# Planar Kinematics of a Rigid Body by rt3463df

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Kinetics (I)

1. Review of Kinetics of Planar Mechanisms
- Inertia (mass and moment of inertia)
- Governing equation (Newton’s Law)
2. Moment/Product of Inertia – spatial rotation

ME 316 Lecture 6          1
Review of Planar Kinetics of a Rigid Body

Kinetics: how does a body move under the force and / or
moment ?

2. If there is no any force or moment applied to a body, the
body will remain its current status. This statement is called
Newton First Law.

3. Force and moment are vector, and they have direction, so
the Newton First Law corresponds to the direction.

Ex: No force in horizontal direction, no motion due
to Newton First Law; however, there is a force
(gravity) in vertical direction, there is motion
ME 316 Lecture 6                 2
Review of Planar Kinetics of a Rigid Body

The basic evidence to support the Newton First Law is the

inertia

1. A property of the body;
2. A measure of how difficult or easy the motional state of
the body can be changed;
3. An inherent resistance to change a body’s motion state

ME 316 Lecture 6                 3
Review of Planar Kinetics of a Rigid Body
Two types of inertia (depending on types of causes)
Cause is force:          inertia is mass (m)
Cause is moment: inertia is the moment of inertia (IP)

Moment causes rotation. Because rotation depends on the location of a
point about which a body rotates, moment of inertia differs with respect to
different points on a body. Thus, in the following, we have

A                                      IA ≠         IB

B                          ME 316 Lecture 6                        4
Review of Planar Kinetics of a Rigid Body

Example (to be filled)
B              A
Find IA and IB

IA ≠ I B

ME 316 Lecture 6              5
Review of Planar Kinetics of a Rigid Body

Parallel – axis theorem
B               G
(to be filled)
To verify the parallel axis
theorem, namely Find IB                  G is the center of gravity
IG+ml2

ME 316 Lecture 6                   6
Review of Planar Kinetics of a Rigid Body

Kinetic energy (K)

1. Translation (T): (to be filled)
2. Rotation (R): (to be filled)
3. General (T + R): (to be filled)

ME 316 Lecture 6   7
Review of Planar Kinetics of a Rigid Body

Kinetics Equation

Translation:
 F  ma                 G

F   x    m(aG ) x       F         y    m(aG ) y

X and Y axes may not be horizontal or vertical;
rather they could be in any direction but not in
parallel. G: center of gravity of a body
ME 316 Lecture 6                    8
Review of Planar Kinetics of a Rigid Body

Rotation:
M   P     ym(aP ) x  xm(aP ) y  I P
Case 1:

G

P

May not be fixed     ME 316 Lecture 6                9
Review of Planar Kinetics of a Rigid Body

Case 2: P=G (mass center)                           P

 M G I G                               G

Case 3: P is fixed point
G
M     P   I P
P

ME 316 Lecture 6                   10
Review of Planar Kinetics of a Rigid Body
Translation and rotation:

F     x    m(aG ) x            F        y    m(aG ) y

Case 1:       M  G    I G

Case 2:       M   P     ym(aP ) x  xm(aP ) y  I P

Case 3:       M   P   I P

ME 316 Lecture 6                    11
Moment/product of Inertia of Spatial Rotation

z
I zz   rz2 dm   ( x 2  y 2 )dm

I yy   ry2 dm   ( x 2  z 2 )dm

I xx   rx2 dm   ( y 2  z 2 )dm

I xy  I yx   xydm

I yz  I zy   yzdm

I xz  I zx   xzdm

ME 316 Lecture 6                                 12
Moment/product of Inertia of Spatial Rotation

Product of Inertia:
The concept of Orthogonal planes

I xy  I yx   xydm

I yz  I zy   yzdm

I xz  I zx   xzdm

ME 316 Lecture 6       13
Moment/product of Inertia of Spatial Rotation

Product of Inertia (special cases)

If either one or both of the orthogonal planes are
planes of symmetry for the mass, the product of
inertia with respect to these planes will be zero.

ME 316 Lecture 6              14
Moment/product of Inertia of Spatial Rotation

Examples of Product of Inertia (special cases)

x

I xy  I xz  0                      I xy  I xz  I yz  0

ME 316 Lecture 6                            15
Moment/product of Inertia of Spatial Rotation

Parallel-axis and parallel-plane theorems

ME 316 Lecture 6           16
Moment/product of Inertia of Spatial Rotation

Inertia Tensor – a compact way to express
{to be filled in classroom}

ME 316 Lecture 6           17
Moment/product of Inertia of Spatial Rotation

Principal axes - principal moments of inertia

If the coordinate axes are oriented such that two
of the three orthogonal planes containing the axes
are planes of symmetry for the body, then all the
products of inertia for the body are zero with
respect to the coordinate planes, and hence the
coordinate axes are principal axes of inertia
ME 316 Lecture 6            18
Moment/product of Inertia of Spatial Rotation

Moment of inertia about an arbitrary axis (to
be filled in classroom}

ME 316 Lecture 6           19
Moment/product of Inertia of Spatial Rotation

Example 1                                  The bent rod
ABCD has a
weight of 1.5 lb/ft

Find: the location of center of gravity G and Ix’, Iy’, Iz’
ME 316 Lecture 6                    20
Moment/product of Inertia of Spatial Rotation

Example 2          The 1.5 Kg rod and 4 Kg disk

Find: Iz’ of the composite body
ME 316 Lecture 6            21
Moment/product of Inertia of Spatial Rotation

Example 3    The bent rod OABC has mass of 4 Kg/m

Find: Ix’x’ of the rod
ME 316 Lecture 6       22

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