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# MACHINE DYNAMICS MANE 4170

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This 86 page Class Notes was uploaded by Hugh Wilkinson on Monday October 19, 2015. The Class Notes belongs to MANE 4170 at Rensselaer Polytechnic Institute taught by Mohammad Poursina in Fall. Since its upload, it has received 301 views. For similar materials see /class/224918/mane-4170-rensselaer-polytechnic-institute in Mechanical and Aerospace Engineering at Rensselaer Polytechnic Institute.

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Date Created: 10/19/15

L Machine Dynamics MANE 417001 Inclass Problem 1 M Poursina Name jli J RIN r o b Nah onm n An 1312 coordinate system which initially coincided with the X YZ coordinate system rst under goes a rotation 01 2 65 about its 392 axis followed by 92 145 about its goods If the position 05 point P in the bodybasis relative to the moving frame is expressed by 2 v 3 4 determine its location in XYZ after these rotations Name RIN An myz coordinate system which initially coincided with the XYZ coordinate system rst under goes a rotation 01 65quot about its 3 axis followed by 62 2 145 about i 7 he position 05 point P in the body basis relative to the moving frame is expressed b determine its location in XYZ after these rotations We know 1 3 quot 1323 2 NCAllACB 393 39 W 1T1 quot 5 Acnf a 6595 a Z CGS o S Mqr CoJl lf O 3 quot o l r 56 0 C6 3 22055 3l03 3m 4 or Nxr2 A Mr A A a1 Cm m gm 2 A S A C A 02 q upsm A m3 7 A A I M m 3 A I 0 Clqgquot S39qr o A N 0 2 SH CW3 CB1 3 he d gm OWLHT GV Mm A A 1 103 A 2 LOWL 5 A Y FM 1 POWquot 223 ML 7 79 50 5quot ago 97C VP 0 7370 0 e 7L 0M NellWWWquot l I l by anal CC Jagquot from TICTWL m h 0 ma 5 gt 93 Kurtakin Im A 7 wk QC AI 0 o I 043 90 t x m oU N IT H 86 U n 9 5 ad lt c C 933021 K 3433 K ni 6 xso lt 4 So qu d wxm G N G3 IIIIl QQ MXN M K lo 39u39 0 717039453 WWWL we we LiaMd 11 MIL L L A 06n7ln7tl39ah a at 43er gym W on brim fth eacA XL amounf 06 mam a om em am 53ng of IhaIMbma w r Name RIN 1 Consider the spherical pendulum shown in the gure whose length can change when the force F ia applied to it The motion of the system can be described by Cartesian coordinates x y z and spherical coordinates L 0 d a Express m y and z in terms of L 0 j b For each set of coordinate systems provide an expression for the constraint equation at the position velocity and acceleration levels if the length of the pendulum is constant Demonstrate that both sets of coordinates give you the same number of degrees of freedom Number of DOF is independent of the choice of the coordinate Three particles are connected together via three rigid rods For this planar system one can use the Cartesian coordinates of each particle to describe the con guration of this system i7y39i7 1723 a Are these six coordinates independent b Provide the expressions for the constraint equations at the position level c Determine the number of degrees of freedom of the system d Suggest a set of independent coordinates to describe the con guration of the system Can one claim that there is only one set of independent coordinates to describe the con guration of the system Why HWO MQClbamp Eyewme H y 152051 439 Z vLng 3 9931 Lug COM lilf 111 2 223C 4 L C P091104 eye a O on t IDO F V39Im3 l2 3 L Number 039 39DOF is inolnfmclwquot r 39Hu CLOr39Cc aquot Coord 39nc ntf b C d 2 Z L L 33 31 139 97 1 I 1L 7Cz39 33a 5 r 2 L L3 1 3373 quot 95vxt m 6 3 7 09 1quot3uw 3 L 2 14 La Hw M AOLCLK W Dynamics 3 O s kkre x1401 39 Y1 l 139 L3 XLLYZ zxLx 49 lfZYLY L m MJ 52 2XZ 22 szm 3935 7gt13 c 3331 6121 02 P 0 On A 9 3 A A O at 0o 4 A A Ian a I 0392 a 0f 0j00 L in 0 1 71 a A 0 NOW 33 NM z n m Same O f JV 2 crow pooM Jquot7 CMrgaS Nap0 A 190 190 N a ap0 prune d wkof 33quot 105 3 35 v 6 c 3035 v0 32 3097 K 3 4334 86 cxw jog 1 63 i J7 if 3 g F E hm at F g w Q h ampV an E h m x f KQQLQX mi 93 395 IV Q L4 4N 9w L4ltw 9k I 8x 2 Hlt gvQAN v v HN N n Machine Dynamics MANE 417001 Inclass Problem M Poursina Name RJN P represents a moving point on the surface of a rigid sphere S39 which is regarded as a reference frame 1 Determine if the vector P is a vector function of q1 q2 q3 in S 2 Determine if the vector r is a vector function of q1 q2 q3 in S 3 Determine if the vector q is a vector mction of q1 q2 q3 in S Name RIN Consider the system of a double gimbal and rotor shown below Reference frames A B and C are assigned to the outer gimbal inner gimbal and rotor C Reference frame D is attached to the ground If P is the position vector from point 0 to a point P of C then determine if P is a function of q1 12 and q3 in 1 Reference frame A 2 Reference frame B 3 Reference frame 039 4 Reference frame D 439 0 q rel 9 01 95 N N N W Name re a 2 Starting from the position shown the box is rotated by angle 40 about the face diagonal AB clockwise as viewed from corner B toward corner A Determine the coordinates of corner 0 relative to the xed reference frame X YZ after this rotation X 15 mm 50 mm 30 mm 00 et joint at corner A It In A to D Find the displacement of Hw Ol v Pmbim 2L pmb39 B Lf C1nSbeY3 A LUUW v1 D L 3 is T A 3 Z 0 0 n3 A 1 quot HIVDIV 0 010101 A A a 39 n 7n 5W91nz A 4 4 1 1 6w m rang 3 A A HM on m w K d hbn Q You Can 3950 um HM A P 9 9 B Lr IGinsberg mambo 034 0433 03 02 w 0 Ut gg 04 0 4133 019 P 2 PageM 26 1621ngch 0 T r 1 A A 13 A A A rmriCjCCJ 50 V 940 a 7 V l 026 