mutual part alignment using elastic vibrations
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 Y 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 ( 0 9 T ) Kaunas 2015 KAUNO UNIVERSITY OF TECHNOLOGY EDVARDAS SADAUSKAS MUTUAL PART ALIGNMENT USING ELASTIC VIBRATIONS Summary of Doctoral Dissertation Technological Sciences, Mechanical Engineering (09T) 2015, Kaunas The research was accomplished during the period of 2010-2014 at Kaunas University of Technology, Faculty of Mechanical Engineering and Design, Department of Production Engineering and Department of Mechatronics. Research was supported by Europe Structural Funds. Scientific supervisor: Prof. Dr. Habil. Bronius BAKŠYS (Kaunas University of Technology, Technological Sciences, Mechanical Engineering – 09T). Dissertation Defense Board of Mechanical Engineering Science Field: Dr. Habil. Algimantas BUBULIS (Kaunas University of Technology, Technological Sciences, Mechanical Engineering – 09T) – chairman; Assoc. Prof. Dr. Giedrius JANUŠAS (Kaunas University of Technology, Technological Sciences, Mechanical Engineering – 09T); Prof. Dr. Habil. Genadijus KULVIETIS (Vilnius Gediminas Technical University, Technological Sciences, Mechanical Engineering – 09T); Prof. Dr. Juozas PADGURSKAS (Aleksandras Stulginskis University, Technological Sciences, Mechanical Engineering – 09T); Prof. Dr. Habil. Arvydas PALEVIČIUS (Kaunas University of Technology, Technological Sciences, Mechanical Engineering – 09T). The official defense of the dissertation will be held at 10 a.m. on 30th of June, 2015 at the Board of Mechanical Engineering Science Field public meeting in the Dissertation Defense Hall at the Central Building of Kaunas University of Technology. Address: K. Donelaičio st. 73 – 403, LT-44029, Kaunas, Lithuania, Phone nr. (+370) 37 300042, Fax. (+370) 37 324144, e-mail: [email protected] The summary of dissertation was sent on 29th of May, 2015. The dissertation is available on internet (http://ktu.edu) and at the library of Kaunas University of Technology (K. Donelaičio st. 20, LT-44239, Kaunas, Lithuania). KAUNO TECHNOLOGIJOS UNIVERSITETAS EDVARDAS SADAUSKAS DETALIŲ TARPUSAVIO CENTRAVIMAS NAUDOJANT TAMPRIUOSIUS VIRPESIUS Daktaro disertacijos santrauka Technologijos mokslai, mechanikos inžinerija (09T) 2015, Kaunas Disertacija rengta 2010-2014 metais Kauno technologijos universitete, Mechanikos ir dizaino fakultete, Gamybos inžinerijos ir Mechatronikos katedrose. Moksliniai tyrimai finansuoti Europos struktūrinių fondų lėšomis. Mokslinis vadovas: Prof. habil. dr. Bronius BAKŠYS (Kauno technologijos universitetas, technologijos mokslai, mechanikos inžinerija – 09T). Mechanikos inžinerijos mokslo krypties daktaro disertacijos gynimo taryba: Habil. dr. Algimantas BUBULIS (Kauno technologijos universitetas, technologijos mokslai, mechanikos inžinerija – 09T) – pirmininkas; Doc. dr. Giedrius JANUŠAS (Kauno technologijos universitetas, technologijos mokslai, mechanikos inžinerija – 09T); Prof. habil. dr. Genadijus KULVIETIS (Vilniaus Gedimino technikos universitetas, technologijos mokslai, mechanikos inžinerija – 09T); Prof. dr. Juozas PADGURSKAS (Aleksandro Stulginskio universitetas, technologijos mokslai, mechanikos inžinerija – 09T); Prof. habil. dr. Arvydas PALEVIČIUS (Kauno technologijos universitetas, technologijos mokslai, mechanikos inžinerija – 09T). Disertacija bus ginama viešame Mechanikos inžinerijos mokslo krypties tarybos posėdyje, kuris įvyks 2015 m. birželio 30 d. 10 val., Kauno technologijos universitete, Centrinių rūmų disertacijų gynimo salėje. Adresas: K. Donelaičio g. 73 – 403, LT-44029, Kaunas, Lietuva Tel. (8 - 37) 300042, faksas (8 - 37) 321444, e. paštas [email protected] Daktaro disertacijos santrauka išsiųsta 2015 m. gegužės 29 d. Disertaciją galima peržiūrėti internete (http://ktu.edu) ir Kauno technologijos universiteto bibliotekoje (K. Donelaičio g. 20, LT-44239, Kaunas, Lietuva). Introduction Automatic assembly systems plays vital role in automating production process. They directly affect production efficiency and quality of the goods. According to the statistical analysis, 30-60% of the tasks in most of the industries branches are assembly operations. Part assembling time takes 35-40% of all manufacture time. Around 33% of all assembly operations are peg to bush assembly operations. Because of that, assembly operation has big potential in reducing manufacture time by improving assembly methods and installing automatic part assembly systems and devices. Assembly processes considering the level of automation sorted to several categories. First is manual assembly when a worker uses tools, worktable, grippers, conveyors etc. to perform traditional assembly operations. Second is mechanised assembly, when workers use variety of power tools (impact wrench, press etc.). In the third category a specialized automatic devices designated only for the particularly assembly operation are used. Devices can be readjusted to produce several types of products. This type of assembly used in making different products in a big series. Assembly type of the fourth category incorporates PLC (programmable logical controller) to control processes of separate assembly line modules. Fifth - is an adaptive assembly system. The process control system uses feedback signal to operate assembly equipment at the different stages of the part assembly. The main progress of automatic assembly is a robotic system, which accommodates programmable assembly 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 may occur. To compensate those inaccuracies manufactures uses passive or active part alignment methods. This work investigates a new approach of passive vibratory part alignment method using elastic vibrations. In this method bush placed on the assembly plane and is free to move in a narrow space. Another component (peg) fixed 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 frequency harmonic excitation 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 trajectory. Because of the friction force between the components, bush moves to the part alignment direction. Parts successfully assembled after the alignment occurs. This passive alignment method allows assembling parts with circular and rectangular cross-section with no chamfers and at their axial misalignment of few millimetres, or makes it possible to use low accuracy robots with repeatability value of ±1-2 mm. A vibratory part alignment device that uses elastic vibrations is more simple 5 technologically since it does not use feedback signals or sophisticated control algorithms. Such alignment system with proper chosen excitation signal parameters provides reliable, more efficient and cost effective part assembly comparing to active alignment systems. Aims and objectives of scientific research. Research objective – theoretically and experimentally investigate vibratory part alignment in automatic assembly when using elastic vibrations of the peg. Determine excitation parameters for the stable and reliable part alignment. To achieve those objectives following tasks has to be fulfilled. Analyse scientific papers about widely used part alignment methods in a now days industry. Carry out experimental 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 excitation signal amplitude and bush-to-peg pressing force. Perform part alignment experiments with circular and rectangular crosssection pegs using their elastic vibrations. Determine influence of excitation and mechanical system parameters to the alignment efficiency and reliability. Compose mathematical model of circular part alignment when the peg excited in axial and transversal direction. Determine excitation signal and mechanical system parameters for the stable and reliable part alignment at impact and non-impact modes. Methods of research. Numerical and experimental methods used in this work. Peg and bush movement expressed by the system of second order differential equations and solved by 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 excitation and dynamic system parameters to the bush motion. Special experimental set-up designed for the vibratory part alignment. Experiments performed with parts of circular and rectangular cross-section and made from steel and aluminium. The peg fixed in a gripper and vibratory excitation done to the upper end of the peg in a longitudinal direction by mean of piezoelectric vibrator. Low frequency generator Г3-56/1 provides excitation signal to the circular shape piezo ceramic CTS-19. Bush and a peg alignment performed at different excitation signal parameters, peg-to-bush pressing force and misalignment distance between the parts. Oscilloscope PicoScope 4424 and computer Compaq nc6000 measures alignment duration. Laser dopler vibrometer OFV512/OFV5000 used in peg’s lateral and longitudinal vibration measurements. Scientific novelty. The new scientific data revealed during preparation of the thesis: 6 1. Technologically easier way for vibratory part alignment using peg’s elastic vibrations was proposed. Piezoelectric vibrator presses upper end of the peg and excites it in longitudinal direction. 