0.4949) converged in 2 iterations.
259. 1.37 with step 0.0000 (0.4949) converged in 2 iterations.
260. 1.37 with step 0.0000 (0.4949) converged in 2 iterations.
261. 1.37 with step 0.0000 (0.4949) converged in 2 iterations.
262. 1.37 with step 0.0000 (0.4949) converged in 2 iterations.
263. 1.37 with step 0.0000 (0.4949) converged in 2 iterations.
264. 1.37 with step 0.0000 (0.4949) converged in 2 iterations.
265. 1.37 with step 0.0000 (0.4949) converged in 2 iterations.
266. 1.37 with step 0.0000 (0.4949) converged in 2 iterations.
267. 1.37 with step 0.0000 (0.4949) converged in 2 iterations.
268. 1.37 with step 0.0000 (0.4949) converged in 2 iterations.
269. 1.37 with step 0.0000 (0.4949) converged in 2 iterations.
270. 1.37 with step 0.0000 (0.4949) converged in 2 iterations.
271. 1.37 with step 0.0000 (0.4949) converged in 2 iterations.
272. 1.37 with step -0.0000 (0.4949) converged in 2 iterations.
273. 1.37 with step -0.0000 (0.4949) converged in 2 iterations.
274. 1.37 with step -0.0000 (0.4949) converged in 2 iterations.
275. 1.37 with step -0.0000 (0.4949) converged in 2 iterations.
276. 1.37 with step -0.0000 (0.4949) converged in 2 iterations.
277. 1.37 with step -0.0000 (0.4949) converged in 2 iterations.
278. 1.37 with step -0.0000 (0.4949) converged in 2 iterations.
279. 1.37 with step -0.0000 (0.4949) converged in 2 iterations.
280. 1.37 with step -0.0000 (0.4949) converged in 2 iterations.
281. 1.37 with step -0.0000 (0.4949) converged in 2 iterations.
282. 1.37 with step -0.0000 (0.4949) converged in 2 iterations.
283. 1.37 with step -0.0000 (0.4949) converged in 2 iterations.
284. 1.37 with step -0.0000 (0.4949) converged in 2 iterations.
285. 1.37 with step -0.0000 (0.4949) converged in 2 iterations.
286. 1.37 with step -0.0000 (0.4949) converged in 2 iterations.
287. 1.37 with step -0.0000 (0.4949) converged in 2 iterations.
288. 1.37 with step -0.0000 (0.4949) converged in 2 iterations.
289. 1.37 with step -0.0000 (0.4949) converged in 2 iterations.
290. 1.37 with step -0.0001 (0.4949) converged in 2 iterations.
291. 1.37 with step -0.0001 (0.4949) converged in 2 iterations.
292. 1.37 with step -0.0001 (0.4949) converged in 2 iterations.
293. 1.37 with step -0.0001 (0.4949) converged in 2 iterations.
294. 1.37 with step -0.0001 (0.4949) converged in 2 iterations.
295. 1.37 with step -0.0001 (0.4949) converged in 2 iterations.
296. 1.37 with step -0.0001 (0.4949) converged in 2 iterations.
297. 1.37 with step -0.0001 (0.4949) converged in 2 iterations.
298. 1.37 with step -0.0001 (0.4949) converged in 2 iterations.
299. 1.37 with step -0.0001 (0.4949) converged in 2 iterations.
300. 1.37 with step -0.0001 (0.4949) converged in 2 iterations.
301. 1.37 with step -0.0001 (0.4949) converged in 2 iterations.
302. 1.37 with step -0.0001 (0.4949) converged in 2 iterations.
303. 1.37 with step -0.0001 (0.4949) converged in 2 iterations.
304. 1.37 with step -0.0001 (0.4949) converged in 2 iterations.
305. 1.37 with step -0.0001 (0.4949) converged in 2 iterations.
306. 1.37 with step -0.0001 (0.4949) converged in 2 iterations.
307. 1.37 with step -0.0001 (0.4949) converged in 2 iterations.
308. 1.37 with step -0.0001 (0.4949) converged in 2 iterations.
309. 1.37 with step -0.0001 (0.4949) converged in 2 iterations.
310. 1.37 with step -0.0001 (0.4949) converged in 2 iterations.
