An Introduction to Three-Dimensional, Rigid Body Dynamics. James W. Kamman, PhD. Volume II: Kinetics. Unit 2. Newton/Euler Equations of Motion

Transcription

1 Suary n ntroducton to Three-ensonal, gd Body ynacs Jaes W. Kaan, Ph Volue : Knetcs Unt Newton/uler quatons of Moton Ths unt presents the Newton/uler equatons of oton for rgd odes. These equatons relate the kneatcs of a ody to the forces and torques actng upon t. The applcaton of these equatons focuses on ndvdual odes wthn a dynac syste through the use of free ody dagras. The analyss requres workng knowledge of force systes, kneatcs of rgd odes, and angular oentu. The equatons that result fro the applcaton of ths ethod ay e algerac equatons, dfferental equatons, or a conaton of oth. Page Count xaples Suggested xercses 5 7 Copyrght Jaes W. Kaan, 06

2 Newton/uler quatons of Moton for a gd Body Usng the theory of systes of partcles, t can e shown that the equatons of oton for rgd ody oton n an nertal frae ay e wrtten as F a M r F H dt Here, F (,, N) are a set of N ernal forces actng on the ody, a s the acceleraton of the ass center of the ody n an nertal frae, and H s the angular oentu of the ody B aout ts ass center. Usng the dervatve rule (presented n Unt of Volue ), the rght sde of the oent equaton ay e rewrtten as follows: or B M H B B B H d d d dt dt dt M B B H Here, s the nerta dyadc (atrx) assocated wth the ody s ass center. quvalent Force Systes d The oent equaton ay e ended for takng oents aout any pont usng the concept of equvalent force systes. Systes of forces are sad to e equvalent f they have the sae resultant and the sae oent aout any pont. The resultant of a force syste s sply the su of all the forces. F. The oent of the syste aout soe pont P s M P p F Note here that the oent of the syste aout another pont Q ay e related to the oent aout P as follows: M Q q F r P/ Q p F r / F p F r / F M P Q P Q P Syste of Forces Copyrght Jaes W. Kaan, 06 Volue, Unt : page /

3 or M M r F / Q P P Q lternate Moent quaton f pont Q n the aove analyss s any pont, and P s the ass center, then a oent equaton for ay e wrtten as follows. or M M r/ a ( s any pont) M B B H r/ a Specal Case: Moton aout a Fxed Pont f soe pont O of the ody s fxed, then the aove equatons of oton take the for d F a M O H O O B B HO dt where H s the angular oentu of the ody aout the fxed pont O. Note the eleents of O O B the nerta dyadc O ay e deterned usng the parallel axes theores for oents and products of nerta. Notes:. f the ody has three-densonal oton, the Newton/uler equatons represent sx scalar equatons, three force equatons and three oent equatons.. f the expressons used n these equatons are vald only at an nstant of te, then the equatons are algerac. f the expressons are vald for all te, then the equatons are dfferental equatons and ay e ntegrated to sulate oton of the syste. xaple : Bearng oads on a Sple Crank Shaft (CS) The fgure shows a sple crank shaft CS consstng of seven segents, each consdered to e a slender ar. ach segent of length has ass. There are sx segents of length and one segent of length (segent 4). The ass center of the syste s located on the axs of rotaton. Copyrght Jaes W. Kaan, 06 Volue, Unt : page /

4 eference fraes: :, j, k (fxed frae) S :, j, k (rotates wth the shaft) Fnd: (neglectng weght forces) B, - the earng loads on the shaft at and B T - the drvng torque aout the Z axs Soluton: s ths syste rotates, dynac loads noral to the drecton of rotaton are generated. The loads appled y the earngs at and B to the syste ay e found y applyng the Newton/uler equatons of oton. The fgure elow shows a free-ody dagra of the syste fro an olque vew. Bearng forces are shown n the X and Y drectons (noral to the axs of rotaton), and a drvng torque s shown along the axs of rotaton. No weght force s ncluded so the calculated earng loads wll e due only to the rotaton of the asyetrc syste. Prevous esults t was noted n Unt of ths volue that only the thrd colun of the nerta atrx of the syste s needed here ecause the syste rotates only aout the Z axs. The three nerta values and the angular oentu vector of the syste aout ts ass center were found to e XZ YZ 0 ZZ 0 H ( ) k 0 Newton/uler quatons The Newton/uler equatons for ths syste ay e wrtten F a 0 / M H r a H CS CS CS CS zero Clearly, assung the ass center les on the axs of rotaton ( a 0) splfes the equatons. Usng the prevously generated results, the ters on the rght sde of the oent equaton ay e wrtten as follows. X X X Y X Z 0 X Z CS Y X Y Y Y Z 0 Y Z Z X Z Y Z Z Z Z CS ( ) k 0 CS H k H j Copyrght Jaes W. Kaan, 06 Volue, Unt : page /

