Meccanica (2011) 46:699–710 DOI 10.1007/s11012-010-9331-y
O R I G I N A L A RT I C L E
Experimental research for robust design of power transmission components Milosav Ognjanovic · Matug Benur
Received: 10 January 2010 / Accepted: 9 June 2010 / Published online: 3 August 2010 © Springer Science+Business Media B.V. 2010
Abstract In the future progress of technical systems it is impossible to avoid the power transmission (PT) components. Mechatronical technical systems will include the innovated PT components with high-level quality indicators. The article proposes the application of the new approaches to those components design in order to challenge innovation and inventions. The main objective is to define the design parameters in terms of reliability, vibration and noise as design constraints in the stage of the Embodiment design of PT components. Robust design is provided by using the axiomatic method in this way. Reliability as the design constraint of PT components is defined and modeled in a specific way suitable for this purpose and application. Also, the model of gear vibrations and gear units noise generation is presented in a new way suitable for applying as the design constraint. Those design constraints provide design parameters definition in an efficient way, with high-level service quality indicators. The presented models are based on a great volume of experimental data about service conditions probability, gear and bearing failure probability, gear units vibration and modal behavior etc. Theoretical knowledge and models are insufficient yet to provide the necessary data. The article contains presentation
M. Ognjanovic () · M. Benur Faculty of Mechanical Engineering, University of Belgrade, Kraljice Marije 16, 11120 Belgrade, Serbia e-mail:
[email protected]
of testing methods and data processing oriented to provide data necessary for the application in the suggested approach to PT components design. Keywords Power transmission · Design · Reliability · Vibration · Noise
1 Introduction Product development and design of technical systems is informational process, in general, which contains data transformation using the achieved level of knowledge. The level of technical systems follows the global level of knowledge in science and technology. Experimental research complements the missing theoretical knowledge and data necessary for the development of certain kinds of products. Natural and technical sciences can’t supply all necessary data and knowledge, now and in the near future. Experimental research provides it or presents the proof/check for the starting theory. Intuition, association and cognitive thinking additionally complete the missing knowledge for product development. Product development and design have been of scientific interest for a long time. The first methodologies were established in the late 70s and early 80s of the last century, by VDI norms and presented in the book by Pahl and Beitz: Engineering design—a systematic approach. In this period there started an intensive development of this area which included Theory
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of Technical Systems and Principles of Engineering Design (Hubka & Eder). In the next period this activity was oriented to finding more effective and more holistic methods and approaches to technical system development and design. This is the search for effective method of data and knowledge transformation in the technical system. In this article some of the methods will be applied to the development of power transmission systems. Robust design and axiomatic design [1–3] are the two of these methods. Their detailed application is presented in Sect. 2 of this paper. The development of technical systems (TS) is the process of technical evolution from one to the next generation of TS. The intention of robust design is to shorten this process and provide maximal functional characteristics in the first attempt, which are insensitive in service conditions variation. Axiomatic design towards Suh axioms [1] has the intention to establish the relation between design parameters and functional requirements of TS in order to create a model of data transformation for a certain kind of TS. Instead of TS evolution, genetic approach with “chromosome mutation” can provide the possibility for a greater jump in the TS innovation. A set of approaches in searching for new ideas is available. Together with robust and axiomatic methods this approach can provide successful innovations or inventions. The new design and product development methods can’t provide innovations and inventions without a great volume of data which is theoretically impossible to obtain. In this regard, the actual article treats reliability, vibration and noise of mechanical systems. Failures of machine parts (wear or fracture) are random processes. Theoretically, these failure processes have not been investigated enough and the exact model for calculation has not been established so far. Statistical approach [4, 5] using experimental data is the only possibility to take in consideration this very important aspect of TS. Vibrations of mechanical systems have been presented by numerous theoretical models, but often there existed the difference between calculated and measured vibrations. Also, calculation of vibrations should include the parameters, such as stiffness, damping, mass etc. which is difficult to measure, and it is easier to measure the vibrations directly. In the area of gear systems, theoretical considerations of dynamic behavior are carried out with a few objectives [6–12]. Dynamic load calculation and theoretical consideration of influences and effects interaction is carried out in the articles [6] and [7]. The effect of gear
