MUTUAL PART ALIGNMENT USING ELASTIC VIBRATIONS

Transcription

1 E D V A R D A S S A D A U S K A S MUTUAL PART ALIGNMENT USING ELASTIC VIBRATIONS S U M M A R O F D O C T O R A L D I S S E R T A T I O N T E C H N O L O G I C A L S C I E N C E S, M E C H A N I C A L E N G I N E E R I N G ( 9 T ) Kaunas 5

2 KAUNO UNIVERSIT OF TECHNOLOG EDVARDAS SADAUSKAS MUTUAL PART ALIGNMENT USING ELASTIC VIBRATIONS Summar of Doctoral Dissertation Technological Sciences, Mechanical Engineering (9T) 5, Kaunas

3 The research was accomplished during the period of -4 at Kaunas Universit of Technolog, Facult of Mechanical Engineering and Design, Department of Production Engineering and Department of Mechatronics. Research was supported b Europe Structural Funds. Scientific supervisor: Prof. Dr. Habil. Bronius BAKŠS (Kaunas Universit of Technolog, Technological Sciences, Mechanical Engineering 9T). Dissertation Defense Board of Mechanical Engineering Science Field: Dr. Habil. Algimantas BUBULIS (Kaunas Universit of Technolog, Technological Sciences, Mechanical Engineering 9T) chairman Assoc. Prof. Dr. Giedrius JANUŠAS (Kaunas Universit of Technolog, Technological Sciences, Mechanical Engineering 9T) Prof. Dr. Habil. Genadijus KULVIETIS (Vilnius Gediminas Technical Universit, Technological Sciences, Mechanical Engineering 9T) Prof. Dr. Juozas PADGURSKAS (Aleksandras Stulginskis Universit, Technological Sciences, Mechanical Engineering 9T) Prof. Dr. Habil. Arvdas PALEVIČIUS (Kaunas Universit of Technolog, Technological Sciences, Mechanical Engineering 9T). The official defense of the dissertation will be held at a.m. on th of June, 5 at the Board of Mechanical Engineering Science Field public meeting in the Dissertation Defense Hall at the Central Building of Kaunas Universit of Technolog. Address: K. Donelaičio st. 7 4, LT-449, Kaunas, Lithuania, Phone nr. (+7) 7 4, Fa. (+7) 7 444, doktorantura@ktu.lt The summar of dissertation was sent on 9 th of Ma, 5. The dissertation is available on internet ( and at the librar of Kaunas Universit of Technolog (K. Donelaičio st., LT-449, Kaunas, Lithuania).

4 KAUNO TECHNOLOGIJOS UNIVERSITETAS EDVARDAS SADAUSKAS DETALIŲ TARPUSAVIO CENTRAVIMAS NAUDOJANT TAMPRIUOSIUS VIRPESIUS Daktaro disertacijos santrauka Technologijos mokslai, mechanikos inžinerija (9T) 5, Kaunas

5 Disertacija rengta -4 metais Kauno technologijos universitete, Mechanikos ir dizaino fakultete, Gambos inžinerijos ir Mechatronikos katedrose. Moksliniai trimai finansuoti Europos struktūrinių fondų lėšomis. Mokslinis vadovas: Prof. habil. dr. Bronius BAKŠS (Kauno technologijos universitetas, technologijos mokslai, mechanikos inžinerija 9T). Mechanikos inžinerijos mokslo krpties daktaro disertacijos gnimo tarba: Habil. dr. Algimantas BUBULIS (Kauno technologijos universitetas, technologijos mokslai, mechanikos inžinerija 9T) pirmininkas Doc. dr. Giedrius JANUŠAS (Kauno technologijos universitetas, technologijos mokslai, mechanikos inžinerija 9T) Prof. habil. dr. Genadijus KULVIETIS (Vilniaus Gedimino technikos universitetas, technologijos mokslai, mechanikos inžinerija 9T) Prof. dr. Juozas PADGURSKAS (Aleksandro Stulginskio universitetas, technologijos mokslai, mechanikos inžinerija 9T) Prof. habil. dr. Arvdas PALEVIČIUS (Kauno technologijos universitetas, technologijos mokslai, mechanikos inžinerija 9T). Disertacija bus ginama viešame Mechanikos inžinerijos mokslo krpties tarbos posėdje, kuris įvks 5 m. birželio d. val., Kauno technologijos universitete, Centrinių rūmų disertacijų gnimo salėje. Adresas: K. Donelaičio g. 7 4, LT-449, Kaunas, Lietuva Tel. (8-7) 4, faksas (8-7) 444, e. paštas doktorantura@ktu.lt Daktaro disertacijos santrauka išsiųsta 5 m. gegužės 9 d. Disertaciją galima peržiūrėti internete ( ir Kauno technologijos universiteto bibliotekoje (K. Donelaičio g., LT-449, Kaunas, Lietuva).