5 The left sde of the oent equaton represents the su of the oents of the ernal forces and torques actng on CS aout pont. eferrng to the free ody dagra, / 4 4 M r F T k B B jt k B B Y X Susttutng these results nto the Newton/uler equatons and separatng the coponents gves the followng fve scalar equatons of oton. X Y B B X Y 0 0 (force equatons) 4 B Y 4 B X T 0 (oent equatons) Solvng these equatons gves the followng results for the support force coponents and the drvng torque. BX X Y B Y (support force coponents) 0 T (drvng torque) xaple : Bearng oads on a Shaft wth a Msalgned sk The syste shown conssts of two odes, shaft B of length and dsk of ass and radus r. s welded to B so that an axs noral to akes an angle wth the shaft axs. non-zero angle ndcates the dsk s not deally algned on the shaft. The shaft and dsk rotate together aout the Z axs at an angular velocty of (r/s) and angular acceleraton (r/s ). eference fraes: ( s the fxed frae) S : (, j, k) (rotates wth the shaft; algned wth the shaft) : (, e, e ) (rotates wth the shaft; algned wth the dsk) Fnd: (neglectng weght forces) B, - the earng loads on the shaft at and B T Soluton: Prevous esults - the drvng torque aout the Z axs n Unt of ths volue the nerta atrx of the dsk aout ts ass center and the angular velocty of the dsk were found to e r k S e C e Copyrght Jaes W. Kaan, 06 Volue, Unt : page 4/

6 ecall the suscrpt on the nerta atrx ndcates the nertas are easured aout the dsk-fxed drectons. Specfcally, n ths case, the nertas are easured around the drectons ndcated y e (,,). Usng these results, the angular oentu of aout ts ass center was found to e 4 4 H r S e C e r S C j C k Newton/uler quatons ssung the nerta of the shaft aout the Z axs s sall copared to that of the dsk, the earng loads at and B ay e found y applyng the Newton/uler equatons of oton to the free-ody dagra of the syste. F a 0 / M H r a H Usng the prevously generated results, the ters on the rght sde of the oent equaton ay e wrtten as r 4 0 S 0 0 C zero 4 r S e C e 4 r S C j C k 4 4 H k r SC j C k r SC The left sde of the oent equaton represents the su of the oents of the ernal forces and torques on the syste aout pont. That s, / M r B T k B B jt k B Y X Susttutng these results nto the Newton/uler equatons and separatng the coponents gves the followng fve scalar equatons of oton. X Y B B X Y 0 0 (force equatons) BY 4 r SC BX 4 r SC 4 T r C (oent equatons) Copyrght Jaes W. Kaan, 06 Volue, Unt : page 5/

7 Solvng these equatons gves the followng results for the support force coponents and drvng torque. B X 8 r SC X BY 8 r SC Y (support force coponents) T r C 4 (drvng torque) Notes on xaples and : o The forces at the earngs at and B are n opposte drectons. f the support earngs were on elastc ounts, the syste would wole as t rotates. Ths effect s due to the ass asyetres of the systes. The crank shaft of xaple s asyetrcal wthn the XZ plane, and the dsk/shaft syste of xaple s asyetrcal wthn the Y Zplane. o The earng forces wthn the XZ plane of the crank shaft ( X and plane of the dsk/shaft syste ( Y and B X ) and those n the YZ BY ) are non-zero even when the systes are rotatng at constant rates and ther agntudes ncrease wth the square of the angular speed. o Because the XZ plane s a plane of syetry of the crank shaft, the earng forces Y and BY are non-zero only when the angular speed of the syste s not constant. Because the YZ plane s a plane of syetry of the dsk/shaft syste, the earng forces the angular speed of the syste s not constant. X and B X are non-zero only when o These systes are exaples of systes that are statcally alanced ut not dynacally alanced. They have statc alance ecause ther ass centers are on the axes of rotaton, ut not dynacally alanced ecause they are asyetrcal. o Bearng loads on opposte ends of a shaft are n the sae drecton f they are due to statc alance, whereas earng loads due to dynac alance are n opposte drectons. o Of course, no engneerng systes are truly statcally or dynacally alanced. ather, they are desgned to nze the loads on the earngs due to statc and dynac alances. xaple : quatons of Moton of a otatng Bar The syste shown conssts of two odes, the frae F and the ar B. Frae F rotates aout the fxed vertcal drecton annotated y the unt vector k. Bar B s pnned to and rotates aout the horzontal ar of F. F rotates relatve to the ground at a rate a rate of (r/s). (r/s) and B rotates relatve to F at eference fraes: ( s the fxed frae) F :( n, n, k ) (rotates wth frae F) B :( e, n, e ) (rotates wth the ar B) Copyrght Jaes W. Kaan, 06 Volue, Unt : page 6/