311. 1.37 with step -0.0001 (0.4949) converged in 2 iterations.
312. 1.37 with step -0.0001 (0.4949) converged in 2 iterations.
313. 1.37 with step -0.0001 (0.4949) converged in 2 iterations.
314. 1.37 with step -0.0001 (0.4949) converged in 2 iterations.
315. 1.37 with step -0.0001 (0.4949) converged in 2 iterations.
316. 1.37 with step -0.0001 (0.4949) converged in 2 iterations.
317. 1.37 with step -0.0001 (0.4949) converged in 2 iterations.
318. 1.37 with step -0.0001 (0.4949) converged in 2 iterations.
319. 1.36 with step -0.0002 (0.4949) converged in 2 iterations.
320. 1.36 with step -0.0002 (0.4949) converged in 2 iterations.
321. 1.36 with step -0.0002 (0.4949) converged in 2 iterations.
322. 1.36 with step -0.0002 (0.4949) converged in 2 iterations.
323. 1.36 with step -0.0002 (0.4949) converged in 2 iterations.
324. 1.36 with step -0.0002 (0.4949) converged in 2 iterations.
325. 1.36 with step -0.0002 (0.4949) converged in 2 iterations.
326. 1.36 with step -0.0002 (0.4949) converged in 2 iterations.
327. 1.36 with step -0.0002 (0.4949) converged in 2 iterations.
328. 1.36 with step -0.0002 (0.4949) converged in 2 iterations.
329. 1.36 with step -0.0002 (0.4949) converged in 2 iterations.
330. 1.36 with step -0.0002 (0.4949) converged in 2 iterations.
331. 1.36 with step -0.0002 (0.4949) converged in 2 iterations.
332. 1.36 with step -0.0002 (0.4949) converged in 2 iterations.
333. 1.36 with step -0.0002 (0.4949) converged in 2 iterations.
334. 1.36 with step -0.0002 (0.4949) converged in 2 iterations.
335. 1.36 with step -0.0002 (0.4949) converged in 2 iterations.
336. 1.36 with step -0.0002 (0.4949) converged in 2 iterations.
337. 1.36 with step -0.0002 (0.4949) converged in 2 iterations.
338. 1.36 with step -0.0002 (0.4949) converged in 2 iterations.
339. 1.36 with step -0.0002 (0.4949) converged in 2 iterations.
340. 1.36 with step -0.0002 (0.4949) converged in 2 iterations.
341. 1.36 with step -0.0003 (0.4949) converged in 2 iterations.
342. 1.36 with step -0.0003 (0.4949) converged in 2 iterations.
343. 1.36 with step -0.0003 (0.4949) converged in 2 iterations.
344. 1.36 with step -0.0003 (0.4949) converged in 2 iterations.
345. 1.36 with step -0.0003 (0.4949) converged in 2 iterations.
346. 1.36 with step -0.0003 (0.4949) converged in 2 iterations.
347. 1.36 with step -0.0003 (0.4949) converged in 2 iterations.
348. 1.36 with step -0.0003 (0.4949) converged in 2 iterations.
349. 1.36 with step -0.0003 (0.4949) converged in 2 iterations.
350. 1.36 with step -0.0003 (0.4949) converged in 2 iterations.
351. 1.36 with step -0.0003 (0.4949) converged in 2 iterations.
352. 1.36 with step -0.0003 (0.4949) converged in 2 iterations.
353. 1.36 with step -0.0003 (0.4949) converged in 2 iterations.
354. 1.36 with step -0.0003 (0.4949) converged in 2 iterations.
355. 1.36 with step -0.0003 (0.4949) converged in 2 iterations.
356. 1.36 with step -0.0004 (0.4949) converged in 2 iterations.
357. 1.36 with step -0.0004 (0.4949) converged in 2 iterations.
358. 1.36 with step -0.0004 (0.4949) converged in 2 iterations.
359. 1.35 with step -0.0004 (0.4949) converged in 2 iterations.
360. 1.35 with step -0.0004 (0.4949) converged in 2 iterations.
361. 1.35 with step -0.0004 (0.4949) converged in 2 iterations.
362. 1.35 with step -0.0004 (0.4949) converged in 2 iterations.
363. 1.35 with step -0.0004 (0.4949) converged in 2 iterations.