6 Introduction Automatic assembl sstems plas vital role in automating production process. The directl affect production efficienc and qualit of the goods. According to the statistical analsis, -6% of the tasks in most of the industries branches are assembl operations. Part assembling time takes 5-4% of all manufacture time. Around % of all assembl operations are peg to bush assembl operations. Because of that, assembl operation has big potential in reducing manufacture time b improving assembl methods and installing automatic part assembl sstems and devices. Assembl processes considering the level of automation sorted to several categories. First is manual assembl when a worker uses tools, worktable, grippers, conveors etc. to perform traditional assembl operations. Second is mechanised assembl, when workers use variet of power tools (impact wrench, press etc.). In the third categor a specialized automatic devices designated onl for the particularl assembl operation are used. Devices can be readjusted to produce several tpes of products. This tpe of assembl used in making different products in a big series. Assembl tpe of the fourth categor incorporates PLC (programmable logical controller) to control processes of separate assembl line modules. Fifth - is an adaptive assembl sstem. The process control sstem uses feedback signal to operate assembl equipment at the different stages of the part assembl. The main progress of automatic assembl is a robotic sstem, which accommodates programmable assembl devices, robots and manipulators. Because of geometrical tolerances of the parts, inappropriate basing of the parts, tolerances of the robot/manipulator positioning, linear and angular mismatch of the assembled parts ma occur. To compensate those inaccuracies manufactures uses passive or active part alignment methods. This work investigates a new approach of passive vibrator part alignment method using elastic vibrations. In this method bush placed on the assembl plane and is free to move in a narrow space. Another component (peg) fied in a gripper, which has piezoelectric vibrator in it. Vibrator presses upper end of the peg. Peg and a bush also pressed to each other with a predetermined force. Piezoelectric vibrator generates high frequenc harmonic ecitation to the peg and creates elastic vibrations of the peg in longitudinal and lateral directions. The lower end of the peg moves in elliptical shape trajector. Because of the friction force between the components, bush moves to the part alignment direction. Parts successfull assembled after the alignment occurs. This passive alignment method allows assembling parts with circular and rectangular cross-section with no chamfers and at their aial misalignment of few millimetres, or makes it possible to use low accurac robots with repeatabilit value of ±- mm. A vibrator part alignment device that uses elastic vibrations is more simple 5

7 technologicall since it does not use feedback signals or sophisticated control algorithms. Such alignment sstem with proper chosen ecitation signal parameters provides reliable, more efficient and cost effective part assembl comparing to active alignment sstems. Aims and objectives of scientific research. Research objective theoreticall and eperimentall investigate vibrator part alignment in automatic assembl when using elastic vibrations of the peg. Determine ecitation parameters for the stable and reliable part alignment. To achieve those objectives following tasks has to be fulfilled. Analse scientific papers about widel used part alignment methods in a now das industr. Carr out eperimental research of the peg s tip vibration while he is in a contact with bush. Determine nature of the peg s vibrations, their relationship to the ecitation signal amplitude and bush-to-peg pressing force. Perform part alignment eperiments with circular and rectangular crosssection pegs using their elastic vibrations. Determine influence of ecitation and mechanical sstem parameters to the alignment efficienc and reliabilit. Compose mathematical model of circular part alignment when the peg ecited in aial and transversal direction. Determine ecitation signal and mechanical sstem parameters for the stable and reliable part alignment at impact and non-impact modes. Methods of research. Numerical and eperimental methods used in this work. Peg and bush movement epressed b the sstem of second order differential equations and solved b Runge-Kuta method in Matlab. Movement of the movable component (bush) is modelled. Obtained results represented in a form of graphs and shows influence of ecitation and dnamic sstem parameters to the bush motion. Special eperimental set-up designed for the vibrator part alignment. Eperiments performed with parts of circular and rectangular cross-section and made from steel and aluminium. The peg fied in a gripper and vibrator ecitation done to the upper end of the peg in a longitudinal direction b mean of piezoelectric vibrator. Low frequenc generator Г-56/ provides ecitation signal to the circular shape piezo ceramic CTS-9. Bush and a peg alignment performed at different ecitation signal parameters, peg-to-bush pressing force and misalignment distance between the parts. Oscilloscope PicoScope 444 and computer Compaq nc6 measures alignment duration. Laser dopler vibrometer OFV5/OFV5 used in peg s lateral and longitudinal vibration measurements. Scientific novelt. The new scientific data revealed during preparation of the thesis: 6

8 . Technologicall easier wa for vibrator part alignment using peg s elastic vibrations was proposed. Piezoelectric vibrator presses upper end of the peg and ecites it in longitudinal direction.. As peg ecitation done on one end, the other end performs elliptical shape motion. Friction forces arise b pressing peg and bush to each other and ensure linear and rotational motion for the bush.. Vibrator part alignment using elastic peg s vibrations allows of centring circular and square cross-section parts at non-impact and impact modes when there is mechanical contact between them. 4. Was done peg-bush alignment simulations at non-impact and impact modes and alignment duration dependencies on ecitation frequenc, amplitude, initial pressing force were determined. Practical value. Part alignment using peg s elastic vibrations allows centring circular and rectangular cross-section parts with chamfers and without it and at aial misalignment error of several mm between the components. The proposed method epands technological capabilities of automatic assembl. Data collected in theoretical and practical research are useful in design and development of vibrator devices and sstems. Scope and structure of the dissertation. Dissertation consists of introduction, three chapters, conclusions, references and the author s publication list. The tet of dissertation comprises 9 pages, 6 figures and two tables. Propositions to be defended:. New vibrator part alignment method is technologicall easier method since it does not require feedback signal.. Peg s end tip moves in elliptical shape trajector and friction forces that rises in a contact point between the parts provides linear and rotational motion to the bush.. Nature of the bush motion and alignment duration depends on frequenc and amplitude of peg s vibrations, phase shift between longitudinal and lateral components, initial peg-to-bush pressing force, and aial misalignment between the parts. 4. Mathematical models sufficientl good describe real vibrator part alignment sstem and theoretical trend of part alignment duration dependencies correlates with eperimental ones.. Literature review More and more companies use robotic or automated assembl lines to ensure product qualit and reduce production costs. For the parts (peg and bush) to be assembled their connection surfaces has to match. 7

