Chapter 21: Three Dimensional Kinetics of a Rigid Body

1. Chapter 21: Three Dimensional Kinetics of a Rigid Body

2. Chapter Objectives • To introduce the methods for finding the moments of inertia and products of inertia of a body about various axes. • To show how to apply the principles of work and energy and linear and angular momentum to a rigid body having three-dimensional motion. • To develop and apply the equations of motion in three dimensions. • To study the motion of a gyroscope and torque-free motion.

3. Chapter Outline • Moments and Products of Inertia • Angular Momentum • Kinetic Energy • Equations of Motion • Gyroscopic Motion • Torque-Free Motion

4. 21.1 Moments and Products of Inertia Moment of Inertia • Consider the rigid body • The moment of inertia for a differential element dm of the body about any one of the three coordinate axes is defined as the product of the mass of the element and the square of the shortest distance from the axis to the element • From the figure,

5. 21.1 Moments and Products of Inertia Moment of Inertia • For mass moment of inertia of dm about the x axis, • For each of the axes,

6. 21.1 Moments and Products of Inertia Moment of Inertia • Moment of inertia is always a positive quantity • Hence, summation of the product of mass dm is always positive

7. 21.1 Moments and Products of Inertia Product of Inertia • Defined with respect to a set of orthogonal planes as the product of the mass of the element and the perpendicular (or shortest) distances from the plane to the element • For product of inertia dIxy for element dm, dIxy = xy dm • Note dIyx = dIxy

8. 21.1 Moments and Products of Inertia Product of Inertia • For each combination of planes,

9. 21.1 Moments and Products of Inertia Product of Inertia • Unlike the moment of inertia, which is always positive, the product of inertia may be positive, negative or zero • Result depends on the sign of the two defining coordinates, which vary independently from one another • If either one or both of the orthogonal planes are planes of symmetry for the mass, the product of inertia with respect to these planes will be zero

10. 21.1 Moments and Products of Inertia Product of Inertia • In such cases, the elements of mass will occur in pairs on each side of the plane of symmetry • On one side of the plane, the product of inertia will be positive while on the other side, the product of inertia will be negative, the sum therefore yielding zero • Consider the y-z plane of symmetry, Ixy = Ixz = 0

11. 21.1 Moments and Products of Inertia Product of Inertia • Calculation of Iyz will yield a positive result since all elements of mass are located using only positive y and z coordinates • Consider the x-z and y-z being planes of symmetry, Ixy = Iyz = Ixz = 0

12. 21.1 Moments and Products of Inertia Parallel Axis and Parallel Plane Theorems • If G has coordinates xG, yG, zG defined from the z, y, z axes, the parallel axis equations used to calculate the moments of inertia about the x, y, z axes are • Products of inertia of a composite body are computed in the same manners as the body’s moments of inertia

13. 21.1 Moments and Products of Inertia Parallel Axis and Parallel Plane Theorems • Parallel axis theorem is used to transfer products of inertia of the body from a set of three orthogonal planes passing through the body’s mass center to a corresponding set of parallel planes passing through some other points O • For parallel axis equations,

14. 21.1 Moments and Products of Inertia Inertia Tensor • Inertial properties of a body are completely characterized by nine terms, six of which are independent of one another • For an inertia tensor, • An inertia tensor has a unique set of values for a body when it is computed for each location of the origin O and orientation of the coordinates origin

15. 21.1 Moments and Products of Inertia Inertia Tensor • For point O, specify a unique axes inclination for which the products of inertia for the body are zero when computed with respect to these axes • For a diagonalized inertia tensor, • For principal moments of inertia for the body

16. 21.1 Moments and Products of Inertia Inertia Tensor • Of these three principal moments of inertia, one will be a maximum and one will be a minimum • If the coordinates axes are orientated such that two of the three orthogonal planes containing the axes are planes of symmetry for the body, then all the products of inertia for the body are zero with respect to the coordinate planes, hence the coordinate axes are principle axes of inertia

17. 21.1 Moments and Products of Inertia Moment of Inertia About an Arbitrary Axis • Consider the body where nine elements of the inertia tensor have been computed for the x, y, z axes having an origin at O • Determine the moment of inertia of the body about the Oa axis, for which the direction is defined by vector ua • IOa = ∫b2 dm where b is the perpendicular distance from dm to Oa