364. 1.35 with step -0.0004 (0.4949) converged in 2 iterations.
365. 1.35 with step -0.0004 (0.4949) converged in 2 iterations.
366. 1.35 with step -0.0004 (0.4949) converged in 2 iterations.
367. 1.35 with step -0.0005 (0.4949) converged in 2 iterations.
368. 1.35 with step -0.0005 (0.4949) converged in 2 iterations.
369. 1.35 with step -0.0005 (0.4949) converged in 2 iterations.
370. 1.35 with step -0.0005 (0.4949) converged in 2 iterations.
371. 1.35 with step -0.0005 (0.4949) converged in 2 iterations.
372. 1.35 with step -0.0005 (0.4949) converged in 2 iterations.
373. 1.35 with step -0.0005 (0.4949) converged in 2 iterations.
374. 1.35 with step -0.0005 (0.4949) converged in 2 iterations.
375. 1.35 with step -0.0005 (0.4949) converged in 2 iterations.
376. 1.35 with step -0.0006 (0.4949) converged in 2 iterations.
377. 1.35 with step -0.0006 (0.4949) converged in 2 iterations.
378. 1.35 with step -0.0006 (0.4949) converged in 2 iterations.
379. 1.34 with step -0.0006 (0.4949) converged in 2 iterations.
380. 1.34 with step -0.0006 (0.4949) converged in 2 iterations.
381. 1.34 with step -0.0006 (0.4949) converged in 2 iterations.
382. 1.34 with step -0.0007 (0.4949) converged in 2 iterations.
383. 1.34 with step -0.0007 (0.4949) converged in 2 iterations.
384. 1.34 with step -0.0007 (0.4949) converged in 2 iterations.
385. 1.34 with step -0.0007 (0.4949) converged in 2 iterations.
386. 1.34 with step -0.0007 (0.4949) converged in 2 iterations.
387. 1.34 with step -0.0007 (0.4949) converged in 2 iterations.
388. 1.34 with step -0.0008 (0.4949) converged in 2 iterations.
389. 1.34 with step -0.0008 (0.4949) converged in 2 iterations.
390. 1.34 with step -0.0008 (0.4949) converged in 2 iterations.
391. 1.34 with step -0.0008 (0.4949) converged in 2 iterations.
392. 1.34 with step -0.0009 (0.4949) converged in 2 iterations.
393. 1.33 with step -0.0009 (0.4949) converged in 2 iterations.
394. 1.33 with step -0.0009 (0.4949) converged in 2 iterations.
395. 1.33 with step -0.0009 (0.4949) converged in 2 iterations.
396. 1.33 with step -0.0010 (0.4949) converged in 2 iterations.
397. 1.33 with step -0.0010 (0.4949) converged in 2 iterations.
398. 1.33 with step -0.0010 (0.4949) converged in 2 iterations.
399. 1.33 with step -0.0011 (0.4949) converged in 2 iterations.
400. 1.33 with step -0.0011 (0.4949) converged in 2 iterations.
401. 1.33 with step -0.0011 (0.4949) converged in 2 iterations.
402. 1.32 with step -0.0012 (0.4949) converged in 2 iterations.
403. 1.32 with step -0.0012 (0.4949) converged in 2 iterations.
404. 1.32 with step -0.0013 (0.4949) converged in 2 iterations.
405. 1.32 with step -0.0013 (0.4949) converged in 2 iterations.
406. 1.32 with step -0.0014 (0.4949) converged in 2 iterations.
407. 1.32 with step -0.0014 (0.4949) converged in 2 iterations.
408. 1.32 with step -0.0015 (0.4949) converged in 2 iterations.
409. 1.31 with step -0.0015 (0.4949) converged in 2 iterations.
410. 1.31 with step -0.0016 (0.4949) converged in 2 iterations.
411. 1.31 with step -0.0016 (0.4949) converged in 2 iterations.
412. 1.31 with step -0.0017 (0.4949) converged in 2 iterations.
413. 1.31 with step -0.0018 (0.4949) converged in 2 iterations.
414. 1.31 with step -0.0019 (0.4949) converged in 2 iterations.
415. 1.30 with step -0.0020 (0.4949) converged in 2 iterations.
416. 1.30 with step -0.0020 (0.4949) converged in 2 iterations.
