kinetics of a rigid body: force and acceleration equation of rotational...
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Equation of Rotational Motion
X
Y
prir
iP
r r
X
Y
prir
( )p i i i i im M r F r f r a
( )i p a a α r ω ω r
( ) [ ( ) ( ( ))]p i i pm M r a r α r r ω ω r
( ) {( ) [( ) ( ) ]
( ) [ ( )]}p i i p x p yM m x y a a
x y x y
k i j i j
i j k i j
2 2
2
( ) [ ( ) ( ) ]
( ) [ ( ) ( ) ]p i i p x p y
p i i p x p y
M m y a x a x y
M m y a x a r
k k
Kinetics of a Rigid Body: Force and Acceleration
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Equation of Rotational Motion
X
Y
prir
2
2
2
2
( ) [ ( ) ( ) ]
( ) [ ( ) ( ) ]
( ) [ ( ) ( ) ]
( )( ) ( )( ) ( )
p i i p x p y
p i p x p y
p i p x p y
p x p y
M m y a x a r
M dm y a x a r
M dm y a x a r
ydm a xdm a r dm
( ) ( )
G i
G G
m dm
mx my xdm ydm
mx my
r r
i j i j
i j
m dm
( ) ( )p p x p y pM ym a xm a I 2( )pI r dm where If point P = point G G GM I
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Equation of General Motion
G GM I
( ) ( )p p x p y pM ym a xm a I 2( )pI r dm where
If point P = point G
The sum of the moments of all the external forces computed about the body’s mass center G is equal to the product of the moment of inertia of the body about an axis passing through G and the body’s angular acceleration.
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Equation of General Motion
( ) ( )p p x p y pM ym a xm a I 2 2 2( )p G GI I mr I m x y
2 2
2 2
( ) ( ) ( )
[ ( ) ] [( ) ]
[ ( ) ] [( ) ]
( ) ( )
p p x p y G
p x p y G
G x G y G
G x G y G
M ym a xm a I m x y
ym a y xm a x I
ym a x xm a y I
ym a xm a I
( ) ( ) ( )p G x G y G k pM ym a xm a I M 2
G P a a r r
The moments of the external forces about P are equivalent to the sum of the “kinetic moments” of the components of about P plus the “kinetic moment” of .
Gma
GI
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General Application of the Equations of Motion
G GM I
( )p k pM M
( )
( )x G x
y G y
F m a
F m a
or
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Equations of Motion: Translation
0GM ( )
( )x G x
y G y
F m a
F m a
Rectilinear Translation
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Equations of Motion: Translation
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Equations of Motion: Translation
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( )
( )n G n
t G t
F m a
F m a
Equations of Motion: Translation
Curvilinear Translation
0GM
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Equations of Motion: Curvilinear Translation
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Equations of Motion: Curvilinear Translation
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2( )
( )n G n G
t G t G
F m a m r
F m a m r
Equations of Motion: Rotation about a Fixed Axis
G GM I
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2( )
( )n G n G
t G t G
F m a m r
F m a m r
Equations of Motion: Rotation about a Fixed Axis
G GM I Moment equation can be replaced by arbitrary point P on or off the body provided one accounts for the moments produced by
about the point.
Do we know this force?
In order to eliminate the unknown force , it is convenient to sum moments about the pin point at O.
( )p k pM M
( )k pM ( ) , ( ) ,G n G t Gm a m a I
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2( )
( )n G n G
t G t G
F m a m r
F m a m r
Equations of Motion: Rotation about a Fixed Axis
In order to eliminate the unknown force , it is convenient to sum moments about the pin point at O.
2
( ) ;
( ) ( )
( )
o k o
o G G t G G G G
G G o
M M
M r m a I r m r I
I mr I
2( )o G G oM I mr I
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Equations of Motion: Rotation about a Fixed Axis
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Equations of Motion: Rotation about a Fixed Axis
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Equations of Motion: Rotation about a Fixed Axis
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Equations of Motion: General Plane Motion
Freebody diagram Kinetic diagram
( )
( )x G x
y G y
F m a
F m a
G GM I
( )p k pM M or
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