8 Fnd: a) Forces and torques transtted through the pn at assung all otons are known. ) Forces and torques transtted through the pn at and the dfferental equaton of oton for the angle f only the oton of the frae F s known. Soluton: Prevous esults n Unt the nerta atrx of the ar aout ts ass center and the angular velocty of the ar were found to e B B n k ( S ) e n ( C ) e ecall, agan, the suscrpt B on the nerta atrx ndcates the nertas are easured aout the ody-fxed drectons. Specfcally, n ths case, the nertas are easured around the drectons ndcated y the unt vector set B :( e, n, e ). Usng these results, the angular oentu vector of B aout ts ass center was found to e S H B 0 0 H n C e 0 0 C Newton/uler quatons Here Usng the prevously generated results, the Newton/uler equatons for the ar ay e wrtten as follows F a M H B B a d n d n d C e S e d n a d C e d n d S e ( S ) e n ( C ) e B B B B B d d d dt dt dt B S C e n C S e The ters on the rght sde of the oent equaton can e wrtten as Copyrght Jaes W. Kaan, 06 Volue, Unt : page 7/

9 0 0 0 S C B 0 0 B n C S e 0 0 C S e n e B H S C B H SC n S e 0 C a) f and are known, the forces and torques requred to antan the oton ay e calculated as follows. See the front and sde vews of the free ody dagra of the ar elow. Unknown forces: F e F d C Unknown torques: F n F d M e T 0 M n T S C F e F d S M e T C S Ths process of calculatng forces and torques assocated wth known (specfed) oton s referred to as an nverse ynac analyss. n ths case, snce the oton of the syste s fully deterned, the Newton/uler equatons are algerac equatons. ) f s known and s not known, the forces and torques requred to antan the oton and the dfferental equaton governng the oton of the ar relatve to the frae ay e calculated as follows. gan, see the front and sde vews of the free ody dagra. fferental equaton for angle : Free Body agras Sde T SC (torque T s assued to e a known drvng torque) Front Unknown forces: (as n part (a)) F e F d C Unknown torques: F n F d M e T 0 M e T C S F e F d S Copyrght Jaes W. Kaan, 06 Volue, Unt : page 8/

10 n ths case, the oton of the frae s known, ut the oton of the ar relatve to the frae s not. The process of fndng the dfferental equaton of oton for the angle that descres the oton of the ar relatve to the frae s called a Forward ynac analyss. The analyss would proceed as follows. a) ven the angular rate () t and the torque T () t, solve the dfferental equaton to fnd () t. ) Usng values of, ( ),, and, fnd the unknown force and torque coponents. f the drvng torque T ( t) 0, the oton of the ar exhts steady-state equlru postons. These postons ay e calculated y settng all te-varyng parts of the dfferental equaton to zero. That s, set SC 0. Ths equaton s satsfed for the values 0,. n general, these equlru postons ay e stale or unstale. f the ar s released n a stale equlru poston, t wll rean n that poston as the frae F rotates. f the ar s released n an unstale equlru poston, t wll ove away fro that poston as the frae F rotates. n ths case, t can e shown that 0 s a stale equlru poston and s an unstale equlru poston. Stalty analyss wll e dscussed n a later unt. xaple 4: rcraft wth Two ngnes The arcraft shown has two engnes, one on each wng. The orentaton of the arcraft relatve to a fxed reference frae s defned y a -- ody-fxed rotaton sequence,,. For the purposes of ths exaple, the arcraft s ade up of three an coponents, the arfrae and the two engnes and. The ter arfrae s used to refer to all the statonary coponents of the arcraft, and the ter engne s used to refer to the rotatng coponents of the engnes. The ponts (, ) are the ass centers of the two engnes, s the ass center of the arfrae, and s the ass center of the arcraft. The arcraft s syetrcal wth respect to the xz plane. The two engnes are assued to e dentcal and placed syetrcally on the arfrae so the poston vector of the ass center of the arfrae and the poston vectors of (, ) the ass centers of the engnes relatve to the ass center of the arcraft can e wrtten as r x z / r x y z / r x y z / Copyrght Jaes W. Kaan, 06 Volue, Unt : page 9/

