Новое в системе Wolfram Mathematica 9 › Расширенные гибридные и дифференциально-алгебраические уравнения

Семи-шарнирная механическая система

In[1]:=

params0 = {Subscript[x, a] -> -0.06934, Subscript[y, a] -> -0.00227, 
   Subscript[x, b] -> -0.03635, Subscript[y, b] -> 0.03273, 
   Subscript[x, c] -> 0.014, Subscript[y, c] -> 0.072};
params1 = {mom -> 0.033, d -> 0.028, Subscript[e, a] -> 0.01421, 
   Subscript[r, r] -> 0.007, Subscript[s, a] -> 0.01874, 
   Subscript[s, d] -> 0.02, Subscript[t, b] -> 0.00916, 
   Subscript[u, b] -> 0.00449, Subscript[d, a] -> 0.0115, 
   Subscript[z, f] -> 0.02, Subscript[r, a] -> 0.00092, 
   Subscript[s, b] -> 0.01043, Subscript[z, t] -> 0.04, u -> 0.04, 
   Subscript[c, 0] -> 4530, e -> 0.02, Subscript[f, a] -> 0.01421, 
   Subscript[s, s] -> 0.035, Subscript[s, c] -> 0.018, 
   Subscript[t, a] -> 0.02308, Subscript[u, a] -> 0.01228, 
   Subscript[L, 0] -> 0.07785};
params2 = {Subscript[m, 1] -> 0.04325, Subscript[m, 2] -> 0.00365, 
   Subscript[m, 3] -> 0.02373, Subscript[m, 4] -> 0.00706, 
   Subscript[m, 5] -> 0.07050, Subscript[m, 6] -> 0.00706, 
   Subscript[m, 7] -> 0.05498};
params3 = {Subscript[I, 1] -> 2.194 10^-6, 
   Subscript[I, 2] -> 4.410 10^-7, Subscript[I, 3] -> 5.255 10^-6, 
   Subscript[I, 4] -> 5.667 10^-7, Subscript[I, 5] -> 1.169 10^-5, 
   Subscript[I, 6] -> 5.667 10^-7, Subscript[I, 7] -> 1.912 10^-5};
andrewsParams = Join[params0, params1, params2, params3];

In[10]:=

In[11]:=

In[12]:=

Animate[
 p1 = {Subscript[r, r] Cos[Subscript[q, 1][tt]], 
   Subscript[r, r] Sin[Subscript[q, 1][tt]]};
 p2 = {Subscript[r, r] Cos[Subscript[q, 1][tt]] - 
    d Cos[Subscript[q, 1][tt] + Subscript[q, 2][tt]], 
   Subscript[r, r] Sin[Subscript[q, 1][tt]] - 
    d Sin[Subscript[q, 1][tt] + Subscript[q, 2][tt]]};
 p3 = {Subscript[x, b] + Subscript[s, c] Sin[Subscript[q, 3][tt]] + 
    Subscript[s, d] Cos[Subscript[q, 3][tt]], 
   Subscript[y, b] - Subscript[s, c] Cos[Subscript[q, 3][tt]] + 
    Subscript[s, d] Sin[Subscript[q, 3][tt]]};
 p3c = {Subscript[x, b] + Subscript[s, a] Sin[Subscript[q, 3][tt]] + 
    Subscript[s, b] Cos[Subscript[q, 3][tt]], 
   Subscript[y, b] - Subscript[s, a] Cos[Subscript[q, 3][tt]] + 
    Subscript[s, b] Sin[Subscript[q, 3][tt]]};
 p4 = {Subscript[x, a] + Subscript[z, t] Cos[Subscript[q, 5][tt]], 
   Subscript[y, a] + Subscript[z, t] Sin[Subscript[q, 5][tt]]};
 p5 = {Subscript[x, a] + Subscript[t, a] Cos[Subscript[q, 5][tt]] - 
    Subscript[t, b] Sin[Subscript[q, 5][tt]], 
   Subscript[y, a] + Subscript[t, a] Sin[Subscript[q, 5][tt]] + 
    Subscript[t, b] Cos[Subscript[q, 5][tt]]};
 p6 = {Subscript[x, a] + u Sin[Subscript[q, 7][tt]], 
   Subscript[y, a] - u Cos[Subscript[q, 7][tt]]};
 p7 = {Subscript[x, a] + Subscript[u, a] Sin[Subscript[q, 7][tt]] - 
    Subscript[u, b] Cos[Subscript[q, 7][tt]], 
   Subscript[y, a] - Subscript[u, a] Cos[Subscript[q, 7][tt]] - 
    Subscript[u, b] Sin[Subscript[q, 7][tt]]};
 