9 The principal problem in automated assembl is the uncertainties between mating parts due to various errors. These are sstematic positioning errors of robot or other manipulating device as well as errors due to insufficient control sstem resolution or due to vibrations in assembl area, etc. Uncertainties also result from random dimensions of assembled parts, their basing and fiing on a worktable or manipulation device. All those errors lead to inaccuracies between relative position of mating parts and prevents them from joining with each other. Alignment is the most important stage in automated assembl process, which compensates position offset between mating parts. Mutual part alignment carried out in man was: b interaction between part and a chamfer or other guiding element, using auto search method or active compliance control devices. Using auto search methods mating parts makes translational and rotational motion in a plane perpendicular to the joining ais until their connecting surfaces matches. Auto search categorized into three tpes: non-directional search (without feedback), directional search with feedback, directional search without feedback (but with vibration assistance). Alignment devices with feedback signals classif as active alignment methods. Passive alignment methods do not use feedback signal. There is no need of feedback signal if directional motion of movable based part done b mean of vibrational ecitation. Principal of vibrational part alignment lies in vibrational displacement effect inherent to the nonlinear asmmetric mechanical sstems. Structural, kinematic, force etc. asmmetr might emerge in nonlinear mechanical sstems. Force and kinematic asmmetr is most common in vibrator alignment. Force asmmetr originates from the rotational motion of the part as the components rest to each other. Vibrational ecitation of the turned part causes kinematic asmmetr. Vibrational non-impact displacement of a mobile-based bod on an inclined plane analsed b researches B. Bakss and N. Puodziuniene [, ] from Kaunas Universit of Technolog. Peg - hole alignment under aial peg vibrational ecitation investigated b B. Bakss and J. Baskutiene []. It was determined that alignment duration depends on ecitation frequenc, amplitude and sstem stiffness parameters. Immovable based bush also might be ecited in aial direction or in two perpendicular directions in a vertical plane. B. Bakss and K. Ramanauskte [4] investigated mutual part alignment using vibrator auto search method. Horizontal vibrating plane provides search motion for the part on it. Plane ecited in a two perpendicular directions can generate circular, elliptical or interwinding heli search paths. Motion of the unconstrained and elasticall and damping constrained part pressed with constant and varing pressing force was investigated. Mutual part alignment b directional vibrational displacement performed in the same wa as linear and rotational motion of the output link in the ultrasonic motors. M. E. Archangelskj [5] investigated oscillations of the cascade steel 8

10 vibrator. It was found that generating high frequenc (7.7 KHz) aial vibrations at the end of the vibrator, other end oscillates in aial and transverse directions. Because of the phase shift between vibration components tip of the vibrator makes elliptical motion. The brass disk starts to rotate around its aes when it touches rear or end surfaces of the vibrator, thus indicates about periodicall intermittent mechanical contact between disk and vibrator. Authors N. Mohri and N. Saito [6] investigated effect of lateral and longitudinal vibrations to the part insertion. It was determined that high frequenc lateral vibrations reduces dr friction coefficient between rear surfaces of the mating parts and facilitates part insertion. If the peg ecited in longitudinal and lateral directions a motive force generated for the parts mating. To epand technological capabilities of automated assembl, to simplif and reduce cost of assembl equipment a new vibrator alignment method presented. The novelt of this method is the use of elastic high frequenc vibrations. A friction force that rises in a contact point between components provides linear and rotational motion to the bush and directs it to the alignment direction. Vibrational displacement makes possible alignment and joining of the parts without chamfers and with aial part misalignment of few millimetres. Method is suitable for the parts with circular and rectangular cross-section. A comprehensive theoretical and eperimental research presented in this work as there is no scientific papers on this alignment method.. Eperiments To investigate peg s vibrations while it is in a contact with bushing the following eperimental equipment has been used (Fig..). Peg 4 fied in a middle cross-section in a gripper. Piezoelectric vibrator pressed to the upper end of the peg with pressing force F and ecites peg in aial direction. Ecitation signal to the vibrator provided b signal generator. The lower end of the peg is pressed to the bushing 5 with initial pressing force F while ais misalignment Δ. One ais laser dopler vibrometer (LDV) used to register peg s vibrations. The interferometer head OFV5 measures vibrations and controller OFV5 coverts signal from interferometer to the voltage signal corresponded to vibration amplitude. Further signal captured with oscilloscope PicoScope 444 and displaed on a computer screen. 9

11 Z π/ Г-56/ OFV5 OFV5 i F i 9 PicoScope444 +Δ 8 4 F 5 O 6 7 z i a) b) Fig.. Eperimental setup: a measurement scheme: gripper piezoelectric vibrator ecitation signal generator Г-56/ 4 peg 5 bushing 6 fiber interferometer OFV5 7 vibrometer controller OFV5 8 oscilloscope PicoScope444 9 personal computer PC z i longitudinal vibrations i, i lateral vibrations b measurement equipment Vibrometer measurements were taken in a three directions,, Z. Where, corresponds to lateral vibrations in a two perpendicular directions and Z are longitudinal vibrations. Peg Aial misalignment of the parts Δ and +Δ lies on - ais (Fig..). Thus mutual part alignment occurs when bushing center coincides with coordinate aes center. Bush Peg s tip vibration magnitude was investigated +Δ Δ under different pressing forces F (vibrator to the Fig.. Bush placement in respect to the peg peg) and F (peg to the bushing) when ecitation frequenc f varies from 65 to 67 Hz, and tip movement trajector was defined in relation with misalignment position Δ. In order to find movement trajector of the peg s end tip, measurements of two perpendicular aes (-, Z-, Z-) were taken. Snchronization signal related to the ecitation signal snchronizes measurement process. As long as vibrations are periodic and stead, vibrations magnitude ( i, i, z i ) of each ais PK

