1樓：野生學渣
% Motion of a capsule.
% % Author: AdamTurner, 2021.05.
% Written in MATLAB R2018a.
clear
all;
close
all;
clc;
% ----- Initial Conditions ----- %% Initial position
xc0=
0.000
;yc0
=0.000
;zc0
=0.900
;% Initial orientationphi0
=0.450*pi
;theta0
=0.750*pi
;% ----- Parameters ----- %globalLr
m1m2gdt
;L=0.100;r
=0.100;m1
=0.010;m2
=0.010;g
=9.80665;dt
=1.000e-2
;% ----- Mesh of Capsule ----- %s1=
linspace
(0.000
,0.500*pi
,6).'
;s2=linspace(-
pi,pi,
21).';
[s1,s2
]=meshgrid(s1
,s2);xCapsule=r
*sin(s1
).*cos(
s2);
yCapsule=r
*sin(s1
).*sin(
s2);
zCapsule=r
*cos(s1
);xCapsule=[
xCapsule
,xCapsule
(:,end
),xCapsule
(:,end
),fliplr
(xCapsule
)];yCapsule=[
yCapsule
,yCapsule
(:,end
),yCapsule
(:,end
),fliplr
(yCapsule
)];zCapsule=[
0.500*L
+zCapsule
,zeros(21
,2),-
0.500*L
-fliplr
(zCapsule
)];[
nS2,
nS1]
=size
(zCapsule
);colorCapsule
=zeros
(nS2
,nS1,3
);colorCapsule
(:,1:7
,1)=
1;colorCapsule
(:,8:14
,3)=
1;% ----- Solve ------ %Q0=[xc0
+0.500*L
*sin
(theta0)*
cos(
phi0
);...
yc0+
0.500*L
*sin
(theta0)*
sin(
phi0
);...
zc0+
0.500*L
*cos
(theta0
);...
xc0-
0.500*L
*sin
(theta0)*
cos(
phi0
);...
yc0-
0.500*L
*sin
(theta0)*
sin(
phi0
);...
zc0-
0.500*L
*cos
(theta0
);...
0.000
;...
0.000
;...
0.000
;...
0.000
;...
0.000
;...
0.000
];tSpan=(
0.000:dt
:10.000);[
t,Q]
=...
ode15s
(@KineticEq
,tSpan,Q0
,...
struct
('RelTol'
,1.000e-5
,...
'AbsTol'
,1.000e-5
,...
'MaxStep'
,1.000e-3
,...
'InitialStep'
,1.000e-3
));[x1,
y1,z1,
x2,y2,
z2,u1,
v1,w1,
u2,v2,
w2]=ParseQ(Q
);hFigure
=figure
();hAxes
=...
axes
(hFigure
,...
'Box'
,'on'
,'BoxStyle'
,'full'
,...
'NextPlot'
,'add'
,...
'XGrid'
,'on'
,'YGrid'
,'on'
,'ZGrid'
,'on'
,...
'XLimMode'
,'manual'
,'XLim',[
-0.500
,0.500
],...
'YLimMode'
,'manual'
,'YLim',[
-0.500
,0.500
],...
'ZLimMode'
,'manual'
,'ZLim',[
0.000
,1.000
]);view([-
30.000
,30.000
]);xlabel
(hAxes
,'$x$'
,'Interpreter'
,'latex'
);ylabel
(hAxes
,'$y$'
,'Interpreter'
,'latex'
);zlabel
(hAxes
,'$z$'
,'Interpreter'
,'latex'
);nFrame
=numel(t
);forii=
1:nFrame
ozCapsule=[
x1(ii)
-x2(ii
);y1(ii
)-y2(
ii);z1(
ii)-z2
(ii)];ozCapsule
=ozCapsule
/sqrt
(ozCapsule.'
*ozCapsule
);oxCapsule
=cross
(ozCapsule,[
0.000
;0.000
;1.000
]);oxCapsuleM
=sqrt
(oxCapsule.'
*oxCapsule
);if
oxCapsuleM
<=eps
oxCapsule=[
1.000
;0.000
;0.000
];oyCapsule=[
0.000
;1.000
;0.000
];ozCapsule=[
0.000
;0.000
;1.000
];else
oxCapsule
=oxCapsule
/oxCapsuleM
;oyCapsule
=cross
(ozCapsule
,oxCapsule
);end
% /* if oxCapsuleM <= eps */xyzCapsule
=...
