Contents. Dynamics and control of mechanical systems. Focuses on

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1 Dnamics and control of mechanical sstems Date Da (/8) Da (3/8) Da 3 (5/8) Da 4 (7/8) Da 5 (9/8) Da 6 (/8) Content Review of the basics of mechanics. Kinematics of rigid bodies - coordinate transformation, angular velocit vector, description of velocit and acceleration in relativel moving frames. Euler angles, Review of methods of momentum and angular momentum of sstem of particles, inertia tensor of rigid bod. Dnamics of rigid bodies - Euler equations, application to motion of smmetric tops and groscopes and problems of sstem of bodies. Kinetic energ of a rigid bod, virtual displacement and classification of constraints. D Alembert s principle. ntroduction to generalized coordinates, derivation of Lagrange's equation from D Alembert s principle. Small oscillations, matrix formulation, Eigen value problem and numerical solutions. Modelling mechanical sstems, ntroduction to MATLAB, computer generation and solution of equations of motion. ntroduction to complex analtic functions, Laplace and Fourier transform. PD controllers, Phase lag and Phase lead compensation. Analsis of Control sstems in state space, pole placement, computer simulation through MATLAB. Contents Focuses on 4 Derivation of Euler angles 4 Review of principle of impulse and angular momentum 4 Angular moments in 3D and inertia tensors 4 Examples

2 Euler angles Application areas RR - Aircraft and aerospace simulation - Robot simulation - Computer graphics - Orientation of mobile phones 3 ROV built b UiS students 6 ntroduction Euler angles 4 Euler angles are the three angles used to represent a rigid bod in 3D rotations. èusing Euler angles an rotation can be described b 3 successive rotations about a linearl independent angles è Speciall in multibod dnamics, Euler angles are useful to express the motion of rotating bodies è Euler angles relate an rotating frame (non-inertial frame) fixed to a rigid bod with the fixed inertial frame through the successive rotations Note: The "inertial frame" is an Earth-fixed set of axes that is used as an unmoving reference. 4 Man different tpes of Euler angles can be driven depending on the sequence of rotations 4

3 Euler angles Let s consider the following sequence of rotation: αà b à g, for rotation about x, Y and Z (à --3 sequence) Determine R RgRbRα 5 Euler angles Multibod danamics for robotics and simulation software in ADAMS uses the 3--3 sequence Show that the following are correct for the 3--3 sequence, i.e. Rotation about z b α (R zα)à rotation about x b b (R x b) à rotation about z b g (R z g) éca - Sa ù é ù R za Sa Ca ; Rx' b Cb - Sb ë û ë Sb Cb û R z'' g écg - Sg Sg Cg ë ù û Resultant Eulerian rotation matrix CαCγ SαSγCβ (CαSγ + SαCγCβ) SγSβ S Sin C Cos SαCγ + CαSγCβ SαSγ + CαCγCβ CαSβ SγSβ CαSβ Cβ 6

4 Review of momentum and angular momentum 4 Principle of impulse and momentum Newotn' s Second Law: dv F ma m ; ntegrating : t ò L F t $!#!" Linear impulse ò v v mdv mv - mv $!#!" Linear momentum 4 The principle sas: impulse applied to an object during a time interval ( t t ) is equal to the change in the object s linear momentum 4 Note: mpulse force is a force of large magnitude acting over a short period of time Average force : Fav t -t 7 t ò t F Review of momentum. 4 Principle of impulse and momentum The linear momentum (L) of a rigid bod is defined as L m v c where v c is velocit of the mass center è The linear momentum vector L has a magnitude equal to (mv c ) and a direction defined b v c. The angular momentum ( c ) of a rigid bod rotating with angular velocit ω about its mass center (c) is defined as c c ω Where the direction of c is perpendicular to the plane of rotation, and c is mass moment of inertia about its mass center. 8

5 Review of momentum 4 Principle of impulse and momentum... When a rigid bod undergoes rectilinear or curvilinear translation, its angular momentum is zero because ω è L m v c and c For a rigid bod motion about a fixed axis passing through point O: - Linear momentum: L mv c - Angular momentum about C: c c ω - Angular momentum about the center of rotation O. O ( r c mv c ) + c ω O ω where o is calculated about O 9 Moment of linear momentum Review of momentum 4 Angular momentum dv r x F r x ma r x m Angular momentum d dr dv about O where ( r x mv) x mv + r x m! o r x mv %"$"# v m( vxv) à Moment of linear momentum Rate of change of angular momentum d d dv ( r x mv r x m r x $!#!" F o ) Rate of change of the moment of momentum about point o Rate of change of angular momentum d o t r x F Þ ò!!" ò ( r xf) do o - o t $!!# Angular impulse Principle of angular impulse and momentum

6 Review of momentum 4 Angular momentum in matrix form Where r xi + j + zk and v (v x, v, v z) Exercise: Show that the following components of the angular momentum are correct Review of momentum. 4 Consrevation of angular momentum and The angular momentum is conserved when i.e. f the resultant moment about a fixed point O is zero, then angular momentum remains constant, or it is said to be conserved. Note: Angular momentum ma be conserved in one coordinate (e.g., x), but not necessaril in others (e.g., or z)

