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کانیکمیسدمهـندادکشنه

Chapter 4 : Kinematics of Rigid Bodies

Advanced Dynamics

Lecturer: Hossein Nejat

• A rigid body is defined to be a collection of particles whose distance of separation is invariant. In this circumstance, any set of coordinate axes xyz that is scribed in the body will maintain its orientation relative to the body. Such a coordinate system forms a body-fixed reference frame. The orientation of xyz relative to the body and the location of its origin are arbitrary.
• The velocity and acceleration of point A on the body are given by:

• Example: Observation of the motion of the block reveals that, at this instant components relative to the body-fixed xyz coordinate system of the velocities of the corners are known to be (vA)x = 15 , (vB)x = 10 , (vc)z = 20 , (vD)x = 10 , and (vE)y = 5 , where all values are in units of meters/second. Determine whether these values are possible, and if so, evaluate the velocity of corner F.

• Example: Observation of the motion of the block reveals that at a certain instant the velocity of corner A is parallel to the diagonal AE. At this instant components relative to the body-fixed xyz coordinate system of the velocities of the other corners are known to be (vB)x = 10 , (vc)z = 20 , (vD)x = 10 , and (vE)y = 5 , where all values are in units of meters/second. Determine whether these values are possible, and if so, evaluate the velocity of corner F.

• Three independent direction angles define the orientation of a set of xyz axes. Eulerian angles treat this matter as a specific sequence of rotations. Chapter 4 : Kinematics of Rigid Bodies, 4.2 Eulerian Angles

• The transformation from may be found from to be
• The second transformation is given by
• The last transformation is given by

• We can use the expressions for 𝝎 and 𝝎′ to obtain angular acceleration as:
• Expressions for unit vectors in terms of body coordinate vector are:
• Thus, the angular velocity and angular acceleration are

• These expressions, particularly the one for angular aceleration, are quite complicated. For that reason, the x"y"z" axes, which do not undergo the spin, are sometimes selected for the representation. Then
• The new expressions for angular velocity and expressions can be derived using those previous formulas by setting 𝜙 = 0.

QUATERNIONS

• The quaternion's basic definition is a consequence of the properties of the direction cosine matrix [A]. It is shown by linear algebra that a proper real orthogonal 3 X 3 matrix has at least one eigenvector with eigenvalue of unity.
• The quaternion is defined as a vector by Hamilton 1866 , Goldstein 1950 , and Dalquist 1990.
• The elements of the quaternions, sometimes called the Euler symmetric parameters, can be expressed in terms of the principal eigenvector e (see Sabroff et al. 1965 ). They are defined as follows:

• A Comparison Between quaternion and Euler angles in the field of speed function dangles_dt=Euler(t,angles) wx= 10 sin(t); wy= 12 cos(sqrt( 2 )t); wz= 11 sin(sqrt( 3 )t+pi/ 4 ); sp=sin(angles( 3 )); cp=cos(angles( 3 )); tant=tan(angles( 2 )); sect= 1 /cos(angles( 2 )); dangles_dt=[(wysp+wzcp)sect;wycp- wzsp;wx+(wysp+wzcp)tant]; function dq_dt=quat(t,q) wx= 10 sin(t); wy= 12 cos(sqrt( 2 )t); wz= 11 sin(sqrt( 3 )t+pi/ 4 ); q=q/norm(q); dq_dt=0.5[wzq( 2 )-wyq( 3 )+wxq( 4 );- wzq( 1 )+wxq( 3 )+wyq( 4 );wyq( 1 )- wxq( 2 )+wzq( 4 );-wxq( 1 )-wyq( 2 )- wz*q( 3 )]; clc clear alle close all q 0 =[ 0 0 0 1 ]'; angles 0 =[ 0 0 0 ]'; TimeSpan=[ 0 10000 ]; t 0 = tic; [t,q]=ode 45 (@quat,TimeSpan,q 0 ); toc(t 0 ) t 2 = tic; [t 1 ,angles]=ode 45 (@Euler,TimeSpan,an gles 0 ); toc(t 2 ) Simulation Time Elapsed Time (Quaternion Method) Elapsed Time (Euler Method) 1000 1.88 3. 2000 2.64 8. 5000 4.46 22. 10000 5.80 46. ZYX or 321

• Example: A free gyroscope consists of a flywheel that rotates relative to the inner gimbal at the constant angular speed of 8,000 rev/min, while the rotation of the inner gimbal relative to the outer gimbal is 𝛾 = 0. sin( 100 𝜋𝑡) rad. The rotation of the outer gimbal is 𝛽 = 0.5sin( 50 𝜋𝑡) rad. Use the Eulerian angle formulas to determine the angular velocity and angular acceleration of the flywheel at t = 4 ms. Express the results in terms of components relative to the body-fixed xyz and space-fixed XYZ reference frames, where the z axis is parallel to the Z axis at t = 0.

4.3 INTERCONNECTIONS

Chapter 4 : Kinematics of Rigid Bodies, 4.3 Interconnections