 link1 = {Thickness[0.05], Black, CapForm["Round"], 
   Line[{{0, 0}, p1}]} ;
 link2 = {Thickness[0.05], Green, EdgeForm[Black], CapForm["Round"], 
   Line[{p1, p2}]} ;
 link3 = {Thickness[0.05], Blue, CapForm["Round"], Line[{p2, p3c}], 
   Line[{{Subscript[x, b], Subscript[y, b]}, p3c}], Line[{p3c, p3}]} ;
 link4 = {Thickness[0.05], Brown, CapForm["Round"], Line[{p2, p4}]};
 link5 = {Thickness[0.05], LightRed, CapForm["Round"], Line[{p2, p6}]};
 link6 = {Thickness[0.05], Magenta, CapForm["Round"], 
   Line[{{Subscript[x, a], Subscript[y, a]}, p5}], Line[{p5, p4}]} ;
 link7 = {Thickness[0.05], Orange, CapForm["Round"], 
   Line[{{Subscript[x, a], Subscript[y, a]}, p7}], Line[{p7, p6}]} ;
 spring = {Thickness[0.02], Purple, CapForm["Round"], 
   Line[{p3, {Subscript[x, c], Subscript[y, c]}}]};
 
 disk0 = {White, EdgeForm[{Black, Thick}], Disk[{0, 0}, 0.0017]};
 disk1 = {White, EdgeForm[{Black, Thick}], Disk[p1, 0.0017]};
 disk2 = {White, EdgeForm[{Black, Thick}], 
   Disk[{Subscript[x, b], Subscript[y, b]}, 0.0017]} ;
 disk3 = {White, EdgeForm[{Black, Thick}], Disk[p2, 0.0017]} ;
 disk4 = {White, EdgeForm[{Black, Thick}], Disk[p4, 0.0017]};
 disk5 = {White, EdgeForm[{Black, Thick}], Disk[p3, 0.001]};
 disk6 = {White, EdgeForm[{Black, Thick}], Disk[p6, 0.0017]};
 disk7 = {White, EdgeForm[{Black, Thick}], 
   Disk[{Subscript[x, a], Subscript[y, a]}, 0.0017]};
 
 Graphics[{link1, disk0, link2, link3, link4, link5, link6, link7, 
     spring, disk1, disk2, disk3, disk4, disk6, disk7, disk5} /. 
    andrewsParams /. solAndrews,
  PlotRange -> {{-0.09, 0.02}, {-0.03, 0.08}}, Frame -> True, 
  ImageSize -> 400]
 , {{tt, 0.008, "time"}, 0.008, 0.0276}, AnimationRunning -> False, 
 SaveDefinitions -> True]

Моделирование отжимающего механизма Эндрю.

X
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Система состоит из семи шарниров различной формы, один из которых прикреплён к пружине. Шарнир наименьшей длины приводится в движение мотором, который в свою очередь, приводит в движение остальные шарниры.

Размеры различных шарниров указаны на следующих рисунках. Семь шарниров обозначены как -. Обозначения параметров, используемых в модели и их описание можно найти в следующей статье.

E. Hairer and G. Wanner, Solving Ordinary Differential Equations II, Stiff and Differential-Algebraic Problems, 2nd ed., Springer Series in Computational Mathematics, Vol. 14, New York: Springer, 1996.