12 Vibration amplitude, µm Snchronization signal, V. i. i z i -. ε z ε -. - τ i.5..5 τ i, ms Fig.. Vibration signals and phase difference ε: snchronization signal vibrations vibrations 4 Z vibrations ε z 4 defined at the same periodic time τ i according to the snchronization signal (Fig..). Plotting those values in a Cartesian coordinate sstem peg s path in all three planes found. Time interval between two vibration signals at the same instantaneous phase gives us phase difference ε between those signals. Ecitation parameters and objects of eperiments presented in Table.. Table., Characteristics of ecitation signal and aligned parts No. I II III Peg Steel S5JR Diameter, mm Length, mm Chamfers No Bush Steel S5JR Hole diameter, mm... Ecitation signal parameters Frequenc, Hz Amplitude, V Influence of forces F and F to vibration amplitude was investigated on peg No. III. Pressing force of piezoelectric vibrator to the peg was graduall increased ever 4 N and corresponding measurements of vibration magnitudes on all three aes were taken. The results presented on figure. and.4. Force F and ecitation frequenc has no impact on vibration magnitude along ais. Vibration amplitude in -ais direction graduall increases when pressing force reaches 49 N and later stabilizes at 5 N. Meanwhile overall vibration magnitude decreases as ecitation frequenc increases. Amplitude of longitudinal vibrations increases more rapidl after F eceeds 9 N until that growth relativel small. Such character of amplitude increment related with contact area changes between peg and piezoelectric vibrator. More force is applied bigger micro deformations between peg and vibrator thus bigger contact area and more ecitation energ transferred to the peg. Since peg ecited with vibrations of high frequenc and small amplitudes, contact area between peg and

13 Vibration amplitude, µm amplitude, µm vibrator plas vital role. This could be seen from graph and, as ecitation frequenc increases pressing force F also has to be increased to keep same longitudinal vibration amplitude. It was also eperimentall set that mutual part alignment starts when force F eceeds 9 N, until that process of part alignment is not stable or it does not work at all F, N F = N 66, Hz 65, Hz 67, Hz F, N Fig.. Vibration amplitude versus force F : longitudinal Z vibrations: f=65 Hz f=66 Hz lateral vibrations: f=65 Hz 4 f=66 Hz 5 f=67 Hz lateral vibrations: 6 f=65 Hz 7 f=66 Hz 8 f=67 Hz Fig..4 vibration amplitude versus force F If force F had no impact on vibrations in -ais, totall different impact had force F. As the peg is pressed to the bushing with ais misalignment Δ=+.5 mm, vibration amplitude graduall increases as force F increases. The same tendenc retains even if ecitation frequenc changes in range from 65 to 67 Hz (Fig..4). In our case, part alignment is most rapid when peg is ecited at 66 Hz frequenc and vibrations in -ais are the biggest. Eperiment results mentioned above in generall shows what influence for the vibrations amplitude has mounting conditions of the peg, and that ecited peg vibrates in three directions perpendicular each other. However, there still no answer wh bushing is slides toward coordinate aes centre. To find out what factors in charge of this effect, motion trajector and direction of peg s tip was determined. After ecitation frequenc for stable and stead part alignment was eperimentall set to all pegs (Table.), motion trajector of the tip was taken in all three coordinate planes. Ecitation frequenc mainl depends from the peg s natural frequenc, design of the gripper and force F. Thus for the grippers with different design or made from different material ecitation frequenc for stead and stable part alignment will be different. In our case, ecitation frequenc for stable and stead part alignment have lied between second and

14 , µm Z, µm third natural bending mode of the peg. Figure.5 shows peg s tip path while forces F = N, F = N.5 Peg I Peg II Peg III Peg I Peg II Peg III Longitudinal vibrations are dominant in all cases and are twice as high as transverse ones. While in O plane the polarized in direction since peg s vibrations in direction are negligible. When the peg is pressed to the bushing with the force F =. N and ais misalignment Δ=-.5 mm, lateral vibrations on ais increases significant and peg s end moves in elliptical shape trajector in all three coordinate planes (Fig..6). Black dots on the path indicate its direction. For the different pegs, direction of rotation is different, that depends from ecitation frequenc, and natural mode gripper peg sstem vibrates. In order the alignment of the parts could occur, bushing has to slide along positive direction. There are two was how bushing aligned. First is direct alignment (Peg II and III). In this case peg s tip moves counter-clockwise in ZO plane (Peg II and III, b), thus direction of the normal force in the contact point lies on the positive direction and bushing is directl pushed toward coordinate aes center. Vibrations along -ais has little effect since their amplitude smaller than, and overall vibrations are more polarized along -ais (Peg II and III, a). Normal peg to bushing pressing force is bigger when longitudinal vibrations amplitude is negative. Thus, propellant force is bigger when peg vibrates along positive -ais rather than negative. Second wa of part alignment is indirect alignment (Peg I). Here peg s motion is clockwise in ZO plane (Peg I, b) and bushing is pushed from the coordinate aes center. However, because of the peg s tip elliptical movement in , µm ,. -., µm a) b) Fig..5 Path trajector of unloaded peg: a in O plane b in ZO plane

15 , µm Z, µm Z, µm, µm Z, µm Z, µm, µm Z, µm Z, µm III II I O plane (Peg I, a), bushing is turned b the angle so the alignment trajector lie on the major ais of the ellipse and then pushed towards coordinate aes center.. ε =-. ε z =-.5 ε z = , µm -., µm -., µm ε =. ε z =. ε z = , µm -.8, µm -.5, µm ε =.9 ε z =.6 ε z = , µm -., µm -., µm a b c Fig..6 Path trajector of loaded peg when Δ=-.5 mm: a in O plane b in ZO plane c in ZO plane Results of peg s tip trajector while Δ=+.5 mm presented in figure.7. As contact conditions between peg and bushing has changed (contact area crescent now faced to opposite side), phases between vibrations also changed. In this case, for the bushing to align with the peg, bushing has to slide along negative direction. Peg s I and II tip moves clockwise in a ZO plane (Peg I and II, b) thus direct alignment is going. Bushing with the Peg III aligned during indirect alignment. The bush is propelled along negative direction, but because of rotation effect in O plane (Peg III, a) bushing is turned and pointed to the coordinate aes center. It is clear that during direct alignment peg s motion in ZO plane plas ke role, meanwhile during indirect alignment there is combination of peg s movement in ZO and O planes.

