Download Plane Kinetics of Rigid Bodies

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PLANE KINETICS OF RIGID BODIES
GENERAL EQUATIONS OF MOTIONS
By replacing the external forces by their equivalent force-couple system in which the resultant force
acts through the mass center, we may visualize the action of the forces and the corresponding dynamic
response.
Equivalent Force-Couple System
Free-Body Diagram
F1
“m”
G
F2
Kinetic Diagram

M
 G


F


HG

“m”

ma
“m”
G
Fn
G
F3


F
 = ma
(c)
(b)
(a)


 MG = H G
a) relevant free-body diagram
b) equivalent force-couple system with resultant force applied through G
c) kinetic diagram which represents the resulting dynamic effects
PLANE MOTION EQUATIONS
Rigid body moves in the xy plane as shown. The mass
center G has an acceleration a , and the body has an angular
velocity and angular acceleration .
y


mi
It is instructive to use an approach to derive the moment
equation by referring directly to the forces which act on the
representative particle of mass mi. The angular momentum
of this particle about G, HGi;
i
G

a




H Gi  i  mi  i

i : position vector relative to G of the representative particle

 

 i  v i     i : velocity of mi





 i   i cos  i  sin  j
  k


 
H Gi  i  mi   i 


Sum of the angular momenta about G of all particles (Angular momentum about G of the rigid body)













 
H G   H Gi    i  m i    i     i cos  i  sin  j  m i k   i cos  i  sin  j







H G  I





H G   m i  i2  cos 2   sin 2  k   m i  i2 k




H G  Ik
1
I
m i  i cos  j  m i  i sin  i
x
2
I is the mass moment of inertia about the z-axis through G.
n 
I  kgm 2
n
I   m i  i2    2 dm
i 1
mi


 MG  HG
 MG 

HG
d
d
 I  I
 I
dt
dt
dt

 M G  I

Mass Moment of Inertia
Mass moment of inertia I of the body about the axis O-O
I   dI   r 2 dm
As the mass m of a body is a measure of the resistance to translational acceleration, the moment of
inertia I is a measure of resistance to rotational acceleration of the body.
Transfer of Axes
If the moment of inertia of a body is known about an axis passing through the mass center, it may be
determined about any parallel axis.
I O  I  md 2
Radius of Gyration
The radius of gyration k of a mass m about an axis for which the moment of inertian is I is defined as
I
k is a measure of the distribution of mass of a given body about the axis
m
question and is called as the radius of gyration.
I  k 2m
k
Slender Bar
y
l/2
z
Circular Thin Plate
y
Ix  0
G
I y  Iz 
l/2
x
1 2
ml
12
G
z
1 2
mr
2
1
I y  I z  mr 2
4
Ix 
r
x
Rectangular Thin Plate
1
1
Ix  m b2  h 2
I y  mb 2
12
12
y
1
I z  mh 2
12
z
h
G

b

x