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Dynamic Analysis of Mechanical Systems Kinematics
创建日期：2024-05-06 17:59:14
所属分类：经济Economics
标签： 3D
Dynamic Analysis of Mechanical Systems Kinematics

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Dynamic Analysis of Mechanical Systems Kinematics

1
3.3 3D Kinematics of Rigid Bodies
3.3.1 Fixed Axis Rotation
A shown in the figure, let’s consider a rigid body that rotates
with angular velocity about a fixed axis n-n. In the Cartesian
frame O XYZ− , the position of a point P on the body can be
located by a position vector r . As you can see, the path of
point P is an arc in the plane normal to
In a small time interval t , the body swings an small angle
t=  . This leads to the change of r
( )
magnitude
direction
sin
sin
sin
t
t t
  
 


=


= = 
r
r r
r
r
r r
r




 


According to the definition of derivative, the velocity of point P
can be expressed as
( )
0
lim
t
t
t t →
 
= = = 
 
rr
v r

 (1)
⊥ ⊥v v r，
Taking time derivative of Eq.(1) by the chain rule leads to the
acceleration of P

( )
t n
= =  +  =  +  
a a
a r r v r r     (2)
where =  ---- the angular acceleration of the body.
t = a r ----the tangential component
( )n =  a r  ----the normal component

Dynamic Analysis of Mechanical Systems Kinematics
2
3.3.2 Fixed Point Rotation
As shown in the Figure, rotation of a body about a fixed point can
be regarded as the rotation about an axis called the instantaneous
axis. Thus, α can be decomposed into the following form
 ⊥= +α α α (3)
where α is aligned with the rotation axis, representing the
change rate of  in magnitude, while ⊥α is normal to the
rotation axis, reflecting the change rate of  in direction.

3.3.3 General Motion
The general motion of a body can be visualized as a combination
of the translation along with a point (the reference point) on the
body and the rotation about an axis passing through the point.
Let A and B are two points on a body. If we take B as the
reference point, the position vector of A can be represented by
A B AB= +r r r , =AB BAr (4)

Taking time derivative of Eq.(4) gives
A B AB= +v v v
ABv --- the velocity of A relative to B.
For pure translation
A B AB=  =v v v 0
For pure rotation with angular velocity about an axis passing
through B
B A AB AB=  = = v v v r0 
For general motion
A B AB= + v v r (5)
Taking time derivative of Eq.(5) leads to the acceleration of A.
( )A B AB AB= +  +  a a r r   (6)



Dynamic Analysis of Mechanical Systems Kinematics
3
For 2D motion in the x y− plane, because kω = and kα = ,
Eqs.(5) and (6) can be simplified as
( )( )
2
A B AB
A B AB AB
B AB AB

  
 
= + 
= +  +  
= +  −
v v k r
a a k r k k r
a k r r
(7)

Eq.(7) can also be expressed by using 2D coordinate vector
notation. Let Ar and Br are the 2D coordinate vectors of points A
and B. Then， AB A B= −r r r can be expressed as
cos
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