2. As peg excitation done on one end, the other end performs elliptical shape motion. Friction forces arise by pressing peg and bush to each other and ensure linear and rotational motion for the bush. 3. Vibratory 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 excitation frequency, 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 axial misalignment error of several mm between the components. The proposed method expands technological capabilities of automatic assembly. Data collected in theoretical and practical research are useful in design and development of vibratory devices and systems. Scope and structure of the dissertation. Dissertation consists of introduction, three chapters, conclusions, references and the author’s publication list. The text of dissertation comprises 90 pages, 61 figures and two tables. Propositions to be defended: 1. New vibratory part alignment method is technologically easier method since it does not require feedback signal. 2. Peg’s end tip moves in elliptical shape trajectory and friction forces that rises in a contact point between the parts provides linear and rotational motion to the bush. 3. Nature of the bush motion and alignment duration depends on frequency and amplitude of peg’s vibrations, phase shift between longitudinal and lateral components, initial peg-to-bush pressing force, and axial misalignment between the parts. 4. Mathematical models sufficiently good describe real vibratory part alignment system and theoretical trend of part alignment duration dependencies correlates with experimental ones. 1. Literature review More and more companies use robotic or automated assembly lines to ensure product quality and reduce production costs. For the parts (peg and bush) to be assembled their connection surfaces has to match. 7 The principal problem in automated assembly is the uncertainties between mating parts due to various errors. These are systematic positioning errors of robot or other manipulating device as well as errors due to insufficient control system resolution or due to vibrations in assembly area, etc. Uncertainties also result from random dimensions of assembled parts, their basing and fixing 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 assembly process, which compensates position offset between mating parts. Mutual part alignment carried out in many ways: by 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 axis until their connecting surfaces matches. Auto search categorized into three types: non-directional search (without feedback), directional search with feedback, directional search without feedback (but with vibration assistance). Alignment devices with feedback signals classify 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 by mean of vibrational excitation. Principal of vibrational part alignment lies in vibrational displacement effect inherent to the nonlinear asymmetric mechanical systems. Structural, kinematic, force etc. asymmetry might emerge in nonlinear mechanical systems. Force and kinematic asymmetry is most common in vibratory alignment. Force asymmetry originates from the rotational motion of the part as the components rest to each other. Vibrational excitation of the turned part causes kinematic asymmetry. Vibrational non-impact displacement of a mobile-based body on an inclined plane analysed by researches B. Baksys and N. Puodziuniene [1, 2] from Kaunas University of Technology. Peg - hole alignment under axial peg vibrational excitation investigated by B. Baksys and J. Baskutiene [3]. It was determined that alignment duration depends on excitation frequency, amplitude and system stiffness parameters. Immovable based bush also might be excited in axial direction or in two perpendicular directions in a vertical plane. B. Baksys and K. Ramanauskyte [4] investigated mutual part alignment using vibratory auto search method. Horizontal vibrating plane provides search motion for the part on it. Plane excited in a two perpendicular directions can generate circular, elliptical or interwinding helix search paths. Motion of the unconstrained and elastically and damping constrained part pressed with constant and varying pressing force was investigated. Mutual part alignment by directional vibrational displacement performed in the same way as linear and rotational motion of the output link in the ultrasonic motors. M. E. Archangelskyj [5] investigated oscillations of the cascade steel 8 vibrator. It was found that generating high frequency (17.7 KHz) axial vibrations at the end of the vibrator, other end oscillates in axial 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 axes when it touches rear or end surfaces of the vibrator, thus indicates about periodically 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 frequency lateral vibrations reduces dry friction coefficient between rear surfaces of the mating parts and facilitates part insertion. If the peg excited in longitudinal and lateral directions a motive force generated for the parts mating. To expand technological capabilities of automated assembly, to simplify and reduce cost of assembly equipment a new vibratory alignment method presented. The novelty of this method is the use of elastic high frequency 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 axial part misalignment of few millimetres. Method is suitable for the parts with circular and rectangular cross-section. A comprehensive theoretical and experimental research presented in this work as there is no scientific papers on this alignment method. 2. Experiments To investigate peg’s vibrations while it is in a contact with bushing the following experimental equipment has been used (Fig. 2.1). Peg 4 fixed in a middle cross-section in a gripper 1. Piezoelectric vibrator 2 pressed to the upper end of the peg with pressing force F2 and excites peg in axial direction. Excitation signal to the vibrator provided by signal generator 3. The lower end of the peg is pressed to the bushing 5 with initial pressing force F1 while axis misalignment Δ. One axis laser dopler vibrometer (LDV) used to register peg’s vibrations. The interferometer head OFV512 measures vibrations and controller OFV5000 coverts signal from interferometer to the voltage signal corresponded to vibration amplitude. Further signal captured with oscilloscope PicoScope 4424 and displayed on a computer screen. 