417. 1.30 with step -0.0021 (0.4949) converged in 2 iterations.
418. 1.30 with step -0.0022 (0.4949) converged in 2 iterations.
419. 1.30 with step -0.0023 (0.4949) converged in 2 iterations.
420. 1.29 with step -0.0025 (0.4949) converged in 2 iterations.
421. 1.29 with step -0.0026 (0.4949) converged in 2 iterations.
422. 1.29 with step -0.0027 (0.4949) converged in 2 iterations.
423. 1.28 with step -0.0029 (0.4949) converged in 2 iterations.
424. 1.28 with step -0.0030 (0.4949) converged in 2 iterations.
425. 1.28 with step -0.0032 (0.4949) converged in 2 iterations.
426. 1.27 with step -0.0034 (0.4949) converged in 2 iterations.
427. 1.27 with step -0.0036 (0.4949) converged in 2 iterations.
428. 1.27 with step -0.0038 (0.4949) converged in 2 iterations.
429. 1.26 with step -0.0040 (0.4949) converged in 2 iterations.
430. 1.26 with step -0.0043 (0.4949) converged in 2 iterations.
431. 1.25 with step -0.0046 (0.4949) converged in 2 iterations.
432. 1.25 with step -0.0049 (0.4949) converged in 2 iterations.
433. 1.24 with step -0.0052 (0.4949) converged in 2 iterations.
434. 1.24 with step -0.0055 (0.4949) converged in 2 iterations.
435. 1.23 with step -0.0058 (0.4949) converged in 2 iterations.
436. 1.23 with step -0.0061 (0.4949) converged in 2 iterations.
437. 1.22 with step -0.0064 (0.4949) converged in 2 iterations.
438. 1.21 with step -0.0065 (0.4949) converged in 2 iterations.
439. 1.21 with step -0.0065 (0.4949) converged in 2 iterations.
440. 1.20 with step -0.0063 (0.4949) converged in 2 iterations.
441. 1.20 with step -0.0055 (0.4949) converged in 2 iterations.
442. 1.19 with step -0.0039 (0.4949) converged in 2 iterations.
443. 1.19 with step -0.0009 (0.4949) converged in 2 iterations.
444. 1.20 with step 0.0046 (0.4949) converged in 2 iterations.
445. 1.23 with step 0.0146 (0.4949) converged in 2 iterations.
446. 1.28 with step 0.0336 (0.4949) converged in 2 iterations.
447. 1.39 with step 0.0740 (0.4949) converged in 3 iterations.
448. 1.52 with step 0.1260 (0.4041) converged in 3 iterations.
449. 1.59 with step 0.0851 (0.3300) converged in 3 iterations.
450. 1.62 with step 0.0438 (0.2694) converged in 2 iterations.
451. 1.65 with step 0.0329 (0.2694) converged in 2 iterations.
452. 1.68 with step 0.0271 (0.2694) converged in 2 iterations.
453. 1.70 with step 0.0242 (0.2694) converged in 2 iterations.
454. 1.73 with step 0.0230 (0.2694) converged in 2 iterations.
455. 1.75 with step 0.0229 (0.2694) converged in 2 iterations.
456. 1.77 with step 0.0236 (0.2694) converged in 2 iterations.
457. 1.80 with step 0.0249 (0.2694) converged in 2 iterations.
458. 1.83 with step 0.0265 (0.2694) converged in 2 iterations.
459. 1.85 with step 0.0282 (0.2694) converged in 2 iterations.
460. 1.89 with step 0.0298 (0.2694) converged in 2 iterations.
461. 1.92 with step 0.0313 (0.2694) converged in 2 iterations.
462. 1.95 with step 0.0323 (0.2694) converged in 2 iterations.
463. 1.98 with step 0.0330 (0.2694) converged in 2 iterations.
464. 2.02 with step 0.0331 (0.2694) converged in 2 iterations.
465. 2.05 with step 0.0328 (0.2694) converged in 2 iterations.
466. 2.08 with step 0.0321 (0.2694) converged in 2 iterations.
467. 2.11 with step 0.0311 (0.2694) converged in 2 iterations.
468. 2.14 with step 0.0298 (0.2694) converged in 2 iterations.
