3-D Kinetics of Rigid Bodies

Transcription

1 3-D Kinetics of Rigid Bodies Angular Momentum Generalized Newton s second law for the motion for a 3-D mass system Moment eqn for 3-D motion will be different than that obtained for plane motion Consider a rigid body in general motion in space : x-y-z are attached to the body with origin at mass center G : Angular velocity ω of body becomes the angular velocity of x-y-z as observed from fixed X-Y-Z 1

2 3-D Kinetics of Rigid Bodies Angular Momentum : Absolute angular momentum of the G v i is the absolute vel of m i ω x ρ i is rel vel of m i wrt G as observed from non-rotating axes x-y-z (By reversing the order of cross product in first term and changing sign) : With the origin at G, first term is zero since 2

3 3-D Kinetics of Rigid Bodies Angular Momentum Substituting dm for m i and ρ for ρ i in the second term gives: HG [ ρ ( ω ρ)] dm If the rigid body rotates about a fixed point O: : x-y-z are attached to the body with origin at O : Both body and axes have an ang vel ω Angular O: for the rigid body v i = ω x r i Substituting dm for m i and r for r i : HO [ r ( ω r)] dm 3

4 3D Kinetics of Rigid Bodies Angular Momentum HG [ ρ ( ω ρ)] dm In both cases, the posn vectors ρ i and r i are given by the same expression: xi + yj + zk Both eqns have same form Symbol H will be used for either case Expanding the two eqns: HO [ r ( ω r)] : Components of ω are invariant wrt the integrals over the body they become constant multipliers of the integrals dm : Applying cross product expansion to triple product: Also, 4

5 3D Kinetics of Rigid Bodies Angular Momentum HG [ ρ ( ω ρ)] dm HO [ r ( ω r)] dm Expression for H: Components of H: : General expression for angular momentum mass center G a fixed point O for a rigid body rotating with instantaneous ang vel ω : In each of the two cases, reference axes x-y-z are attached to the rigid body moment of inertia integrals and product of inertia integrals become invariant with time. 5

6 3D Kinetics of Rigid Bodies Angular Momentum : If x-y-z axes were to rotate with respect to an irregular body - inertia integrals = f(t) - complexity of angular momentum equations : If a rigid body its axis of symmetry - inertia integrals are not affected by the angular position of the its spin axis one axis of the reference system x-y-z may be chosen to coincide with the axis of rotation, and the other two axes are not allowed to turn with the body Two components of angular momentum are required to be accounted for: (1) due to angular velocity Ω of the reference axes (2) along the spin axis due to the relative the axis 6

7 3D Kinetics of Rigid Bodies Angular Momentum Principal Axes The array of moments and products of inertia in the above eqn is called Inertia Matrix or Inertia Tensor :: A unique orientation of the reference axes x-y-z for a given origin for which the products of inertia = 0 For this orientation, the inertia matrix takes the form: The axes x-y-z are called Principal Axes of Inertia The Principal Moments of Inertia for a given origin represent the maximum, the minimum, and an intermediate value of MI If the coordinate axes coincide with the principal axes of inertia, angular the mass center a fixed point: H and ω have different directions in general except (a) when the body one of the principal axes of inertia or when (b) I xx = I yy = I zz.

8 3D Kinetics of Rigid Bodies Angular Momentum Transfer Principle for Angular Momentum Momentum properties of a rigid body may be represented by: Resultant linear momentum vector G = mv through the mass center, and Resultant angular momentum vector H the mass center H G has the properties of a free vector (for convenience it is represented through mass center G). Vectors G and H G have properties analogous to those of a force and a couple Angular any point P = the free vector H G + moment of the linear momentum vector P Transfer theorem for Angular Momentum Applicable for any point on or off the body 8

9 3D Kinetics of Rigid Bodies Kinetic Energy For any general system of mass, rigid or non-rigid: v is the vel of mass center G, and ρ i is the vel of mi wrt translating reference frame moving with G Total KE of a mass system = KE due to translation of the system as a whole + KE due to motion of all particles relative to the mass center. The translational term: r is the vel v of mass center and G is the linear momentum of the body For a rigid body, the relative term becomes KE due to mass center ω is the angular velocity of the body Since dot and cross may be interchanged in triple scalar product, i.e., P x Q R = P Q x R 9

10 Example on 3-D Kinetics Solution: Mass Calculations: m A = (0.1)(0.125)70 = kg m B = (0.075)(0.15)70 = kg 10

11 Example on 3-D Kinetics Solution: m A = kg, m B = kg Part A: Angular momentum is given by: ω x = 0 ω y = 0 ω z = 30 rad/s Calculating the moments and products of inertia (neglect thickness of the plate): For MI and PI: Consider Area (face) of the plate normal to the axis under consideration. Plate A 11

12 Example on 3-D Kinetics Solution: m A = kg, m B = kg Part A: Angular momentum Plate A Since the plate is symmetric and thickness is neglected Plate B 12

13 Example on 3-D Kinetics Solution: m A = kg, m B = kg Part A: Angular momentum Plate B Summing the respective terms for the two plates: 13

14 Example on 3-D Kinetics Solution: m A = kg, m B = kg Part A: Angular momentum Part B: KE Since the body is fixed point O: 14

15 ME101 Announcements : Collect all previous corrected tutorials (11 tutorials) and quizzes (2 quizzes) from your respective tutors - Collect corrected Tutorial 12 answer script during endsem week : Check attendance records from TAs. : End-sem Exam on 29 April 2015 (Wednesday) during 2:00-5:00 pm : Corrected answer sheets will be shown on 08 May 2015 (time and venue will be informed later) On-spot re-evaluation! : Entire syllabus for end-sem 20-25% :: pre-midsem 80-75% :: post-midsem 15

16 ME101 Announcements : No class on 23 April 2015 (Thursday) morning - Make-up class on 20 April 2015 (Monday) evening 6:00-6:55pm in L1 End Sem Exam : Do not ask any questions/doubts as questions are self explanatory. In case of any confusion, use appropriate assumptions and state clearly in the answer sheet. : Write page numbers on the answer sheet Indicate page numbers of all answers on cover page of answer sheet : Start answering new questions from fresh page. 16