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CHAPTER 7 Basic Concepts and Kinematics of Rigid Body Motion

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CHAPTER 7 Basic Concepts and Kinematics of Rigid Body Motion CHAPTER 7 Basic Concepts and Kinematics of Rigid Body Motion 7.1 Degrees-of-freedom: F2 F1 (of a rigid body) m f m2 1 12 f Consider three 21 Z f r 23 unconstrained particles 1 C m f32 rC 3 O m r3 3 X F3 Y 1 The positions are defined by ri xi i yi j zi k i = 1, 2, 3. degrees of freedom = n = 9 = 3N Now, constrain the particles (three particles placed l m m1 1 2 at the corners of a Z triangle whose sides r1 l2 are formed by rigid l3 massless rods) O m r3 3 X Y 2 Now: there are three constraints r1 r 2 l 1;; r 3 r 2 l 2 r 1 r 3 l 3 2 2 2 2 or(x2 x 1 )( y 2 y 1 )( z 2 z 1 ) l 1 0, etc . n = 3(N) - 3 = 6 degrees of freedom. Now: as another particle is introduced, its position is specified by 3 additional coordinates, but also have 3 additional constraints. four rigidly connected particles also have only 6 degrees of freedom. 3 In general: a rigid body (any collection of particles whose relative positions are fixed) has 6 degrees of freedom. • translational motion of a point on the body - specified by 3 translational degrees of freedom. • rotational motion about the specified point - 3 rotational degrees of freedom. 4 Laws of motion for a system of particles (extended to a rigid body) a) F mrc translational motion of the C.M. (3 degrees of freedom) d b) MH about C.M. or an inertially dt fixed point rotational motion about the C.M. (3 rotational degrees of freedom) 5 7.2 Moments of Inertia Z Recall: the notation P and definitions rP C i C m fi2 r mi The equation for O C ri moment about an X Fi arbitrary point P is: Y d M H mr p p c p dt 6 Now, the angular momentum about the point P is N H m p i i i1 i where - velocity of as viewed by a i mi non-rotating observer translating with P. (relative velocity in an inertial frame) For a rigid body - suppose that P is fixed in the body = constant, and i i i where - angular velocity of the body. 7 Thus, the angular momentum about P is N Z H m p i i i t1 dm By analogy: for a rigid body r rotating with r angular velocity O P P X V Y H (r) ( )dV ; dm (r)dV V 8 We now consider the various cases: Reference point P is at origin: Z x i yj zk dm xi y j z k Then 22 ( ) [(yz ) x O=P xyy xz z][ i xy x 22 X V ()]x zyz yz j Y 22 [()] xzx yz y x y z k 9 Let us define: I (y2 z2 )dV xx V I (x2 z2 )dV ; I (x2 z2 )dV yy zz V V These are the moments of inertia Ixx about x axis Iyy about y axis through the reference point O P Izz about z axis 10 Similarly, we define products of Inertia: I I xy dV xy yx V I I xz dV xz zx V I I yzdV yz zy V Hp Hx i Hy j Hz k (angular momentum vector for the body, or angular momentum about P) 11 Here, HIIIx xx x xy y xz z HIIIy yx x yy y yz z HIIIz zx x zy y zz z In compact notation: 33 HP Iij j ei i 1 j 1 where e1 i , e2 j, e3 k and symbols 1, 2, 3 x, y, z. 12 The components of the angular momentum are 3 Hi I ij j , i 1,2,3 j1 Note that I ii is the second moment of the mass distribution with respect to a Cartesian axis. Radius of gyration Iii (effective location of kii , 1,2,3 m a mass point) or 2 Iii mk i 13 Three moments of inertia and three products of inertia specify the inertia properties of a rigid body with regards to rotational motion. • Note that P is fixed in the rigid body and is the angular velocity of the rigid body. • No assumption is made concerning the rotational motion of the xyz system. The angular velocity in terms of the xyz system, i j k x y z is valid at the instant considered. 