469. 2.17 with step 0.0284 (0.2694) converged in 2 iterations.
470. 2.19 with step 0.0270 (0.2694) converged in 2 iterations.
471. 2.22 with step 0.0255 (0.2694) converged in 2 iterations.
472. 2.24 with step 0.0240 (0.2694) converged in 2 iterations.
473. 2.26 with step 0.0226 (0.2694) converged in 2 iterations.
474. 2.28 with step 0.0212 (0.2694) converged in 2 iterations.
475. 2.30 with step 0.0199 (0.2694) converged in 2 iterations.
476. 2.32 with step 0.0187 (0.2694) converged in 2 iterations.
477. 2.34 with step 0.0175 (0.2694) converged in 2 iterations.
478. 2.35 with step 0.0163 (0.2694) converged in 2 iterations.
479. 2.37 with step 0.0153 (0.2694) converged in 2 iterations.
480. 2.38 with step 0.0142 (0.2694) converged in 2 iterations.
481. 2.40 with step 0.0132 (0.2694) converged in 2 iterations.
482. 2.41 with step 0.0123 (0.2694) converged in 2 iterations.
483. 2.42 with step 0.0114 (0.2694) converged in 2 iterations.
484. 2.43 with step 0.0106 (0.2694) converged in 2 iterations.
485. 2.44 with step 0.0097 (0.2694) converged in 2 iterations.
486. 2.45 with step 0.0089 (0.2694) converged in 2 iterations.
487. 2.45 with step 0.0082 (0.2694) converged in 2 iterations.
488. 2.46 with step 0.0074 (0.2694) converged in 2 iterations.
489. 2.47 with step 0.0067 (0.2694) converged in 2 iterations.
490. 2.47 with step 0.0060 (0.2694) converged in 2 iterations.
491. 2.48 with step 0.0053 (0.2694) converged in 2 iterations.
492. 2.48 with step 0.0046 (0.2694) converged in 2 iterations.
493. 2.49 with step 0.0039 (0.2694) converged in 2 iterations.
494. 2.49 with step 0.0033 (0.2694) converged in 2 iterations.
495. 2.49 with step 0.0026 (0.2694) converged in 2 iterations.
496. 2.49 with step 0.0020 (0.2694) converged in 2 iterations.
497. 2.49 with step 0.0013 (0.2694) converged in 2 iterations.
498. 2.49 with step 0.0007 (0.2694) converged in 2 iterations.
499. 2.49 with step 0.0000 (0.2694) converged in 2 iterations.
500. 2.49 with step -0.0006 (0.2694) converged in 2 iterations.
501. 2.49 with step -0.0012 (0.2694) converged in 2 iterations.
502. 2.49 with step -0.0019 (0.2694) converged in 2 iterations.
503. 2.49 with step -0.0025 (0.2694) converged in 2 iterations.
504. 2.48 with step -0.0032 (0.2694) converged in 2 iterations.
505. 2.48 with step -0.0038 (0.2694) converged in 2 iterations.
506. 2.47 with step -0.0045 (0.2694) converged in 2 iterations.
507. 2.47 with step -0.0052 (0.2694) converged in 2 iterations.
508. 2.46 with step -0.0059 (0.2694) converged in 2 iterations.
509. 2.45 with step -0.0066 (0.2694) converged in 2 iterations.
510. 2.45 with step -0.0073 (0.2694) converged in 2 iterations.
511. 2.44 with step -0.0081 (0.2694) converged in 2 iterations.
512. 2.43 with step -0.0088 (0.2694) converged in 2 iterations.
513. 2.42 with step -0.0096 (0.2694) converged in 2 iterations.
514. 2.41 with step -0.0104 (0.2694) converged in 2 iterations.
515. 2.40 with step -0.0113 (0.2694) converged in 2 iterations.
516. 2.38 with step -0.0122 (0.2694) converged in 2 iterations.
517. 2.37 with step -0.0131 (0.2694) converged in 2 iterations.
518. 2.36 with step -0.0140 (0.2694) converged in 2 iterations.
519. 2.34 with step -0.0150 (0.2694) converged in 2 iterations.
520. 2.32 with step -0.0160 (0.2694) converged in 2 iterations.
521. 2.31 with step -0.0171 (0.2694) converged in 2 iterations.