16 , µm Z, µm Z, µm, µm Z, µm Z, µm, µm Z, µm Z, µm III II I. ε =.5. ε z =-.5. ε z = , µm -., µm -., µm.8 ε z =-.97 ε z = ε = , µm -.8, µm -.8, µm.5 ε ε z =.9 =.6 ε z = , µm -., µm -.5, µm a b c Fig..7 Path trajector of loaded peg when Δ=+.5 mm: a in O plane b in ZO plane c in ZO plane Eperimental setup designed and made to investigate part alignment when elastic vibrations applied to the peg (Fig..8). The peg fied in a gripper 8. Gripper can move in vertical direction in order to insert peg into the bush when alignment occurs. Spring 7 works as gravit force compensator for the gripper and helps to capture the moment as the peg falls into the bush hole. Verticall, moving table 6 adjusts pressing force of mating parts. The table moved horizontall in order to change ais misalignment, which is measured with indicator 9. Low frequenc signal generator provides signal to the piezoelectric vibrator. The amplitude and frequenc of the signal are measured b multimeter. Switch 5, oscilloscope 4 and personal computer are used for alignment event triggering and alignment duration measurement respectivel. 5

17 Fig..8 Eperimental setup: multimeter FLUKE computer Compaq nc6 signal generator Г 56/ 4 oscilloscope PicoScope switch 6 table 7 spring 8 - gripper 9 indicator BDS Technics The peg is hold in a middle cross-section b the clamps of the gripper (Fig..9). Piezoelectric vibrator is implemented in a housing. F Threaded end of the housing can freel rotate in a gripper at the same time performing linear 9 motion towards the peg. 4 Δ As the piezoelectric vibrator F 5 lean to the peg, further torque increment sets pressing force for 6 7 the piezoelectric vibrator to the 8 peg. Bush 4 is mobile based on the electricall conductive plate Fig..9 Measurement circuit: Peg piezoelectric vibrator housing 4 bush 5 while the latter is located on 5 plate 6 force sensor 7 9 V power the force sensor 6. The bush, suppl 8 switch 9 light-emitting diode plate, and force sensor fied to (LED) oscilloscope signal the table and moves together. generator computer table The following electrical circuit was designed to measure the alignment time. Anode of the power suppl 7 connected to the electricall 6

18 conductive plate. Cathode first connected to the switch 8 and LED 9 and later to the gripper. Oscilloscope is measuring voltage signal on the LED. When the switch closes electrical circuit, the voltage jump on the LED occurs. At the same time, ecitation signal from generator connected to the piezoelectric vibrator 6. As the bush slides to the peg s center, contact resistance alternating and electrical signal has unstable manner. When alignment between peg and the bush occurs, there is no mechanical contact between them and the voltage jump on the LED is the lowest. Measured signal transferred to the computer and b mean of the software alignment time is calculated. During investigation, the peg is ecited in aial direction b mean of clindrical shape piezoelectric vibrator with mm in diameter and mm in height. Pressing force vibrator-to-peg is set to N and kept constant throughout the eperiments. Harmonic ecitation signal generated b low frequenc generator. Each time eperiment repeated four times and a mean value of four trials is taking as a result. Influence of ais misalignment Δ, ecitation frequenc f, ecitation signal amplitude U and initial peg-to-bush pressing force F to the alignment duration Δt is investigated. Eperiments were carried out with steel and aluminium pegs with circular (C) and rectangular (R) crosssections and their counterparts steel and aluminium bushings. The alignment of rectangular parts was done along short side of the peg. The parts were both tpe with chamfers and with no chamfers. Measurements of the parts used in eperiments are given in Table. Table., Material and geometrical data on specimens No. Peg Bush Crosssection Chamfers Steel S5JR Diameter, mm Lengh, mm Diameter, mm No I C II C III C IV C V C Aliuminium SAPA68-T6 VI C No VII C.554º Steel S5JR Lengh Widh Heigh, mm Lengh Widh, mm VIII R Nėra I R.49º The dependencies of alignment duration Δt on ais misalignment Δ is presented in figure.. Steel peg I ecited under different ecitation frequenc 7

19 and initial pressing force F. Ecitation signal amplitude U=4 V is same to all investigated pegs. It is determined that alignment duration increases as ais misalignment increases. The character of a graph is linear and do not depend on ecitation frequenc. We can also see that alignment duration depends on misalignment direction. In a direct alignment case the alignment duration is shorter (Fig., a). However, ecitation frequenc has significant influence to the alignment duration (Fig..).,5.5 F =. N Δt, s,., , 5. F =. N Δt, s,.,7.7,.,..4,4,4.4,5.5 Δ, mm,5.4,4,4.4,5.5 Δ,,5 mm a) b) Fig.. Alignment duration dependencies on ais misalignment Δ: a) bush placement +Δ, b) bush placement Δ f=7 Hz f=75 Hz f=7 Hz 4 f=75 Hz 5 f=7 Hz 4,5.5 Δt, s,.,5.5, f, Hz F =. N 4 5, 5. a) b) F =. N Δt, s 9, ,7.7 4, f, 7 Hz Fig.. Alignment duration dependencies on ecitation frequenc f: a) bush placement +Δ, b) bush placement Δ Δ=,4 mm Δ=,6 mm Δ=,8 mm 4 Δ=, mm 5 Δ=,5 mm 6 Δ=, mm 7 Δ=,5 mm 8 Δ=, mm 9 Δ=,5 mm

