Условия и точки стабильности порядка 1/2 механической системы выполнил: Чан Туан Чунг,
Московский энергетический институт(ТУ), 2 курс
2004 |
Архив разработки (115 Кб, Maple)
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Решение задачи с одной обобщённой координатой
Данные значения
> |
MDz:=Mo-k*omega3:
Mnz:=-mu*omega4:
Fnx:=-nu*v5x: |
Начальные условия
> |
Yslo1:=m2=16,m3=34,m4=26,nu=8000,mu=13,k=15,Mo=10,R1=0.38,I1=9,r1=0.27,R3=0.28,
R4=0.2,r4=0.12,i4=0.15:
Yslo2:=m2=16,m3=34,m4=26,nu=8000,mu=13,k=k,Mo=10,R1=0.38,I1=9,r1=0.27,R3=0.28,
R4=0.2,r4=0.12,i4=0.15:
phi1o:=1.3:omega1o:=0.5:
NaYs:=phi1(0)=phi1o,D(phi1)(0)=omega1o: |
Имеем
> |
omega1:=diff(phi1(t),t):
v2x:=-r1*omega1*sin(phi1(t)):
omega3:=omega1*R1/R3:
omega4:=v2x/(2*R4):
v4x:=omega4*R4:
v5x:=omega4*(R4-r4): |
Кинетическая энергия системы
> |
T1:=0.5*I1*omega1^2:
T2:=0.5*m2*v2x^2:
T3:=0.25*(m3*R3^2)*omega3^2:
T4:=0.5*m4*v4x^2+0.5*m4*i4^2*omega4^2:
T:=T1+T2+T3+T4; |


Обобщенные силы
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Q:=(MDz*omega3+Mnz*omega4+Fnx*v5x)/omega1; |


Уравнение Лагранжа 2-го рода
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eps:= (Q-0.5*B*omega1^2*sin(2*phi1(t)))/(A+B*sin(phi1(t))^2); |



Коэффициенты :
> |
A:=evalf(coeff(algsubs(omega1^2*sin(phi1(t))^2=x,T*2),omega1^2));
B:= evalf(coeff(algsubs(omega1^2*sin(phi1(t))^2=x,T*2),x));
Qo := evalf(subs(omega1=omega1o,phi1(t)=phi1o,Q));
Epso := evalf(subs(omega1=omega1o,phi1(t)=phi1o,eps)); |






Дифференциальное уравнение
> |
Urav:=diff(omega1,t) - eps; |




Коэффициент а1 вариации omega1
> |
a1:= factor(coeff(D(Urav),D(omega1))); |





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Найдём коэффициенты и нарисуем графики зависимости угла и коэффициента a1(t) от времени
Коэффициенты
> |
A:=subs(Yslo1,A);
B:=subs(Yslo1,B);
Qo:=subs(Yslo1,Qo);
Epso:=subs(Yslo1,Epso); |




Нарисуем графики зависимости угла и коэффициента a1(t) от времени
> |
Urav1:=subs(Yslo1,Urav);
F1:= dsolve({Urav1,NaYs},phi1(t),type = numeric);
a11:=subs(Yslo1,a1);
plots[odeplot](F1,[[t,phi1(t)],[t,a11]],t=0..2*Pi,title="График",legend=["Phi(t)","a1(t)"],thickness=2); |




![[Maple Plot]](images/%C3%8A%C3%B3%C3%B0%C3%B1%C3%AE%C3%A2%C3%AE%C3%A9%20%C3%AF%C3%B0%C3%AE%C3%A5%C3%AA%C3%B231.gif)
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Найдём условие нестабильности при значении t
> |
Urav2:=subs(Yslo2,Urav); |
> |
F2:=dsolve({Urav2,NaYs},phi1(t),type = numeric,output=listprocedure); |
> |
phi1:=subs(F2,phi1(t)): phi1tk:=phi1(tk); |
> |
Dphi1:=subs(F2,diff(phi1(t),t)): Dphi1tk:=Dphi1(tk); |
> |
k:=evalf(subs(diff(phi1(t),t)=Dphi1tk,phi1(t)=phi1tk,k)); |
> |
atk=evalf(subs(diff(phi1(t),t)=Dphi1tk,phi1(t)=phi1tk,ak)); |
> |
plots[odeplot](F2,[t,ak],0..2*Pi,title="коэффициента a(t), при касании cо осью при t",color=blue,thickness=2,legend="Коэффициент a1"); |
> |
plots[odeplot](F2,[t,ak],5.5..2*Pi,title="коэффициента a(t), при касании cо осью при t",color=blue,thickness=2,legend="Коэффициент a1"); |