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teeth parameters and bearings at the level of dynamic loads produced by teeth meshing in relation to teeth mesh frequency [6] is modeled, calculated and analyzed theoretically. Similar analyses in [7] are carried out in order to include the effect of transmission errors in gear teeth mesh in calculated vibrations. Also, the effect of nonlinear damping [8] and effect of sliding friction [9] are theoretically analyzed in order to obtain a better covering between calculated and measured results of gear vibration and dynamic forces. Covering is successful in the sub-critical and in critical teeth mesh frequency range, but in super-critical range the differences are significant. According to theoretical models, in this frequency range, vibration level decreases. The results of gear vibrations measurement show that in super-critical teeth mesh the frequency range vibrations slightly increases. Possible explanation and model for theoretical explanation of that gear phenomenon is presented in article [10]. Gear vibrations are treated as restorable free vibrations which restore with every gear tooth entering in conjugation. The presented model based on measured gear vibrations result gives a full covering for both results measured and calculated. The calculation model is developed towards the equilibrium of disturbance energy caused by teeth meshing and kinetic energy of gear vibration. The energetic balance of gear vibrations is also treated in the articles [11] and [12] but in another way and for other purposes. Results of gear transmission housing dynamic behavior analysis are presented in articles [13] and [14]. The objective of this research was to investigate mechanism of disturbance energy transmission trough design structure in gear transmission surroundings. The articles [15] and [16] contain design synthesis of specific mechanical systems taking in consideration specific approaches. Furthermore, articles [17–19] are oriented to conditions analysis towards limited values of operating indicators. In design process those indicators can be used as design constraints. Power transmission components are the main group of function carriers of numerous machines. Reliability, vibration and noise level of those components are the main indicators of service quality. Increase of those indicators is in strong contradiction with the others, such as compatibility, cost etc. Increase of some of quality indicators without disturbing the others presents a very complex problem. On the other hand, in the future it is not possible to avoid power transmission systems in
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machine structures, and it is necessary to pay attention to design innovations and inventions in this direction. In addition to theoretical and design research, the experimental data search presents the most important activity for robust design of those components.
2 Robust design of power transmission components Robust design means the technical products insensitive to variation of operating conditions and also products which are successfully designed in the first attempt. Power transmission components operate in extremely random conditions. Operating regimes (loads, speeds, etc.), production conditions and failure processes are random. It is known that random processes are possible to identify, present and analyze only using the experimental approach supported by statistical indicators. Power transmission operating regime for a certain machine system (vehicle, dredge, etc.) is possible to identify by systematic measurement in service conditions. Possibility of power transmission components (gears, bearings, sealing sets, etc.) failure can be identified by failure probability which needs extensive laboratory tests of the listed components. The relation between service regime and failure probability leads to component reliability. Using these elementary reliabilities as design limitations (constraints) for design parameters definition, the robustness of power transmission unit is fulfilled. Similar situation occurs with the vibration and noise of power transmission components. The level and frequency structure of vibration and noise produced by these components can be used as design constraints for design parameters definition and harmonization of their interaction. The relations between operating conditions and component parameters and dynamic responses have to be harmonized using experimental approach. For the purpose of power transmission system design the specific procedure is established and presented in Fig. 1 in the form of the design model. The PTSD (Power Transmission Systems Design) model consists of four modules which respect general design procedure of technical systems and specific needs for the design of PT systems. The first is Solution Module which is oriented to the creation of conceptual solutions for certain service conditions. Power transmission systems in conceptual sense are a variation structure i.e. conceptual solutions are the result of various
Fig. 1 Power Transmission Systems Design—PTSD model