11 Furtherore, the engnes (rotatng coponents) are assued to e solds of revoluton algned wth the x axs (eanng they are rotatonally syetrcal aout that axs). Fnally, the velocty of the ass center of the arcraft s specfed n ody and ground fraes as v X N Y N Z N v u v w eference fraes: : N, N, N (nertal or ground frae) :,, (frae fxed n the arcraft). Fnd: (assung all ernal forces and torques and engne speeds are known) quatons of oton of the arcraft Soluton: Prevous results n Volue, Unt 5 the angular velocty and acceleraton of the arfrae were found to e (usng a -- ody-fxed, orentaton angle sequence) S C C S S C C B d d dt dt S C C C S C C S S C C S C S S S C C n Unt of ths volue, the nerta atrx and angular oentu of the arfrae were wrtten as follows. Note the zero values for soe of the products of nerta due to the assued syetry of the arfrae. x 0 x x z 0 0 y y x 0 z zz H x x x z y y x z z z The angular veloctes, nerta atrces, and angular oenta of the engnes were also found n Unt. ( ) (,) x x y y z z 0 0 xx 0 y 0 y 0 0 zz H ( ) (,) (,) Copyrght Jaes W. Kaan, 06 Volue, Unt : page 0/

12 ecall here that the nertas and yy are equal and they are constant relatve to drectons fxed n the zz arfrae ecause of the assued rotatonal syetry of the engnes aout the drecton. Newton/uler quatons ngnes For the purposes of ths exaple, the engnes are assued to e algned wth the arcraft reference frae. The dagra to the rght shows the ody-fxed coponents of the total forces and torques actng on an engne. The Newton/uler equatons of oton for the engnes ay then e wrtten as Total Forces and Torques on ngnes F F F F arfrae a (,) M T T T H (,) arfrae Or, f oents are taken aout the ass center of the arcraft, then M T r F H r a (,) / / The forces and torques actng on an engne have een dvded nto two categores. The suscrpt refers to loads appled to the engnes y ernal sources such as gravty, thrust, and aerodynac drag, and the suscrpt arfrae refers to the loads appled to the engnes y the arfrae. The forces and torques assocated wth ernal sources are assued to e known, whle the forces and torques assocated wth the arfrae are unknown. rfrae The Newton/uler equatons of oton for the arfrae are arfrae F F F F a arfrae / arfrae M T T r F H Or, f oents are taken aout the ass center of the arcraft, then / arfrae / arfrae M T r F T r F / H r a Copyrght Jaes W. Kaan, 06 Volue, Unt : page /

13 Here, all ernal forces and ernal torques (lft, drag, weght, etc.) on the arfrae have een replaced y a sngle force F actng at the ass center of the arfrae and a correspondng torque T ernal. The effects of a set of forces and torques equal and opposte to those the arfrae apples to the engnes are also ncluded n the equatons. rcraft Force quatons The force equatons for the arfrae and the two engnes can e added to produce a sngle set of force equatons for the arcraft as follows. arfrae F F F F F F arfrae T a a a T F F a The left sde of the equaton represents all the ernal forces on the arfrae and oth engnes, s the total ass of the arcraft, and a s the acceleraton of the ass center of the T arcraft. Note that the nternal forces etween the engnes and the arfrae cancel out of the equaton. The acceleraton of can e expressed n the ground frae or n the ody frae. Specfcally, a X N Y N Z N Or, d d a v v v u v w dt dt u v w a u w v v u w w v u rcraft Moent quatons The oent equatons for the arfrae and the two engnes can also e added together to produce a set of oent equatons for the arcraft. r / arfrae F T r F T T T / arfrae r / F r / Farfrae arfrae / / H r a H r a Copyrght Jaes W. Kaan, 06 Volue, Unt : page /

14 / / T r F T r F / / H r a H r a The left sde of the equaton represents the su of the oents of all ernal forces aout the ass center of the arcraft. The ters on the rght sde of the equaton can also e resolved aout the ass center of the arcraft. To do ths, frst consder the ters nvolvng the acceleratons of and (, ) the ass centers of the arfrae and the two engnes. xpressng these acceleratons n ters of the acceleraton of and usng the defnton of center of ass gves / / r a r a r a a r a a / / / / / / / / / / / r r r a r a r a zero xpressng the acceleratons of and (, ) relatve to n ters of the angular oton of the arcraft gves / / / / / r a r a r r r / / / r r r / / / / / / r a r a r r r r r r r r / / / / Susttutng ths result nto the rght sde of the oent equaton and rearrangng ters gves Copyrght Jaes W. Kaan, 06 Volue, Unt : page /

