maple非线性振动

1、1、已知：I，m, c, k, F=Fosin(cot)求：系统的振动微分方程，伙）， =0时质点的振幅 解：建模：质点作圆周运动，杆作定轴转动。 Maple> restart: >J0:=m*lA2:> Fe:=-k*3*l*theta(t):>Fd:=-c*2*l*diff(theta(t),t)：> F:=FO*sin(omega*t):>eq:=J0*diff(theta(t)/t$2)=Fd*2*I+Fe*3*l+F*3*l:>eq:=subs(diff(theta(t),t$2)=DDt heta,diff(theta(t),t)=Dthe

2、ta, thQta(t)=thQta,eq):>eq:=m*|A2*DDtheta+4*c*|A2*Dtheta+9*k*r2*theta= 3*F0*sin(omega*t)*l:> eq:=expand(eq/(m*lA2);cc .4 c Dtheta 9 k 8 o FO sin( co /)eq := DDtheta += 3m mm I> b:=h/sqrt(omega0A2-omegaA2)A2+4*deltaA2*omegaA2):> bl:=subs(omega=omegaO,b);> omega0:=sqrt(9*k/m);> delta

3、:=(2*c)/m:> h:=(3*f0)/(m*l):> B:=simplify(l*bl,symbolic);« A答：系统振动微分方程为0 + ?9k0Fosin?(cot)+ = 3jmmlf丽> D-o4c'k2、已知：m=, h=, k=m, 0 = 30求：系统的固有频率和振幅，物块的运动方程。 Maple程序> restart:> deltaO:=m*g*sin(beta)/k:>eq:=m*diff(x(t)/t$2)= m*g*sin(beta)- k*(deltaO+x):> eq:=lhs(eq)-rhs(eq

4、)=O:>eq:=subs(diff(x(t),t$2)=DDx, eq)：> eq:=simplify(eq);eq := m DDx +xk = 0> X:=A*sin(omega0*t+theta):> omega0:=sqrt(k/m):> xO:=-deltaO:> v0:=sqrt(2*g*h):> A:=sqrt(x0A2+(v0/omega0)A2):> theta:=arctan(omega0*x0/v0):> m:=:> h:=:> k:=:> beta:=Pi/6:>g：=：> co。：=

5、 40.00 omQgaO:=Qvalf(omQga0,4);> A:=evalf(A,4);A := .03513> thetaevalfftheta);0 := -.08724> X:=eval(X);X := .03513 sin(40.001 - .08724)答：(oo = 40rad's, A=,物块的运动方程为x=mmo3、吸引子的仿真。以杜芬方程为例，杜芬方程表示如下x +cx+ax+ b x3 =Acos 31 Maple程序> restart:> with(plots):> del:=diff(x(t)/t)=y(t):>d

6、e2:=diff(y(t)/t)=-a*x(t)-b*x(t)A3-c*y(t)+A*cos(Omega*t):> a:="l:b:=l:c:=:A:=:Omega:=l: duffing:=dsolve(del/de2,y(0)=/x(0)="l,x(t)/y(t)/type=numeric,method=lso de):> duffplot:=odeplot(duffing/x(t)/y(t)/0.200/numpoints=4000):> duffplot;答：杜芬方程相图如图所示。4、已知：闫0Vt求：摆的运动方程解：建模：小球作平面运动 自由度f

7、=l,取广义坐标(P Maple程序> restart:> xrho:=l:>xphi:=l*phi:> xrho:=subs(l=l(t),xrho):> xphi:=subs(phi=phi(t),xphi):> vrho:=diff(xrho,t):> vphi:=diff(xphi,t):> V:=vector(MrhoMphi):> vA:=sqrt(vrhoA2+vphiA2): >T:=l/2*m*vAA2;> T:=subs(diff(phi(t)/t)=Dphi/phi(t)=phi/T):> Tcoll

8、ectfLDphi):> TDphi:=diff(T,Dphi):> Tphi:=diff(T,Dphi);7=0©> TDphi:=subs(l=lO-v*t/Dphi=Dphi(t)/TDphi);TDphi = /H(/0-vr)2Dphi(r)> V:=-m*g*(l0-v*t)*cos(phi):> Qphi:=diff(6phi);。产 g(4 - y f) sin(<|>)> eq:=diff(TDphiLt)TphiQphi=O;2(3eq := -2 m (/ - v r) Dphi(O v + m (I -v t)

9、Dphi(r) + m g (I - v t) sin(6)=0 0° ydt )0> eq:=subs(diff(Dphi(t),t)=DDphi,Dphi(t)=Dphi,eq):> eq:=(l0-v*t)*DDphi-2*v*Dphi+g*sin(phi)=0;eq := (/() - v r) DDphi - 2 v Dphi + g sin(|) =0答：摆的运动微分方程为(lo - vl)(p - 2v<p + gsin (p = 0。5、己知：m, r求：系统的微分方程解：建模：圆环作平面运动(纯滚动), 质点作曲线运动自由度f=l,取广义坐标(P。

10、 Maple程序> restart:> J0:=m*rA2:> xO:=r*phi:> xO:=subs(phi=phi(t),xO):>vO:=diff(xObt):>vA:=sqrt(v0A2+v0A2-2*v0A2* cos(phi):>T:=l/2*J0*diff(phi(t),t)A2+l/2*m*v0A2+l/2*m*vAA2:> T:=simplify(T):> T:=factor(T);(-2 + cos(0)> V:=-m*g*r*cos(phi);> L:=T-V;V := t g r cos( 4)L :=

