Dynamics and control of mechanical systems. Content

Transcription

1 Dynmics nd control of mechnicl systems Dte Dy 1 (01/08) Dy (03/08) Dy 3 (05/08) Dy 4 (07/08) Dy 5 (09/08) Dy 6 (11/08) Content Review of the bsics of mechnics. Kinemtics of rigid bodies plne motion of rigid bodies, ngulr velocity vector, description of velocity nd ccelertion in reltively moving frmes. Euler ngles, Review of methods of momentum nd ngulr momentum of system of prticles, inerti tensor of rigid body. Dynmics of rigid bodies - Euler's eqution, ppliction to motion of symmetric tops nd gyroscopes nd problems of system of bodies. Kinetic energy of rigid body, virtul displcement nd clssifiction of constrints. D lembert s principle. Introduction to generlized coordintes, derivtion of Lgrnge's eqution from D lembert s principle. Smll oscilltions, mtrix formultion, Eigen vlue problem nd numericl solutions. Modelling mechnicl systems, Introduction to MTLB, computer genertion nd solution of equtions of motion. Introduction to complex nlytic functions, Lplce nd Fourier trnsform. PID controllers, Phse lg nd Phse led compenstion. nlysis of Control systems in stte spce, pole plcement, computer simultion through MTLB. 1 4 Introduction 4 Symbols nd designtions Content 4 Kinemtics of rigid bodies in plne 4 Instntneous center of motion/velocity 4 nlysis of rigid body motion (velocity nd ccelertion) q Grphicl methods to determine velocities nd ccelertions q nlyticl methods to (briefly): References 1. F. P. Ber, E.R. Johnston nd P.J. Cornwell; Vector Mechnics for Engineers, Dynmics, 10th Edition.. Bedford nd W. Fowler, Engineering Mechnics Sttics nd dynmics principles 3. D.T. Greenwood, dvnced dynmics, Cmbridge University Press (006) 4. G. Gry, et l.,-engineering Mechnics - Dynmics-McGry (010)

2 Introduction 4 Study of the Kinemtics of rigid bodies (mechnisms) : 3 Kinemtics of rigid bodies 4 Study of the Kinemtics of rigid bodies (mechnisms) : 4

3 j Symbols & designtions Position, velocity nd ccelertion s position vector of point (size nd orienttion) s v size of position vector s velocity vector of point (size nd orienttion) v speed (size of the velocity vector v) n t ceelertion vector of point (size nd orienttion) size of n ccelertion vector norml component of the cc. vector (norml to pth of motion) tngentil component of cc. vector (tngent to pth of motion) j ngulr position w dj/dt j ngulr speed à tme derivtive dw/dt w ngulr ccelertion 5 Symbols & designtions Position figure, velocity polygon nd cc- polygon, B, C, etc. Points in the position polygon, i.e. line segment to B in position polygon gives the distnce between point nd B, B, C, etc. Points in the velocity polygon, i.e.. Line segment from to B n velocity polygon gives the reltive velocity of point B wrt point ( v B), while the bsolute velocity of point (v v O) is given by the line segment from O to., B, C, etc. Points in the ccelertion polygon, i.e.. Lije segment from to B in the cc. polygon gives the reltive cc. of B wrt ( B), while bsolute cce. Of point ( ) is given by the line segment O to. 6

4 Symbols & designtions Some exmples on use of indices r B v B distnce from point to point B reltive velocity of point wrt point B v B reltive velocity of point B wrt point ( -v B ) v bsolute velocity of point reltive velocity of wrt sttionry point ( v O ) v rel reltive velocity Bn norml comp. Of reltive cc. B O ij instntneous center for reltive motion of prt i nd prt j Kinemtics of rigid bodies 4 Kinemtics of rigid bodies: Reltions between time nd positions, velocities nd ccelertions of prticles composing the rigid body 4 Clssifictions of rigid body motion - direction of ny stright line inside the body is constnt, - ll prticles forming the body move in prllel lines. - ll prticles hve the sme velocity nd the sme ccelertion.!!!! v v nd B B - ll prticles in the body hve the sme ngulr velocity nd ngulr ccelertion!! dr!! v w r dt!!! w wk " q k is ng. velocity d! w!!! k " wk "" q k! dt 8 - Sum of trnsltion nd rottion

5 Plne motion of rigid body 4 Generl plne motion: motion where ll prticles in the body move in prllell plne 4 generl plne motion is composed of two motions 9 4 Position vector of prticle in motion Plne motion rigid body 4 Grphicl representtion of velocity nd cc. vectors v ω xr Dr dr V lim Dt dt DV dv lim Dt dt DV n DV t + DV n n DVn V lim Dt r t + n og der n α xr t ω x ω xr der 10 D Vn ( V + DVt ) Dt r Dr Dt! V

6 Instntneous center of motion Instntneous center of zero velocity For the cr to be ble to drive long curve without sliding, ll the xles of the wheels indicte towrds the sme point, which is n instntneous center of motion (point O) è The cr (t the moment) is undertking rottionl motion bout point O 11 Instntneous center of motion Fig. (): Prt 3 moves wrt prt 1 (which is sttionry prt) Point C nd D follow circulr pths bout nd B respectively Point O (t the moment) hs no velocity in ny direction à It is Instntneous Center point where prt 3 (t the given instnt) undergoes rottionl motion bout the point à Curve OO is the motion pth tht the ICV follows 1