combinations of gear pairs, shafts, bearings etc. In interactive communication the module offers all possible combinations and stores in the Conceptual Base. The next LAHP-module has the task to adapt limitations and constraints to every conceptual solution and to every design component. Limitation Analytic Hierarchy Processing—LAHP module divides the conceptual design in sub-conceptions or function carriers, and for that structure the processing limitations and constraints in hierarchy order. General transmission ratio of the system is decomposed to the level of every transmission stage. General value of the system reliability (chosen in advance) is hierarchically decomposed to the level of possible failure. The LAHP-module is the key part of that approach which supports reverse calculation in order to fulfill one of the features of design robustness i.e. to design the system with a chosen general level of reliability. The next is Design Parameters Definition—DPD module based on the calculation of design parameters, especially dimensions, using axiomatic approach. By observing the axiomatic rules and by inclusion of design constrains this module fulfills the features of robustness. The last module is Priority Module whose task is to check which design solution satisfies service limitations, such as volume, weight, efficiency, cost etc. in a better way. The calculation of priority indicators is interactive and gives possibility for additional corrections and adoptions. The main feature of design robustness contains DPD module based on axiomatic rules. These are the two axioms, the axiom of independency and the axiom of information minimum. In Fig. 2 the principle
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Fig. 2 Relations in DPD module
Fig. 3 Example of DP minimization and variation
of DPD module based on axiomatic rules is presented. Functional Requirements FR of every design component are defined by service conditions, where the system operates. This FR is necessary to transform into Design Parameters—DP of design component. Transformation matrix [A] established by Suh [1] for the relation in Fig. 2 is inverse matrix [A]−1 . Numeric values of matrix members define the relations in design component, which are constrained by numerous limitations, such as safety or reliability, stiffness, standards, rules, etc. These limitations and constraints are the result of service conditions, which is deducted by LAHP module to the level of design component. For the purpose to present this relation more clearly, the following example is processed. In Fig. 3 the example is presented. The assembly of the gear, shaft and bearing is defined by a great collection of design parameters, especially dimensions. The calculation of dimensions is reduced to the three dimensions, gear diameter d, gear width b and shaft diameter dhs . In this way, the axiom of information minimum is fulfilled. Other dimensions are in relation
with those calculated. Matrix [G] (Fig. 3) is the shape vector which defines transformation of parameters in the all shape dimensions. This is the shape parameterization where varying of the shape parameters varies the complete shape and dimensions. In Fig. 3 are presented the two shapes of the same assembly obtained in this way. Similar approach is incorporated in CAD tools for the shape modeling. The structure of the matrix [A] according to Suh can be uncoupled, coupled and decoupled. The ideal situation is with uncoupled matrix where one DP is responsible for one FR. Real situation is more complex. In order to obtain the decoupled matrix of transformation, the matrix [A]−1 is presented in the form of matrix [C] in the following form. ⎧ ⎫ ⎡ ⎤ ⎧ 1/3 ⎫ d ⎪ c11 0 0 0 ⎪ ⎪ ⎪T ⎪ ⎪ ⎪ ⎪ ⎨ ⎬ ⎢ ⎥⎨ T ⎬ dhs 0 0 0 c 22 ⎢ ⎥ =⎣ . (1) C ⎪ 0 0 c33 0 ⎦ ⎪ T ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎩ ⎭ ⎭ SE 0 0 0 c44 1 The axiom of independency is conditionally fulfilled. The members of matrix [C] and design parameters d, dhs , carrying capacity of the bearings C and seal type indicator SE have to be calculated successively. The member c11 is in relation with elementary reliability R1 against wear failure of gear pair, c11 = f (R1 ). After the calculation of gear diameter d, it is possible to calculate the shaft diameter because the shaft loads depend of the gear diameter, i.e. c22 = f (d, R2 ), and also of the shaft reliability R2 . The next step is calculation of bearing carrying capacity using matrix member c33 which is in relation with the both diameters d and dhs , the total number of bearing revolutions along the service life n , and of bearing reliability R3 , i.e. c33 = f (d, dhs , n , R3 ). At the end, the matrix member c44 is in relation to the shaft diameter dhs , to the total number of the shaft revolutions n and to seal reliability R4 , i.e. c44 = f (dhs , n , R4 ). The main feature of robustness is covered by reliability. The values of calculated parameters have to be insensitive to service conditions varying. For example, gear diameter calculation is in the form of T 1/3 d = k11 3 = c11 (σHdes )Tmax . (2) 2 ϕσHdes Design available gear teeth flank stress σHdes is in direct relation to reliability against teeth flank failure R1 . Unreliability Fp = 1−R1 is the complex func-
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Fig. 5 Load spectrums and service regimes
Fig. 4 Relation between service stress varying and gear wear probability
tion of service conditions probability p and of failure probability PF under these service conditions, i.e. Fp = pPF . If in the service life the gear pair is exposed to the flank stress σH 1 with n1 cycles and to stress σH 2 with n2 cycles and to σH 3 with n3 cycles (Fig. 4), gear wear unreliability can be calculated as Fp =
3
pi PF i ;
i=1
PF i = 1 − e
−(
pi =
ni ; n
σH i β i ηi )
(3)
Teeth wear (failure) probability PF i is presented by Weibull’s functions, where the parameters of those functions ηi and βi are defined for every σH i and ni (see Fig. 4). For this purpose, it is necessary to have the area of failure probability for a certain gear pair, which can be obtained by extensive gear wear testing. If calculated unreliability is close to unreliability which is defined as design constraint, the maximal stress can be chosen as design available stress, i.e. σHdes = σH 1 . If it is not, it is necessary to change the relations in Fig. 4 and to repeat the calculation. However, that definition of σHdes includes in this way all randomness and variations of service conditions, the calculated design parameters are insensitive to service conditions varying.