[oxCapsule
,oyCapsule
,ozCapsule
]...*[
xCapsule
(:),
yCapsule
(:),
zCapsule
(:)].';
xyzCapsule(1
,:)=xyzCapsule(1
,:)+0.500*(
x1(ii)
+x2(ii
));xyzCapsule(2
,:)=xyzCapsule(2
,:)+0.500*(
y1(ii)
+y2(ii
));xyzCapsule(3
,:)=xyzCapsule(3
,:)+0.500*(
z1(ii)
+z2(ii
));xxCapsule
=reshape
(xyzCapsule(1
,:).',
nS2,
nS1);
yyCapsule
=reshape
(xyzCapsule(2
,:).',
nS2,
nS1);
zzCapsule
=reshape
(xyzCapsule(3
,:).',
nS2,
nS1);
ifii~=1
hSurfCapsule
.XData
=xxCapsule
;hSurfCapsule
.YData
=yyCapsule
;hSurfCapsule
.ZData
=zzCapsule
;else
hSurfCapsule
=...
surf
(xxCapsule
,yyCapsule
,zzCapsule
,colorCapsule
);end
% /* if ii ~= 1 */
currentFrame
=getframe
(hFigure);[
colorIndex
,colorMap]=
rgb2ind
(currentFrame
.cdata,32
);if
ii~=
1imwrite
(colorIndex
,colorMap
,'Capsule.GIF'
,...
'WriteMode',,
'DelayTime'
,0.050
);else
imwrite
(colorIndex
,colorMap
,'Capsule.GIF'
,'Loopcount'
,Inf
,...
'WriteMode'
,'overwrite'
,'DelayTime'
,0.050
);end
% /* if ii ~= 1 */
end% /* for ii */
% ----- Function(s) ----- %function
dQ =
KineticEq
(t, Q)
globalLr
m1m2gdt
;[x1,
y1,z1,
x2,y2,
z2,u1,
v1,w1,
u2,v2,
w2]=ParseQ(Q
);M=diag
([m1
*ones(3
,1);m2
*ones(3
,1)]);
PhiQ
=PHI_Q(t
,Q);contactFrameBottom
=...
[1.000
,0.000
,0.000
;...
0.000
,1.000
,0.000
;...
0.000
,0.000
,1.000
];contactFrameLeft
=...
[0.000
,0.000
,1.000
;...
1.000
,0.000
,0.000
;...
0.000
,1.000
,0.000
];contactFrameRight
=...
[0.000
,0.000,-
1.000
;...
-1.000
,0.000
,0.000
;...
0.000
,1.000
,0.000
];contactFrameFront
=...[-
1.000
,0.000
,0.000
;...
0.000
,0.000
,1.000
;...
0.000
,1.000
,0.000
];contactFrameBack
=...
[1.000
,0.000
,0.000
;...
0.000
,0.000,-
1.000
;...
0.000
,1.000
,0.000];f
=[ContactForce(x1
,y1,z1
-r,u1
,v1,w1
,contactFrameBottom
);...
ContactForce(x2
,y2,z2
-r,u2
,v2,w2
,contactFrameBottom
)]...+[
ContactForce(x1
+0.500-r
,y1,z1
,u1,v1
,w1,contactFrameLeft
);...
ContactForce(x2
+0.500-r
,y2,z2
,u2,v2
,w2,contactFrameLeft
)]...+[
ContactForce(x1
-0.500+r
,y1,z1
,u1,v1
,w1,contactFrameRight
);...
ContactForce(x2
-0.500+r
,y2,z2
,u2,v2
,w2,contactFrameRight
)]...+[
ContactForce(x1
,y1+0.500-r
,z1,u1
,v1,w1
,contactFrameFront
);...
ContactForce(x2
,y2+0.500-r
,z2,u2
,v2,w2
,contactFrameFront
)]...+[
ContactForce(x1
,y1-0.500+r
,z1,u1
,v1,w1
,contactFrameBack
);...
ContactForce(x2
,y2-0.500+r
,z2,u2
,v2,w2
,contactFrameBack
)]...+[
0.000
;0.000;-
m1*g;
0.000
;0.000;-
m2*g];
dq=[u1
;v1;w1
;u2;v2
;w2];Phi
=PHI(t
,Q);PhiT
=PHI_T(t
,Q);PhiQQ
=PHI_QQ(t
,Q);PhiQT
=PHI_QT(t
,Q);PhiTT
=PHI_TT(t
,Q);alpha
=1.000e3
;beta
=1.000e3
;gamma=(
-dq.'
