Dynamics 12e. Copyright 2010 Pearson Education South Asia Pte Ltd. Chapter 20 3D Kinematics of a Rigid Body

Transcription

1 Engineering Mechanics: Dynamics 12e Chapter 20 3D Kinematics of a Rigid Body

2 Chapter Objectives Kinematics of a body subjected to rotation about a fixed axis and general plane motion. Relative-motion analysis of a rigid body using translating and rotating axes.

3 Chapter Outline 1. Rotation About a Fixed Point 2. The Time Derivative of a Vector Measured from Either a Fixed or Translating- Rotating System 3. General Motion 4. Relative-Motion Analysis Using Translating and Rotating Axes

4 20.1 Rotation About a Fixed Point Euler s Theorem two infinitesimal rotations about different axes passing through a point are equivalent to a single resultant rotation about an axis passing through the point. If more than two rotations are applied, they can be combined into pairs, and each pair can be further reduced to combine into one rotation.

5 20.1 Rotation About a Fixed Point Finite Rotations If component rotations used in Euler s theorem are finite, it is important that the order in which they are applied be maintained. Finite rotations do not obey law of vector addition, and hence cannot be classified as vector quantities.

6 20.1 Rotation About a Fixed Point Finite Rotations Each rotation has a magnitude of 90 and a direction defined by the right-hand rule. The resultant orientation of the block is θ 2 + θ 1 θ 1 +θ 2 as shown, the resultant position of the block is not the same as in previous diagram.

7 20.1 Rotation About a Fixed Point Finite Rotations Finite rotations do not obey the commutative law of addition (θ 1 + θ 2 θ 2 + θ 1 ) and cannot be classified as vectors. Infinitesimal Rotations When defining 3D angular motions, only rotations ti which h are infinitesimally it i small will be considered. Such rotations may be classified as vectors, since they can be added vectorially in any manner.

8 20.1 Rotation About a Fixed Point Infinitesimal Rotations For two infinitesimal rotations dθ 1 +dθ 2 on the body, point P can moves along the path dθ 1 x r +dθ 2 x r to P For two successive rotations ti occurred in the order dθ 2 + dθ 1, the resultant t displacements of P is dθ 2 x r + dθ 1 x r

9 20.1 Rotation About a Fixed Point Angular Velocity When body is subjected to angular rotation dθ, the angular velocity of the body is defined by ω= d θ dt The line specifying the direction of ω, which is collinear with dθ is the instantaneous axis of rotation.

10 20.1 Rotation About a Fixed Point Angular Velocity Since dθ is a vector quantity, so too is ω, and it follows from vector addition that if the body is subjected to two component angular motions, and The resultant angular velocity is ω = ω 1 + ω 2 Angular Acceleration The body s angular acceleration is determined from time derivative of angular velocity, α = ω&

11 20.1 Rotation About a Fixed Point Angular Acceleration As the direction of the instantaneous axis of rotation changes in space, the locus of points defined by the axis generates a fixed space cone. If the change in this axis is viewed with respect to the rotating body, the locus of the axis generates a body cone.

12 20.1 Rotation About a Fixed Point Angular Acceleration As the direction of the instantaneous axis of rotation changes in space, the locus of points defined by the axis generates a fixed space cone. The locus of the axis generates a body cone. Velocity Once ω is specified, the velocity of any point P on a body is v = ω r

13 20.1 Rotation About a Fixed Point Acceleration If ω and α are known, the acceleration of any point P on the body is a= α r+ ω ( ω r) The equation defines the acceleration of a point located on a body subjected to rotation about a fixed axis.

14 20.2 The Time Derivative of a Vector Measured from Either a Fixed or Translating-Rotating System* The angular velocity ω is specified in terms of its component angular motions. The disk spins about the horizontal y axis at ω s while it rotates about the vertical z axis at ω p Resultant angular velocity is ω = ω s + ω p Consider the x, y, z axes of the moving frame to have angular velocity Ω

15 20.2 The Time Derivative of a Vector Measured from Either a Fixed or Translating-Rotating System* We have A= A i+ A j+ Ak x Since the directions of the components do not change with respect to the moving reference, ( A) = A i+ A j Ak xyz x y & & & + Directions of i, j, k change only on account of the rotation, Ω,, of the axes, in general y z & z A & = A & i+ A& j+ A& k+ A & i+ A & j+ Ak& x y z x y z

16 20.2 The Time Derivative of a Vector Measured from Either a Fixed or Translating-Rotating System* For both the magnitude and direction of di, we can define i using the cross product, i = Ω x I & i =Ω i & j=ω j k& =Ω k Therefore we can obtain A & = & xyz +Ω ( A ) +Ω A Ω x A is the change of A caused by the rotation of the of the x, y, z frame.

17 20.3 General Motion A translating coordinate system will be used to define relative motion. A rigid body can be subjected to general motion in 3D for which the angular velocity is ω and the angular acceleration is α Instantaneous axis of rotation is v B/A = ω x r B/A a B/A = α x r B/A + ω x (ω x r B/A )

18 20.3 General Motion For translating axes the relative motions are related to absolute motions by v = v + ω r / B A B A a = a + α r + ω ( ω r ) B A B / A B / A For general plane motion, α and ω are always parallel or perpendicular to the plane of motion.

19 20.4 Relative-Motion Analysis Using Translating and Rotating Axes* The locations of points A and B are specified relative to the X, Y, and Z frame of reference by position vectors r A and r B. The base point A represents the origin of the x, y, z coordinate system, which is translating and rotating with respect to X, Y, Z. All vectors are measured w.r.t the X, Y, Z frame of reference.

20 20.4 Relative-Motion Analysis Using Translating and Rotating Axes* Position If the position of B w.r.t A is specified by the relative-position vector r B/A, thus Velocity r = r + r B A B / A The velocity of point B measured from X, Y, Z is r & = ( r& ) +Ω r = ( v +Ω r B / A B/ A xyz B/ A B/ A) xyz B/ A Here (v B/A ) xyz is the relative velocity of B wrta w.r.t A measured from x, y, z. Thus, v = v +Ω r + ( v ) B A BA / BA / xyz

21 20.4 Relative-Motion Analysis Using Translating and Rotating Axes* Acceleration The acceleration of point B measured from X, Y, Zis where d dt d v & = v +Ω & B & A r & + dt B / A+Ω r B / A ( v B / A ( v B / A) xyz = ( v& B / A) xyz + Ω ( vb / = ( a B / A) xyz + Ω ( vb / ) xyz A A ) ) xyz xyz

22 20.4 Relative-Motion Analysis Using Translating and Rotating Axes* Acceleration (a B/A ) xyz is the relative acceleration of B w.r.t A measured from x, y and z, thus a = a + Ω & r + B A B / A + Ω ( Ω rb / A) + 2Ω ( v B / A) xyz ( ab / A ) xyz

23 20.4 Relative-Motion Analysis Using Translating and Rotating Axes* Procedure for Analysis Coordinate Axes Solutions are obtained if at the instant when 1. The origins are coincident 2. The axes are collinear 3. The axes are parallel

24 20.4 Relative-Motion Analysis Using Translating and Rotating Axes* Procedure for Analysis Kinematics Equations After the origin of the moving reference, A, is defined and the moving point B is specified, the equations should be written in symbolic form as v + a B = v A +Ω r B / A+ ( v B / A ) xyz B = a A +Ω & r +Ω ( Ω r B/ A B/ A + 2 Ω ( v B / A) xyz+ ( ab/ A) xyz )