Download presentation

1
**Kinetics of Rigid Bodies:**

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

2
**(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

3
**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

4
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}.

5
**(Q) Simplified Version Plane Motion**

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

6
product of inertia moment of inertia Using

7
**(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,

8
If (symmetry) + (acceleration of the point A = 0) If (symmetry) + (A = G)

9
**(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

10
**There are various ways of choosing this small mass element for integration.**

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

11
**You may treat the rigid body as a system of particles.**

12
2nd moment of area

13
**If the density of the body is uniform,**

14
**a rigid body summation of several simple shape rigid bodies**

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

15
**= 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 !!!!!

16
**(Q) What is the Parallel-Axis Theorem for Moments of Inertia?**

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

18
z’

19
**(Q) More about the Product of Inertia**

dm In 2-D space y Rz x y x

20
**(Q) What is the effect of symmetry on the product of inertia?**

x y z x y x z z y

21
z y x y z z x y x

22
**(Q) What is the Parallel-Axis Theorem for Product of Inertia?**

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

23
**(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

24
or old new [R] rotation transformation matrix from old to new frame a vector in the old frame a vector in the new frame

25
(Example) y’ y x Θ Θ x’ Rotation about z’-axis It means that [R] is an orthonormal matrix.

26
z’ z y Θ Θ y’ Rotation about x’-axis x’ x z Θ Θ z’ Rotation about y’-axis

27
**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

28
**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

29
a c G b

30
z’ a c G b y’ x’ By using the parallel axis theorem,

31
z z’ b a c y Θ y’ Θ x’ x

32
Slender rod

33
**Thin rectangular plate**

34
Thin circular plate

40
Quiz #1 Y’ X’ Z’ {x’y’z’} 좌표 시스템에 대해 표현된 Inertia matrix를 구하시오.

41
**(Q) How to analyze the General Plane Motion of NonSymmetric Bodies?**

For Plane Motion

42
For Plane Motion

43
**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.

45
**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

47
Or next page

48
x x’ z’ z Sym.

49
**(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?

50
**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

51
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}.

52
**If we use the Cartesian coordinate system,**

In vector-matrix form,

53
Or

55
= 75 rad/s constant = 25 rad/s constant

56
= 75 rad/s constant = 25 rad/s constant or more mathematically

57
= 75 rad/s constant = 25 rad/s constant ∴ Solvable!

61
Therefore,

Similar presentations

OK

Mechanics for Engineers: Dynamics, 13th SI Edition R. C. Hibbeler and Kai Beng Yap © Pearson Education South Asia Pte Ltd 2013. All rights reserved. MOMENT.

Mechanics for Engineers: Dynamics, 13th SI Edition R. C. Hibbeler and Kai Beng Yap © Pearson Education South Asia Pte Ltd 2013. All rights reserved. MOMENT.

© 2018 SlidePlayer.com Inc.

All rights reserved.

To make this website work, we log user data and share it with processors. To use this website, you must agree to our Privacy Policy, including cookie policy.

Ads by Google