![[Maple Plot]](images/%C3%8A%C3%B3%C3%B0%C3%B1%C3%AE%C3%A2%C3%AE%C3%A9%20%C3%AF%C3%B0%C3%AE%C3%A5%C3%AA%C3%B244.gif)
![[Maple Plot]](images/%C3%8A%C3%B3%C3%B0%C3%B1%C3%AE%C3%A2%C3%AE%C3%A9%20%C3%AF%C3%B0%C3%AE%C3%A5%C3%AA%C3%B245.gif)
Выыод : При t=5.7[c] система не стабильна, если k=2.4623
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Решение задачи с двумя обобщёнными координатами
Данные значения
> |
MDz:=Mo-k*omega3:
Mnz:=-mu*omega4:
Fnx:=-nu*v5x: |
Начальные условия
> |
Yslo2:=m2=16,m3=34,m4=26,nu=8000,mu=13,k=15,Mo=10,R1=0.38,I1=9,r1=0.27,R3=0.28,R4=0.2,
r4=0.12,i4=0.15,c=100:
phi1o:=1.3:omega1o:=0.5:
phi3o:=1.3:omega3o:=0.5:
NaYs:=phi1(0)=phi1o,D(phi1)(0)=omega1o,phi3(0)=phi3o,D(phi3)(0)=omega3o: |
Имеем
> |
omega1:=diff(phi1(t),t):
v2x:=-r1*omega1*sin(phi1(t)):
omega3:=diff(phi3(t),t):
omega4:=v2x/(2*R4):
v4x:=omega4*R4:
v5x:=omega4*(R4-r4): |
Кинетическая энергия системы
> |
T1:=0.5*I1*omega1^2:
T2:=0.5*m2*v2x^2:
T3:=0.25*(m3*R3^2)*omega3^2:
T4:=0.5*m4*v4x^2+0.5*m4*i4^2*omega4^2:
T:=T1+T2+T3+T4: |
Потенциальная энергия системы:
> |
П := 0.5*c*(R1*phi1(t)-R3*phi3(t))^2: |
Энергия системы



Система уравнений
> |
eq1:= diff(subs(w1=omega1,diff(subs(omega1=w1,L),w1)),t) - subs(f1=phi1(t),diff(subs(phi1(t)=f1,L),f1));
eq2:= diff(subs(w3=omega3,diff(subs(omega3=w3,L),w3)),t) - subs(f3=phi3(t),diff(subs(phi3(t)=f3,L),f3)); |





Вариация:
> |
eq11:= subs(sin(phi1(t))=sin@phi1(t),cos(phi1(t))=cos@phi1(t),eq1):
eq21:= subs(sin(phi1(t))=sin@phi1(t),cos(phi1(t))=cos@phi1(t),eq2):
eq11a:= D(eq11):
eq21a:= D(eq21):
eq11:=subs(sin@phi1(t)=sin(phi1(t)),(-sin)@phi1(t)=-sin(phi1(t)),cos@phi1(t)=cos(phi1(t)),eq11a):
eq21:=subs(sin@phi1(t)=sin(phi1(t)),(-sin)@phi1(t)=-sin(phi1(t)),cos@phi1(t)=cos(phi1(t)),eq21a): |
Найдём точки нестабильности
> |
a1:= coeff(eq11,D(phi3(t)));
a2:= coeff(eq11,D(phi1(t)));
a3:= coeff(eq21,D(phi1(t)));
a4:= coeff(eq21,D(phi3(t))); |