7 Instntneous center of motion Summry ICV is: 1) point on prt 1 (sttionry prt) where prt 3 rottes bout t the moment Exmple: for point E on prt 3 Speed : V E w x I ccelertion, tngentil : ccelrtion, norml: 3 ) point on prt 3 (sttionry prt) where prt 1 rottes bout t the moment 3) crossing point where both prt 1 nd prt 3 hve the sme bsolute motion, i.e. null reltive motion, (vlid when both re in motion) 13 E-O 13 Et En x r V r E E-O 13 E-O 13 ; w x r 3 E-O 13 Number of instntneous center of velocity Severl prts in mechnism: è severl instntneous centers (ICV) # instntneous centers: # different wys of selecting to prts F. ex. The mechnism shown below hs 4 prts joined together t joints, B, C nd D Ech joint hs two prts hving reltive motion ginst ech other. à Ech joint is n ICV 14

8 Types of Instntneous Centers Three min types of ICVs Type 1: Fixed instntneous centers fixed point on body where nother prt is rotting bout f. ex. O 1 nd O 14 Type : Permnent instntneous centers - common point of two prts tht re in motion with the sme speed. This is vlid for joints f. ex. O 3 nd O 34 NB: Type 1 nd re known (identified) ICVs Type 3: Imginry instntneous centers - n imginry point on or outside of mechnism where the prt cn be imgined to rotte bout t the moment f. ex. O 13 nd O 4 15 Types of Instntneous Centers Three types of known/identified instntneous centers Pin joint mechnism Ech joint is known instntneous center Sliding contct - Two points nd B on rigid body tht is in sliding motion undertkes trnsltion motion, thus it hs known instntneous center - Q: where is the instntneous center O B? Rolling contct gives known instntneous center - Where is the instntneous center O 1? ssume pure rolling motion! 16

9 Instntnous Centers of Velocity, exmples Find nd locte the ICVs for the following mechnisms Exmple 1 Exmple Exmple 3 17 Kennedy s Theorem 4 Kennedy s Theorem Gives the bsis for systemtic decisions of the number nd loction of instntneous center of velocity The Theorem is expressed s follows For reltive motion of three rndomly selected prts, the three ICVs re locted on stright line segment NB: The Theorem is vlid if only one of the prts is in motion or both re in motion Exmple: Kennedy s theorem for two prts in sliding contct 18

10 Kennedy s Theorem, n exmple 19 Kennedy s theorem nd grphicl solution, exmple systemtic procedure of using Kennedy s theorem to find nd locte ICVs:. ænö n( n -1) 1. Determine number of prts in the mechnism: n, nd # ICV: ç èø. Drw circle with rndom dimeter size. 3. Divide the circumference into n equl sections nd number them ( 1,, n). 4. Locte the known ICVs on the circle by drwing chord between the mrked points 1 nd, nd 3, etc. to get O 1, O 3, O 34, etc. Loction of ICVs t joints re known ICVs. 5. Serch for chord tht cretes the lst side of two tringles (common side to two tringles). 6. Drw this chord in the circle using dshed line until the loction of the ICV is found on the mechnism. 7. ccording to Kennedy s theorem, the sides of ech tringle correspond to ICVs tht re on stright line. This mens tht ech tringle represents stright line. 8. The intersection between two lines (corr. to two tringles) determines the serched ICV. 9. When the ICV is loclized, drw the chord with solid line. 10. Go bck nd repet the procedure from step 5 until ll ICVs re loclized on the mechnism 0

11 Kennedys teorem og grfiske løsninger 1 Grphicl methods of determining velocities nd ccelertions B Velocity polygon + B Bn + Bt n + t Bn + Bt sr r sr sr sr r V O : perpendiculr to C V B O B : perpendiculr to DB V B B : perpendiculr to B t ccelertion polygon

12 Illustrtive exmple How will the motion of point P on prt 3 look like, when prt rottes? 3 Exmple exercise In this mechnism, the crnk B is rotting with constnt ngulr speed so tht the speed of point B is V B 5 m/s. t the given position, use grphicl method nd determine ccelertion of point D, which is corner point of the rigid body 3. B V B O 5,0 V C V CB 5,0 C 5,0 4

13 Exmple exercise C B + CB B + ( CBn + CBt ) ( r) ( rs) ( rs) ( r) D Þ D Þ Dn Dn + B C Dt Dt DB DC C + ( DCn + DCt ( rs) ( rs) ( r) B + ) ( + ) DBn DBt ( rs) ( rs) ( r) 5 nlyticl pproch Reltive motion of point B wrt : - Velocity: V B V + V B - ccelertion: B + B - cc. Components: Bn + Bt n+ t + Bn+ Bt 4 br linkge mechnism Vector exp ression of velocity nd cc. components!! V w x r, ( mgn. v wr)!!! w x V w x ( w x r), ( mgn. w r) n!!!. α x r w x r V t w x r! w k x - w y ( mgn. r) t ( x i + y j) i + w x n j i x i i x j x k x i j x j - j j k x k 0 x i k k - k x j i - i x k j i j k V w x r 0 0 w - w yi + w x j x y 0 ( w x r) n w x w k x 0 0 w x y 0 i 6 j k - w ( x i + w y j) - r w

14 Summry nd questions The following re covered in tis prt of the lecture 4 Kinemtics of rigid bodies nlysis nd synthesis 4 Clssifiction of rigid body motion 4 Prmeters to describe generl plne motion 4 Instntneous center of motion/velocity 4 nlysis of rigid body motion (velocity nd ccelertion) q Grphicl methods to determine velocities nd ccelertions q nlyticl methods to determine velocities nd ccelertions:? Next: Review of methods of momentum nd ngulr momentum of system of prticles, Euler ngles, inerti tensor of rigid body. 7 Exercise n oil pumping rig is shown in the figure below. The flexible pump rod D is fstened to the sector t E nd is lwys verticl s it enters the fitting below D. The link B cuses the bem BCE to oscillte s the crnked weight O revolves. If O hs constnt CW speed of 1 rev every 3 s, determine the ccelertion of the pump rod D when the bem nd the crnk O re both in the horizontl position s shown. 8