Dynamics and control of mechanical systems

Transcription

1 JU 18/HL Dnamics and control of mechanical sstems Date Da 1 (3/5) 5/5 Da (7/5) Da 3 (9/5) Da 4 (11/5) Da 5 (14/5) Da 6 (16/5) Content Revie 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, Revie 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 generalied coordinates, derivation of Lagrange's equation from D Alembert s principle. Small oscillations, matri formulation, Eigen value problem and numerical solutions. Modelling mechanical sstems, ntroduction to MATLAB, computer generation and solution of equations of motion. ntroduction to comple 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. 1

2 Contents Focuses on 4 Derivation of Euler angles 4 Revie of principle of impulse and angular momentum 4 Angular moments in 3D and inertia tensors 4 Eamples JU 18/HL

3 Euler angles Application areas RR Aircraft and aerospace simulation Robot simulation Computer graphics Orientation of mobile phones JU 18/HL 3 ROV built b UiS students 16

4 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 epress the motion of rotating bodies è Euler angles relate an rotating frame (noninertial frame) fied to a rigid bod ith the fied inertial frame through the successive rotations Note: The "inertial frame" is an Earthfied set of aes that is used as an unmoving reference. JU 18/HL 4 Man different tpes of Euler angles can be driven depending on the sequence of rotations 4

5 Euler angles Let s consider the folloing sequence of rotation: αà b à g, for rotation about, Y and Z (à 13 sequence) JU 18/HL Determine R RgRbRα 5

6 JU 18/HL 6 Euler angles Multibod danamics for robotics and simulation softare in ADAMS uses the 313 sequence Sho that the folloing are correct for the 313 sequence, i.e. Rotation about b α (R α )à rotation about b b (R b ) à rotation about b g (R g ) û ù ë é û ù ë é û ù ë é 1 1 ; 1 '' ' g g g g b b b b a a a a g b a C S S C R C S S C R C S S C R Resultant Eulerian rotation matri CαCγ SαSγCβ (CαSγ + SαCγCβ) SγSβ SαCγ + CαSγCβ SαSγ + CαCγCβ CαSβ SγSβ CαSβ Cβ S Sin C Cos

7 Revie of momentum and angular momentum 4 Principle of impulse and momentum Neotn' s Second La: F ma m dv ; dt L ntegrating : t F dt t1 $!#!" ò Linear impulse ò v v1 mdv mv mv1 $!#!" Linear momentum 4 The principle sas: impulse applied to an object during a time interval ( t 1 t ) is equal to the change in the object s linear momentum JU 18/HL 4 Note: mpulse force is a force of large magnitude acting over a short period of time Average force : 1 Fav t t 7 1 t ò t 1 Fdt

8 Revie of momentum. 4 Principle of impulse and momentum The linear momentum (L) of a rigid bod is defined as L m v c here 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 (H c ) of a rigid bod rotating ith angular velocit ω about its mass center (c) is defined as JU 18/HL H c c ω Where the direction of H c is perpendicular to the plane of rotation, and c is mass moment of inertia about its mass center. 8

9 Revie of momentum 4 Principle of impulse and momentum... When a rigid bod undergoes rectilinear or curvilinear translation, its angular momentum is ero because ω è L m v c and H c JU 18/HL For a rigid bod motion about a fied ais passing through point O: Linear momentum: L mv c Angular momentum about C: H c c ω Angular momentum about the center of rotation O. H O ( r c mv c ) + c ω O ω here o is calculated about O 9 Moment of linear momentum

10 Revie of momentum 4 Angular momentum r here Rate of dh dt F r ma d dt ( r mv) change of angular momentum d dv ( r mv r m r $!#!" F dt dt o ) dv r m dt dr dv mv + r m! dt dt %"$"# v m( vv) Angular momentum about O H o r mv à Moment of linear momentum Rate of change of the moment of momentum about point O JU 18/HL Rate of change of angular momentum dh dt 1 o t r F Þ ò!!" ò H ( r F) dt dho Ho H o1 t1 $!!# Angular impulse H1 Principle of angular impulse and momentum

11 Revie of momentum 4 Angular momentum in matri form Where r i + j + k and v (v, v, v ) Eercise: Sho that the folloing components of the angular momentum are correct JU 18/HL 11

12 Revie of momentum. 4 Consrevation of angular momentum and JU 18/HL The angular momentum is conserved hen i.e. f the resultant moment about a fied point O is ero, then angular momentum remains constant, or it is said to be conserved. Note: Angular momentum ma be conserved in one coordinate (e.g., ), but not necessaril in others (e.g., or ) 1

13 JU 18/HL 13 Revie of momentum 4 Angular momentum of a rigid bod in 3D Let s define an arbitrar angular velocit vector as ( ) Equation of angular momentum: H r mv r m(r ) m(r (r )), here r (,, ) 4 Using equation for double cross product: A (B C) B(A.C) C(A.B) r (r ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ω) r(r r) ω(r ( ) ( ) ( ) m m m m m m m m m!! "! $! # $!#!" $!#!" $!#!"!! "! $! # $!#!" $!#!" $!#!"!! "! $! # ). ( ). ( ). ( ). ( ). ( ). ( H H H