15 / / H r a H r a r r r r / / / / H H r / r / / / r r nalyses of the ters on the rght sde of ths equaton are presented elow. The frst racketed ter on the rght sde of ths result can e splfed y notng that d / / / / r r r r dt d dt Ths result s ased on a result developed n Unt of ths volue. Specfcally, / / / / H r r r r The second and thrd racketed ters for the engnes can e splfed n a anner lke that for the arfrae ters aove. To do ths, frst relate the angular acceleraton of the engnes to the angular acceleraton of the arfrae. Or, d ( ) dt d dt ˆ wth ˆ Usng these results, the second and thrd racketed ters assocated wth the engnes reduce as follows. Copyrght Jaes W. Kaan, 06 Volue, Unt : page 4/

16 ˆ r / r / / / / / r r r r ˆ ˆ Susttutng these results nto the rght sde of the oent equaton gves / / H r a H r a ˆ ˆ H H r / r / r / r / ˆ ˆ arcraft H H / / / r r r r / The last three ters n the expresson aove can e further splfed y usng the vector dentty ac a c a c. Splfcaton of the arfrae ter s shown elow and slar results are true for the engne ters. r / / r r / r / r / r / r / Susttutng these results nto the oent equaton aove gves the fnal for of the vector oent equaton. / / T r F T r F ˆ arcraft H H / / / / r r r r Copyrght Jaes W. Kaan, 06 Volue, Unt : page 5/

17 arcraft s n Unt, represents the nerta tensor for the arcraft (arfrae and two engnes) aout ts ass center. Scalar Moent quatons for the rcraft The ters on the rght sde of the vector oent equaton presented aove are expanded here ter-yter nto ther coponent fors. x 0 x x z 0 y 0 arcraft y x 0 z zz x x x z y y z z x z arcraft 0 0 zz xx 0 0 ˆ 0 0 yy ˆ (, ) x x y y z z H x x x z y y z z x z z z x z y y xx xz zz xz y y x x x z y y x x x z H z z y y x z x z x x z z H zz y y x x y y z z x x z z y y x x Copyrght Jaes W. Kaan, 06 Volue, Unt : page 6/

18 y y xx H z z y y x x z z / / r r x z x 0 z r / r / x z z z x x / / r r x y z x y z / / r r x y z y z z x x y / / r r x y z x y z / / r r x y z y z z x x y Sung the last two ters gves / / / / r r r r x y z x y x x y z y z x y z y z x y z z x x y z z x / / / / r r r r y z x y x z y z x z z x x y x z Copyrght Jaes W. Kaan, 06 Volue, Unt : page 7/

19 Conng all of the aove ters gves the detaled rght sde of the oent equaton. The splfcatons of the three scalar coponents are shown elow. Frst coponent: coponent: z z y y x z x x x z xx zz y y z x z x z y z x x x z xx xz z x x z z z y y z zz y y y z ddng and sutractng the ters x and x to the aove result gves coponent: z z x y y x z z z x y y y x z x x x z x x xz x z x z nvokng the parallel axes theore gves coponent: z z y y z z y y x x x z x x x z x z ddng the nertas of the arfrae and engnes aout the ass center of the arcraft gves the fnal result. coponent: x x x z x x xz zz y y Copyrght Jaes W. Kaan, 06 Volue, Unt : page 8/

20 Second coponent: coponent: y y y y x z x x zz x x zz = x z z x x z z x earrangng ters, addng and sutractng the ter y, and nvokng the parallel axes theore gves coponent: x z x z x x z z y y x z x z z x y z x y x z x x z z yy x x z z y y x z xx xx zz zz x x x z Fnally, addng the nertas of the arfrae and engnes aout the ass center of the arcraft gves coponent: y y x z x x z z x x Thrd coponent: coponent: y y x x x z zz x z zz y y x x x z x x y x z zz x z y y x x y y x x x x y xz x z x z y y x x z z ddng and sutractng the ters z and z to the aove result gves Copyrght Jaes W. Kaan, 06 Volue, Unt : page 9/

