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  1. ClearAll["Global`*"];
  2. W[1] = c[1]*Sin[b*x1] + c[2]*Cos[b*x1] + c[3]*Sinh[b*x1] +
  3.    c[4]*Cosh[b*x1];
  4. W[2] = c[5]*Sin[b*(x1 - z[1])] + c[6]*Cos[b*(x1 - z[1])] +
  5.    c[7]*Sinh[b*(x1 - z[1])] + c[8]*Cosh[b*(x1 - z[1])];
  6. W[3] = c[9]*Sin[b*(x1 - z[2])] + c[10]*Cos[b*(x1 - z[2])] +
  7.    c[11]*Sinh[b*(x1 - z[2])] + c[12]*Cosh[b*(x1 - z[2])];
  8. w = Piecewise[{{W[1], x1 <= z[1]}, {W[2], z[1] <= x1 <= z[2]}, {W[3],
  9.      x1 >= z[2]}}];
  10.  
  11. boundary[i_, j_] :=
  12.  Module[ {bc},
  13.   bc1 = {W[i] /. {x1 -> 0}, (D[W[i], {x1, 2}]) /. {x1 -> 0},
  14.     W[j] /. {x1 -> L1}, ((D[W[j], {x1, 2}]) /. {x1 -> L1})}; bc = bc1]
  15.  
  16. countinuity[i_, j_] :=
  17.  Module[{eq},
  18.   eq1 = {((W[i] /. x1 -> z[i]) - (W[j] /.
  19.         x1 -> z[i])), (((D[W[i], {x1}]) /.
  20.         x1 -> z[i]) - ((D[W[j], {x1}]) /.
  21.         x1 -> z[i])), (((D[W[i], {x1, 2}]) /.
  22.         x1 -> z[i]) - ((D[W[j], {x1, 2}]) /.
  23.         x1 -> z[i])), (((D[W[i], {x1, 3}]) /.
  24.          x1 -> z[i]) - ((D[W[j], {x1, 3}]) /. x1 -> z[i])) + (K[i]*
  25.         W[i] /. x1 -> z[i])}; eq = eq1 ]
  26.  
  27. e1 = boundary[1, 3];
  28. e2 = countinuity[1, 2];
  29. e3 = countinuity[2, 3];
  30. comb = Tuples[{0, 1*^12}, 3];
  31. eq = Flatten[{e1, e2, e3}];
  32. var = Table[c[i], {i, 1, Length[eq]}];
  33. R = Normal@CoefficientArrays[eq, var][[2]];
  34. R = R /. {K[1] -> K1, K[2] -> K2};
  35. MatrixForm[R];
  36. P = -b^10 (Sinh[
  37.        b L1] (16 b^3 ((K1 + K2) Cos[b L1] - K1 Cos[b (L1 - 2 z[1])] -
  38.            K2 Cos[b (L1 - 2 z[2])]) + 2 (32 b^6 - K1 K2) Sin[b L1] +
  39.         K1 K2 (4 Sin[b (L1 - 2 z[1])] -
  40.            2 I Sin[b (L1 - 2 (z[1] + I z[2]))] -
  41.            4 Sin[b (L1 - 2 z[2])] +
  42.            4 Sin[b (L1 + 2 z[1] - 2 z[2])] + (1 + 2 I) Sin[
  43.              b (L1 - 2 I z[2])] + (1 - 2 I) Sin[b (L1 + 2 I z[2])] +
  44.            2 I Sin[b (L1 - 2 z[1] + 2 I z[2])] +
  45.            4 Cosh[b z[
  46.                2]] ((1 - 2 Cosh[2 b z[1]]) Cosh[b z[2]] Sin[b L1] +
  47.               8 Sin[b z[1]] Sin[b (L1 - z[2])] Sinh[b z[1]])) +
  48.         4 K1 (K2 Cos[b L1] - K2 Cos[b (L1 - 2 z[2])] +
  49.            4 b^3 Sin[b L1]) Sinh[2 b z[1]] +
  50.         4 K2 Sin[b L1] (4 b^3 + K1 Sinh[2 b z[1]]) Sinh[2 b z[2]]) -
  51.      Cosh[
  52.        b L1] (-K1 K2 (10 Cos[b L1] - 6 Cos[b (L1 - 2 z[1])] -
  53.            4 Cos[b (L1 - 2 z[2])] - (1 - 2 I) Cos[b (L1 - 2 I z[2])] +
  54.             Cos[b (L1 - 2 z[1] - 2 I z[2])] - (1 + 2 I) Cos[
  55.              b (L1 + 2 I z[2])] + Cos[b (L1 - 2 z[1] + 2 I z[2])]) +
  56.         4 K1 Cosh[b z[1]]^2 (K2 Cos[b L1] - K2 Cos[b (L1 - 2 z[2])] +
  57.            4 b^3 Sin[b L1]) +
  58.         2 (2 b^3 (-2 (2 K1 + 3 K2) Sin[b L1] +
  59.               K2 (Sin[b (L1 - 2 I z[2])] + Sin[b (L1 + 2 I z[2])])) +
  60.            2 K2 Cosh[b z[2]]^2 (K1 Cos[b L1] -
  61.               K1 Cos[b (L1 - 2 z[1])] + 4 b^3 Sin[b L1] +
  62.               K1 Sin[b L1] Sinh[2 b z[1]]) +
  63.            K1 (2 (K2 Cos[b L1] - K2 Cos[b (L1 - 2 z[2])] +
  64.                  4 b^3 Sin[b L1]) Sinh[b z[1]]^2 +
  65.               16 K2 Sin[b z[1]] Sin[b (L1 - z[2])] Sinh[b z[1]] Sinh[
  66.                 b z[2]] +
  67.               K2 Sin[b L1] ((-3 + Cosh[2 b z[2]]) Sinh[2 b z[1]] -
  68.                  2 Cosh[2 b z[1]] Sinh[2 b z[2]])))));
  69.  
  70. f[z1_, z2_, l1_, k1_, k2_, beta_] :=
  71.   Module[{m}, z[1] = z1; z[2] = z2 ; L1 = l1; K1 = k1; K2 = k2;
  72.    r = beta; s1 = P; s2 = NSolve[s1 == 0 && 0 < b < 30];
  73.    s3 = N[b /. s2];
  74.    s4 = s3[[r]]; {uu, ww, vv} =
  75.     SingularValueDecomposition[R /. b -> s4];
  76.    NN = Last[Transpose[vv]];   sub1 = Flatten[{var, b}];
  77.    sub2 = Flatten[{NN, s4}];    
  78.    m = w /. Table[sub1[[i]] -> sub2[[i]], {i, 1, Length[sub1]}];
  79.    Return[m]];
  80. n = 2;
  81. comb = Tuples[{0, 1*^12}, 2];
  82. g[i_, r_] :=
  83.  Module[{s5}, spring = comb[[i]]; n1 = spring[[1]];
  84.   n2 = spring[[2]];  n3 = r; s5 = f[0.25, 0.75, 1, n1, n2, n3]]
  85. beammodes1 = Table[g[i, 1], {i, 1, 2^n}]
  86. beammodes2 = Table[g[i, 2], {i, 1, 2^n}]
  87. beammodes = Flatten[{beammodes1, beammodes2}];
  88. Table[Plot[beammodes[[i]], {x1, 0, L1}, PlotRange -> All], {i, 1,
  89.   Length[beammodes]}]
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