Kinetics of Rigid Bodies:

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Kinetics of Rigid Bodies:
From Riley’s Dynamics Chapter 16 Kinetics of Rigid Bodies: Newton’s Laws

(Q) What are the Euler’s Equations of Motion?
Newton’s Law  applies only to the motion of a single particle translation R R G G only translation translation + rotation Newton’s 2nd Law Euler’s Equations of Motion

Euler’s Equations of Motion
Rotation of a Rigid Body moment ∴ Starting Point Moment of F & f about A Newton’s 2nd Law Substitution yields

What’s this? After integration, we can get the general form of the Euler’s equations of motion. Very general equation about rotation.  Need to unify the coordinate systems to {Axyz}.

(Q) Simplified Version  Plane Motion
Mass center G lies in the xy-plane. dm r Now, After the similar calculation, we have

product of inertia moment of inertia Using

(Note) The 1st 2 equations are required to maintain the plane motion about z-axis,
especially for non-symmetrical geometry case. If the body is symmetric about the plane of motion,

If (symmetry) + (acceleration of the point A = 0) If (symmetry) + (A = G)

(Q) More about the Moment of Inertia
For the particle dm IF widely distributed THEN larger moment of inertia For the entire body It uses the information about its geometry. ∴ THE SAME MASS BUT DIFFERENT GEOMETRY DIFFERENT MOMENT OF INERTIA

There are various ways of choosing this small mass element for integration.
A specific mass element may be easier to use than other elements.

You may treat the rigid body as a system of particles.

2nd moment of area

If the density of the body is uniform,

a rigid body  summation of several simple shape rigid bodies
Practical approach a rigid body  summation of several simple shape rigid bodies composite body

= I : moment of inertia about the axis (the moment of inertia about
gyration [ʤaiəréiʃən] n. U,C 선회, 회전, 선전(旋轉); 〖동물〗 (고둥 따위의) 나선. ㉺∼al [-ʃənəl] ―a. 선회의, 회전의. (Q) What is the radius of gyration? = k m m I : moment of inertia about the axis (the moment of inertia about the axis) = mk2 NO useful physical interpretation!! Maybe baseball  Home Run !!!!!

(Q) What is the Parallel-Axis Theorem for Moments of Inertia?
measurement of the location of the mass center from the mass center = m

z’

(Q) More about the Product of Inertia
dm In 2-D space y Rz x y x

(Q) What is the effect of symmetry on the product of inertia?
x y z x y x z z y

z y x y z z x y x

(Q) What is the Parallel-Axis Theorem for Product of Inertia?
From definition or But, mass center from the mass center and Therefore,

(Q) What is the Rotation Transformation of Inertia Properties?
z’ Consider z y x’ y’ x We know that We can represent i’, j’, and k’ w.r.t. i, j, and k. Substitution yields

or old new [R] rotation transformation matrix from old to new frame a vector in the old frame a vector in the new frame

(Example) y’ y x Θ Θ x’ Rotation about z’-axis It means that [R] is an orthonormal matrix.

z’ z y Θ Θ y’ Rotation about x’-axis x’ x z Θ Θ z’ Rotation about y’-axis

Now, the rotational kinetic energy is
This term will be derived in the next chapter. Since energy is invariant Let : known old frame Let : unknown new frame old new from old to new

Claim: [I] = ? (Example) z a = 240 mm b = 120 mm m = 60 kg y c = 90 mm
(Idea) z’ z y G y’ x’ x

a c G b

z’ a c G b y’ x’ By using the parallel axis theorem,

z z’ b a c y Θ y’ Θ x’ x

Slender rod

Thin rectangular plate

Thin circular plate

Quiz #1 Y’ X’ Z’ {x’y’z’} 좌표 시스템에 대해 표현된 Inertia matrix를 구하시오.

(Q) How to analyze the General Plane Motion of NonSymmetric Bodies?
For Plane Motion

For Plane Motion

Claim: 5 reactions & T ? 30 mm dia. m = 1.2 kg l = 220 mm
600 rpm ccw increasing in speed at the rate of 60 rpm per second 30 mm dia. m = 1.2 kg l = 220 mm = /2-40/2 40 mm dia. 8.5 kg Claim: 5 reactions & T ? 120 mm dia. m = 7.5 kg Bearing A resists any motion in the z-direction.

For the entire system The same result for this sphere since
zG and xG are minus sign. 120 mm dia. m = 7.5 kg 30 mm dia. m = 1.2 kg l = 220 mm = /2-40/2 The same result for this bar since zG and xG are minus sign. 40 mm dia. 8.5 kg For the entire system

Or next page

x x’ z’ z Sym.

(Q) How to analyze the 3-D Motion of a Rigid Body?
x Y O X All vectors are represented w.r.t. the body-fixed {xyz}. Recall How?

Euler’s Equations of Motion
Rotation of a Rigid Body moment ∴ Starting Point Moment of F & f about A Newton’s 2nd Law Substitution yields

What’s this? After integration, we can get the general form of the Euler’s equations of motion. Very general equation about rotation.  Need to unify the coordinate systems to {Axyz}.

If we use the Cartesian coordinate system,
In vector-matrix form,

Or