20 The alignment of the parts is most rapid when ecitation frequenc is between 75-7 Hz and this trend visible under different ais misalignment. As frequenc changes from these values, alignment duration increases. It was also determined that for a small ais misalignment (up to mm) the influence of ecitation frequenc is negligible. Ecitation frequenc at which alignment of the parts is most rapid increases as geometrical dimensions of the peg decreases. However, size of the peg is not the onl reason of frequenc changes. Contact qualit between peg and piezoelectric vibrator plas significant role in an ecitation frequenc. More is the area the end surface of the peg touches vibrator, more acoustic energ transferred to it, as well as ecitation frequenc is lower. During eperiments was noticed that end surface of smaller diameter peg was harder to make parallel to the end surface of the vibrator. That circumstance should be taken in consideration making an conclusions on ecitation frequenc using smaller diameter pegs.,.,5.5 9 f=75 Hz Δt, s 7 f=75 Hz Δt, s 8,.,7.7, ,.,5.5,..5,5 F, N 6,5.5,., ,.,5.5.,,5.5 a) b) F, N Fig.. Alignment duration dependencies on force F : a) bush placement +Δ, b) bush placement Δ Δ=,4 mm Δ=,6 mm Δ=,8 mm 4 Δ=, mm 5 Δ=,5 mm 6 Δ=, mm 7 Δ=,5 mm 8 Δ=, mm 9 Δ=,5 Figure. represents dependencies of alignment duration Δt on initial pressing force F under different ais misalignment. Influence of initial pressing force on alignment duration is relativel small when ais misalignment is up to mm. In a case when Δ> mm alignment duration decreases as force F increases.. Numerical simulation of part alignment at non-impact and impact modes During vibrator alignment, two solid bodies like peg and a bush interact with each other. Peg presses bush with predetermined force and its elastic vibrations are ecited. Friction forces that rise during interaction of those two 9

21 bodies guide bushing to the ais alignment direction. It is necessar to make sstems consisting of two interactive bodies dnamical modelling in order to eamine alignment process further. In general case peg is fied in a speciall designed gripper while bush is based on a plane. Piezoelectric vibrator presses the top end of the peg and ecites its elastic vibrations. Eperimental research has showed that longitudinal and lateral vibrations of the peg are created. There is a phase shift between them thus, peg s tip moves in elliptical trajector on a vertical plane. Alignment process is modelled b two-mass dnamical sstem in a reference frame O (Fig..). A mass m depicts peg that oscillates in two Asinω t K Δ N m perpendicular directions while mass m is a bush that has to be aligned to the peg. Alignment process is possible onl when peg press bush with predetermined force and oscillation amplitude is on the proper level. Vibration amplitude depends on ecitation frequenc and is the biggest when sstem oscillates close to their natural mode. Natural mode itself depends on the geometrical characteristics of piezoelectric vibrator and a peg, the wa in which peg fied in a gripper, peg-to-bush pressing force magnitude. Tpical ecitation frequenc is in a range of kilohertz and amplitude of few micrometres. Since vibration amplitude is at the same measurable level as roughness of the surfaces, rheological properties of the bodies should be taken into account. At the contact point, the surface teture deforms in a normal and tangential directions. During high frequenc elastic vibrations not onl elastic but also elasto-plastic deformations ma occur. To evaluate deformations of this kind we use rheological Kelvin Voigt model. Thus, the surface of the mass m in normal and tangential directions constructed b stiffness (K, K ) and damping (H, H ) elements connected in parallel. Surface deformations induce reaction forces R, R : R H Bsin(ωt+ε) K H K m K R H K Fig.. Dnamical model H, K H. H (.)

22 where deformation of the contact surface because of the initial pressing force, ( - ) relative deformation speed in direction, deformation speed in direction. Mass s m tip longitudinal t A sin and lateral t Bsin vibration amplitudes var according to the law of the sinus. Stiffness and damping forces restricts mass movement in and directions.., H K R H K R (.) When two bodies are in contact, friction forces arise in their contact zone. Its magnitude epressed using dr friction model. Force F fr affects mass m. Force F fr that is sum of friction forces peg-bush and bush-base acts on mass m. sign N F fr (.). sign N sign N F fr (.4) where μ coefficient of friction between mass m and m, μ coefficient of friction between m and base, N normal pressing force. All friction forces formulated taking into account that the are not affected b relative speed. We get equation of motions for mass m and m b projecting all acting forces in and aes:., sin, sin sign N sign N K H m t A K K K H H m t B K sign N K H m where / / dt d dt d We are using following dimensionless parameters to have generalized results of simulation:. m l k n l l l p m N n p l A a l B b m m k k k p m K k p m K k K K k h h h p m H h p m H h p m H h p m H h l l l m K p pt (.5)

23 Then motion equations written in a dimensionless form: h h k h n sign k k bsin, asin k, n sign n sign k. (.6) where d / dt d / d We used a program code written in MATLAB environment to obtain numerical simulation results. Solver ode5s was used in calculating stiff differential equation. Since our dnamic sstem is described b second order differential equations, we had to rewrite them to a pair of simultaneous first order differential equations (Esfandiari, ) to obtain the solution. The following initial values of the parameters of the dnamic sstem were used b, a, k 5, k,, k,, h h h h,7,,,,,,,,,,,,. Initial conditions alignment conditions,,,,,. As τ= normal reaction force is equal to initial pressing force n k. After the ecitation signal is applaed, force n alternates and it s value depends on mass s m coordinate, thus n k. Alignment occurs when. During simulation, we analsed the effect of each parameter on the part alignment process b adjusting onl one parameter and keeping all the others constant. Alignment duration dependenc on ecitation frequenc at different phase shifts between longitudinal and lateral vibrations represented in figure.4. Alignment duration is lower at the lower frequenc values if phase shift is between and π/. As ecitation frequenc increases, alignment duration also increases. When phase shift is grater then π/, alignment duration decreases as ecitation frequenc increases since ellipsis of peg movement trajector have changed inclination angle and short ais of the ellipsis have shortened. There is also a peak in alignment duration at the ecitation frequenc ν= no matter the phase shift between vibration components. This is because ecitation frequenc became equal to the natural frequenc of the bush along -ais. Figure.5 shows alignment duration dependenc on the phase shift between vibrations at different ecitation frequencies. Dependencies have parabolic character, thus, there eist phase shift at which alignment process is the most rapid. All curves have intersection points in the region between π/4 to 5π/ and it means that ecitation frequenc has little effect on alignment duration in this region.