14 Revie of momentum JU 18/HL 4 nertia Tensors of rigid bodies Epressed in matri form éh H ëh ù û Angular momentum é ë é ë ù û nertia Tensor of the bod about its mass center ù û Note: The term tensor refers to a higherorder vector. A vector is ritten as a column, hile a tensor is ritten as a matri. The inertial tensor has the propert that ij ji è it is a smmetirc tensor of nd order and T Angular speed Alternativel: Letting 11,, 33, 1, etc. nertia Tensors transform the vector into the vector H c 14 éh H ë H 1 3 ù û é ë H 1 3 ij ij ù û j é ë 1 3 ù û

15 Revie of momentum JU 18/HL 4 nertia Tensors of rigid bodies in 3D nertia tensor is calculated through a point independent of the ais of rotation, and once that is established, the angular momentum about an ais through that point can be determined. à f a ne sstem of aes is used a ne inertia tensor is obtained the angular momentum H c f() remains the same because it is independent of the choice of coordinate sstem f the coordinate sstem coincides ith the principal aes (aes here the angular momentum and the angular speed coincide), calculation of inertia tensors is especiall simple. é ë ù û Where, and are principal centroidal moments f, H c and are collinear, otherise (in general) the principal moments are different and hence H c and are in different directions 15

16 Revie of momentum 4 Eample: Consider that the cube of dimension a has a uniform distribution of mass ith densit r m/a 3. (1) Find the inertia tensor for rotation about one of the corners as given in the figure. () Assume that the cube rotates about ais and find the angular momentum JU 18/HL 16

17 Eample, nertia Tensor for Cube 4 The mass of the cube is evenl distributed è the summation can be converted to integration. For eample, the upperleft element becomes: a a a d d dr + ò ò ò! ( ), here! r M / a 3 denotes the mass densit. 4 Thus, ( a a a a a a ) ò ò ò ò ò ò! r d d d+ d d d! ra Ma Smmetr condition give that, and similarl for the offdiagonal elements. JU 18/HL

18 Eample, Cont d 4 The offdiagonal elements have the form a d a d a d! r,! r a d a d a d! ra Ma. 4 B smmetr, all the offdiagonal elements have the same value. Thus, the moment of inertia tensor is é 1 1 3Ma 4Ma 4Ma ù é 8 3 3ù 1 1 Ma 4Ma 3Ma 4Ma [about a corner] Ma 4Ma 3Ma ë û ë3 3 8û 4 The angular momentum for rotation about an ais through this corner. Eamples: ò ò ò ò ò ò Rotation about ais ( (,, )): L Ma /1 (8, 3, 3) Ma (/3, 1/4, 1/4). L not in same direction as rotation ais JU 18/HL 4 Rotation about diagonal through O ( ω / 3 (1,1,1)): é 8 3 3ùé1ù éù Ma Ma Ma L ω ω 6 ë3 3 8 ûë1û ëû L is in same direction as rotation ais

19 Eample, 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/ òa/ òa/ òa/ òa/ òa/! r d d d+ d d d! ra ( a/) Ma. but the offdiagonals are all odd functions, so the change the limits leads them to go to ero, e.g. a / a / a / a / a / a / d d d! r,! r d d d. ò ò ò ò ò ò a/ a/ a/ a/ a/ a/ 4 The inertia tensor is then diagonal, i.e é1 ù Ma Ma ë 1û JU 18/HL 4 Note that, no matter hat direction is, L is alas parallel to it: Ma L ω ω. 6

20 Eample 4 Eample 1 B M D A To masses A and B of,4 kg each ith initial velocit of m/s eperience a moment M,6 Nm. Find the speeds of the masses hen time t 4 s. C Sol: Appling the principle of angular impulse and momentum t t1 M dt Change of angular momentum of mass A and B (r.m(v v 1 )) (,3)(,4) (v m/s),4v,48 Where 4 Mdt,6 (4),4 JU 18/HL è,4,4v,48 è v 1 m/s

21 Eample, Cont d 4 Eample A rod assembl rotates around its ais. The mass C is 1 kg and its initial velocit is m/s. A moment of M 8t + 5 Nm and a force of F 6 N act as shon. Find the velocit of mass C after seconds. Sol: Appling the principle of angular impulse and momentum about ais (the ais of rotation) Angular impulse: Mdt (r F) dt 5 (8t 5 + 5)dt + (,75 A 6) dt B 85,33 nm.s Change of angular momentum: H H 1 r.m(v v 1 ),75(1)(v ) 7,5v + 15 JU 18/HL B the principle of angular impulse and momentum 7,5v ,33 è v 13,4 m/s

22 Summar and questions n this lecture, the folloing are covered 4 Euler angles: Eample applications and derivations 4 Revie of the principle of angular impulse and moment 4 Revie of angular momentum and its conservation 4 Angular moments in 3D and inertia tensors? JU 18/HL Net: Dnamics of rigid bodies Euler equations, application to motion of smmetric tops and groscopes