0403 0 12 0 3390 04037 aqgw Oo330 U39 06517 0330 072669 1 3400 53692 Hw 3903 Prob evw 3 Machine Dynamics MANE 4170 Spring 2013 TuesdayFriday 12150 Instructor Mohammad Poursina Prereguisites ENGR 2090 Engineering Dynamics MATH 2400 Intro to Differential Equations or equivalent and Linear Algebra Students must be proficient in all of following basic mathematics a Basic vector algebra and calculus eg dot product cross product chain rule for differentiation b Basic LinearAIgebra Matrix multiplication matrix inversion etc o Trigonometric Identities and definitions plus the Law of Sines and the Law of Cosines d Basic Differential Equations Formulation and solution of simple homogeneous nonhomogeneous linear differential equations exposure to simple d Eigenvalue problems and their solution e Solving double triple integrals Pdf notes Course notes quotMachine Dynamics available in pdf form on LMS Required Grade Summary Homeworkquizzes Tests 15 20 20 Final Exam Project Bonus 15 55 20 10 5 Homeworkquiz Students must solve all the problems assigned however one or more than one problem in the homework assignment will be generally graded and will be returned Students will be penalized for any unsolved problem For instance if you are assigned 4 problems and you don t solve one problem you will automatically loose 2510 Quizzes may be takehome or inclass The takehome quizzes will be collected the next lecture after it is distributed Tests Three 110minute inclass closedbook closednote format Crib sheet is allowed Final Exam A Closedbook Closednote inolass final exam given during the final exam period Crib sheet is allowed The exam will cover the entire course Bonus Problems Bonus problems are assigned and will be collected These problems are designed for those students who want to face more challenges After posting the solutions bonus problems are treated as homework After posting the solution all students are expected to work these problems since they may appear in the tests According to the number of correctly answered bonus problems 05 points will be added to the final last exam Course Conduct and Academic Inteqrity Students are expected to conduct themselves in a professional manner at all times eg being attentive being on time not sleeping not disrupting the class with talking or cell phones not using Iaptop computerssmart phones unless instructed to do so etc Web surfing other than using class materials and sending nonclass related emails during class are not permitted Repeat offenders may be asked to leave Dynamics Branch of mechanics Deals with the motion of bodies under the action of forces Science of motion We look at dynamics in two distinct parts Kinematics The description of motion without reference to the forces which cause motion Relate position velocity acceleration and time without reference to the cause of the motion GEOMERTY OF MOTION Interested in finding the relations between 91929394 Kinematics is MATH NO physical laws 61626364 l 2 3 4 Kinetics Studies the relations between The forces and moments acting on the body The mass distribution of the body The change in motion of the body You need to apply the Physics Laws Newton s Laws of motion Kinetics Studies the motion and its causes Interested in finding the followings due to applying F Forces at joints 61626364 91929394 919293 4 Particle a body of negligible dimensions When the dimensions of a body are irrelevant to the description of its motion it can be treated as a particle I 22 I I Rigid body a body whose changes in shape are negligible compared with the overall dimensions of the body Constraint Restriction on motion 2yAyB 2yB lconstL Generalized Coordinates A set of coordinates which describes the configurationposition of the system It can be an angle length r 9 01 Jay Independent coordinates If you freeze all but any one there still exists a range of values for that coordinate which do not violate the kinematic constraints lb Degrees of freedom Minimum number of independent manners generalized coordiantes in which the system can move The number of DOF is a characteristic of a system and independent of choice of coordinates DOFn m n number of coordinates used to specify the configuration of the system m number of independent constraint equations Coordinates chosen yAayB ZyA yB 2yB lconstL DOF211 7 u D 314 gig Are x and y dependent coordinates u39 Determine the DOF of the system G yA What do you suggest as XE independent coordinates to describe the configuration of this system 113 A set of coordinate is called complete if their values can be used to specify any and all admissible configurations of all parts of the system y2 a M M II HQ 0 quot6 rn on 21 are raf 7149 A W m w bo km 10053 a l l IO SDQ 1093 I11 4WD A A A n 1 n2 0 N bo hm an A A CHCZ ICJ body t ed MENMa mnz 01 s h q w Co kg39cka A A mH k qcza 77 Body I 16m Rn39l39a39hbns 5 VbKpH mRS Occur QAOWL 4X65 WCW 7 y baay I Kabh m sgwmbsn MbIx IL 1 Mums 3 J W 4 KN Ncc Km L qu 9 gain 3 Z N C I C J a 0 xncfhbm F 0W0 a1 3 g Moe eayfxxea 12 3 09 wn 39 A aquot J mama 39 A 059 quotu I A 3 gt l D ch AampZgt 3 1 d g 624w r C H20 1 a tr 1 NC 2 M 7L0 ra I7LDAfI ad 4M 0 S pqu 3 39Wc 4 SIWWQ 9 Ib fxamV f pbdbr A gt CalmleC a E rquot oolr Camd r 43 r73 bar A quot RA JRJDLR boil I C A9 w oia 03d 0quot E61 398 MMSLJ n a P A 4 F m 4 9 gt Wed 4 quot 0 9P x 375 93 s

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