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Движение системы можно задать с помощью семи углов, обозначаемые здесь как -. Уравнения движения выводятся при помощи Гамильтоновых принципов, что даёт систему уравнений формы , где - это матрица массы, - это вектор сил, - это вектор связей, и - это множитель Лагранжа.

In[2]:=

Зададим матрицу массы , определяющую систему.

In[3]:=

M = SparseArray[{
    {1, 1} -> 
     Subscript[m, 1] Subscript[r, a]^2 + 
      Subscript[m, 
       2] (Subscript[r, r]^2 - 
         2 Subscript[d, a] Subscript[r, r] Cos[Subscript[q, 2][t]] + 
         Subscript[d, a]^2) + Subscript[I, 1] + Subscript[I, 2],
    {1, 2} -> 
     Subscript[m, 
       2] (Subscript[d, a]^2 - 
         Subscript[d, a] Subscript[r, r] Cos[Subscript[q, 2][t]]) + 
      Subscript[I, 2],
    {2, 1} -> 
     Subscript[m, 
       2] (Subscript[d, a]^2 - 
         Subscript[d, a] Subscript[r, r] Cos[Subscript[q, 2][t]]) + 
      Subscript[I, 2],
    {2, 2} -> 
     Subscript[m, 2] Subscript[d, a]^2 + Subscript[I, 2], {3, 3} -> 
     Subscript[m, 3] (Subscript[s, a]^2 + Subscript[s, b]^2) + 
      Subscript[I, 3], {4, 4} -> 
     Subscript[m, 4] (e - Subscript[e, a])^2 + Subscript[I, 4],
    {5, 4} -> 
     Subscript[m, 
       4] ((e - Subscript[e, a])^2 + 
         Subscript[z, 
          t] (e - Subscript[e, a]) Sin[Subscript[q, 4][t]]) + 
      Subscript[I, 4],
    {4, 5} -> 
     Subscript[m, 
       4] ((e - Subscript[e, a])^2 + 
         Subscript[z, 
          t] (e - Subscript[e, a]) Sin[Subscript[q, 4][t]]) + 
      Subscript[I, 4],
    {5, 5} -> 
     Subscript[m, 
       4] (Subscript[z, t]^2 + 
         2 Subscript[z, 
          t] (e - Subscript[e, a]) Sin[Subscript[q, 4][t]] + (e - 
           Subscript[e, a])^2) + 
      Subscript[m, 5] (Subscript[t, a]^2 + Subscript[t, b]^2) + 
      Subscript[I, 4] + Subscript[I, 5],
    {6, 6} -> 
     Subscript[m, 6] (Subscript[z, f] - Subscript[f, a])^2 + 
      Subscript[I, 6], {6, 7} -> 
     Subscript[m, 
       6] ((Subscript[z, f] - Subscript[f, a])^2 - 
         u (Subscript[z, f] - Subscript[f, a]) Sin[
           Subscript[q, 6][t]]) + Subscript[I, 6],
    {6, 7} -> 
     Subscript[m, 
       6] ((Subscript[z, f] - Subscript[f, a])^2 - 
         u (Subscript[z, f] - Subscript[f, a]) Sin[
           Subscript[q, 6][t]]) + Subscript[I, 6],
    {7, 6} -> 
     Subscript[m, 
       6] ((Subscript[z, f] - Subscript[f, a])^2 - 
         u (Subscript[z, f] - Subscript[f, a]) Sin[
           Subscript[q, 6][t]]) + Subscript[I, 6],
    {7, 7} -> 
     Subscript[m, 
       6] ((Subscript[z, f] - Subscript[f, a])^2 - 
         2 u (Subscript[z, f] - Subscript[f, a]) Sin[
           Subscript[q, 6][t]] + u^2) + 
      Subscript[m, 7] (Subscript[u, a]^2 + Subscript[u, b]^2) + 
      Subscript[I, 6] + Subscript[I, 7]}
   ];

Зададим различные силы , действующие на систему.