21 coponent: x x z z x z y y x z z x x x z y z x z y y x x x z nvokng the parallel axes theore gves coponent: zz x z y y y y xx xx x z x x Conng the nertas of the arfrae and the engnes aout the ass center of the arcraft gves coponent: z z x z y y x x x z x x Fnal esults Force quatons: Or T F F u w v v u w w v u () T F F X N Y N Z N () Moent quatons: / / T r F T r F x x x z x x x z zz y y y y x z x x zz x x zz x z y y xx xz xx () Copyrght Jaes W. Kaan, 06 Volue, Unt : page 0/

22 f the acceleraton of the ass center of the arcraft s resolved n (local) arfrae coponents, a set of sx kneatcal dfferental equatons ust e ncluded to track changes n the poston and orentaton of the arcraft. Specfcally, to track changes n the -- ody-fxed orentaton angles, use the followng equatons that relate the angular velocty coponents to the orentaton angles and ther dervatves. S C C S S C C Or, these equatons can also e easly nverted to gve S C C C S S S C C (4) To track changes n the gloal poston of the arcraft, use the transforaton atrx for a -- orentaton angle sequence to convert the local velocty coponents nto gloal velocty coponents. X u C C S C S u T Y v C S S S C S S S C C C S v Z w C S C S S S SC C S C C w T (5) Together, quatons (), (), (4), and (5) represent a set of twelve frst-order, ordnary dfferental equatons n as any unknowns ( X, Y, Z, u, v, w,,,,,, ) that ay e used to sulate the oton of the arcraft. The three second-order equatons () can e rewrtten as sx frst-order equatons, so quatons (), () and (4) also represent a set of twelve frst-order, ordnary dfferental equatons n as any unknowns. n ths case, the unknowns are ( X, Y, Z, X, Y, Z,,,,,, ). Clearly, the usefulness of these odel equatons depends on the analyst s alty to fnd equatons to represent the ernal forces actng on the arfrae and engnes. Ths ncludes a eans to fnd expressons for the aerodynac loads on the arcraft and the thrusts generated y the engnes. xaple 5: oule Pendulu or r The syste shown s a three-densonal doule pendulu or ar. The frst lnk s connected to ground and the second lnk s connected to the frst wth all and socket jonts at O and. The orentaton of each lnk s defned relatve to the ground usng a -- ody-fxed rotaton sequence. The lengths of the lnks are and. The lnks are assued to e slender ars wth ass centers at ther dponts. Copyrght Jaes W. Kaan, 06 Volue, Unt : page /

23 eference fraes: : N, N, N (fxed frae) : n, n, n (,) (fxed n the two lnks) Fnd: The equatons of oton descrng the free oton of the doule pendulu under the acton of gravty. ssue the N drecton s vertcal. Soluton: Prevous esults The transforaton atrces and the ody-fxed coponents of the angular veloctes of the lnks (,) are as gven n Unt 5 of Volue. C C S C S S C C C S S S C S S C C S S C C C S C S S C S C S S C S C S C (,) n Unt of ths volue, the nerta atrces and angular oenta of the lnks aout ther ass centers were found to e B 0 0 H n n (,) n addton, the nerta atrx and angular oentu of the frst lnks aout ts end O were found to e 0 0 O H O n n (lnk ) Newton/uler quatons for nk : The Newton/uler equatons of oton for the frst lnk ay e wrtten as follows. Here, advantage s taken of the fact that lnk s rotatng aout the fxed pont Free Body agras O. F N O a N F N O W a N F N O a N Copyrght Jaes W. Kaan, 06 Volue, Unt : page /

24 / / O O O O O M r W N r N N N H The acceleraton of ay e calculated n ody-fxed coponents as follows. a a / O r/ O r / O n n n n n n n n n n n n a n n n These coponents ay e transfored nto the ase frae usng the transforaton atrx for the lnk. a N T a N a N The left sde of the oent equaton ay e calculated as follows. Here, the ters,, and represent the entres n the second row of the transforaton atrx. These are the ase-frae coponents of the unt vector n. See Unt 5 of Volue. r / O W N r / O N N N n W N n N N N N N N N N N W 0 0 r / O W N r / O N N N W N W N N The ase-frae coponents of the ters on the rght sde of the oent equaton ay e calculated as follows. Here the ters are frst calculated n the ody-frae and then oved to the ase-frae usng the lnk s transforaton atrx. See Unt 5 of Volue. Copyrght Jaes W. Kaan, 06 Volue, Unt : page /