522. 2.29 with step -0.0182 (0.2694) converged in 2 iterations.
523. 2.27 with step -0.0194 (0.2694) converged in 2 iterations.
524. 2.25 with step -0.0206 (0.2694) converged in 2 iterations.
525. 2.22 with step -0.0218 (0.2694) converged in 2 iterations.
526. 2.20 with step -0.0231 (0.2694) converged in 2 iterations.
527. 2.18 with step -0.0243 (0.2694) converged in 2 iterations.
528. 2.15 with step -0.0255 (0.2694) converged in 2 iterations.
529. 2.12 with step -0.0265 (0.2694) converged in 2 iterations.
530. 2.09 with step -0.0274 (0.2694) converged in 2 iterations.
531. 2.07 with step -0.0281 (0.2694) converged in 2 iterations.
532. 2.04 with step -0.0284 (0.2694) converged in 2 iterations.
533. 2.01 with step -0.0284 (0.2694) converged in 2 iterations.
534. 1.98 with step -0.0278 (0.2694) converged in 2 iterations.
535. 1.96 with step -0.0268 (0.2694) converged in 2 iterations.
536. 1.93 with step -0.0253 (0.2694) converged in 2 iterations.
537. 1.91 with step -0.0233 (0.2694) converged in 2 iterations.
538. 1.89 with step -0.0208 (0.2694) converged in 2 iterations.
539. 1.87 with step -0.0180 (0.2694) converged in 2 iterations.
540. 1.86 with step -0.0150 (0.2694) converged in 2 iterations.
541. 1.85 with step -0.0120 (0.2694) converged in 2 iterations.
542. 1.84 with step -0.0089 (0.2694) converged in 2 iterations.
543. 1.84 with step -0.0059 (0.2694) converged in 2 iterations.
544. 1.84 with step -0.0030 (0.2694) converged in 2 iterations.
545. 1.84 with step 0.0001 (0.2694) converged in 2 iterations.
546. 1.84 with step 0.0034 (0.2694) converged in 2 iterations.
547. 1.85 with step 0.0074 (0.2694) converged in 2 iterations.
548. 1.87 with step 0.0124 (0.2694) converged in 2 iterations.
549. 1.89 with step 0.0195 (0.2694) converged in 2 iterations.
550. 1.93 with step 0.0306 (0.2694) converged in 2 iterations.
551. 2.00 with step 0.0500 (0.2694) converged in 2 iterations.
552. 2.12 with step 0.0881 (0.2694) converged in 3 iterations.
553. 2.28 with step 0.1304 (0.2200) converged in 3 iterations.
554. 2.44 with step 0.1476 (0.1796) converged in 2 iterations.
555. 2.61 with step 0.1654 (0.1796) converged in 2 iterations.
556. 2.78 with step 0.1730 (0.1796) converged in 2 iterations.
557. 2.96 with step 0.1763 (0.1796) converged in 2 iterations.
558. 3.14 with step 0.1778 (0.1796) converged in 2 iterations.

Plotting

Now we plot the response harmonics. It can be seen that two primary and 3 secondary (super harmonic) resonances have been picked up. The nonlinearity level is quite strong, as evidenced by the non-trivial shapes of the response curves.

his = [1, 3, 5];

fsz = 24;
fig = Figure(fontsize=fsz, size=(1000, 600));

axs = [];
for i in eachindex(his[his.<=maximum(h)])
    ax = Axis(fig[1, i],
        ylabel=L"$H_%$(his[i])$ Response (m)", yscale=log10);
    lines!(ax, Oms, abs.(uh[his[i].+1, :, 1]), label="x1");
    lines!(ax, Oms, abs.(uh[his[i].+1, :, 2]), label="x2");
    push!(axs, ax);

    ax = Axis(fig[2, i], xlabel=L"Excitation Frequency $\Omega$",
        ylabel=L"$H_%$(his[i])$ Phase (rad)");
    lines!(ax, Oms, unwrap(angle.(uh[his[i].+1, :, 1])), label="x1");
    lines!(ax, Oms, unwrap(angle.(uh[his[i].+1, :, 2])), label="x2");
    push!(axs, ax)
end

linkxaxes!(axs...);

    fig
Example block output

This page was generated using Literate.jl.