Oпределитель






Численно
> |
eq1:=subs(Yslo2,eq1);
eq2:=subs(Yslo2,eq2);
det:=subs(Yslo2,det); |





Pешение системы дифференциальных уравнений:
> |
F:=dsolve({eq1,eq2,NaYs},{phi1(t),phi3(t)},type=numeric,output=listprocedure): |
> |
plots[odeplot](F,[[t,phi1(t)],[t,phi3(t)],[t,det]],0..2*Pi,title="График",legend=["Phi1(t)","phi3(t)","Det12(t)"],
thickness=2); |
![[Maple Plot]](images/%C3%8A%C3%B3%C3%B0%C3%B1%C3%AE%C3%A2%C3%AE%C3%A9%20%C3%AF%C3%B0%C3%AE%C3%A5%C3%AA%C3%B273.gif)
Вывод : На промежутке времени [0,2Pi] система имеет 2 точки нестабильности порядка [1/2]
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Моделирование механизма в движении
> |
restart;
pi:= evalf(Pi):
with(plots): with(plottools): |
Warning, the name changecoords has been redefined Warning, the name arrow has been redefined
Процедура изображения окружности с центром в точке i
> |
Circle:= proc(i,R) PLOT(circle([x[i],y[i]],R,color=blue)): end: |
Процедура изображения точек на вращающемся колесе радиуса R c центром в точке i
> |
Rot:= proc(i,R,f) local j:
PLOT(seq(POINTS([x[i]+R*cos(f+pi/3*j),y[i]+R*sin(f+pi/3*j)]),j=0..5)):
end: |
Процедура изображения линии от точки i к j
> |
Line:= proc(i,j,c) PLOT(CURVES([[x[i],y[i]],[x[j],y[j]]]),COLOR(HUE,c/10.)): end: |
Процедура изображения неподвижной опоры в точке i
> |
Rest:= proc(i,R) local x0,x1,y0,h,N:
x0:=x[i]-R*0.8: x1:= x[i]+R*0.8:
y0:=y[i]-R*0.8: h:= 3*R: N:= 2:
display(PLOT(circle([x[i],y[i]],R,color=blue)),PLOT(CURVES([[x0,y0],[x[i]-h,y[i]-h]],[[x1,y0],
[x[i]+h,y[i]-h]],[[x0-h,y[i]-h],[x1+h,y[i]-h]],seq([[x0-h*(-j/N+1),y[i]-h],[x0-h*(-j/N+1)-h/4,y[i]-h*1.3]],j=0..2*N+1)))):
end: |
Процедура изображения горизонтальной опорной поверхности (вверху h<0 , внизу h>0)
> |
Surf:= proc(x,y,L,h) local N:
N:= round(abs(L/h))-2: display(PLOT(CURVES([[x,y],[x+L,y]],seq([[x+L*j/N/2,y],[x+L*j/N/2-h,y-h]],j=0..2*N)))):
end: |
Процедура изображения прямоугольника a x b c центром в точке i
> |
Box:= proc(i,A,B) PLOT(POLYGONS([[x[i]-A,y[i]+B],[x[i]+A,y[i]+B],[x[i]+A,y[i]-B],[x[i]-A,y[i]-B]],COLOR(HUE,0.2))): end: |
Начальные значения
> |
nam:=[O3,O1,A,E]; Nnam:= [1,6,7,10];
R1:= 38: r1:=27: R3:= 28: R4:= 20: r4:=12: l:=50: |
![nam := [O3, O1, A, E]](images/%C3%8A%C3%B3%C3%B0%C3%B1%C3%AE%C3%A2%C3%AE%C3%A9%20%C3%AF%C3%B0%C3%AE%C3%A5%C3%AA%C3%B274.gif)
![Nnam := [1, 6, 7, 10]](images/%C3%8A%C3%B3%C3%B0%C3%B1%C3%AE%C3%A2%C3%AE%C3%A9%20%C3%AF%C3%B0%C3%AE%C3%A5%C3%AA%C3%B275.gif)
> |
phio:= 1.3: omegao:= 0.5: |
Координаты основных точек
> |
x[1]:= 0: y[1]:= 0:
x[2]:= 0: y[2]:= R3:
x[3]:= -R1-R3-l: y[3]:= R1:
x[4]:= 0: y[4]:= -R3:
x[5]:= x[3]: y[5]:= -R1:
x[6]:= x[5]: y[6]:=0:
y[8]:= 0:
y[9]:=0:
y[10]:= y[9]+R4:
y[11]:= y[9]+R4+r4:
y[12]:= y[11]: |
Количество кадров
Создаем все кадры
> |
for i from 0 to N do
t:= 2*pi*i/N;
phi:= phio + t:
x[7]:= x[6] + r1*cos(phi): y[7]:= y[6] + r1*sin(phi):
x[8]:= x[7]:
x[9]:= x[8] - 5*l:
x[10]:= x[9]/2-3*l:
x[11]:= x[10]*(R4-r4)/R4-6*l:
x[12]:= x[11] + 5*l:
F:= arrow([x[11]-10,y[11]],evalm([30,0]),2,6,0.3):
M:=arc([x[10],y[10]],R4-5,Pi/2..Pi,color=red):
Mm:=arrow([x[10],y[10]+R4-5],evalm([7,-1]),1,5,0.5,color=red):
M1:=arc([x[1],y[1]],R4-5,Pi/2..Pi,color=red):
M1m:=arrow([x[1]-R4+5,y[1]],evalm([-1,-7]),1,5,0.5,color=red):
P[i]:= display(F,Box(8,2,r1),Circle(1,R3), Rot(1,R3,phi*R1/R3),
Circle(6,R1), Rot(6,R1,phi),
Circle(10,R4),Rot(10,R4,x[10]/R4),
Circle(10,r4),Rot(10,r4,x[10]/R4),
Circle(10,1),
Mm,M,M1m,M1,
seq(Line(2*k,2*k+1,1),k=1..2),
Line(6,7,0),
Line(8,9,0),
Line(11,12,0),
seq(TEXT([x[Nnam[j]]+8,y[Nnam[j]]],nam[j]),j=1..4));
od: |
Изображение механизма в движении
> |
PP:=display(seq(P[i],i=0..N),insequence=true,thickness=2,axes=none): |
> |
display( PP,Rest(1,2),Rest(6,2),
Surf(-8.4*l,y[10]+R4,4*l,-6),
Surf(-3.5*l,y[8]+2,10,-2),
Surf(-3.5*l,y[8]-2,10,2),
Surf(-5*l,y[11]-2,10,2),
Surf(-5*l,y[11]+2,10,-2),
Surf(-7.8*l,y[11]-2,10,2),
Surf(-7.8*l,y[11]+2,10,-2)); |
![[Maple Plot]](images/%C3%8A%C3%B3%C3%B0%C3%B1%C3%AE%C3%A2%C3%AE%C3%A9%20%C3%AF%C3%B0%C3%AE%C3%A5%C3%AA%C3%B276.gif)
Наверх
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