25 0 0 O O n n 0 0 n n n 0 H O H O n n O N T O N 0 O N H O N T H O N 0 H O N Susttutng these results nto the Newton/uler equatons gves the followng scalar equatons a N O T O W a N O a N W W T 0 (6) (7) quatons (6) and (7) represent a set of sx frst-order, ordnary dfferental and algerac equatons n twelve unknowns sx force coponents: O, O, O,,, ; three angular velocty coponents:,, ; and three angles,,. Newton/uler quatons for nk : The Newton/uler equatons of oton for the second lnk ay e wrtten as follows. Unlke the frst lnk, ths lnk has general oton. F N a N F N W a N F N a N M r N N N / H Copyrght Jaes W. Kaan, 06 Volue, Unt : page 4/

26 The acceleraton of ay e calculated usng the concept of relatve acceleraton as follows. a a a / Here, a a / O a / O and the result for a can e easly found fro the result for / a y / O sply replacng all lnk references to the second lnk. See also Unt 7 of Volue. Usng ths approach, the ase-frae coponents of a ay e wrtten as follows. a N a N a / N T T a N a N a / N a a N N a / N The ase-frae coponents of the ters on the left sde of the oent equaton ay e calculated as follows. The ters,, and represent the entres n the second row of the transforaton atrx. These are the ase-frae coponents of the unt vector r N N N n N N N / N N N n. See Unt 5 of Volue. N N N The ase-frae coponents of the ters on the rght sde of the oent equaton ay e calculated as follows. s efore, the ters are frst calculated n the ody-frae and then oved to the ase-frae usng the lnk s transforaton atrx. See Unt 5 of Volue n n 0 0 n n n 0 H H n n Copyrght Jaes W. Kaan, 06 Volue, Unt : page 5/

27 N T N 0 N H N T H N 0 H N Susttutng these results nto the Newton/uler equatons gves the followng sx scalar equatons. T T W T T 0 0 (8) (9) quatons (8) and (9) represent a set of sx frst-order, ordnary dfferental and algerac equatons n ffteen unknowns Three force coponents:,, ; Sx angular velocty coponents:,,,,, ; and sx orentaton angles:,,,,,. Together, quatons (6), (7), (8), and (9) represent a set of twelve frst-order, ordnary dfferental and algerac equatons n eghteen unknowns sx force coponents: O, O, O,,, ; sx angular velocty coponents:,,,,, ; and sx orentaton angles:,,,,,. These equatons ust e suppleented wth sx kneatcal dfferental equatons that assocate the angular velocty coponents wth the te dervatves of the orentaton angles. The equatons for a -- ody-fxed orentaton angle sequence were provded n xercse 5.. S C S C S C S C S (,) (0) quatons (6)-(0) represent a set of eghteen frst-order, ordnary dfferental and algerac equatons that can e solved to sulate the oton of the doule pendulu under the acton of gravty. Copyrght Jaes W. Kaan, 06 Volue, Unt : page 6/

28 xercses:. thn rectangular plate P of ass s welded to a shaft so that t rotates aout ts dagonal as shown. ven that the plate has angular velocty and angular acceleraton, fnd: (a) the earng loads at the ends of the plate, and () the drvng torque T. ssue that the earng loads are concentrated near the corners of the plate. xpress the results n the X, Y and Z drectons that rotate wth the shaft. nswers: a ( a ) a) FB F jk ( a ) ) T a 6( a ). The syste shown conssts of two -shaped ars welded to a shaft of length a. ach length a has ass. The planes of the ars are at rght angles to the shaft. ven that the syste has angular velocty and angular acceleraton, fnd: (a) the earng loads at the ends of the shaft, and () the drvng torque T. ssue that all parts are ade of slender ars. xpress the results n shaft-fxed the X, Y and Z drectons that rotate wth the shaft. nswers: a) FB F a 6 ( ) ( ) j ). The syste shown conssts of a ar B of length and ass that s pnned through the center of a shaft of length a. s the shaft rotates aout the Z axs at a constant rate (rad/sec), B rotates aout the Y axs at a rate (rad/sec). (a) Fnd the oents that are transtted (n the X and Z drectons) through the pn at. () Fnd the dfferental equaton of oton governng the angle. nswers: a) M C C S k ) 6 T 0 S C 0 a Copyrght Jaes W. Kaan, 06 Volue, Unt : page 7/