7 Review of momentum 4 Angular momentum of a rigid bod in 3D - Let s define an arbitrar angular velocit vector as w (w x w w z) - Equation of angular momentum: r x mv r x m(r x w) m(r x (r x w)), where r (x,, z) 4 Using equation for double cross product: A x (B x C) B(A.C) C(A.B) r x (r x w) x ω(r r) -r(r ω) + z w - x w - x z w m( + z ) wx - m x w m $!!#!!" $!#!" (. ) - ( - - ( ) x ( ) ( ) z ( x) wx + ( z + x ) w - ( z) wz ( z x) w x - ( z ) w + ( x + ) w z x zx xx z x ( z + x ) - m(. x) wx + m w - m(. z) wz $!#!" $!!#!!" $!#!" x. z) wz $!#!" xz ( x + ) w z z - m( z. x) wx - m( z. ) w + m $!#!" $!#!" $!!#!!" zz z 3 Review of momentum 4 nertia Tensors of rigid bodies - Expressed in matrix form é ë x z ù é û ë Angular momentum xx x zx x z ù éw w û ëw xz z zz nertia Tensor of the bod about its mass center x z ù û Note: The term tensor refers to a higher-order vector. A vector is written as a column, while a tensor is written as a matrix. The inertial tensor has the propert that ij ji è it is a smmetirc tensor of nd order and T Angular speed Alternativel: Letting xx,, zz 33, x, etc. nertia Tensors transform the vector w into the vector c 4 é ë 3 ù é û ë 3 3 ij ij w ù éw w û ë w j 3 ù û

8 Review of momentum 4 nertia Tensors of rigid bodies in 3D - nertia tensor is calculated through a point independent of the axis of rotation, and once that is established, the angular momentum about an axis through that point can be determined. à f a new sstem of axes is used - a new inertia tensor is obtained - the angular momentum c f(w) remain the same because it is independent of the choice of coordinate sstem - f the coordinate sstem coincides with the principal axes (axes where the angular momentum and the angular speed coincide), calculation of inertia tensors is especiall simple. é x ë ù û z Where x, and z are principal centroidal moments f x z, c and w are collinear, otherwise (in general) the principal moments are different and hence c and w are in different directions 5 Review of momentum 4 Example: Consider that the cube of dimension a has a uniform distribution of mass with densit r m/a 3. () Find the inertia tensor for rotation about one of the corners as given in the figure. () Assume that the cube rotates about x-axis and find the angular momentum 6

9 Example, nertia Tensor for Cube 4 The mass of the cube is evenl distributed è the summation can be converted to integration For example, the upper-left element becomes: xx where 4 Thus, a a a ò ò ò % dx d dzr + z % r M / a xx 3 ( ), denotes the mass densit. a a a a a a ( ò ò ò ò ò ò ) % r dx d dz+ dx d z dz % ra Ma Smmetr condition give that xx zz, and similarl for the offdiagonal elements. Example, Cont d 4 The off-diagonal elements have the form 5 - a dx a d a dz% rx, - % r a xdx a d a dz - % ra - Ma. 4 B smmetr, all the off-diagonal elements have the same value. Thus, the moment of inertia tensor is é 3Ma - 4Ma - 4Ma ù é ù Ma 4Ma 3Ma 4Ma [about a corner] - - 4Ma 4Ma 3Ma ë- - û ë-3-3 8û 4 The angular momentum for rotation about an axis through this corner. Examples: x ò ò ò ò ò ò Rotation about x axis (w (w,, )): L w Ma / (8w, -3w, -3w) Ma w (/3, -/4, -/4). L not in same direction as rotation axis 4 Rotation about diagonal through O ( ω w / 3 (,,)): é ùéù éù Ma w Ma w Ma L ω ω 6 ë ûëû ëû L is in same direction as rotation axis

10 Example, Cont d 4 f the origin is shifted to the center of the cube, the diagonal element integrals are just as eas, simpl change the limits, e.g. a/ a/ a/ a/ a/ a/ ( ò-a/ ò-a/ ò-a/ ò-a/ ò-a/ ò-a/ ) % r dx d dz+ dx d z dz % ra ( a/) Ma. xx but the off-diagonals are all odd functions, so the change the limits leads them to go to zero, e.g. a / a / a / a / a / a / - dx d dz% rx, - % r xdx d dz. x ò ò ò ò ò ò -a/ -a/ -a/ -a/ -a/ -a/ 4 The inertia tensor is then diagonal, i.e é ù Ma Ma. 6 6 ë û 4 Note that, no matter what direction w is, L is alwas parallel to it: Ma L ω ω. 6 Summar and questions n this lecture, the following are covered 4 Euler angles: Example applications and derivations 4 Review of the principle of angular impulse and moment 4 Review of angular momentum and its conservation 4 Angular moments in 3D and inertia tensors? Next: Dnamics of rigid bodies - Euler equations, application to motion of smmetric tops and groscopes

11 Example B M D A 4 Example Two masses A and B of,4 kg each with initial velocit of m/s experience a moment M,6 Nm. Find the speeds of the masses when time t 4 s. C Sol: Appling the principle of angular impulse and momentum t t M Change of angular momentum of mass A and B (r.m(v v )) (,3)(,4) (v m/s),4v,48 Where 4 M,6 (4-),4 è,4,4v,48 è v m/s Example, Cont d 4 Example A rod assembl rotates around its z-axis. The mass C is kg and its initial velocit is m/s. A moment and force both act as shown, where M 8t + 5 Nm and F 6 N. Find the velocit of mass C after seconds. Sol: Appling the principle of angular impulse and momentum about z-axis (the axis of rotation) Angular impulse: M + 46 (r x F) 46 5 (8t ) + (,75 x A 6) B 85,33 nm.s Change of angular momentum: z z r.m(v v ),75()(v ) 7,5v + 5 B the principle of angular impulse and momentum 7,5v ,33 è v 3,4 m/s