18. 21.1 Moments and Products of Inertia Moment of Inertia About an Arbitrary Axis • Position of dm is located using r, b = rsinθ, which represents the magnitude of the cross-product ua x r • For moment of inertia, • Provided

19. 21.1 Moments and Products of Inertia Moment of Inertia About an Arbitrary Axis • Hence, • For moment of inertia,

20. 21.1 Moments and Products of Inertia Moment of Inertia About an Arbitrary Axis • If the inertia tensor is specified for the x, y, z axes, the moment of inertia of the body about the inclined Oa axis can be found • Direction cosines ux, uy and uz can be determined • Direction angles α, β and γ made between the positive Oa axis and the positive x, y, z axes can be determined from the direction cosines ux, uy and uz, respectively

21. 21.1 Moments and Products of Inertia Example 21.1 Determine the moment of inertia of the bent rod about the Aa axis. The mass of each of the three segments are shown in the figure.

22. 21.1 Moments and Products of Inertia View Free Body Diagram Solution • For moment of inertia of a slender rod, I = 1/12 ml2 • For each segment of the rod,

23. 21.1 Moments and Products of Inertia Solution • For unit vector of the Aa axis, • Thus, • Hence,

24. 21.2 Angular Momentum • Consider the rigid body having a mass m and center of mass at G • X, Y, Z coordinate system represent an inertial frame of reference and its axes are fixed or translating with a constant velocity • Angular momentum as measured from this reference will be computed relative to the arbitrary point A

25. 21.2 Angular Momentum • Position vectors rA and ρA are drawn from the coordinates to point A and from A to the ith particle of the body • If the particle’s mass is mi, for angular momentum about point A, (HA)i = ρA x mivi where vi represent the particle’s velocity measured from the X, Y, Z coordinate system • If the body has an angular velocity ω, vi = vA + ω x ρA

26. 21.2 Angular Momentum • Hence, (HA)i = ρA x mi(vA + ω x ρA ) = (ρAmi) x vA + ρA x (ω x ρA)mi • Summing all the particles of the body, HA = (∫mρA dm) x vA + ∫m ρA x (ω x ρA) dm

27. 21.2 Angular Momentum Fixed Point O • If A becomes fixed point O in the body, vA = 0 HO = ∫m ρO x (ω x ρO) dm

28. 21.2 Angular Momentum Center of Mass G • If A is located at the center of mass G, ∫mρA dm = 0 HG = ∫m ρG x (ω x ρG) dm

29. 21.2 Angular Momentum Arbitrary Point A • In general, A may be some point other than O or G HA = ρG/A x mvG + HG • Angular momentum consists of two parts – the moment of the linear momentum mvG of the body about point A added (vectorially) the angular momentum HG

30. 21.2 Angular Momentum Rectangular Components of H • Choosing a second set of x, y, z axes having an arbitrary orientation relative to the X, Y, Z axes, H= ∫m ρ x (ω x ρ) dm • Expressing in terms of x, y, z components, Hxi + Hyj+ Hzk= ∫m (xi + yj+ zk) x [(ωxi + ωyj+ ωzk) x (xi + yj+ zk)]dm • Expanding and equating the i, j and k components, Hx = Ixxωx - Ixyωy - Ixzωz Hy = Iyyωy - Iyxωx - Iyzωz Hz = Izzωz - Izxωx - Izxωx

31. 21.2 Angular Momentum Rectangular Components of H • If the x, y, z coordinate axes are oriented such as they become the principal axes of inertia for the body at that point • If these axes are used, for products of inertia, Ixy = Iyz = Izx = 0 • If the principal moments of inertia about the x, y, z axes are represented as Ix = Ixx, Iy = Iyy, Iz = Izz, for components of angular momentum, Hx = Ixωx, Hy = Iyωy, Hz = Izωz

32. 21.2 Angular Momentum Principle of Impulse and Momentum • For principle of impulse and momentum, • In 3D, each vector term can be represented by 3 scalar components and 6 scalar equations • 3 equations relate the linear impulse and momentum in the x, y, z directions and the other 3 equations relate the body’s angular impulse and momentum about the x, y, z axes