3 Reliability as design constraint The presented approach in the design of power transmission components towards the new approaches as
axiomatic and robust design can be successful with a great volume of experimental data only. The new methods can succeed in the design process but remain to be carried out towards experimental data. However, the data have to be oriented and adapted to be suitable for application in a new way. Reliability is the term with a very wide area of applications. For the purpose of applying the new methods and approaches, the reliability is defined in a specific way (3). The main features of reliability as design constraint are the following. Firstly, elementary reliability is connected to possible failure, not to the component of the system. In one possible failure a few components can participate or one component can be exposed to more than one failure. Secondly, this elementary reliability is composed of cause probabilities which produce failure in order to avoid possible failures by design activities (Design Parameters Definition). In this regard, the elementary reliability is complex probability composed of operating stress probability and failure probability under that operating stress. Both of these probabilities are the result of extensive experimental research. For the purpose of power transmission components design, the experiments and the data processing are as follows. 3.1 Operating load (stress) probability identification The power transmission components in operating conditions are exposed to extreme random loads caused by resistances in service conditions. For example, bucket wheel excavator (Fig. 5a) has the wide variations of resistant forces which are the result of numerous causes. Extensive torque measurements at the wheel traction shaft are one of the possibilities for load probability identification. Measurements have to be carried out for a set of specific operating conditions, for example, for various properties of processed material, various quantities of material per time unit, various handling
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etc. By estimation of those conditions’ participation along the service life, and by combination with the measured torque, the load spectrum (Fig. 5b) can be obtained. This spectrum presents the participation of the load levels in the service life of power transmission component. The spectrum can be also presented in the continual form. The shapes of continual lines can represent the characteristics of service (exploitation) regime. The light (easy) service regime is represented by line e. In this regime, low levels of torque participate dominantly and high levels of torque arise from time to time. Heavy regime represented by line h is with a great participation of high levels of torque. The middle regime m consists of relatively equal participation of low and high levels of torque. The exploitation (service) regime is the result of the load measurements, resistant calculations, long-time experience in monitoring service conditions of the system etc. The load (service stress) spectrum represents the operating stress probability which comes from empirical and experimental sources. 3.2 Gear wear probability identification Failure probability of power transmission components is the result of extensive laboratory testing. The collection of the same kind of components is exposed to the same load (stress) until failure occurs. The stress cycles numbers until failure occurrence is random and is distributed in a wide range. By the corresponding method for data processing the results of testing are transformed into the function of probability distribution. In Fig. 6a the testing rig with power circulation (back-to-back system) is presented. This is the known principle for loading of rotational components for a long-time testing until failure occurs. Power circulation provides economic long-time testing. At the beginning of gear teeth wear research and for definition of failure probability range distribution, it is necessary to make a few decisions. Firstly, in the range of finite fatigue (wear) life it is necessary to choose the two stress levels, so as to identify this range bounded by inclined strict lines (Fig. 6c). The distance between these stresses levels has to be large enough in order to obtain a more precise angle of line inclination. Higher level σH 1 has to be less than the stress which can cause plastic deformation of surface layer, or load has to be less than the load which can cause teeth fracture by bending stress. Less stress level σH 2 has to be high
Fig. 6 Gear wear probability testing
enough to avoid the range of failure distribution in infinite fatigue life range. The second decision refers to the number of gear pairs, which it is necessary to test at each of the two stress levels. For probability calculation, a minimal statistical set is eight. For precise definition of failure probability functions PF (N ) in the range of finite life range, it is necessary to make wear testing the two times of eight gear pairs (for both stress levels σH 1 and σH 2 ). Independent variable N is the number of gear revolutions (teeth stress cycles number) until failure occurs. Infinite fatigue life range for teeth wear is beyond 5 × 107 to 109 (see Fig. 7), depending of the teeth surface layer hardness. To identify the function of failure probability distribution PF (σH ), it is necessary to choose the stress cycles number NDV in this range (Fig. 6d). This is the
