上海交通大学：《复杂系统动力学计算机辅助分析》课程教学资源_Chapter 3_CHAP3.2-Absolute constraints

3.2.Constraints between a body and ground (Absolute constraints) Absolute x constraint Absolute y constraint Absolute angle constraint Absolute distance constraint

• Absolute x constraint • Absolute y constraint • Absolute angle constraint • Absolute distance constraint 3.2. Constraints between a body and ground (Absolute constraints)

Absolute position constraint .Absolute x(y)constraint Point P of the rigid body B is constrained to move along the axis parallel to axis x or axis y of the inertial frame Absolute x constraint Absolute y constraint

Absolute position constraint • Absolute x(y) constraint – Point P of the rigid body Ba is constrained to move along the axis parallel to axis x or axis y of the inertial frame Absolute x constraint Absolute y constraint

Absolute x constraint B, Inertial frame O-e Body-fixed frame C,-e ,=(9,'sP=x”P Since=C =元+.匠+）-c=0 C x def x6+4s）-G=0x=[0] S=1 DOF=n-S=3-1=2 pax()=x,+x cosp-y sinpi-c=0 Parameters: i,sP=(xy）,c

• Absolute x constraint Bi y  x  O i x  i y  Ci i r i P P ir  P i s  1 c e  Inertial frame O  Ci i e  Body-fixed frame    T T i i i q  r   T P i P i P i s  x  y  1 x r c P  i    P i i P ir r s      def ( )  ax i    0 x  r  s  c1  P i i      0 1 T    c  P i i i x r A s cos sin 0 1 ( )  x  x   y  i  c  P i i P i i ax i    Since Parameters:   1 T i, x y , c P i P i P i s      T x  1 0 s 1 DOF  n  s  31 2

Φ)=xTG+AsP）-c=0 B 本0=xr(+AsP）=0 y Velocity constraint equation 市o=xr(G+RAsf0,)=0 市w=西，meg,-vaw=0,g,=上，J →Dm=erx'r4g]va0=0 Acceleration constraint equation Cx X p)=x"(+RAs",+RAs)=O A=RAO 本0=xT+RAs,-As2)=0 RR=-1 市m0=Φ，0i,-y0=0,4= →ym0=xAs02

• Velocity constraint equation   1 0 ( ) T      c  P i i i ax i x r As   0 ( ) T     P i i i ax i x r A s      0 ( ) T    i  P i i i ax i     x r RA s i ii  ( )  T        0 A  RA i P i i i P i i i ax i        x r RA s RA s RR  I   0 ( ) T 2      i  P i i i P i i i ax i      x r RA s A s • Acceleration constraint equation   T i T i i ax i i ax i ax i v i       q q r  q   0,  ( ) ( ) ( )    P i i ax i i x x RA s q   ( ) T T    T i T i i ax i i ax i ax i i        q q r  q   0,  ( ) ( ) ( )  ( ) T 2 i P i i ax i   x A s  Bi y  x  O i x  i y  Ci i r i P P ir  P i s  x c 0 ( )  ax i v

Absolute y constraint Inertial frame o-e B Body-fixed frame C-e C2 sP=(PP）J Since =元+5”，(+5）-02=00 X B0)=y+A,s;")-c2=0 y=[0 1 pardi)=y,+x sin +y cosp-c2=0 Parameters:,s=c

Bi y  x  O i x  i y  Ci i r i P P ir  P i s  2 c • Absolute y constraint e  Inertial frame O  Ci i e  Body-fixed frame    T T i i i q  r   T P i P i P i s  x  y  P i i P ir r s      def ( )  ay i    0 y  r  s  c2  P i i      0 2 T    c  P i i i y r A s sin cos 0 2 ( )  y  x   y  i  c  P i i P i i ay i    Since   T y  0 1 Parameters:   2 T i, x y , c P i P i P i s   

Φ0=yT(+AsP）c2=0 B 市0=y(+AsP）=0 Velocity constraint equation D0=y(G+RAs0,)=0 市0=Da0豆-vm0=0,g,=上0 →( )=yT yRA,s;()=0 Acceleration constraint equation 市o=y+R4sy,+RAsp,)=0 A,=RA 市0=y+RAs-As2)=0 RR=-1 市0=Φ，0，-y0=0,,=, →y0=yAs02

• Velocity constraint equation   0 2 ( ) T     c  P i i i ay i  y r A s   0 ( ) T     P i i i ay i y r A s      0 ( ) T    i  P i i i ay i     y r RA s i ii    0 A  RA ( ) T      i  P i i i P i i i ay i        y r RA s RA s   0 RR  I ( ) T 2      i  P i i i P i i i ay i      y r RA s A s • Acceleration constraint equation Bi y  x  O i x  i y  Ci i r i P P ir  P i s  y c   T i T i i ax i i ax i ax i i        q q r  q   0,  ( ) ( ) ( )  ( ) T 2 i P i i ax i   y A s    T i T i i ay i i ay i ay i v i       q q r  q   0,  ( ) ( ) ( )    P i i ay i i y y RA s q   ( ) T T  0 ( )  ay i v

Absolute angle constraint B The rigid body is constrained to undergo translational motion without rotation p,=C0 def Φao() =,-C=0 g,=(p,) DOF=n-s=2 Velocity constraint equation Acceleration constraint equation 本a90=0,=0 iao0）=0,=0 Φ0=[00] vap(i)=0 (=O Parameter: i Co

The rigid body is constrained to undergo translational motion without rotation i  c  0 def ( )  a i    c i  • Velocity constraint equation 0 a ( )    i i    • Acceleration constraint equation 0 a ( )    i i     Parameter: i, c i r Bi y  x  O i x  i y  Ci i DOF  n  s  2   T T i i i q  r Absolute angle constraint 0 0 1 ( )  a i i   q 0 a ( )   i 0  ( )  a i v 

Absolute distance constraint The distance between point P of B,and a fixed point on the ground is constant Absolute distance constraint

• The distance between point P of Bi and a fixed point on the ground is constant Absolute distance constraint Absolute distance constraint

Description of the constraint Inertial frame o-e B Body-fixed frame of Bi C-e 4,=Tp,） h P Point P of B:sP=y) 9 Fixed Point r=C=(C C2) Defining a vector h=⑨丽=P-=方+p-0 h=n+As-C According to the definition of the constraint,the distance between point P and Q is equal to c def h.h-c2=0 Φadi=hTh-c2=0

• Description of the constraint Bi y  x  O i x  i y  Ci i r  P P ir P i s e  Inertial frame O  Ci i e  Body-fixed frame of Bi    T T i i i q  r Point P of Bi P i s   T P i P i P i s  x  y  Fixed Point Q Q r   T C1 C2 Q r  C  Q h  Q r h  QP  P Q ir r     P Q i i r r        h  r  A s C P i i i 0 2 h  h  c     0 T 2 h h  c  def ( )  ad i  Defining a vector According to the definition of the constraint, the distance between point P and Q is equal to c

Constraint equation B Φad(=hTh-c2=0 h P h=n+AsP-C s=1 9 g,=g,) D0F8=3-1=2 r X Parameters of the constraint i,s=”y',r2=C=(CC2)》',c

Constraint equation  0 h  r  A s C P i i i ( ) T 2 c ad i   h h  s  1 DOF   31  2   T T i i i q  r Parameters of the constraint i x y  C C  c P Q i P i P i , , , T 1 2 T s    r  C  Bi y  x  O i x  i y  Ci i r  P P ir P i s Q h  Q r