24 τ 5.7 τ ,.,6.6,.,5.5 ν,. Fig..4 Alignment duration dependencies on ecitation frequenc ν ε= ε=.79 ε=.57 4 ε=.6 5 ε=.4.,.,5.5,9.9 ε,4.4 Fig..5 Alignment duration dependenc on phase shiftε ν=. ν=. ν=.5 6 ν=.7 - ε= ε=π/6 ε=π/ ε=π/ a - a a a b b b b a) b) c) d) ε=π/ - - ε=5π/6 - ε=π - ε=7π/ a - a - a a b b b b e) f) g) h) Fig..6 Peg s tip motion trajector on phase ε Motion trajector of the peg s end tip depends on the phase shift between lateral and longitudinal vibrations (Fig..6). In our case for the bush to be aligned with the peg, necessar that peg s end tip moves counter clockwise direction (Fig..6, b-f). Phase shift is between π/6 and 5π/6 radians in this case. Peg s end tip moves in a clockwise direction when the phase shift is grater then π (Fig..6, h). The bush does not align with the peg anmore, but rather moves awa from it. In case when phase shift is or π radians (Fig..6, a, g) -4-4

25 alignment process has unstable manner and depending on ecitation frequenc alignment of the parts ma occur or not. τ 5 τ b 5 7 a 9 Fig..7 Alignment duration dependencies Fig..8 Alignment duration dependencies on lateral vibration amplitude b a= on longitudinal vibration amplitude a a= a=4 4 a=6 5 a=8 b= b= b=9 Another important parameter that has direct influence on the alignment time is amplitude of lateral and longitudinal vibrations of the peg figure.7,.8 respectivel. When lateral vibrations reach certain limit (in our case b=) their influence on the alignment duration becomes negligible. Much bigger influence on the process time has longitudinal vibrations. As amplitude increases, alignment duration constantl decreases. τ 4,5,9 4, 5,6 k 7, Fig..9 Alignment duration dependencies on initial deformation k b= b= b= 4 b=4 5 b= τ ,..,,5.5.7,7 Fig.. Alignment duration dependencies on dr friction coefficient μ a=6: μ =,6 μ =,8 μ =, 4 μ =, 5 μ =,6 6 μ =,8 Because repulsive force created due to the friction between a peg and a bush, initial pressing force between those parts plas ke role in the part alignment (Fig..9). For the process to be stable and reliable initial pressing force has to be at a certain limit, but not <.5. If it is less, the sstem runs into the impact.4. 4 μ 6 5

26 mode and our model cease to be valid. We have to increase initial pressing force to rule out sstem from the impact mode. However to obtain the shortest alignment duration we have to keep it as low as possible but avoiding sstem to fall into the impact mode. As initial pressing force increases, alignment duration also increases. In case when lateral vibration is lower, alignment duration increases more rapidl in comparison to the cases when vibration is higher. Two friction forces acts on the bush during part alignment. One is the friction force between peg and the bush that moves bush to the alignment direction. Second one is friction force between bush and the base which causes bush movement to slow down. Friction forces directl proportional to the coefficient of dr friction between acting surfaces (Fig..). Alignment duration keeps stable as coefficient μ increases, there is onl small decrease of alignment duration at μ =.. There is rapid increase in the alignment duration when coefficient of friction is more than.5. Dr friction coefficient between bush and the base has to be as small as possible. When μ >. alignment process starts at μ >.. When μ >.9 alignment process stops. τ δaa 5 4 τ 5.,.4,4,7.7 ν a) b) Fig.. Alignment duration dependencies a) on ais misalignment δ ν=. ν=. ν=.6 4 ν=.8 5 ν=., b) on ecitation frequenc ν δ=4 δ=8 δ=5 4 δ=5 5 δ=5 Alignment take place at different ais misalignment between the parts and at different ecitation frequenc (Fig.., a, b). The alignment duration increases as ais misalignment increases. Dependencies have linear character no matter the ecitation frequenc. We can see that ecitation frequenc has low impact on the alignment duration when ais misalignment is small (Δ<8). Onl when misalignment increases the influence of ecitation parameter becomes apparent. Work pieces align most rapidl when mass m vibrates at resonant frequenc. As ecitation frequenc rangers from resonant, alignment duration constantl increases until process becomes impossible. Qualitativel dependencies have a

27 good match with eperimental results thus confirms validit of our mathematical model. During eperimental alignment of the parts was observed part alignment at impact mode when contact disappears between bush and the peg tip. At this moment peg breaks awa from the bush and later hits it at certain speed level. Such alignment regime forms when longitudinal oscillations has bigger Asinω m I N Δ K m H Bsin(ωt+ε) H K I I Fig.. Model of the contact interaction amplitude or pressing bush-to-peg force is not sufficient. During impact part alignment a recurrent interaction between peg and a bush is going. Peg breaks from the bush when normal component of ecitation force higher then pegto-bush pressing force. To simulate impact part alignment it is necessar to form equations describing motion of the impact bod before the impact with the bush and the impact interaction. Peg rendering mass oscillates in normal and tangential directions. Resultant bod motion trajector depends from ecitation amplitude components and their phase. The motion trajector in the vertical plane could be circular, elliptical or linear inclined at the certain angle to the horizontal ais (Fig..). Motion of the mass m when it breaks from the bush m defines equations: m H K K Bsin t m H K K Asint., (.7) Diagonal impacts of mass m causes motion of the mass m defines b equation: m N sign (.8) We use dimensionless parameters to get generalised version of motion equation: 6