9 Z Г3-56/1 OFV512 OFV5000 π/2 1 xi 2 3 yi F2 PicoScope4424 9 +Δ 8 F1 4 5 O Y X 6 7 PK zi a) b) Fig. 2.1 Experimental setup: a – measurement scheme: 1 – gripper; 2 – piezoelectric vibrator; 3 – excitation signal generator Г3-56/1; 4 – peg; 5 – bushing; 6 – fiber interferometer OFV512; 7 – vibrometer controller OFV5000; 8 – oscilloscope PicoScope4424; 9 – personal computer PC; zi – longitudinal vibrations; xi, yi – lateral vibrations; b – measurement equipment Vibrometer measurements were taken in a three directions X, Y, Z. Where X, Y corresponds to lateral vibrations in a two perpendicular directions and Z are longitudinal vibrations. Peg Axial misalignment of the parts –Δ and +Δ lies on Xaxis (Fig. 2.2). Thus mutual part alignment occurs when bushing center coincides with coordinate axes center. Peg’s tip vibration Bush magnitude was investigated Y under different pressing +Δ −Δ X forces F2 (vibrator to the Fig. 2.3 Bush placement in respect to the peg peg) and F1 (peg to the bushing) when excitation frequency f varies from 6523 to 6723 Hz, and tip movement trajectory was defined in relation with misalignment position Δ. In order to find movement trajectory of the peg’s end tip, measurements of two perpendicular axes (X-Y, Z-Y, Z-X) were taken. Synchronization signal related to the excitation signal synchronizes measurement process. As long as vibrations are periodic and steady, vibrations magnitude (xi, yi, zi) of each axis 10 εzy 1 1 xi 0.1 yi 2 4 3 0 0 zi -0.1 εxy Synchronization signal, V Vibration amplitude, µm 0.2 εzx -0.2 -1 0 τi 0.05 0.1 0.15 τ i, ms Fig. 2.3 Vibration signals and phase difference ε: 1 – synchronization signal; 2 – Y vibrations; 3 – X vibrations; 4 – Z vibrations defined at the same periodic time τi according to the synchronization signal (Fig. 2.3). Plotting those values in a Cartesian coordinate system 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. Excitation parameters and objects of experiments presented in Table 2.1. Table 2.1, Characteristics of excitation signal and aligned parts No. I Peg Diameter, mm 10 Length, mm 59.8 Chamfers Bush Hole diameter, mm 10.1 Excitation signal parameters Frequency, Hz 8475 Amplitude, V II Steel S235JR 10 79.65 No Steel S235JR 10.1 6711 132 III 10 99.75 10.1 6623 Influence of forces F1 and F2 to vibration amplitude was investigated on peg No. III. Pressing force of piezoelectric vibrator to the peg was gradually increased every 14 N and corresponding measurements of vibration magnitudes on all three axes were taken. The results presented on figure 2.3 and 2.4. Force F2 and excitation frequency has no impact on vibration magnitude along axis X. Vibration amplitude in Y-axis direction gradually increases when pressing force reaches 49 N and later stabilizes at 115 N. Meanwhile overall vibration magnitude decreases as excitation frequency increases. Amplitude of longitudinal vibrations increases more rapidly after F2 exceeds 90 N until that growth relatively 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 excitation energy transferred to the peg. Since peg excited with vibrations of high frequency and small amplitudes, contact area between peg and 11 vibrator plays vital role. This could be seen from graph 1 and 2, as excitation frequency increases pressing force F1 also has to be increased to keep same longitudinal vibration amplitude. It was also experimentally set that mutual part alignment starts when force F2 exceeds 90 N, until that process of part alignment is not stable or it does not work at all. 0.09 2 1 3 0.2 4 6 7 5 8 0 29 49 69 89 F2=10 1N 6623, Hz 0.08 109 F2, N Fig. 2.3 Vibration amplitude versus force F2: longitudinal Z vibrations: 1 – f=6523 Hz; 2 – f=6623 Hz; lateral Y vibrations: 3 – f=6523 Hz; 4 – f=6623 Hz; 5 – f=6723 Hz; lateral X vibrations: 6 – f=6523 Hz; 7 – f=6623 Hz; 8 – f=6723 Hz; X amplitude, µm Vibration amplitude, µm 0.4 0.07 0.06 0.05 6523, Hz 0.04 6723, Hz 0.03 0.02 0 0.5 1.0 1.5 2.0 F1, N Fig. 2.4 X vibration amplitude versus force F1 If force F2 had no impact on vibrations in X-axis, totally different impact had force F1. As the peg is pressed to the bushing with axis misalignment Δ=+1.5 mm, vibration amplitude gradually increases as force F1 increases. The same tendency retains even if excitation frequency changes in range from 6523 to 6723 Hz (Fig. 2.4). In our case, part alignment is most rapid when peg is excited at 6623 Hz frequency and vibrations in X-axis are the biggest. Experiment results mentioned above in generally shows what influence for the vibrations amplitude has mounting conditions of the peg, and that excited peg vibrates in three directions perpendicular each other. However, there still no answer why bushing is slides toward coordinate axes centre. To find out what factors in charge of this effect, motion trajectory and direction of peg’s tip was determined. After excitation frequency for stable and steady part alignment was experimentally set to all pegs (Table 2.1), motion trajectory of the tip was taken in all three coordinate planes. Excitation frequency mainly depends from the peg’s natural frequency, design of the gripper and force F1. Thus for the grippers with different design or made from different material excitation frequency for steady and stable part alignment will be different. In our case, excitation frequency for stable and steady part alignment have lied between second and 12 third natural bending mode of the peg. Figure 2.5 shows peg’s tip path while forces F2=101 N, F1=0 N 0.05 Peg II 0.03 Peg III 0.02 Y, µm 0.01 0 Peg I 0.05 0 0 -0.01 Peg II Peg III 0.1 0.1 0.2 0.05 0.05 0.1 0 0 0 -0.05 -0.05 -0.1 -0.1 -0.1 -0.2 Z, µm Peg I -0.02 -0.05 -0.05 -0.03 -0.05 0 0.05 -0.05 0 0.05 -0.1 0 0.1 Y, µm 0.02 0 -0.01 0.01 0 -0.01 -0,02 0.02 0 -0.01 X, µm a) b) Fig. 2.5 Path trajectory of unloaded peg: a – in YOX plane; b – in ZOY plane. Longitudinal vibrations are dominant in all cases and are twice as high as transverse ones. While in YOX plane they polarized in Y direction since peg’s vibrations in X direction are negligible. When the peg is pressed to the bushing with the force F1=2.2 N and axis misalignment Δ=-1.5 mm, lateral vibrations on X axis increases significant and peg’s end moves in elliptical shape trajectory in all three coordinate planes (Fig. 2.6). Black dots on the path indicate its direction. For the different pegs, direction of rotation is different, that depends from excitation frequency, and natural mode gripper–peg system vibrates. In order the alignment of the parts could occur, bushing has to slide along positive X direction. There are two ways how bushing aligned. First is direct alignment (Peg II and III). In this case peg’s tip moves counter-clockwise in ZOX plane (Peg II and III, b), thus direction of the normal force in the contact point lies on the positive X direction and bushing is directly pushed toward coordinate axes center. Vibrations along Y-axis has little effect since their amplitude smaller than X, and overall vibrations are more polarized along X-axis (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 X-axis rather than negative. Second way of part alignment is indirect alignment (Peg I). Here peg’s motion is clockwise in ZOX plane (Peg I, b) and bushing is pushed from the coordinate axes center. However, because of the peg’s tip elliptical movement in 13 YOX plane (Peg I, a), bushing is turned by the angle so the alignment trajectory lie on the major axis of the ellipse and then pushed towards coordinate axes center. εxy=-0.02 -0.1 -0.15 Z, µm 0 X, µm a -0.15 0 Y, µm εzx=-1.11 0.1 0 0 -0.1 -0.1 -0.1 II -0.08 -0.08 0 X, µm εzy=2.26 0.1 0 0 III 0.1 -0.1 Z, µm Z, µm 0 -0.15 0 X, µm εxy=2.91 -0.1 0 Y, µm εzx=-2.95 0.08 0.15 -0.1 Y, µm 0 0 X, µm -0.1 0 X, µm εxy=0.22 εzy=3.11 0 I Z, µm -0.1 0.1 Y, µm 0 Z, µm -0.1 εzx=-2.07 0.1 0.1 0 -0.1 εzy=-2.05 Z, µm Y, µm 0.1 0 X, µm -0.1 b 0 Y, µm -0.1 c Fig. 2.6 Path trajectory of loaded peg when Δ=-1.5 mm: a – in YOX plane; b – in ZOX plane; c – in ZOY plane Results of peg’s tip trajectory while Δ=+1.5 mm presented in figure 2.7. 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 X direction. Peg’s I and II tip moves clockwise in a ZOX 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 X direction, but because of rotation effect in YOX plane (Peg III, a) bushing is turned and pointed to the coordinate axes center. It is clear that during direct alignment peg’s motion in ZOX plane plays key role, meanwhile during indirect alignment there is combination of peg’s movement in ZOX and YOX planes. 