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Fig. 7 The ranges of the teeth flank ultimate stress and wear probability distribution
stress cycles number limit for sample testing in infinite fatigue life. Samples can be failed before this limit or not failed until NDV . In this range, the testing is carried out for a few levels of the stress σH . For each of them, the testing set of gear pairs is minimum the eight. The result of tested results processing are parameters of Weibull’s function for wear probability distribution PF (σH ). When all of three failure probability functions (Fig. 6d) are obtained (the two functions PF (N ) and function PF (σH )) it is possible to complete the total area of failure probability distribution bounded by lines for PF = 0.1 and PF = 0.9. Using this area of probability distribution for every stress level or for every stress cycles number it is possible to define Waibull’s function, necessary for unreliability calculation (see Fig. 4). Testing of failure probability is the procedure developed and standardized for some of frequent failures and some machine components. This is predominantly for fracture damages, not for wear, especially not for gear teeth wear. This failure is very interesting for identification of reliability as the design constraint for gears, but a few phenomena in this wear process are the reason why this probability wasn’t tested in
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this way before. Gear wear is a complex process which consists of a set of the wear entities. The main of them is the flank surface layer pitting (Fig. 6b). This is the result of fatigue which fails surface layer after random stress cycles number. The results of pitting failure probability testing follow the presentation in Fig. 6d. The next entity of gear flank wear is abrasive or adhesive sliding wear. This group of wear entities contains micro-pitting and scuffing. These are permanent processes which remove material from the teeth flanks (Fig. 6c). For these entities a series of question arises. The first one is if there exists infinite wear life, i.e. if a horizontal part of failure probability distribution area (Fig. 6d) exists for these wear entities. The second is which wear entity is possible to cause by laboratory testing. It depends of stress level, gear flank thermal treatment, speed of rotation, lubrication, etc. Also, there arises the problem that teeth fracture instead of wear is possible to cause with the high load. That is the reason why the gear presented in Fig. 6c was in the contact with one half of the wide. Furthermore, for wear probability testing the problem is that it is not possible to effectuate the same conditions in this sense for each of the three functions of failure probability presented in Fig. 6d. In real situation all wear entities are mixed and combinations can be numerous. The load (the flank stress) can be various and combinations with the various teeth flank hardness are not the same. From this point of view, randomness of the teeth wear is much higher compared to randomness of individual teeth wear entities. An additional phenomenon with gear flank wear is that gear pairs can operate with the partly failed flanks by wear but with significantly reduced service quality. The limits of teeth wear parameters are not precisely identified. All mentioned problems open up a few directions of experimental research. One of them is the definition of statistical parameters of wear probability distribution for individual teeth wear entities. The second is interaction between the entities and possible parameters of wear probability distribution. Then follows, to find the relations between gear operation quality indicators and wear level and the others. Summarized experimental results of the gear teeth flank ultimate stresses are presented in the DIN 3990 and ISO 6336 standards. Variation of the steel type, teeth flanks thermal treatment, i.e. teeth flanks hardness effectuates all indicators of ultimate flank stress distribution. In Fig. 7 some of the chosen examples
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are presented. Increase of the teeth flank hardness significantly increases ultimate flank stress, i.e. translates the range of failure probability distribution in the area of higher stresses. 3.3 Failure probability identification of bearings and seals Failure probability testing of the bearings is carried out according to the principle similar to the gear wear probability testing. In Fig. 8a one of the possible testing rigs is presented. The range of failure probability distribution is bounded by inclined straight lines in a double logarithmic coordinate system (Fig. 8b). For the straight lines definition, it is necessary to make the tests at the two levels of force acting at the bearing. Similar to the gear testing at the both force levels, it is necessary to test the collection of the same bearing type. Using the revolution numbers to the bearing failure N , the Waibull’s functions of failure distribution PF (N ) are defined. For those functions, in the next step, the range of bearing failure probability distribution is bound. Boundary lines for failure probability PF = 0.1 and PF = 0.9 give possibility to identify Waibull’s functions of failure probability for bearing revolutions from the load spectrum similar to gears (Fig. 4) and to calculate elementary bearing reliability for a certain service regime in exploitation.