28 7. m l l l l p g d g m N l p m N n p l A a l B b m m m p K k K K k m p H h m p H h l l l m K p pt Motion of the bouncing mass m in dimensionless form:. sin, sin k a k k h b k h (.9) Dimensionless equation of motion of mass m :. sign d (.) Interaction of the bodies at the moment of the diagonal impact defines impact equations. Describing diagonal impact, we assume that velocit components of normal impact var according to the linear impact law and do not depend on tangential velocit component. When impact is linear, normal velocit of the mass m after the impact defined b equation:. R (.) where mass m velocit before the impact, R impact restitution coefficient. To define impact interaction we use hpothesis of dr friction that determines link between normal and tangential impact impulses:. I I (.) where I, I impact impulses, μ coefficient of dr friction. Studing diagonal impact, we assume that slipping velocit between the bodies in the impact interval is alwas positive. Such impact called a sliding impact. Normal velocit restitution equation valid for the sliding impact onl. There are two phases of the sliding impact. First is a load phase. It starts from the moment of the contact between bodies and continues until reaches maimum surface deformation. Second is load reduction phase. It starts at the moment of the deformation end until break of the contact between the bodies. When m hits m during load phase in accordance with impulse hpothesis, we can write:. m I (.)

29 8 m. (.4) I where, bod m velocit before impact, absolute sliding velocit at the end moment of the first impact phase. B inserting (.) and (.4) to (.) we get:. (.5) Direction of the tangential velocit cannot be changed at the impact moment, thus. From (.5) we get: / That is a self-stop condition for the bod m. Tangential displacement of bod m stops at the first stage of the impact and it bounces from the bod m in a normal direction if this condition not fulfilled. Because we investigate case of the sliding impact, the bod m does not bounce at the end of the first impact phase and impact process continues. Thus at the end of the second impact mode, we can write:. I m. (.6) m. (.7) I where, bod m velocities after the impact. Epressions (.6) and (.7) linking with (.) and taking in to account (.) and (.5), we can calculate m tangential velocit after the impact:. (.8) R Tangential impulses of the bod m make bod m to slide towards ais misalignment direction. Bodies m and m impact impulses according to the impulses hpothesis is: I I m m,. (.9) Impact impulse to the bod m transferred b the dr friction thus we can write: I I. (.) Composing impulse epressions (.9) to the (.) and taking in to the account (.8), we get equation for calculating velocit of the bod m after the sliding impact:

30 R. (.) Epressions (.), (.8) and (.) used to calculate bodies m and m velocities after the impact. During numerical simulation we used the following constant values: b, a, k 5, k,, h h,7,,57,,,,,,,,,4,, n, R, 7. Initial conditions:,,,. Part alignment condition:., Ecited peg oscillates in longitudinal and lateral directions and hits the bush. Restitution coefficient R valuates deformation of the bush. At the impact moment, the impact energ transferred to the bush and it slides to the ais misalignment direction. Peg bounces from the bush after the energ transferred meanwhile a bush keep sliding because of inertia until the net impact. Proper settings of the ecitation and mechanical sstem parameters must be chosen to have alignment process stable and reliable. The peg ecited in the frequenc range from to.7 to have alignment of the bush reliable (Fig..). in this frequenc range alignment duration do not depend on the phase shift between vibration components and is easil predictable. As ecitation frequenc increases, alignment duration decreases and reaches minimal value at.4. Subsequent increase of the ecitation frequenc makes alignment duration to increase. When ν< or ν>.4 alignment duration is hardl predictable and changes rapidl if small ecitation τ frequenc changes applied. 4 7 As ais misalignment,8 increases, alignment duration also 6 increases. Dependencies has linear,6 character and do not depend on 5 ecitation frequenc (Fig..4, b).,4 Influence of ecitation frequenc to the alignment duration is minimal when δ>8 (Fig..4, a)., Onl when ecitation frequenc,,7,4 ν, increases we can observe Fig.. Alignment duration frequenc range at which part dependencies on ecitation frequenc ν: alignment is the fastest. If - ε= - ε=.6 - ε= ε= ε= ε=. 7 - ε=.57 ecitation frequenc is more than.7 alignment process stops. As ν.9 alignment process recurs again, but alignment duration rapidl decreases as ecitation frequenc increases. 9

31 τ τ ,.,,7.7., δ 5 a) b) Fig..4 Alignment duration dependencies on: a) ecitation frequenc ν: - δ=4 - δ=6 - δ=8 4 - δ= 5 - δ=5 6 - δ= 7 - δ=5 8 - δ= 9 - δ=5 b) ais misalignment δ: - ν= - ν=. - ν= ν= ν= ν= As longitudinal vibration amplitude increases, alignment duration decreases eponentiall (Fig..5, a). Lateral vibration amplitude if it is not equal to zero, has no influence to alignment duration at all (Fig..5, b). τ.6.4.,,, 4, 5.,,..7,7 a 5. 5, τ 4.,,.,7.7 b 5, 5. a) b) Fig..5 Alignment duration dependencies on: a) longitudinal vibration amplitude a: - b= - b= - b= 4 - b=4 5 - b=5 b) lateral vibration amplitude b: - a= - a= - a=4 4 - a=5 Friction forces between bush and peg and between bush and base also have influence to the process duration. Their influence evaluates dr friction coefficients μ and μ. As friction force between bush and peg increases, alignment duration decreases eponentiall (Fig..6, a). Meanwhile if friction forces between bush and base increases, alignment duration increases linearl (Fig..6, b)..6.4.