14 0 εzy=-0.25 0 -0.1 -0.1 0 X, µm Z, µm Y, µm 0.06 0 0 X, µm 0 -0.1 0 Y, µm -0.1 εzy=-1.97 εzx=0.59 0.08 Z, µm -0.1 -0.1 0.08 εzx=1.28 0.1 I Z, µm Y, µm 0.1 0 II εxy=1.53 Z, µm 0.1 0 -0.15 -0.15 0 -0.2 0 X, µm a -0.2 0 X, µm b III 0 Z, µm Z, µm Y, µm -0.08 -0.06 εxy=2.55 -0.06 0 X, µm -0.08 0 Y, µm -0.08 -0.08 0 X, µm 0.15 ε =2.60 εzy=0.49 εzx=3.09 xy 0.2 0.2 0 -0.2 -0.2 0 Y, µm c Fig. 2.7 Path trajectory of loaded peg when Δ=+1.5 mm: a – in YOX plane; b – in ZOX plane; c – in ZOY plane Experimental setup designed and made to investigate part alignment when elastic vibrations applied to the peg (Fig. 2.8). The peg fixed 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 gravity force compensator for the gripper and helps to capture the moment as the peg falls into the bush hole. Vertically, moving table 6 adjusts pressing force of mating parts. The table moved horizontally in order to change axis misalignment, which is measured with indicator 9. Low frequency signal generator 3 provides signal to the piezoelectric vibrator. The amplitude and frequency of the signal are measured by multimeter 1. Switch 5, oscilloscope 4 and personal computer 2 are used for alignment event triggering and alignment duration measurement respectively. 15 1 2 3 4 5 6 7 8 9 10 Fig. 2.8 Experimental setup: 1 – multimeter FLUKE 110; 2 – computer Compaq nc6000; 3 – signal generator Г3 – 56/1; 4 – oscilloscope PicoScope 4424; 5 – switch; 6 – table; 7 – spring; 8 - gripper; 9 – indicator BDS Technics The peg 1 is hold in a middle cross-section by the clamps of 3 the gripper (Fig. 2.9). Piezoelectric vibrator 2 is implemented in a housing 3. Threaded end of the housing can F 2 2 freely rotate in a gripper at the same time performing linear 11 1 motion towards the peg. 9 As the piezoelectric vibrator 4 Δ 10 lean to the peg, further torque 12 F1 5 increment sets pressing force for 7 6 the piezoelectric vibrator to the 13 8 peg. Bush 4 is mobile based on the electrically conductive plate Fig. 2.9 Measurement circuit: 1 – Peg; 2 – 5 while the latter is located on piezoelectric vibrator; 3 – housing; 4 – bush; the force sensor 6. The bush, 5 – plate; 6 – force sensor; 7 – 9 V power plate, and force sensor fixed to supply; 8 – switch; 9 – light-emitting diode the table 13 and moves together. (LED); 10 – oscilloscope; 11 – signal The following electrical generator; 12 – computer; 13 table circuit was designed to measure the alignment time. Anode of the power supply 7 connected to the electrically 16 conductive plate. Cathode first connected to the switch 8 and LED 9 and later to the gripper. Oscilloscope 10 is measuring voltage signal on the LED. When the switch closes electrical circuit, the voltage jump on the LED occurs. At the same time, excitation signal from generator 11 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 12 and by mean of the software alignment time is calculated. During investigation, the peg is excited in axial direction by mean of cylindrical shape piezoelectric vibrator with 30 mm in diameter and 13 mm in height. Pressing force vibrator-to-peg is set to 101 N and kept constant throughout the experiments. Harmonic excitation signal generated by low frequency generator. Each time experiment repeated four times and a mean value of four trials is taking as a result. Influence of axis misalignment Δ, excitation frequency f, excitation signal amplitude U and initial peg-to-bush pressing force F1 to the alignment duration Δt is investigated. Experiments 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 type with chamfers and with no chamfers. Measurements of the parts used in experiments are given in Table 2.2 Table 2.2, Material and geometrical data on specimens No. I II III IV V VI VII VIII IX Peg Bush Steel S235JR Diameter, mm Lengh, mm Diameter, mm 10 99.75 10.1 10 79.65 10.1 10 59.8 10.1 7.95 99.85 8.05 5.95 99.6 6 Aliuminium SAPA6082-T6 10 99.95 10.05 10 99.95 10.05 Steel S235JR Lengh x Widh x Heigh, mm Lengh x Widh, mm 10.1x5.3x99.3 10.4x5.45 10.05x5.1x99.5 10.4x5.45 Crosssection Chamfers No C C C C C C C No 0.55x43º R R Nėra 0.33x49º The dependencies of alignment duration Δt on axis misalignment Δ is presented in figure 2.10. Steel peg I excited under different excitation frequency 17 and initial pressing force F1. Excitation signal amplitude U=142 V is same to all investigated pegs. It is determined that alignment duration increases as axis misalignment increases. The character of a graph is linear and do not depend on excitation frequency. We can also see that alignment duration depends on misalignment direction. In a direct alignment case the alignment duration is shorter (Fig 2.10, a). However, excitation frequency has significant influence to the alignment duration (Fig. 2.11). 1,5 1.5 Δt, s 5,0 5.0 F1=2.2 N 1 1,0 1.0 2 Δt, s 4 5 0,5 0.5 F1=2.2 N 3,3 3.3 1 1,7 1.7 3 3 0.0 0,0 0.4 0,4 2 4 1.4 1,4 2.5 2,5 3,5 Δ, mm 0.0 0,0 0.4 0,4 1.4 1,4 2.5 2,5 a) Δ, 3,5 mm b) Fig. 2.10 Alignment duration dependencies on axis misalignment Δ: a) bush placement +Δ, b) bush placement –Δ; 1 – f=7000 Hz; 2 – f=7050 Hz; 3 – f=7100 Hz; 4 – f=7150 Hz; 5 – f=7200 Hz 5,0 5.0 1.5 1,5 F1=2.2 N Δt, s F1=2.2 N Δt, s 9 8 5 6 7 8 9 1.0 1,0 3,3 3.3 0.5 0,5 1,7 1.7 1 2 3 4 0,0 0.0 0.0 0,0 7000 7 6 5 4 3 2 1 7050 7100 7150 a) 7200 f, Hz 7000 7050 7100 7150 f,7200 Hz b) Fig. 2.11 Alignment duration dependencies on excitation frequency f: a) bush placement +Δ, b) bush placement –Δ; 1 – Δ=0,4 mm; 2 – Δ=0,6 mm; 3 – Δ=0,8 mm; 4 – Δ=1,0 mm; 5 – Δ=1,5 mm; 6 – Δ=2,0 mm; 7 – Δ=2,5 mm; 8 – Δ=3,0 mm; 9 – Δ=3,5 mm 18 The alignment of the parts is most rapid when excitation frequency is between 7050-7100 Hz and this trend visible under different axis misalignment. As frequency changes from these values, alignment duration increases. It was also determined that for a small axis misalignment (up to 1 mm) the influence of excitation frequency is negligible. Excitation frequency 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 only reason of frequency changes. Contact quality between peg and piezoelectric vibrator plays significant role in an excitation frequency. More is the area the end surface of the peg touches vibrator, more acoustic energy transferred to it, as well as excitation frequency is lower. During experiments 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 any conclusions on excitation frequency using smaller diameter pegs. 1,0 1.0 2.5 2,5 Δt, s f=7050 Hz 9 8 0.7 0,7 7 6 0,3 0.3 5 4 1.5 1,5 6 1.0 1,0 5 0.5 0,5 2 0,0 0.0 1.5 1,5 f=7050 Hz Δt, s 7 2.0 2,0 2.0 2,0 1 1 2 3 4 3 2.5 2,5 a) F1, N 0.0 0,0 1.5 1,5 2.0 2,0 2.5 2,5 F1, N b) Fig. 2.12 Alignment duration dependencies on force F1: a) bush placement +Δ, b) bush placement –Δ; 1 – Δ=0,4 mm; 2 – Δ=0,6 mm; 3 – Δ=0,8 mm; 4 – Δ=1,0 mm; 5 – Δ=1,5 mm; 6 – Δ=2,0 mm; 7 – Δ=2,5 mm; 8 – Δ=3,0 mm; 9 – Δ=3,5 Figure 2.12 represents dependencies of alignment duration Δt on initial pressing force F1 under different axis misalignment. Influence of initial pressing force on alignment duration is relatively small when axis misalignment is up to 1 mm. In a case when Δ>1 mm alignment duration decreases as force F1 increases. 3. Numerical simulation of part alignment at non-impact and impact modes During vibratory alignment, two solid bodies like peg and a bush interact with each other. Peg presses bush with predetermined force and its elastic vibrations are excited. Friction forces that rise during interaction of those two 19 Y1 bodies guide bushing to the axis alignment direction. It is necessary to make systems consisting of two interactive bodies dynamical modelling in order to examine alignment process further. In general case peg is fixed in a specially designed gripper while bush is based on a plane. Piezoelectric vibrator presses the top end of the peg and excites its elastic vibrations. Experimental 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 trajectory on a vertical plane. Alignment process is modelled by two-mass dynamical system in a reference frame XOY (Fig. 3.13). A mass m1 depicts peg that oscillates in two Y perpendicular directions while mass m2 is a bush that K1y H1y has to be aligned to the peg. Asinω Alignment process is possible t Bsin(ωt+ε) only when peg press bush H1x with predetermined force and m1 X oscillation amplitude is on K1x N 0 the proper level. Vibration 1 H2x Y0 amplitude depends on excitation frequency and is Δ the biggest when system K2y H2y K2x m2 oscillates close to their natural mode. Natural mode 0 X X2 itself depends on the geometrical characteristics of Fig. 3.13 Dynamical model piezoelectric vibrator and a peg, the way in which peg fixed in a gripper, peg-to-bush pressing force magnitude. Typical excitation frequency 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 texture deforms in a normal and tangential directions. During high frequency elastic vibrations not only elastic but also elasto-plastic deformations may occur. To evaluate deformations of this kind we use rheological Kelvin–Voigt model. Thus, the surface of the mass m2 in normal and tangential directions constructed by stiffness (K2X, K2Y) and damping (H2X, H2Y) elements connected in parallel. Surface deformations induce reaction forces Rx, RY: R2 X K 2 X X 2 X 1 H 2 X X 2 X 1 , R2Y K 2Y Y1 Y0 H 2Y Y1. 