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  • 2103-617 Advanced Dynamics

    Thitima Jintanawan

  • 2cCopyright 2005Thitima Jintanawan

  • Contents

    1 Kinematics 51.1 Evolution of Kinematics . . . . . . . . . . . . . . . . . . . . . 51.2 Position Vector, Velocity, and Acceleration . . . . . . . . . . . 51.3 Angular Velocity . . . . . . . . . . . . . . . . . . . . . . . . . 71.4 Rate of Change of a Constant-Length Vector . . . . . . . . . 81.5 Moving Coordinate Systems . . . . . . . . . . . . . . . . . . . 101.6 Coordinate Transformation . . . . . . . . . . . . . . . . . . . 17

    1.6.1 First set of Euler anglesprecession-nutation-spin () 201.6.2 Second set of Euler anglesyaw-pitch-row () . . . . 20

    1.7 Angular velocity related to Euler angles . . . . . . . . . . . . 231.8 A Finite Motion . . . . . . . . . . . . . . . . . . . . . . . . . 25

    1.8.1 Transformation matrices for a nite rotation . . . . . 261.8.2 Transformation matrices for a general motion . . . . . 30

    2 Linear and Angular Momentums 352.1 Dynamics of a System of Particles: a Review . . . . . . . . . 35

    2.1.1 Total mass . . . . . . . . . . . . . . . . . . . . . . . . 352.1.2 First moment of mass . . . . . . . . . . . . . . . . . . 352.1.3 Linear momentum . . . . . . . . . . . . . . . . . . . . 372.1.4 Angular momentum . . . . . . . . . . . . . . . . . . . 372.1.5 Moment of force . . . . . . . . . . . . . . . . . . . . . 382.1.6 Laws of linear and angular momentum . . . . . . . . . 38

    2.2 Angular Momentum of a Rigid Body . . . . . . . . . . . . . . 402.3 Mass Moment of Inertia . . . . . . . . . . . . . . . . . . . . . 42

    3 Dynamics of a Rigid Body 453.1 Newton-Euler Equations of a rigid body . . . . . . . . . . . . 453.2 Modied Eulers equations . . . . . . . . . . . . . . . . . . . . 493.3 Introduction to stability of a spin body . . . . . . . . . . . . 53

    3

  • 4 CONTENTS

    4 Multi-Body Mechanical System 574.1 Degrees of Freedom (DOF) . . . . . . . . . . . . . . . . . . . 574.2 Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . 574.3 Constraint Equations . . . . . . . . . . . . . . . . . . . . . . . 574.4 Classication of Constraints . . . . . . . . . . . . . . . . . . . 604.5 Number of DOF vs. Driving Forces . . . . . . . . . . . . . . . 614.6 Dynamic Analysis of Multi-Body Mechanical Systems . . . . 614.7 Example Problem: Dynamics of Two-Link Arms . . . . . . . 62

    5 Principle of Virtual work 695.1 Virtual Displacement and Virtual Work . . . . . . . . . . . . 695.2 Holonomic and Nonholonomic Constraints . . . . . . . . . . . 695.3 Generalized Coordinates and Jacobian . . . . . . . . . . . . . 715.4 Principle of Virtual Work . . . . . . . . . . . . . . . . . . . . 755.5 DAlembert principle . . . . . . . . . . . . . . . . . . . . . . . 76

    6 Lagrange Mechanics 816.1 Kinetic Energy . . . . . . . . . . . . . . . . . . . . . . . . . . 816.2 Potential Energy . . . . . . . . . . . . . . . . . . . . . . . . . 826.3 Remarks on Properties of Generalized Coordinates for the

    System with Holonomic constraints . . . . . . . . . . . . . . . 846.4 Derivation of Lagranges equations . . . . . . . . . . . . . . . 856.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 876.6 Lagrange Multiplier . . . . . . . . . . . . . . . . . . . . . . . 94

    7 Stability Analysis 997.1 Equilibrium, Quasi-Equilibrium, and Steady States . . . . . . 997.2 Stability of Equilibrium or Steady State . . . . . . . . . . . . 99

  • Chapter 1

    Kinematics

    In this chapter various coordinate systems, such as cartesian and cylindri-cal coordinates, are introduced. Position vector, velocity and accelerationof particles and rigid bodies are formulated using dierent reference coordi-nates. Each coordinate system is related to the other through the coordinatetransformation. For the 3-D transformation, two dierent sets of Euler an-gles: precession-nutation-spin and yaw-pitch-roll, are conventionally used.Finally the transformation matrix used to describe a nite motion of rigidbodies is revealed.

    1.1 Evolution of Kinematics

    Prior to 1950s: Express velocity v and acceleration a in terms of scalarcomponents and use graphical method to determine total magnitudeand direction

    1950s and later: Express velocity v and acceleration a using vector ap-proach

    Recent years: Express the rotation with a matrix and utilize the matrixoperation for calculating the cross product. The matrix approach canbe simply implement in a computer simulation program.