11、 -in(-2 + cos(巾)+ ? g r cos(|)> L:=subs(diff(phi(t),t)=Dphi/phi(t)=phi,L):> L:=collect(L,Dphi):> LDphi:=diff(L,Dphi);七必产 一2 m r2 Dphi (-2 + cos( )>Lphi:=m*rA2*Dphi*sin(phi)-m*g*r*sin(phi):> LDphi:=subs(Dphi=diff(phi(t),t)/LDphi):> eq:=diff(LDphi,t)-Lphi=O:>eq:=subs(diff(phi(t),t

12、)=Dphi(t), diff(Dphi(t),t)=DDphi,eq);eq := -2 m r DDphi (-2 +cos(|) -m r Dphi + m g r sin(|) =0答：系统的微分方程为2(2-cos(p)0 + ?)sin (p = 0o6、已知：R, r, m, Jo，圆球C作纯滚动求：系统的运动微分方程解：建模：圆槽绕O作定轴转动, 小球C作平面运动，自由度f=2,取广义坐标8, (p Maple程序> restart:>xP:=r*theta:>xP:=subs(theta=theta(t), xP)：> vP:=diff(xPLt);r

13、d 、3=心町w (枭)序)> vOP:=r*omega;%p=(飙)> l:=phi:> l:=subs(phi=phi(t),l):> omegal:=diff(bt):> vO:=omegal*(R-r);(d %=除婷)J(R_r)> eq:=vO-vP-vOP=0:> SOLsolvelfeqomega):> omegasubsfSOUomega);-信幅)R+信婷)信。)CD ：=-r>JO:=2/5*m*rA2:>T:=l/2*JC*omegalA2+l/2*JO*omegaA2+l/2*m*vOA：>V:=-m*

14、g*(R-r)*cos(phi);V := -tn g (R -r) cos(|)> L:=T-V;+rMH(o+(<l>(r)r+rte0(z)+ 篇仲)(R-r)2 +m g(R-r)co)> L:=simplify(L):>L:=subs(diff(theta(t),t)=Dtheta,theta(t)=theta,diff(phi(t),t)=Dphi/phi(t)=：phi/L):> L:=collect(L,Dtheta):> HDthetaJdifflUDtheta):> LlthetaJdifffUtheta):> LDth

15、eta:=subs(Dtheta=Dtheta(t)/LDtheta):> eql:=diff(LDthQta,t)LthQta=O:> eql:=subs(diff(DthQta(t),t)=DDtheta,Qql);2,eql := - m 广 DDtheta = 0> Lphi:=diff(L,phi):> LDphi:=diff(L,Dphi):>LDphi:=subs(Dtheta=Dtheta(t),theta=theta(t)/Dphi=Dphi(t)/LDphi):> eq2:=diff(LDphi/t)-Lphi=0:>eq2:=su

16、bs(diff(Dtheta(t)/t)=DDtheta,Dtheta(t)=Dtheta,diff(Dphi(t),t)=DDphi/Dphi(t)=Dphi,eq2):> eq2:=simplify(eq2):答：系统运动微分方程为gmR(R-r)。(Jo + ?nR2)6 = 07. 2 mR(R-r)(p-R0 + gsin(p=07、已知：mi, H12, R, l=2R,轮A作纯滚动求：F为多大方可使B端离开地面，保证纯滚动时理解：建模：轮A作平面运动，杆C作平行移动。 Maple程序> restart:> alpha:=a/R:> FIC:=m2*a:&g

17、t; FIA:=ml*a:> MIA:=J*alpha:>eql:=FIC*l/2*sin(phi)-m2*g*|/2*cos(phi)=0:> SOLl:=solve(eql,a):>eq2:=FA*R-FIA*R-MIA-FIC*l/2*sin(phi)-m2*g*l/2*cos(phi):> eq3:=FA-Fs-FIA-FIC=0:> eq4:=FN-ml*g-m2*g=0:> SOL2:=solve(eq2,eq3,eq4,FA,FN,Fs):> a:=subs(SOLl,a):> FA:=subs(SOL2,FA):> F

18、N:=subs(SOL2,FN):> Fs:=subs(SOL2,Fs):> a:=subs(phi=Pi/6/l=2*RJ=l/2*ml*RA2/a):> a:=simplify(a);a := g J 3>FA:=subs(phi=Pi/6/l=2*RJ=l/2*ml*RA2/FA): > FA:=simplify(FA);>FN:=subs(phi=Pi/6/l=2*R/J=l/2*ml*RA2/FN);Fn =，g+m2g> Fs:=subs(phi=Pi/6j=2*RJ=l/2*ml*RA2/Fs):> Fs:=simplify(Fs)

19、;> fs:=FN/Fs:> fs:=simplify(fs);2 (m+ "与)、；亏广.=s 一 3 ni答：使B端刚好离开地面的F二蛉mi 2m2)保证纯滚动时Fsn/%8、已知：m, 2a,恢复因数k求：轴承的碰撞冲量，碰撞中心的位置解：建模：杆在碰撞中作定轴转动，满足动量定理和动量矩守恒定理，碰撞前机械能守恒。 Maple程序> restart:>J0:=l/3*m*(2*a)A2:>T1:=O:> T2:=l/2*J0*omegaA2:> V1:=O:> V2:=-m*g*a:> eql:=T2+V2=Tl+Vl:&g

20、t; LOl:=-J0*omega:> LO2:=J0*omegal:> eq2:=LO2-LOl=Y*l:> v:=l*omega:> vl:=l*omegal:> eq3:=k=vl/v:> SOLl:=solve(eql/eq2/eq3/Y/omegaA2/omegal):> Y:=subs(SOLl,Y);> omegal:=subs(SOLl,omegal);col := k 3> omegasubslSOLomega):> omega:=sqrt(omega);> Pxl:=m*omega*a:> Px2:=-m*omegal*a:> eq4:=Px2-Pxl=YOx-Y:> eq5:=0=YOy:> SOL2:=solv