Fig. 8 Bearing failure probability testing
4 Vibration and noise as design constraints Vibration and noise of machine systems, i.e. power transmission components is the result of interaction between disturbances caused by components in motion and design parameters of components. This interaction in gear teeth meshes and bearings are in the form of machine parts collision, sliding, rolling etc. In gear pair meshes between the meshed teeth all of these processes exist. A few kinds of teeth impacts arise in the mesh. More important is the addendum impact which is the result of gear pitch difference caused by elastic deformations. 4.1 Gear system restorable free vibrations Teeth deformations are proportional to teeth load and teeth stiffness. Deformations replace the first point of contact from the right position A, to position A which is ahead of point A. The contact of teeth pair starts with intensive addendum impact (Fig. 9a). Collision speed vc is proportional to teeth deformation, speed of rotation n and gear design parameters. By analyzing teeth geometry, deformations and speeds, collision speed at the first point of teeth contact is defined and presented in [10]. Every individual teeth impact produces natural free vibration of the gears (Fig. 9b) with natural frequency fn . By strong inside damping these vibrations attenuate in a short time. The next teeth pair entering the mesh collides again and again. Vibrations become restorable after every teeth impact. For relatively slow gear rotation the measured time function of restorable free vibrations is presented in Fig. 10a. The teeth mesh frequency f corresponds to gear revolutions and to gear teeth number. If the speed of the gear rotation is slow enough, the time between the two
Fig. 9 Gear teeth addendum collision (a) and free vibration caused by teeth impact (b)
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The results of those gear vibration measurement are presented in the form of calculation model [10]. The objective of modeling was to prove the hypothesis that gear vibration is a restorable natural one and then to present the vibrations in the form suitable for use as constraints in design parameters definition in gear drive design process. The total level of gear vibration x¨ is divided into two parts, continual x¨a and transient x¨b (Fig. 10b). x¨ = x¨a + x¨b = x¨a (1 + ζT sin ϕ) cγ x0 =A f (1 + ζT sin ϕ) me
Fig. 10 Restorable free vibration of gear pair: (a) the time function, (b) result of vibration measurement and modeling
teeth impacts 1/f is higher than the time necessary for the free vibration attenuation. As the speed of rotation increases the time between the two impacts gets lesser and lesser with the same natural frequency fn . In the full resonance these two frequencies get equal f = fn . For the case of gear vibration measurement (Fig. 10b) (gear teeth number z = 32) the resonant speed of gear rotation is 9860 rpm, and resonant teeth mesh frequency is 5258 Hz. Before the main resonance a few sub-resonances arise. This is the result of equality of additional disturbances and natural frequencies. After (behind) the main resonance arises, numerous super-critical resonances arise and total vibration level slightly increase. This is an interesting phenomenon, especially in the gear meshing process. In super-critical mesh frequency range the absorbed disturbance energy by teeth impact relishes natural free vibrations. Higher speed of rotation produces higher level of absorbed disturbance energy and relished vibration energy has to be of higher level. Also, in supercritical mesh frequency range the modal structure of gear system fluctuates. In some parts of this range energy attenuation is lesser and relished energy by natural vibration is higher. By extensive measurement of gear vibrations with speed of rotation variation [10] and with frequency spectrum analysis these processes are identified.
