ME 316 Lecture 6 1

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 2

Review of Planar Kinetics of a Rigid Body

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

If there is no any force applied to a body then the body will

remain its current status

Total resultant force in a direction

Motion or no-motion in a specific direction

The above is called “Newton First Law”

ME 316 Lecture 6 3

Review of Planar Kinetics of a Rigid Body

The basic evidence to support the Newton First Law is the

inertia

1. A property of an object;

2. A measure of how difficult or easy the motion of a body can be changed;

3. An inherent resistance to change

ME 316 Lecture 6 4

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)

A

B

IA ≠ IB

ME 316 Lecture 6 5

Review of Planar Kinetics of a Rigid Body

IA ≠ IB

B AExample (to be filled)

ME 316 Lecture 6 6

Review of Planar Kinetics of a Rigid Body

B GParallel – axis theorem

(to be filled)G is the center of gravity

ME 316 Lecture 6 7

Review of Planar Kinetics of a Rigid Body

1. Translation (T) (to be filled)

2. Rotation (R) (to be filled)

3. General (T + R) (to be filled)

Kinetic energy (K)

ME 316 Lecture 6 8

Review of Planar Kinetics of a Rigid Body

Translation:

GmaF

xGx amF )( yGy amF )(X and Y axes may not be horizontal or vertical; rather they could be in any direction but not in parallel.

Kinetics Equation

ME 316 Lecture 6 9

Review of Planar Kinetics of a Rigid Body

Rotation:

Case 1:

PMay not be

fixed

G

ME 316 Lecture 6 10

PyPxPP IamxamyM )()(

Review of Planar Kinetics of a Rigid Body

Case 2: P=G (mass center)

GG IM

ME 316 Lecture 6 11

Review of Planar Kinetics of a Rigid Body

PP IM

Case 3: when P is fixed point

ME 316 Lecture 6 12

Translation and rotation

xGx amF )( yGy amF )(

Review of Planar Kinetics of a Rigid Body

GG IM or

PyPxPP IamxamyM )()(

PP IM

or

ME 316 Lecture 6 13

Moment/product of Inertia of Spatial Rotation

z

dmzydmrI xxx )( 222

dmzxdmrI yyy )( 222

dmyxdmrI zzz )( 222

dmxyII yxxy

dmyzII zyyz

dmxzII zxxz

ME 316 Lecture 6 14

Product of Inertia

dmxyII yxxy

dmyzII zyyz

dmxzII zxxz

Orthogonal planes

ME 316 Lecture 6 15

Product of Inertia

0 xzxy II

x

0 yzxzxy III

ME 316 Lecture 6 16

Product of Inertia

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 17

Moment /Product of Inertia

Parallel-axis and parallel-plane theorems

ME 316 Lecture 6 18

Moment /Product of Inertia

Inertia Tensor – a compact way to express

ME 316 Lecture 6 19

Moment /Product of Inertia

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 20

Moment /Product of Inertia

Moment of inertia about an arbitrary axis

ME 316 Lecture 6 21

Moment /Product of Inertia

Example 1

Find: the location of center of gravity G and Ix’, Iy’, Iz’

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

ME 316 Lecture 6 22

Moment /Product of Inertia

Example 2

Find: Iz’ of the composite body

The 1.5 Kg rod and 4 Kg disk

ME 316 Lecture 6 23

Moment /Product of Inertia

The bent rod OABC has mass of 4 Kg/m

Find: Ix’x’ of the rod

Example 3