In[4]:=

In[5]:=

F = ({
     {mom - 
       Subscript[m, 2] Subscript[d, a] Subscript[r, r]
         Subscript[q, 2]'[
         t] (Subscript[q, 2]'[t] + 2 Subscript[q, 1]'[t]) Sin[
         Subscript[q, 2][t]]},
     {Subscript[m, 2] Subscript[d, a] Subscript[r, r]
        Subscript[q, 1]'[t]^2 Sin[Subscript[q, 2][t]]},
     {Subscript[f, 
        x] (Subscript[s, c] Cos[Subscript[q, 3][t]] - 
          Subscript[s, d] Sin[Subscript[q, 3][t]]) + 
       Subscript[f, 
        y] (Subscript[s, d] Cos[Subscript[q, 3][t]] + 
          Subscript[s, c] Sin[Subscript[q, 3][t]])},
     {Subscript[m, 4] Subscript[z, 
       t] (e - Subscript[e, a]) Subscript[q, 5]'[t]^2 Cos[
        Subscript[q, 4][t]]},
     {-Subscript[m, 4] Subscript[z, 
       t] (e - Subscript[e, a]) Subscript[q, 4]'[
        t] (Subscript[q, 4]'[t] + 2 Subscript[q, 5]'[t]) Cos[
        Subscript[q, 4][t]]},
     {-Subscript[m, 6]
         u (Subscript[z, f] - Subscript[f, a]) Subscript[q, 7]'[
        t]^2 Cos[Subscript[q, 6][t]]},
     {Subscript[m, 6]
        u (Subscript[z, f] - Subscript[f, a]) Subscript[q, 6]'[
        t] (Subscript[q, 6]'[t] + 2 Subscript[q, 7]'[t]) Cos[
        Subscript[q, 7][t]]}
    })  // Flatten;

Зададим вектор связей , связанный с различными шарнирами.

In[6]:=

g = ({
     {Subscript[r, r] Cos[Subscript[q, 1][t]] - 
       d Cos[Subscript[q, 1][t] + Subscript[q, 2][t]] - 
       Subscript[s, s] Sin[Subscript[q, 3][t]] - Subscript[x, b]},
     {Subscript[r, r] Sin[Subscript[q, 1][t]] - 
       d Sin[Subscript[q, 1][t] + Subscript[q, 2][t]] + 
       Subscript[s, s] Cos[Subscript[q, 3][t]] - Subscript[y, b]},
     {Subscript[r, r] Cos[Subscript[q, 1][t]] - 
       d Cos[Subscript[q, 1][t] + Subscript[q, 2][t]] - 
       e Sin[Subscript[q, 4][t] + Subscript[q, 5][t]] - 
       Subscript[z, t] Cos[Subscript[q, 5][t]] - Subscript[x, a]},
     {Subscript[r, r] Sin[Subscript[q, 1][t]] - 
       d Sin[Subscript[q, 1][t] + Subscript[q, 2][t]] + 
       e Cos[Subscript[q, 4][t] + Subscript[q, 5][t]] - 
       Subscript[z, t] Sin[Subscript[q, 5][t]] - Subscript[y, a]},
     {Subscript[r, r] Cos[Subscript[q, 1][t]] - 
       d Cos[Subscript[q, 1][t] + Subscript[q, 2][t]] - 
       Subscript[z, f] Cos[Subscript[q, 6][t] + Subscript[q, 7][t]] - 
       u Sin[Subscript[q, 7][t]] - Subscript[x, a]},
     {Subscript[r, r] Sin[Subscript[q, 1][t]] - 
       d Sin[Subscript[q, 1][t] + Subscript[q, 2][t]] - 
       Subscript[z, f] Sin[Subscript[q, 6][t] + Subscript[q, 7][t]] + 
       u Cos[Subscript[q, 7][t]] - Subscript[y, a]}
    }) // Flatten;

Построим уравнения и уравнения связей для определённых значений параметров.

In[7]:=

Зададим начальные условия системы, где система пребывает в покое и имеет определённые значения углов.

In[8]:=

Находим решение системы и строим график углов и сил. Движение шарниров становятся более быстрыми с течением времени. Цвета кривых соответствуют цветам шарниров.

In[9]:=

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