(4)
The gear pair is modeled as a single mass oscillator, where me is equivalent mass reduced in collision direction (Fig. 9a), cγ is the mean gear teeth stiffness, x0 is teeth deformation amplitude in the moment of teeth collision, ζT is transfer function between force transferred to the gear masses and collision force Fc with phase angle ϕ, A—is coefficient of energy absorption inside the vibration system, and f is the gear teeth mesh frequency. The model presented by (4) and by the diagram in Fig. 10b is developed using the theory of singular systems. According to this theory, the singular process consists of the two processes, continual and transient. Vibration energy emission is presented as the continual process (a) and transient (b) which contains resonances (the main, sub-critical and super-critical). Analytical model for continual part is established based on equilibrium between disturbance energy caused by teeth impacts and kinetic energy of restorable free vibrations. A significant part of disturbance energy is attenuated by numerous damping. The ratio between these energies is presented in the form of constant value which is calculated for measured vibrations level. The model for additional (transient) vibration values (b) is developed using analytical relations for resonance responses. These formulas include the difference between the teeth mesh (impact) frequency f and natural frequency fn , dimensionless damping coefficient and phase angle. By variation of damping coefficient the response is calculated to be equal measured vibrations. In Fig. 10b the calculated line which approximates the measured one is presented. This line contains the main resonance only. The program for these values calculation can also include sub-critical and super-critical resonances. The developed analytical model gives possibility to
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present disturbance energy transformation, and to explain the inside processes of gear teeth mesh. Besides, the model gives possibility for design parameters harmonization in order to avoid resonances and reduce gear vibration level, i.e. use gear vibration as design constraints. 4.2 The noise generation caused by gear system vibration The part of disturbance energy absorbed by teeth repeatable collisions is relished in the form of kinetic energy of the gear system vibrations. The part of this kinetic energy is transformed, via free surfaces, into the noise energy inside the gear system housing. Another part of energy is transmitted through the supports (bearings) to the housing walls. This energy can be emitted from the wall surface in the surroundings in the form of noise waves. Also, the housing inside noise can penetrate through the housing walls into the surroundings. Much more important effect of the gear system vibration is disturbance of gear housing natural vibrations. This vibration can significantly intensify the noise that housing surfaces emit into the surroundings. From this point of view, the housing walls have a triple role: transmitter of vibration energy in surrounding, insulator of inside noise and generator of additional noise by own natural vibrations. Detailed analysis of the gear system housing behavior and its effects are presented in the articles [13, 14]. Disturbing of a certain modal shape of elastic deformation with a certain natural frequency fni is the result of a few conditions, equality of natural and disturbing frequency, the same direction and type of elastic deformation of disturbance elastic waves and modal elastic deformations, low level of dimensionless damping coefficient of those modal elastic deformations (elastic waves). In Fig. 11a the experimental result of chosen gearbox housing modal testing is presented. By modal hammer impact a relatively great collection of natural frequencies are disturbed. Four of them fn1 . . . fn4 have dominant effect. If inside the housing the running gear system with mesh (teeth impact) frequencies f1 , f2 and f3 increase with speed of rotation, those frequencies proportionally increase. Equality of disturbance fi and natural fni frequencies produces natural vibrations of housing walls and noise with this frequency. The relation between disturbance and natural frequencies presented in Fig. 11b is a principle one and
Fig. 11 Relation between frequencies of gear system vibration and gear housing natural frequencies
presents Campell’s diagram for this purpose. A real gear system which is settled in the gearbox housing generates vibrations with a much more complex frequency spectrum. Except the presented teeth mesh frequencies f1 , f2 and f3 (Fig. 11b) natural frequencies of the gear system dominate in those spectrums. Furthermore, the gear vibrations as presented in Sect. 4.1 are restorable free ones. These natural vibrations of the gear system (inside the housing) are disturbances for natural vibrations of housing walls. This additionally makes the process of vibration and noise generation more complex. However, all natural vibrations are independent of service conditions (speed of rotation) and are in close relation with design parameters. This is the reason why some of the gear system natural frequencies can permanently disturb some of the housing natural frequencies and produce permanent high level of noise with a certain frequency. It can be changed by some kind of design parameters variation. Disturbance power transmission through the gear unit assembly shows a few relations as follows. Kxc me x˙ 2 f; Wv = Ek f = f; 2 2 Wv Ws ζv = ; ζs = Wc Wv Wc =
(5)
The teeth collision power Wc is proportional to constant K proportional to teeth collision force (gear design parameters and collision speed) and fluctuation of teeth displacement xc and teeth mesh frequency f . The gear vibration power Wv is proportional to the rotating equivalent mass me , the vibration speed x˙ (obtained via (4)) and the teeth mesh frequency f . Disturbance power transfer function from collision power to vibration power is marked by ζv and transfer function from vibration power to the sound power is marked by ζs . All of them are experimental values.