14 • as well as are, in general, x , y ,z Iij functions of time depending on the orientation of xyz relative to the rigid body. • To avoid the difficulties associated with treating as functions of time, often one chooses xyz system that is fixed to the rigid body and rotates with it. • Called a body-fixed coordinate system 15 7.3 MATRIX NOTATION: Consider H Hx i H y j H z k If i , j , k are known, the three scalar components H x , H y , H z can be used to represent H H x • We write H H y as a column vector. H z • A force F can be represented as , a row vector. F Fx , Fy , Fz 16 Fx or F F y as a column vector. Fz A square matrix is a n n array of elements: e.g. the elements of inertia I ij can be written as a square matrix I xx I xy I xz I I yx I yy I yz I zx I zy I zz This is called an inertia matrix for the body. 17 The angular velocity vector can be represented as x , y a column vector. z T • x y z ,the transpose of a column vector gives a row vector, etc. H • consider , the product of a row vector with a column vector HHHHH x x y y z z 18 In scalar components, it is easy to see that H Iij j ei can be expressed as H I Clearly, multiplication of with I transforms into the vector H , usually with a different magnitude as well as direction. In general, The angular velocity and the angular momentum vectors for a rigid body are in different directions 19 Assignment: Complete the review of matrix operations in 7.3. 7.4 Kinetic Energy For a system of n particles, the kinetic energy n is 112 T mvc m i i i 22i1 where • vC - speed of the center of mass th • i - velocity of the i particle as viewed from the C.M. 20 For a set of particles rigidly connected and the assemblage rotating with angular velocity , ii i i()()() i i i i 11nn Trot m i i i i m i i 22ii11 (using permutation in scalar triple product) Now, for a continuous mass distribution 21 11 T dV dV rot 22VV HC 1 dm or T H rot2 C If P is a fixed point: 1 TH rot2 P O=P In vector matrix notation V 11T THHrot P P 22 22 1 T Since HITI , P rot 2 If one uses xyz coordinate system located at the center of mass of the rigid body, 1 T TIIII [ 2 2 2 rot2 xx x yy y zz z 2IIIxyxy 2 xzxz 2 yzyz ]/ 2 In summation, notation 1 33 TIrot ij i j 2 ij11 23 Ex: If has the direction of one of the coordinate axes at an instant, 1 TI 2 rot 2 Here, I - moment of inertia about the axis of rotation, - instantaneous angular velocity of the rigid body. 24 7-6 Translation of Coordinate Axes Z’ Z O’x’y’z’- coordinate system located at the C=O’ centroid of the body; Y’ O Oxyz-any other coord Y system; X X’ O’=C-the centroid O: center of mass: the coordinates of the center of mass in Oxyz system are (xc , yc , zc ) 25 Let I xx , I yy , I zz - moments of inertia about xyz axes system I xx , I yy , I zz = moments of inertia about centroidal axes m - mass of the body Easy to show (read 7.6) that for moments of inertia 2 2 I xx I xx m(yc zc ) 2 2 I yy I yy m(xc zc )parallel axis theorem 2 2 I zz I zz m(xc yc ) 26 For products of inertia I xy I xy mxc yc I xz I xz mxc zc I yz I yz myc zc • Note that the moments of inertia about the centroidal axes are the smallest. • The products of inertia may increase or decrease compared to those about centroidal axes depending on the particular case. 27 7.7 Rotation of Coordinate Axes Consider two different coordinate systems. We assume that the origins for the two systems coincide. z’ z P r consider a vector: O y’ r xi y j zk x xi y j zk x’ y These are two ways of expressing the same vector r . 28 The two systems are characterized by unit vectors i, j,k and i, j,k Let i lxx ilyx jlzx k • are the cosines of the angles lxx , lyx , lzx made by the x axis with the x, y and z directions, respectively. Note that, i lx x i l y x j l z x k 1 l2 l 2 l 2 1 xx y'' y z z 29 Consider the example: z’(k’) xx' direction x(i) z’x angles of yx' x’x x-axis y’x O zx' x’(i’) y’(j’) Also, lx'' x cosx'' x , l y x cos y x , lzx' coszx' 30 Similarily, we can write j lx y i l y y j l z y k k lx z i l y z j l z z k Thus, the vector can be written as r x() lx x i l y x j l z x k r xi y j zk y() lx y i l y y j l z y k z() lx z i l y z j l z z k x i y j z k 31 In vector-matrix notation x' lx''' x l x y l x z x y' ly''' x l y y l y z y z' lz''' x l z y l z z z In more compact notation r lr where r and r are representations of r and r in the two coordinate systems.
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