*PhiQQ*dq
-2*PhiQT*dq
-PhiTT
)...-2
*alpha*(
PhiQ*dq
+PhiT)-
beta
*beta
*Phi
;ddq=[
M,PhiQ.'
;PhiQ
,0.000]\
[f;gamma
];dQ=[
dq;ddq(1:
6)];
end% /* KineticEq */
function
varargout =
ParseQ
(Q)[~,
nColumn]=
size(Q
);if
nColumn==1
Q=Q.
';end% /* if nColumn == 1 */x1=Q(:,1);
y1=Q(:,2);
z1=Q(:,3);
x2=Q(:,4);
y2=Q(:,5);
z2=Q(:,6);
u1=Q(:,7);
v1=Q(:,8);
w1=Q(:,9);
u2=Q(:,
10);v2=
Q(:,
11);w2=
Q(:,
12);
varargout=;
end% /* ParseQ */
function
Phi =
PHI(t, Q)
globalL;
[x1,y1
,z1,x2
,y2,z2
,~,~
,~,~
,~,~
]=ParseQ(Q
);deltaQ=[
x1;y1;
z1]-[
x2;y2;
z2];
Phi=
deltaQ.'
*deltaQ-L
*L;end
% /* PHI */
function
PhiQ =
PHI_Q
(t, Q)[x1
,y1,z1
,x2,y2
,z2,~
,~,~
,~,~
,~]=
ParseQ(Q
);deltaQ=[
x1;y1;
z1]-[
x2;y2;
z2];
PhiQ=[
2*deltaQ;-
2*deltaQ].'
;end
% /* PHI_Q */
function
PhiT =
PHI_T
(t, Q)
PhiT
=0.000
;end
% /* PHI_T */
function
PhiQQ =
PHI_QQ
(t, Q)
PhiQQ
=...
[2.000
*eye(3
,3),-
2.000
*eye(3
,3);...
-2.000
*eye(3
,3),2.000
*eye(3
,3)];end
% /* PHI_QQ */
function
PhiQT =
PHI_QT
(t, Q)
PhiQT
=zeros(1
,6);end
% /* PHI_QT */
function
PhiTT =
PHI_TT
(t, Q)
PhiTT
=0.000
;end
% /* PHI_TT */
function
y =Step3
(x, x1, y1, x2, y2)xi=2.000*(
x-0.500*(
x1+x2))/(
x2-x1);
ifxi
<=-1.000y=
-1.000
;else
ifxi
>=1.000y=
1.000
;elsey=
-0.500*xi
*(xi*
xi-3.000
);end
% /* if xi >= 1.000 */end% /* if xi <= -1.000 */y=0.500*(
y1+y2)
+0.500*y
*(y2-
y1);
end% /* Step3 */
function
f =Impact
(x, v, x1, k, e, cm, d)ifx>=x1
f=0.000
;elsef=
k*(x1
-x)^e-
Step3(x
,x1-d
,cm,x1
,0.000)*
v;f=
max(f,
0.000
);end
% /* if x >= x1 */
end% /* Impact */
function
mu =
FrictionCoefficient
(v, mus, mud)ifv
<=-0.000100mu=
Step3(v
,-0.000150,-
mud,
-0.000100,-
mus);
elseifv
<=0.000100mu=
Step3(v
,-0.000100,-
mus,
0.000100
,mus
);elsemu=
Step3(v
,0.000100
,mus
,0.000150
,mud
);end
% /* if v <= 0.000100 */end% /* if v <= -0.000100 */end% /* FrictionCoefficient */function
f =ContactForce
(x, y, z, u, v, w, contactFrame)r=[x;y
;z];dr=[
u;v;
w];r=
contactFrame.'
*r;dr
=contactFrame.'
*dr;x
=r(1
);y=r
(2);z
=r(3
);u=dr
(1);v
=dr(2
);w=dr
(3);N
=Impact(z
,w,0.000
,1.000e3
,1.000
,1.000e-1
,1.000e-4
);if
N>0.000
vAngle
=angle(u
+v*1
i);vMag
=abs(u
+v*1
i);mu=
FrictionCoefficient
(vMag
,0.200
,0.150);T
=-mu*
N;f=
[T*cos
(vAngle);T
*sin
(vAngle);N
];elsef=
[0.000
;0.000
;0.000
];end
% /* if N > 0.000 */f=contactFrame*f
;end
% /* ContactForce */
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