20 (3.1) where Y0 – deformation of the contact surface because of the initial pressing force, (X′2-X′1) – relative deformation speed in X direction, Y′1 – deformation speed in Y direction. Mass’s m1 tip longitudinal Asin t and lateral Bsint vibration amplitudes vary according to the law of the sinus. Stiffness and damping forces restricts mass movement in X and Y directions. R1 X K1 X X 1 H1 X X 1, R1Y K1Y Y1 H1Y Y1. (3.2) When two bodies are in contact, friction forces arise in their contact zone. Its magnitude expressed using dry friction model. Force F1fr affects mass m1. Force F2fr that is sum of friction forces peg-bush and bush-base acts on mass m2 F1 fr N1sign X 1 X 2 . (3.3) F2 fr N1sign X 2 X 1 N 2 signX 2 . (3.4) where μ1 – coefficient of friction between mass m1 and m2, μ2 – coefficient of friction between m2 and base, N – normal pressing force. All friction forces formulated taking into account that they are not affected by relative speed. We get equation of motions for mass m1 and m2 by projecting all acting forces in X and Y axes: m1 X 1 H1 X X 1 K1 X X 1 N1sign X 1 X 2 K1 X B sint , (3.5) m1Y1 H1Y H 2Y Y1 K1Y K 2Y Y1 Y0 K1Y A sin t , m X H X X K X X N sign X X N signX 0. 2X 2 1 2X 2 1 1 2 1 2 2 2 2 where X d / dt; X d 2 / dt 2 We are using following dimensionless parameters to have generalized results of simulation: pt; p h2 y K 2x X X Y H H H ; x1 1 ; x 2 2 ; y1 1 ; h1x 1X ; h2 x 2 X ; h1 y 1Y ; m2 l l l m1 p m2 p m1 p H 2Y K K K ; h y h1 y h2 y ; k1x 1X ; k1 y 1Y 2 ; k 2 y 2Y 2 ; k y k1 y k 2 y ; m1 p K2X m1 p m1 p Y m1 B A N ; b ; a ; ; n ; y 0 0 ; ; n k y y 0 y1 ; m2 l l p l l m2 p 2 l l 1m. 21 Then motion equations written in a dimensionless form: x1 h1x x1 k1x x1 n1 signx1 x 2 k1x b sin , y1 h y y1 k y y1 k1 y a sin k y y0 , x h x x x x n signx x n signx 0. 2 1 1 2 1 2 2 2 2x 2 1 (3.6) where x d / dt; x d 2 / d2 We used a program code written in MATLAB environment to obtain numerical simulation results. Solver ode15s was used in calculating stiff differential equation. Since our dynamic system is described by second order differential equations, we had to rewrite them to a pair of simultaneous first order differential equations (Esfandiari, 2013) to obtain the solution. The following initial values of the parameters of the dynamic system were used b 3, a 2, k1x 5, k1 y 0,3, k 2 y 0,3, h1x h1 y h2 x h2 y 0,7, 1,3, 1 0,2, 2 0,1, 0,32, 1000, 0,2, y0 3. Initial conditions alignment conditions x1 0, x1 0, y1 0, y1 0, x2 0, x 2 0 . As τ=0 normal reaction force is equal to initial pressing force n k y y 0 . After the excitation signal is applayed, force n alternates and it‘s value depends on mass‘s m 1 coordinate y1, thus n k y y1 . Alignment occurs when x2 . During simulation, we analysed the effect of each parameter on the part alignment process by adjusting only one parameter and keeping all the others constant. Alignment duration dependency on excitation frequency at different phase shifts between longitudinal and lateral vibrations represented in figure 3.14. Alignment duration is lower at the lower frequency values if phase shift is between 0 and π/2. As excitation frequency increases, alignment duration also increases. When phase shift is grater then π/2, alignment duration decreases as excitation frequency increases since ellipsis of peg movement trajectory have changed inclination angle and short axis of the ellipsis have shortened. There is also a peak in alignment duration at the excitation frequency ν=1 no matter the phase shift between vibration components. This is because excitation frequency became equal to the natural frequency of the bush along X-axis. Figure 3.15 shows alignment duration dependency on the phase shift between vibrations at different excitation frequencies. Dependencies have parabolic character, thus, there exist phase shift at which alignment process is the most rapid. All curves have intersection points in the region between π/4 to 5π/12 and it means that excitation frequency has little effect on alignment duration in this region. 22 τ∙105 τ∙105 2 4 2 1.7 1.2 3 1 1.2 0.9 5 0.6 0.5 4 0 0.1 0,1 0.6 0,6 1.1 1,1 0 -1 ε=0 0 -2 -2 0 5 b a) 0 -1 a 10 b -4 -6 -20 -10 0 b) 10 -6 -20 -10 b c) 0 -1 ε=5π/6 ε=2π/3 ε=π/2 -2 -4 0 0 0 ε=π -1 -1 -2 -2 -2 -2 a -3 a-3 a -3 -4 -4 -4 -4 -5 -10 -5 -5 -15 -10 -5 0 5 10 -5 -10 -5 e) b f) b 0 10 b d) a -3 0 5 10 3.14 3,14 0 ε=π/3 a -6 -20 -10 2.09 ε 2,09 -2 -4 -5 -15 -10 -5 0 1.05 1,05 Fig. 3.15 Alignment duration dependency on phase shiftε; 1 – ν=0.1; 2 – ν=0.3; 3 – ν=0.5; 6 – ν=0.7 ε=π/6 a -4 1 0.2 0.00 0,00 1.5 ν 2,0 2.0 1,5 Fig. 3.14 Alignment duration dependencies on excitation frequency ν; 1 – ε=0; 2 – ε=0.79; 3 – ε=1.57; 4 – ε=2.36; 5 – ε=3.14 a -3 3 0 5 10 b -5 -20 -10 g) ε=7π/6 0 10 b h) Fig. 3.16 Peg’s tip motion trajectory on phase ε Motion trajectory of the peg’s end tip depends on the phase shift between lateral and longitudinal vibrations (Fig. 3.16). In our case for the bush to be aligned with the peg, necessary that peg’s end tip moves counter clockwise direction (Fig. 3.16, 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. 3.16, h). The bush does not align with the peg anymore, but rather moves away from it. In case when phase shift is 0 or π radians (Fig. 3.16, a, g) 23 alignment process has unstable manner and depending on excitation frequency alignment of the parts may occur or not. τ∙105 τ∙105 1.4 1.4 1 1 0.9 0.9 2 3 4 0.5 0 0.5 5 11 44 10 77 b 10 Fig. 3.17 Alignment duration dependencies on lateral vibration amplitude b; 1 – a=1; 2 – a=2; 3 – a=4; 4 – a=6; 5 – a=8 2 3 0 11 33 55 77 a 99 Fig. 3.18 Alignment duration dependencies on longitudinal vibration amplitude a; 1 – b=1; 2 – b=3; 3 – b=9 Another important parameter that has direct influence on the alignment time is amplitude of lateral and longitudinal vibrations of the peg figure 3.17, 3.18 respectively. When lateral vibrations reach certain limit (in our case b=2) their influence on the alignment duration becomes negligible. Much bigger influence on the process time has longitudinal vibrations. As amplitude increases, alignment duration constantly decreases. τ∙104 0.7 τ∙104 1 2 0.5 1 3 6 3.4 2 3 4 2.2 4 0.4 5 1.1 5 0.2 0.9 1,5 1.7 2,9 2.6 4,3 3.4 5,6 kyy7,0 0 Fig. 3.19 Alignment duration dependencies on initial deformation kyy0; 1 – b=1; 2 – b=2; 3 – b=3; 4 – b=4; 5 – b=5 0 0.1 0,1 0.3 0,3 0.5 0,5 μ1 0.7 0,7 Fig. 3.20 Alignment duration dependencies on dry friction coefficient μ1; a=6: 1 – μ2=0,06; 2 – μ2=0,08; 3 – μ2=0,1; 4 – μ2=0,12; 5 – μ2=0,16; 6 – μ2=0,18 Because repulsive force created due to the friction between a peg and a bush, initial pressing force between those parts plays key role in the part alignment (Fig. 3.19). For the process to be stable and reliable initial pressing force has to be at a certain limit, but not < 1.5. If it is less, the system runs into the impact 24 mode and our model cease to be valid. We have to increase initial pressing force to rule out system from the impact mode. However to obtain the shortest alignment duration we have to keep it as low as possible but avoiding system 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 rapidly 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 directly proportional to the coefficient of dry friction between acting surfaces (Fig. 3.20). Alignment duration keeps stable as coefficient μ1 increases, there is only small decrease of alignment duration at μ1=0.3. There is rapid increase in the alignment duration when coefficient of friction is more than 0.5. Dry friction coefficient between bush