    1.2 Position Vector, Velocity, and Acceleration

    Fig. 1.1 shows a particle moving in a 3-dimensional (3-D) space. Letsintroduce a cartesian or rectangular coordinate system XY Z as shown in

    5

  • 6 CHAPTER 1. KINEMATICS

    X

    Y

    Z

    ij

    k

    rx

    ry

    r

    rz

    particle

    moving path

    O

    Figure 1.1: A cartesian coordinate system

    Fig. 1.1 in which all coordinates are orthogonal to each other and its axesdo not change in direction. If we choose XY Z in Fig. 1.1 as an inertial orxed reference frame1, the absolute motion of the particle in Fig. 1.1 can bedescribed by a position vector r as follows

    r = rxi+ ryj+ rzk (1.1)

    where i, j, and k are the unit vectors of XY Z and rx, ry, and rz are scalarcomponents of r in X, Y , and Z coordinates. The position vector r can bealternatively presented in a matrix form as a 3 1 column matrix given by

    r = [ rx ry rz ]T (1.2)

    Note that the position vector r must be measured from the origin O of thechosen inertial frame.

    Figure 1.2 shows another set of coordinate system so called the cylindri-cal coordinates z, with their unit vectors eeez. In Fig. 1.2, the positionvector r expressed in terms of eeez is

    r = e + zez (1.3)

    The absolute velocity v is dened as a time derivative of the position vectorr given by

    v =drdt

    (1.4)

    = rxi+ ryj+ rzk= e + e + zez

    1An inertial or xed reference frame is the coordinate system whose origin O is xedin space

  • 1.3. ANGULAR VELOCITY 7

    X

    Y

    Z

    z

    e

    e

    r

    ez

    z

    Figure 1.2: A cylindrical coordinate system

    The absolute acceleration a is dened as a time derivative of the velocity vgiven by

    a =dvdt

    (1.5)

    = rxi+ ryj+ rzk

    =( 2

    )e +

    ( + 2

    )e + zez

    1.3 Angular Velocity

    Figure 1.3 shows a rigid cylinder having a rotation about n axis. The abso-lute angular velocity of the rigid body is dened as

    =d

    dtn (1.6)

    = 1e1 + 2e2 + 3e3

    where 1, 2, and 3 are components of the angular velocity in an arbitraryrectangular coordinate system with unit vectors e1, e2, and e3. The angularvelocity can be expressed in a matrix form as

    =

    0 3 23 0 12 1 0

    (1.7)

  • 8 CHAPTER 1. KINEMATICS

    v

    n

    r

    (t)A e1

    e2

    e3

    Figure 1.3: Angular velocity

    The velocity at point A in Fig. 1.3 is then

    v = r (1.8) r

    Equation (1.9) indicates that the cross product can be represented by thematrix multiplication or

    v =

    v1v2

    v3

    =

    0 3 23 0 12 1 0

    r1r2

    r3

    (1.9)

    Note that for the matrix multiplication in (1.9), components of and rmust be expressed in the same coordinate system.

    1.4 Rate of Change of a Constant-Length Vector

    The Theorem in the Vector of Calculus states that The time derivative ofa xed length vector c is given by the cross product of its rotation rate and the vector c itself.

    dcdt

    = c (1.10)Example 1.1 :The defense jet plane as shown in Fig. 1.4 operates in a roll maneuver withrate of and simultaneously possesses a yaw maneuver (turn to left) witha rate of . Determine a relative velocity of point C on the horizontalstabilizer at coordinates (b, a, 0), observed from the C.G. of the plane.

  • 1.4. RATE OF CHANGE OF A CONSTANT-LENGTH VECTOR 9

    X

    Y

    Z

    G

    C

    ez

    ey

    ex

    Figure 1.4: A defense jet plane

    SolutionFrom Fig. 1.4, the position vector of point C relative to the C.G. is a

    xed length vector given in terms of the body coordinate system as

    r(rel)c = bex + aey = [ b a 0 ]T (1.11)

    Hence the relative velocity of C is

    v(rel)c = r(rel)c r(rel)c (1.12)where = ey + ez is the rotation rate or the angular velocity of thereference coordinates moving with the body. can be written in a matrixform as

    =

    0 0 0 0 0

    Therefore

    v(rel)c =

    0 0 0 0 0

    ba

    0

    = [ a b b]T

    As another example, the unit vectors ijk for any rotating system ofcoordinates xyz is also the xed length vector. Hence the rate of change ofthese ijk vectors can be determined from the same theorem as i = i,j = j, and k = k, where is the angular velocity of such rotatingcoordinate system xyz.

  • 10 CHAPTER 1. KINEMATICS

    X

    Y

    Z

    x

    z

    yo

    path of origin o

    Figure 1.5: Translating coordinate systems

    X

    Y

    Z

    x

    z

    y

    o

    Figure 1.6: Rotating coordinate systems

    1.5 Moving Coordinate Systems

    Any moving coordinate system xyz used to describe the motion can bedivided into 3 types depending on its motion with respect to the inertialframe XY Z. They are

    1. Translating coordinate systems (Fig. 1.5)

    2. Rotating coordinate systems (Fig. 1.6)

    3. Translating and rotating coordinate systems (Fig. 1.7)

    A moving coordinate system, chosen such that it is attached to a movingbody, is normally used as a reference frame to describe kinematics of the

  • 1.5. MOVING COORDINATE SYSTEMS 11

    X

    Y

    Z

    x

    z

    y

    o

    path of origin o

    Figure 1.7: Translating and rotating c