29 .4 The syste shown conssts of a ar B of length and ass that s pnned to the otto of a dsk. s the dsk rotates at a constant rate (rad/sec) aout the Z axs, the ar s free to rotate aout the X axs. The unt vector e s algned wth the X axs, and the angle descres the free rotaton of B relatve to. When the syste s at rest, the ar hangs downward under the acton of gravty. (a) Fnd the ar-fxed force and oent coponents transtted through the pn at P. () Fnd the dfferental equaton of oton governng the angle. nswers: a) M C e ) S C g S C 0.5 The syste shown s a three-densonal doule pendulu or ar. The frst lnk s connected to ground and the second lnk s connected to the frst wth unversal jonts at O and, respectvely. The ground frae s : N, N, N and the lnk fraes are : n, n, n (, ). The orentaton of s defned relatve to and the orentaton of s defned relatve to each wth a - odyfxed rotaton sequence. nk O s orented relatve to the ground frae y frst rotatng through an angle aout the N drecton, and then rotatng aout an angle aout the lnk O y rotatng frst through an angle aout the aout the F F e F e F e n drecton. The lengths of the lnks are and n drecton. nk B s orented relatve to n drecton, and then through an angle wth ass centers are at ther dponts. Fnd the equatons of oton of the free oton of the doule pendulu under the acton of gravty. ssue the N drecton s vertcal. C e S C g S e ( ) ( S ) S g C e Copyrght Jaes W. Kaan, 06 Volue, Unt : page 8/

30 Hnt: Unknown reacton torques are transtted through the two-axs connectng jonts n drectons that are perpendcular to the two hnge drectons, that s, TO TONn and T T n n. nswers: (all coponents resolved n the ase frae syste) ghteen frst-order, ordnary dfferental and algerac equatons n eghteen unknowns T O, O, O, O, T,,,,,,,,,,,, and. O T O W O 0 0 W T TOC T C S S W T T 0 0 T W T T 0 T T C S T T T T 0 0 C S C C S S Copyrght Jaes W. Kaan, 06 Volue, Unt : page 9/

31 .6 The syste shown conssts of three odes, the colun C, the cross-shaped frae and the dsk. C s connected to the ground y a revolute jont allowng oton aout a fxed vertcal axs. s connected to C and s connected to y revolute jonts each allowng rotaton aout the rotatng n drecton. The three relatve rotatons are descred y the angles, and, respectvely. The syste s drven y three known torques T, T and T located at each of the revolute jonts. The unt vector set :( n, n, n ) s fxed n the frae. The ponts and represent the ass centers of and. Usng the Newton/uler equatons, fnd the equatons of oton for a) the frae, and ) for the dsk. ssue the ass and nerta of C are neglgle. nswers: (6 Unknowns: 6 force coponents, torque coponents,,,,,,, ) sk: (6 equatons) F g S n C n n n n T n T n T n n n Frae: (6 equatons) n FC F g S n C n n n n TOC T S n T n TOS T C n T n T n T n n FC n F n n n Or TOC T S n T n TO S T C n T n T n T n n F O O O O O O O O O n n n Kneatcs: (4 equatons) S C Copyrght Jaes W. Kaan, 06 Volue, Unt : page 0/

32 .7 n the syste of xercse.6, the ass center of the syste s located a dstance d to the rght of and a dstance d to the left of. Usng the Newton/uler equatons, fnd the equatons of oton for the syste followng a process lke that descred for the arcraft of xaple 4. ssue the ass and nerta of C are neglgle. nswers: ( Unknowns: force coponents, torque coponent,,,,,,, ) eferences: Force quatons: ( equatons) C F g S n C n g S n C n T d n n n Moent quatons: ( equatons) T S T C d d gc gc O T T T C T S d d gs gs O T Moent equaton for sk n n drecton: ( equaton) T Kneatcs: (4 equatons) S C Note: The frst and thrd oent equatons can e coned to elnate the unknown torque T O reducng the syste to 0 equatons for 0 unknowns.. T.. Kane, P.W. kns, and.. evnson, Spacecraft ynacs, Mcraw-Hll, 98. T.. Kane and.. evnson, ynacs: Theory and pplcaton, Mcraw-Hll, Huston, Multody ynacs, Butterworth-Heneann, H. Baruh, nalytcal ynacs, Mcraw-Hll, H. Josephs and.. Huston, ynacs of Mechancal Systes, CC Press, C. Heler, ngneerng Mechancs: ynacs, th d., Pearson Prentce Hall, 0 7. J.. Mera and.. Crag, ngneerng Mechancs: ynacs, rd d, F.P. Beer and.. Johnston, Jr. Vector Mechancs for ngneers: ynacs, 4 th d, 984 Copyrght Jaes W. Kaan, 06 Volue, Unt : page /