and the base has to be as small as possible. When μ2>0.12 alignment process starts at μ1>0.2. When μ2>0.19 alignment process stops. τ∙104 τ∙105 5 4 1 5 1.5 4 12.0 2 3 3 6.0 0 400 1.0 0.5 1175 1950 a) 2725 δaa 3500 0 0.1 0,1 2 0.4 0,4 0.7 0,7 1 ν1 b) Fig. 3.21 Alignment duration dependencies a) on axis misalignment δ; 1 – ν=0.1; 2 – ν=0.3; 3 – ν=0.6; 4 – ν=0.8; 5 – ν=1.0, b) on excitation frequency ν; 1 – δ=400; 2 – δ=800; 3 – δ=1500; 4 – δ=2500; 5 – δ=3500 Alignment take place at different axis misalignment between the parts and at different excitation frequency (Fig. 3.21, a, b). The alignment duration increases as axis misalignment increases. Dependencies have linear character no matter the excitation frequency. We can see that excitation frequency has low impact on the alignment duration when axis misalignment is small (Δ<800). Only when misalignment increases the influence of excitation parameter becomes apparent. Work pieces align most rapidly when mass m1 vibrates at resonant frequency. As excitation frequency rangers from resonant, alignment duration constantly increases until process becomes impossible. Qualitatively dependencies have a 25 good match with experimental results thus confirms validity of our mathematical model. During experimental 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 away from the bush and later hits it at certain speed level. Such alignment regime forms when longitudinal oscillations has bigger amplitude or pressing bush-to-peg Y force is not sufficient. During impact part alignment a recurrent interaction between peg and a bush K1y H1y Asinω is going. Peg breaks from the bush Bsin(ωt+ε) when normal component of H1x excitation force higher then pegto-bush pressing force. To simulate m1 K1x impact part alignment it is I1Y I1X I2 necessary to form equations Y0 describing motion of the impact body before the impact with the N Δ bush and the impact interaction. m2 Peg rendering mass oscillates in X normal and tangential directions. Resultant body motion trajectory Fig. 3.22 Model of the contact interaction depends from excitation amplitude components and their phase. The motion trajectory in the vertical plane could be circular, elliptical or linear inclined at the certain angle to the horizontal axis (Fig. 3.22). Motion of the mass m1 when it breaks from the bush m2 defines equations: m1 X 1 H1 X X 1 K1X X 1 K1 X B sint , m1Y1 H1Y Y1 K 1Y Y1 Y0 K1Y A sin t. (3.7) Diagonal impacts of mass m1 causes motion of the mass m2 defines by equation: m2 X 2 N2 signX 2 0 (3.8) We use dimensionless parameters to get generalised version of motion equation: 26 pt ; p k1x K1x X X Y H H ; x1 1 ; x2 2 ; y1 1 ; h1x 1 X ; h1 y 1Y ; m1 l l l m1 p m1 p K1 X K m B A ; k1 y 1Y 2 ; 1 ; b ; a ; K2X m2 l l m1 p Y N g ; n ; N m2 g ; d 2 ; y0 0 ; ; l 1m. p l l m2 p 2l pl Motion of the bouncing mass m1 in dimensionless form: x1 h1x x1 x1 k1x b sin , y1 hy y1 k y y1 k1 y a sin k y y0 . (3.9) Dimensionless equation of motion of mass m2: x2 d 2 signx2 0. (3.10) Interaction of the bodies at the moment of the diagonal impact defines impact equations. Describing diagonal impact, we assume that velocity components of normal impact vary according to the linear impact law and do not depend on tangential velocity component. When impact is linear, normal velocity of the mass m1 after the impact defined by equation: y1 Ry1 . (3.11) where y1 – mass m1 velocity before the impact, R – impact restitution coefficient. To define impact interaction we use hypothesis of dry friction that determines link between normal and tangential impact impulses: I1x I1 y . (3.12) where I1x, I1y – impact impulses, μ – coefficient of dry friction. Studying diagonal impact, we assume that slipping velocity between the bodies in the impact interval is always positive. Such impact called a sliding impact. Normal velocity restitution equation valid for the sliding impact only. 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 maximum 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 m1 hits m2 during load phase in accordance with impulse hypothesis, we can write: I11y m1 y1 . (3.13) 27 I11x m1 x0 x1 . (3.14) where x1 , y1 – body m1 velocity before impact, x 0 – absolute sliding velocity at the end moment of the first impact phase. By inserting (3.13) and (3.14) to (3.12) we get: x0 x1 y1 . (3.15) Direction of the tangential velocity cannot be changed at the impact moment, thus x0 0 . From (3.15) we get: x1 / y1 . That is a self-stop condition for the body m1. Tangential displacement of body m1 stops at the first stage of the impact and it bounces from the body m2 in a normal direction if this condition not fulfilled. Because we investigate case of the sliding impact, the body m1 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: I1y2 m1 y1 . (3.16) I1x2 m1 x1 x0 . (3.17) where x1 , y1 – body m1 velocities after the impact. Expressions (3.16) and (3.17) linking with (3.12) and taking in to account (3.11) and (3.15), we can calculate m1 tangential velocity after the impact: x1 x1 y1 1 R. (3.18) Tangential impulses of the body m1 make body m2 to slide towards axis misalignment direction. Bodies m1 and m2 impact impulses according to the impulses hypothesis is: I1x m1 x1 x1 , I 2 x m2 x 2 x 2 . (3.19) Impact impulse to the body m2 transferred by the dry friction thus we can write: I 2 x I1x . (3.20) Composing impulse expressions (3.19) to the (3.20) and taking in to the account (3.18), we get equation for calculating velocity of the body m2 after the sliding impact: 28 1 x 2 x 2 2 1 R y1 . (3.21) Expressions (3.11), (3.18) and (3.21) used to calculate bodies m1 and m2 velocities after the impact. During numerical simulation we used the following constant values: b 3, a 2, k1x 5, k1 y 0,3, h1x h1 y 0,7, 1,57, 1 0,2, 2 0,1, 0,32, 1000, 1,4, y0 1, n 1, R 0,7 . Initial conditions: x1 0, y1 0, y1 0, x2 1000, x 2 0 . Part alignment condition: x2 0 . Excited 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 energy transferred to the bush and it slides to the axis misalignment direction. Peg bounces from the bush after the energy transferred meanwhile a bush keep sliding because of inertia until the next impact. Proper settings of the excitation and mechanical system parameters must be chosen to have alignment process stable and reliable. The peg excited in the frequency range from 1 to 1.7 to have alignment of the bush reliable (Fig. 3.23). in this frequency range alignment duration do not depend on the phase shift between vibration components and is easily predictable. As excitation frequency increases, alignment duration decreases and reaches minimal value at 1.4. Subsequent increase of the excitation frequency makes alignment duration to increase. When ν<1 or ν>1.4 alignment duration is hardly predictable and changes rapidly if small excitation τ∙103 frequency changes applied. 3 4 As axis misalignment 7 0,8 increases, alignment duration also increases. Dependencies has linear 6 0,6 character and do not depend on 5 1 excitation frequency (Fig. 3.24, b). Influence of excitation frequency 2 0,4 to the alignment duration is minimal when δ>800 (Fig. 3.24, a). 0,2 Only when excitation frequency ν 2,0 0,1 0,7 1,4 increases we can observe frequency range at which part Fig. 3.23 Alignment duration alignment is the fastest. If dependencies on excitation frequency ν: excitation frequency is more than 1 - ε=0; 2 - ε=0.26; 3 - ε=0.52; 4 ε=0.79; 5 - ε=1.05; 6 - ε=1.31; 7 1.7 alignment process stops. As ε=1.57 ν≥1.9 alignment process recurs again, but alignment duration rapidly decreases as excitation frequency increases. 