33. 21.3 Kinetic Energy • Consider the rigid body which has a mass m and center of mass at G • For kinetic energy of the ith particle of the body having a mass mi and velocity vi measured relative to the inertial X, Y, Z frame of reference, • Provided the velocity of an arbitrary point A of the body is known,

34. 21.3 Kinetic Energy • For kinetic energy of the particle, • The kinetic energy for the entire body is obtained by summing the kinetic energies of the body • The last term on the right can be re-written as

35. 21.3 Kinetic Energy • Hence,

36. 21.3 Kinetic Energy Fixed Point O • If A is a fixed point O in the body,

37. 21.3 Kinetic Energy Center of Mass G • If A is located at the center of mass G of the body, • Kinetic energy consists of the translational kinetic energy of the mass center and the body’s rotational kinetic energy

38. 21.3 Kinetic Energy Principle of Work and Energy • Used to solve problems involving force, velocity and displacement • Only one scalar equation can be written for each body T1 + ∑U1-2 = T2

39. 21.4 Equations of Motion Equations of Translation Motion • Defined in terms of acceleration of the body’s mass center, which is measured from an inertial X, Y, Z reference • For equations of translation motion in vector form, • For scalar equations of translation motion, • For sum of all external forces acting on the body,

40. 21.4 Equations of Motion Equations of Rotational Motion which states that sum of moments about a fixed point O of all the external forces acting on a system of particles (contained in a rigid body) is equal to the time rate of change of the total angular momentum of the body about point O • When moments of external forces acting on the particles are summed about the system’s mass center G, one again obtain summation ∑MG to the angular momentum HG

41. 21.4 Equations of Motion Equations of Rotational Motion • Consider a system of particles where X, Y, Z represents an inertial frame of reference and the x, y, z axes with origin at G, translate with respect to this frame • In general, G is accelerating, so by definition, the translating frame is not an inertial reference

42. 21.4 Equations of Motion Equations of Rotational Motion • For the angular momentum of the ith particle with respect to this frame, • Taking the time derivatives, • By definition, • Thus, the first term on the right side is zero since the cross-product of equal vectors equals zero.

43. 21.4 Equations of Motion Equations of Rotational Motion • Since • Hence, • When the results are summed, for the time change of the total angular momentum of the body computed relative to point G,

44. 21.4 Equations of Motion Equations of Rotational Motion • For relative acceleration for the ith particle, where ai and aG represent accelerations of the ith particle and point G measured with respect to the inertial frame of reference • By vector cross-product, • By definition of the mass center, since position vector r relative to G is zero,

45. 21.4 Equations of Motion Equations of Rotational Motion • Using the equation of motion, • For the rotational equation of motion for the body, • If the scalar components of the angular momentum HO or HG are computed about x, y, z axes that are rotating with an angular velocity Ω, which may be different from the body’s angular velocity ω, then the time derivative

46. 21.4 Equations of Motion Equations of Rotational Motion must be used to account for the rotation of the x, y, z axes as measured from the inertial X, Y, Z axes • For the time derivative of H, • There are 3 ways to define the motion of the x, y, z axes

47. 21.4 Equations of Motion x, y, z Axes having motion Ω = 0 • If the body has general motion, the x, y, z axes may be chosen with origin at G, such that the axes only translate to the inertial X, Y, Z frame of reference • However, the body may have a rotation ω about these axes, and therefore the moments and products of inertia of the body would have to be expressed as functions of time

48. 21.4 Equations of Motion x, y, z Axes having motion Ω = ω • The x, y, z axes may be chosen with origin at G, such that they are fixed in and move with the body • The moments and the products of inertia of the body relative to these axes will be constant during the motion

49. 21.4 Equations of Motion x, y, z Axes having motion Ω = ω • For a rigid body symmetric with respect to the x-y reference plane, and undergoing general plane motion,

50. 21.4 Equations of Motion x, y, z Axes having motion Ω = ω • Hence, • If the x, y and z axes are chosen as principal axes of inertia, the products of inertia are zero, Ixx = Iz • For the Euler equations of motion,