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**Kinetics of Rigid Bodies:**

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

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**(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

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**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

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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}.

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**(Q) Simplified Version Plane Motion**

Mass center G lies in the xy-plane. dm r Now, After the similar calculation, we have

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product of inertia moment of inertia Using

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**(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,

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

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**(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

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**There are various ways of choosing this small mass element for integration.**

A specific mass element may be easier to use than other elements.

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

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2nd moment of area

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**If the density of the body is uniform,**

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**a rigid body summation of several simple shape rigid bodies**

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

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**= 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 !!!!!

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**(Q) What is the Parallel-Axis Theorem for Moments of Inertia?**

measurement of the location of the mass center from the mass center = m

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z’

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**(Q) More about the Product of Inertia**

dm In 2-D space y Rz x y x

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**(Q) What is the effect of symmetry on the product of inertia?**

x y z x y x z z y

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z y x y z z x y x

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**(Q) What is the Parallel-Axis Theorem for Product of Inertia?**

From definition or But, mass center from the mass center and Therefore,

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**(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

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

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

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

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**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

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**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

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a c G b

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

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z z’ b a c y Θ y’ Θ x’ x

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Slender rod

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**Thin rectangular plate**

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Thin circular plate

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

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

For Plane Motion

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For Plane Motion

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**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.

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**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

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Or next page

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x x’ z’ z Sym.

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**(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?

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**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

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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}.

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**If we use the Cartesian coordinate system,**

In vector-matrix form,

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Or

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= 75 rad/s constant = 25 rad/s constant

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= 75 rad/s constant = 25 rad/s constant or more mathematically

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= 75 rad/s constant = 25 rad/s constant ∴ Solvable!

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Therefore,

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