29 τ∙103 τ∙103 2.6 2.4 5 1.7 9 8 7 6 1 5 4 1.7 3 0.9 0.9 0 1.0 1,0 6 1 1.3 1,3 3 2 1.7 1,7 2.0 2,0 0.2 400 1433 a) 2467 2 4 δ 3500 b) Fig. 3.24 Alignment duration dependencies on: a) excitation frequency ν: 1 - δ=400; 2 δ=600; 3 - δ=800; 4 - δ=1000; 5 - δ=1500; 6 - δ=2000; 7 - δ=2500; 8 - δ=3000; 9 δ=3500; b) axis misalignment δ: 1 - ν=1; 2 - ν=1.2; 3 - ν=1.5; 4 - ν=1.7; 5 - ν=1.9; 6 - ν=2; As longitudinal vibration amplitude increases, alignment duration decreases exponentially (Fig. 3.25, a). Lateral vibration amplitude if it is not equal to zero, has no influence to alignment duration at all (Fig. 3.25, b). τ∙103 τ∙103 0.6 0.6 1 2 0.4 0.4 1, 2, 3, 4, 5 3 0.2 0.2 4 0 1.0 1,0 3.3 2,3 3.7 3,7 a) a 5.0 5,0 0 1.0 1,0 3.3 2,3 3.7 3,7 b 5.0 5,0 b) Fig. 3.25 Alignment duration dependencies on: a) longitudinal vibration amplitude a: 1 b=1; 2 - b=2; 3 - b=3; 4 - b=4; 5 - b=5; b) lateral vibration amplitude b: 1 - a=1; 2 - a=2; 3 - 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 dry friction coefficients μ1 and μ2. As friction force between bush and peg increases, alignment duration decreases exponentially (Fig. 3.26, a). Meanwhile if friction forces between bush and base increases, alignment duration increases linearly (Fig. 3.26, b). 30 τ∙103 6000 6000 τ∙103 8 7 6 5 4.0 4000 4.0 4000 4 3 2.0 2000 2 1 2000 2.0 4 5 6 1 2 00 0.1 0,1 0.3 0,3 0.6 0,6 a) μ1 3 00 0.8 0,8 0.1 0,1 0.3 0,3 0.6 0,6 μ2 0.8 0,8 b) Fig. 3.26 Alignment duration dependencies on: a) dry friction coefficient μ1: 1 – μ2=0,1; 2 – μ2=0,2; 3 – μ2=0,3; 4 – μ2=0,4; 5 – μ2=0,5; 6 – μ2=0,6; 7 – μ2=0,7; 8 – μ2=0,8; b) dry friction coefficient μ2: 1 – μ1=0,1; 2 – μ1=0,2; 3 – μ1=0,3; 4 – μ1=0,4; 5 – μ1=0,5; 6 – μ1=0,6; Initial deformation y0 and longitudinal peg vibration amplitude has influence to the alignment duration in the close relation to each other. When deformation y0 increases alignment duration slightly decreases, but process stops if longitudinal vibration amplitude becomes insufficient and peg no longer hits the bush (Fig. 3.27, a). As longitudinal vibration amplitude increases, alignment duration decreases exponentially. However alignment process possible only when amplitude is bigger than initial deformation (Fig. 3.27, b). τ∙103 700 700 τ∙103 1 1 0.53 532,5 0.52 525 2 0.35 350 3 0.36 365 3 4 4 0.20 197,5 0.17 175 5 00 1.0 1,0 2 5 6 0.03 30 3.7 3,7 6.3 6,3 a) y0 9.0 9,0 1.0 1,0 2.7 2,7 4.3 4,3 a 6.0 6,0 b) Fig. 3.27 Alignment duration dependencies on: a) initial pressing deformation y0: 1 - a=1; 2 - a=2; 3 - a=3; 4 - a=4; 5 - a=5; 6 - a=6; b) peg‘s longitudinal vibrations a: 1 - y0=1; 2 y0=2; 3 - y0=3; 4 - y0=4, 5 - y0=5 When impact restitution coefficient R increases, alignment duration constantly decreases because increases amount of energy bush receives during 31 impact. During pure elastic impact when R=1 alignment duration would be the shortest (Fig. 3.28, a). Friction forces between bush and base restricts bush motion. Thus as normal pressing force between bush and base increases, alignment duration increases (Fig. 3.28, b). If impact force is less then friction force, alignment process does not occur. 1300 τ 103 1300 τ∙103 1 9 10001.0 2 8 10001.0 4 7 7000.7 7000.7 6 4000.4 1 5 2 3 4 0.4 0,4 7 6 4000.4 5 1000.1 0.2 0,2 3 0.6 0,6 a) R 0.8 0,8 1000.1 0.0 0,0 1.7 1,7 3.3 3,3 N 5.0 5,0 b) Fig. 3.28 Alignment duration dependencies on: a) impact restitution coefficient R: 1 N=0,2; 2 - N=0,4; 3 - N =0,6; 4 - N =0,8; 5 - N =1; 6 - N =2; 7 - N =3; 8 - N =4; 9 - N =5; b) normal pressing force N: 1 - R=0,2; 2 - R=0,3; 3 - R=0,4; 4 - R=0,5; 5 - R=0,6; 6 R=0,7; 7 - R=0,8 Conclusions 1. Proposed and investigated part alignment method when using elastic vibrations of the peg. piezoelectric vibrator pressed to the upper end of the peg provides high frequency excitation oscillations. The lower end of the peg starts to vibrate in longitudinal and lateral directions. Part alignment occurs only when mechanical contact ensured between mating parts. Such method compensates axial part misalignment of 1-1.5 mm for the chamferless parts with circular and rectangular cross-section. Vibratory part alignment when using elastic vibrations of the peg enhances productivity and reliability of automatic part assembly operations like: insertion of the shaft to the bearing, tooth wheel, electric motor rotor etc. 2. Experiments have proved that lower end of the peg moves in elliptical shape trajectory in all three coordinate planes while excitation done in longitudinal direction to the upper end. When mating parts pressed to each other, friction force propels bush to the part alignment direction. During part alignment bush makes not only linear motion, but also rotates about contact point to the peg. 3. In all vibratory part alignment experiments, alignment process fastest when peg’s oscillation frequency is closest to the third natural bending mode. As excitation frequency ranges from it, alignment duration increases until process 32 stops. As longitudinal vibration amplitude increases, alignment duration decreases. The shortest alignment duration is at excitation signal level of 142 V. Alignment regime depends on longitudinal peg vibration and part-to-part pressing force. Impact alignment regime starts when longitudinal peg’s vibrations are at high level and part-to-part pressing force is not sufficient. However, for the practical usage non-impact regime is more suitable since it is more stable especially during indirect part alignment. If pressing force higher then 2.9 N lateral vibrations are supressed and alignment stops. If pressing force is less then 1.5 N, alignment process falls in to the impact mode. 4. The mathematical models of part alignment at impact and non-impact regimes were constructed. Computer simulations revealed that alignment duration and reliability mostly depends on part-to-part pressing force, excitation frequency and amplitude, phase shift between vibration components. Mathematical models and simulation results were verified by the experimental alignment of the cylindrical and rectangular cross-section parts. Alignment duration increases as excitation frequency increases if phase shift between vibration components is 0-π/2 at non-impact alignment regime. When phase shift is more than π/2, alignment duration decreases as excitation frequency increases. Alignment duration increases rapidly or alignment process stops at all when excitation frequency equal to bush’s natural frequency. Stable part alignment at impact mode goes when excitation frequency is from 1 to 1.7. Alignment duration, depending on a phase shift between vibration components could differ 7 %. Literature 1. Baksys, B.; Puodziuniene, N. Modeling of Vibrational Non-Impact Motion of Mobile-Based Body. International Journal of Non-Linear Mechanics, 2005, 40(6), p. 861-873. 2. Baksys, B.; Puodziuniene, N. Modelling of Vibrational Impact Motion of Mobile-Based Body. International Journal of Non-Linear Mechanics, 2007, 42, p. 1092-1101. 3. Baksys, B.; Baskutiene, J. The Directional Motion of the Compliant Body Under Vibratory Excitation. International Journal of Non-Linear Mechanics, 2012, 47, p. 129-136. 4. Baksys, B.; Ramanauskyte, K. Motion of a Part on a Horizontally Vibrating Plane. Mechanika, 2005, 55(5), p.20-26. 5. Архангелский, М. Е. О Превращение ультравуковых колебаний поверхности во вращательное и поступательное движение тела. Акустический журнал, 1963, 9(3), p. 275-278. 33 6. Mohri, N.; Saito, N. Some Effects of Ultrasonic Vibration on the Inserting Operation. The International Journal of Advanced Manufacturing Technology, 1994, 9(4), p. 225-230. List of author‘s publications Articles in publications from the master Journal List of the Institute for Scientific Information (ISI) 1. Sadauskas, Edvardas; Bakšys, Bronius. Alignment of the parts using high frequency vibrations // Mechanika / Kauno technologijos universitetas, Lietuvos mokslų akademija, Vilniaus Gedimino technikos universitetas. Kaunas : KTU. ISSN 1392-1207. 2013, Vol. 19, no. 2, p. 184-190. DOI: org/10.5755/j01.mech.19.2.4164. [Science Citation Index Expanded (Web of Science); INSPEC; Compendex; Academic Search Complete; FLUIDEX; Scopus]. [0,500]. [IF (E): 0,336 (2013)] 2. Sadauskas, Edvardas; Bakšys, Bronius; Jūrėnas, Vytautas. Elastic vibrations of the peg during part alignment // Mechanika / Kauno technologijos universitetas, Lietuvos mokslų akademija, Vilniaus Gedimino technikos universitetas. Kaunas : KTU. ISSN 1392-1207. 2013, Vol. 19, no. 6, p. 676680. DOI: 10.5755/j01.mech.19.6.6014. [Science Citation Index Expanded (Web of Science); INSPEC; Compendex; Academic Search Complete; FLUIDEX; Scopus]. [0,333]. [IF (E): 0,336 (2013)] 3. Sadauskas, Edvardas; Bakšys, Bronius. Peg-bush alignment under elastic vibrations // Assembly Automation. Bradford : Emerald. ISSN 0144-5154. 