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Kinetics (I) 1. Review of Kinetics of Planar Mechanisms - Inertia (mass and moment of inertia) - Governing equation (Newton’s Law) 2. Moment/Product of Inertia – spatial rotation ME 316 Lecture 6 1 Review of Planar Kinetics of a Rigid Body Kinetics: how does a body move under the force and / or moment ? 2. If there is no any force or moment applied to a body, the body will remain its current status. This statement is called Newton First Law. 3. Force and moment are vector, and they have direction, so the Newton First Law corresponds to the direction. Ex: No force in horizontal direction, no motion due to Newton First Law; however, there is a force (gravity) in vertical direction, there is motion ME 316 Lecture 6 2 Review of Planar Kinetics of a Rigid Body The basic evidence to support the Newton First Law is the inertia 1. A property of the body; 2. A measure of how difficult or easy the motional state of the body can be changed; 3. An inherent resistance to change a body’s motion state ME 316 Lecture 6 3 Review of Planar Kinetics of a Rigid Body Two types of inertia (depending on types of causes) Cause is force: inertia is mass (m) Cause is moment: inertia is the moment of inertia (IP) Moment causes rotation. Because rotation depends on the location of a point about which a body rotates, moment of inertia differs with respect to different points on a body. Thus, in the following, we have A IA ≠ IB B ME 316 Lecture 6 4 Review of Planar Kinetics of a Rigid Body Example (to be filled) B A Find IA and IB IA ≠ I B ME 316 Lecture 6 5 Review of Planar Kinetics of a Rigid Body Parallel – axis theorem B G (to be filled) To verify the parallel axis theorem, namely Find IB G is the center of gravity IG+ml2 ME 316 Lecture 6 6 Review of Planar Kinetics of a Rigid Body Kinetic energy (K) 1. Translation (T): (to be filled) 2. Rotation (R): (to be filled) 3. General (T + R): (to be filled) ME 316 Lecture 6 7 Review of Planar Kinetics of a Rigid Body Kinetics Equation Translation: F ma G F x m(aG ) x F y m(aG ) y X and Y axes may not be horizontal or vertical; rather they could be in any direction but not in parallel. G: center of gravity of a body ME 316 Lecture 6 8 Review of Planar Kinetics of a Rigid Body Rotation: M P ym(aP ) x xm(aP ) y I P Case 1: G P May not be fixed ME 316 Lecture 6 9 Review of Planar Kinetics of a Rigid Body Case 2: P=G (mass center) P M G I G G Case 3: P is fixed point G M P I P P ME 316 Lecture 6 10 Review of Planar Kinetics of a Rigid Body Translation and rotation: F x m(aG ) x F y m(aG ) y Case 1: M G I G Case 2: M P ym(aP ) x xm(aP ) y I P Case 3: M P I P ME 316 Lecture 6 11 Moment/product of Inertia of Spatial Rotation z I zz rz2 dm ( x 2 y 2 )dm I yy ry2 dm ( x 2 z 2 )dm I xx rx2 dm ( y 2 z 2 )dm I xy I yx xydm I yz I zy yzdm I xz I zx xzdm ME 316 Lecture 6 12 Moment/product of Inertia of Spatial Rotation Product of Inertia: The concept of Orthogonal planes I xy I yx xydm I yz I zy yzdm I xz I zx xzdm ME 316 Lecture 6 13 Moment/product of Inertia of Spatial Rotation Product of Inertia (special cases) 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. ME 316 Lecture 6 14 Moment/product of Inertia of Spatial Rotation Examples of Product of Inertia (special cases) x I xy I xz 0 I xy I xz I yz 0 ME 316 Lecture 6 15 Moment/product of Inertia of Spatial Rotation Parallel-axis and parallel-plane theorems ME 316 Lecture 6 16 Moment/product of Inertia of Spatial Rotation Inertia Tensor – a compact way to express {to be filled in classroom} ME 316 Lecture 6 17 Moment/product of Inertia of Spatial Rotation Principal axes - principal moments of inertia If the coordinate axes are oriented 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, and hence the coordinate axes are principal axes of inertia ME 316 Lecture 6 18 Moment/product of Inertia of Spatial Rotation Moment of inertia about an arbitrary axis (to be filled in classroom} ME 316 Lecture 6 19 Moment/product of Inertia of Spatial Rotation Example 1 The bent rod ABCD has a weight of 1.5 lb/ft Find: the location of center of gravity G and Ix’, Iy’, Iz’ ME 316 Lecture 6 20 Moment/product of Inertia of Spatial Rotation Example 2 The 1.5 Kg rod and 4 Kg disk Find: Iz’ of the composite body ME 316 Lecture 6 21 Moment/product of Inertia of Spatial Rotation Example 3 The bent rod OABC has mass of 4 Kg/m Find: Ix’x’ of the rod ME 316 Lecture 6 22