2014, Vol 34, no. 4, p. 349-356. DOI: 10.1108/AA-05-2014-031. [Science Citation Index Expanded (Web of Science); EMERALD; Compendex]. [0,500]. [IF (E): 0,711 (2013)] Articles in other referred publications from list of the Institute for Science Information (ISI proceedings) 1. Sadauskas, Edvardas; Bakšys, Bronius. Alignment of cylindrical parts using elastic vibrations // Mechanika 2012 : proceedings of the 17th international conference, 12, 13 April 2012, Kaunas University of Technology, Lithuania / Kaunas University of Technology, Lithuanian Academy of Science, IFTOMM National Committee of Lithuania, Baltic Association of Mechanical Engineering. Kaunas : Technologija. ISSN 1822-2951. 2012, p. 267-270. [Conference Proceedings Citation Index]. [0,500] 34 Information about author of the dissertation Name, Surname: Edvardas Sadauskas Date and place of birth: 13 October 1980, Kaunas, Lithuania. E-mail: [email protected] Education and training 2010-09 – 2014-08 2004-09 – 2006-07 2000-09 – 2004-07 Doctoral student at Kaunas University of Technology in the field of Mechanical Engineering Sciences. Kaunas University of Technology, Master of Sciences in Mechanical engineering, Mechanical engineering. Kaunas University of Technology, Bachelor of Sciences in Mechanical engineering, Mechanical engineering. Reziumė Detalių tarpusavio centravimas yra vienas svarbiausių automatinio rinkimo etapų, kurio metu kompensuojamos renkamų detalių tarpusavio padėties paklaidos. Tik diegiant efektyvius automatizuotus rinkimo metodus, galima sumažinti rinkimo darbų sąnaudas, užtikrinti stabilią renkamų gaminių kokybę, palengvinti darbo sąlygas, likviduoti varginančius monotoniškus rankinio rinkimo veiksmus. Rinkimo darbų automatizavimo pažanga daugiausiai priklauso nuo naujų rinkimo technologinių procesų, pagrįstų efektyviais komponentų centravimo metodais sukūrimo. Vienas perspektyvių, iki šiol mažai nagrinėtų yra vibracinis centravimo metodas, pagrįstas strypo tampriaisiais virpesiais. Vieno iš komponentų (įvorės) kryptingas poslinkis ir posūkis užtikrinamas frikcine sąveika su virpančiu strypo laisvuoju galu. Eksperimentiškai ištirtas cilindrinių ir stačiakampio skerspjūvio detalių be nuožulnų centravimas, žadinant strypą. Sudarytos centravimo trukmės priklausomybės nuo detalių pradinio prispaudimo jėgos, ašių nesutapimo, žadinimo dažnio ir amplitudės. Šiame darbe nagrinėjamas vibracinis detalių tarpusavio centravimas besmūgiais ir smūginiais režimais. Sudaryti detalių tarpusavio centravimo įtaisų dinaminis bei matematinis modeliai, įvertinantys detalių sąveiką viso centravimo proceso metu. Ištirtas detalių tarpusavio centravimas kuomet velenas kinematiškai žadinamas sujungimo ašies kryptimi, o įvorė bazuojama paslankiai. Teoriniai tyrimai patvirtinti atliktais eksperimentais. Pateiktos tarpusavio centravimo trukmės priklausomybės nuo dinaminės sistemos bei žadinimo parametrų. Nustatyta, kad didžiausią įtaką centravimo besmūgiais ir smūginiais režimais trukmei turi detalių pradinio prispaudimo jėga 35 bei žadinimo dažnis, išilginių veleno virpesių amplitudė, trinties jėga tarp strypo ir įvorės. Atlikti teoriniai ir eksperimentiniai vibracinio centravimo tyrimai patvirtino, kad vibraciniu metodu galima centruoti automatiškai renkamas cilindrinio ir stačiakampio skerspjūvio detales, nenaudojant jutiklių, vykdymo įtaisų, specialių valdymo algoritmų. Gauti teorinių bei eksperimentinių tyrimų rezultatai gali būti pritaikyti rinkimo įrenginių projektavimui bei automatinio rinkimo technologijų tobulinimui. Darbo struktūra ir apimtis Disertaciją sudaro įvadas, trys skyriai, išvados, autoriaus publikacijų disertacijos tema ir naudotos literatūros sąrašai bei priedai. Disertacijos apimtis 90 puslapių, 61 paveikslas ir 2 lentelės. Literatūros sąrašą sudaro 78 šaltiniai. Pirmame skyriuje, remiantis moksline literatūra išanalizuoti automatiškai renkamų detalių centravimo metodai, jų privalumai ir trūkumai. Pateikta su disertacijos tema susijusių tyrimų apžvalga. Suformuluoti pagrindiniai tyrimų uždaviniai. Antrame skyriuje pateikti eksperimentiniai virpančio strypo galo tyrimai, kai jis liečiasi su įvore. Nustatytos strypo judesio trajektorijos bei strypo–įvorės prispaudimo jėgos ir žadinimo signalo dažnio įtaka išilginių ir lenkimo virpesių amplitudėms. Atlikti apvalaus ir keturkampio skerspjūvio strypo ir įvorės centravimo tyrimai, kai strypas žadinamas sujungimo ašies kryptimi. Sudarytos centravimo trukmės priklausomybės nuo detalių pripaudimo jėgos, ašių nesutapimo, strypo žadinimo parametrų. Trečiame skyriuje pateikti vibracinio centravimo, naudojant tampriuosius strypo virpesius, dinaminiai modeliai, esant nesmūginiam ir smūginiam centravimo režimui. Pateiktos įvorės sąveikaujančios su dviem statmenomis kryptimis judančiu strypo galu, judesio lygtys bei centravimo proceso skaitmeninio modeliavimo rezultatai. Išaiškinta dinaminės sistemos ir žadinimo parametrų įtaka centravimo procesui, Sudarytos parametrų derinių sritys, kai centravimas būna sėkmingas. Mokslinių tyrimų tikslas ir uždaviniai Tyrimų tikslas – teoriškai ir eksperimentiškai ištirti renkamų komponentų vibracinio centravimo, naudojant tampriuosius strypo galo virpesius, procesą. Nustatyti žadinimo parametrų ir mechaninės sistemos įtaką centravimo efektyvumui. Siekiant įgyvendinti šį tikslą, reikia išspręsti šiuos uždavinius: Atlikti mokslinės literatūros apžvalgą apie šiuo metu plačiai pramonėje naudojamus automatiškai renkamų komponentų centravimo metodus. 36 Atlikti apvalaus skerspjūvio strypo galo virpesių eksperimentinius tyrimus, kai strypas liečiasi su įvore. Nustatyti virpesių pobūdį, jų priklausomybę nuo žadinimo signalo amplitudės bei nuo įvorės ir strypo prispaudimo jėgos. Atlikti apvalaus ir stačiakampio skerspjūvio detalių centravimo tyrimus, naudojant tampriuosius strypo virpesius, išsiaiškinti žadinimo bei mechaninės sistemos parametrų įtaką centravimo efektyvumui ir patikimumui Sudaryti cilindrinių komponentų centravimo matematinį modelį, kai naudojami tamprieji strypo galo virpesiai, kuomet strypas žadinamas sujungimo ašies kryptimi, esant nesmūginiam ir smūginiam centravimo režimui. Atlikti nesmūginio ir smūginio centravimo proceso modeliavimą, išsiaiškinti žadinimo bei mechaninės sistemos parametrų įtaką centravimo efektyvumui ir patikimumui. Mokslinis naujumas Rengiant disertaciją buvo gauti šie mechanikos inžinerijos mokslui nauji rezultatai: 1. Pasiūlytas naujas technologiškai paprastesnis detalių centravimo metodas, panaudojant vienos iš centruojamų detalių (strypo) tampriuosius virpesius, kai virpesių žadinimas vyksta sujungimo ašies kryptimi iš galo prispaustu pjezokeraminiu vibratoriumi. 2. Iš galo išilgine kryptimi žadinamo strypo laisvasis galas juda elipsine trajektorija erdvėje. Tokiu dėsniu virpantį strypo galą prispaudus prie įvorės atsiradusi trinties jėga užtikrina įvorei poslinkį ir posūkį 3. Pasiūlytu metodu galima centruoti nesmūginiu ir smūginiu režimu apvalaus ir stačiakampio skerspjūvio strypines detales su įvorės tipo detalėmis nepriklausomai nuo jų tarpusavio padėties, esant mechaniniam kontaktui tarp jų. 4. Sudaryti strypo ir įvorės centravimo nesmūginiu ir smūginiu režimu matematiniai modeliai bei nustatytos centravimo trukmės priklausomybės nuo žadinimo dažnio ir amplitudės, komponentų tarpusavio prispaudimo jėgos. Darbo rezultatų praktinė vertė Taikant tampriuosius strypo virpesius, galima centruoti įvorės tipo detales turinčias apvalaus ir stačiakampio profilio skyles su atitinkamo profilio strypais, kai komponentai yra su nuožulnomis ir be jų, o komponentų tarpusavio padėties paklaida siekia kelis milimetrus, taip praplečiant automatizuoto rinkimo technologines galimybes. Tyrimo rezultatai leidžia nustatyti ir parinkti žadinimo ir įtaiso parametrus ir juos suderinti, kad centravimas būtų sėkmingas, o jo trukmė mažiausia. 37 UDK 621.717 – 658.515] (043.3) SL344. 2015-05-12, 2,5 leidyb. apsk. l. Tiražas 70 egz. Užsakymas 150185. Išleido leidykla „Technologija“, Studentų g. 54, 51424 Kaunas Spausdino leidyklos „Technologija“ spaustuvė, Studentų g. 54, 51424 Kaunas 38
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