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4.3 Design parameters definition constraint by vibration and noise According to the main axiom of axiomatic approach in design, i.e. the axiom of independency, design parameters definition using reliability, vibration and noise as design constraints are independent. Using reliability as design constraint defines gear diameter d and gear width b. Usage of vibration and noise as design constraints provides the possibility to reduce vibration and noise level by variation of gear design parameters (gear teeth number z, gear module m, teeth offset factor x, teeth helical angle β etc. without variation of d and b). The reduction of vibration and noise level with the aim to satisfy limited levels is the main task of vibration and noise design constraint. This constraint can be satisfied with a set of design actions and solutions. It is not a general design solution which can provide this result. One group of these partial activities has the purpose to harmonize gear parameters in order to avoid equivalence between teeth mesh frequencies with natural frequencies of the gear pear and the housing. The second group of design actions is the gears parameters variation and the shape and dimensions variation of housing with the aim to reduce disturbance power, vibration power and noise power including the corresponding transfer functions (5). The main of these activities oriented to satisfying the limited level of vibration and noise are as follows. • By variation of gear teeth design parameters z, m, x, β (without variation of gear dimensions) the teeth stiffness and gear pair natural frequencies fni vary. Also, these variations effectuate the variation of gear teeth mesh frequency f . This is the way to avoid the equilibrium between disturbance and natural frequencies presented in Fig. 11b. It provides significant reduction of vibration and noise of gear unit and satisfaction of design constraint. • By variation of gear teeth parameters z, m, x, β (without variation of gear dimensions) and by flank correction possibility is provided for the reduction of disturbances, especially teeth impacts in the gear pair mesh i.e. collision power Wc . It is the direct way to reduce vibration power Wv and sound power Ws (4) and (5) and satisfy vibration and noise constraint. • By variation of design parameters of gear system housing (shape, thickness of walls, distribution of
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ribs, material characteristics, etc.) gives possibility for harmonization of natural (modal) behavior of housing with natural frequencies and teeth mesh frequencies of gear system. This is the way to reduce the value of transfer function ζs (4) and (5) as one step more in the noise reduction and satisfying of vibration and noise constraint. • Isolation is a passive approach to satisfying the constraints by limited level of vibration and noise. The two groups of isolation principles are available. The first is oriented to interrupting the disturbance transmission through the system, i.e. from one to another machine part. For example, to interrupt disturbance transmission from the gear system to the housing via bearings (reduction of transmission factor ζs (4) and (5). The second group is the coating of housing surfaces or the application of layered housing walls in order to increase the isolation properties and reduce emission activity of the housing walls, also with the aim to reduce ζs and additionally reduce the emitted noise and satisfy the noise constraint. • Active approach to the noise reduction is also available and possible to apply when the passive solutions are insufficient. This approach implies the electronic control system which measures vibration or noise and produces the same vibrations or noise with the phase displacement which, in the sum, provides full elimination of vibration or noise.
5 Conclusions As the power transmission components are a part of the future mechatronic systems, the new design and principles, innovations and inventions of PT components are the key objective. The article offers an innovated design process of PT components, especially in the stage of embodiment design in the sense of the following three contributions. • Power transmission components robust design by applying axiomatic method. The proposed design process is more effective and provides good solution in the first attempt insensitive to service conditions variation. Design constraints by reliability, vibration and noise are the main design process improvement. • Reliability as design constraint of power transmission components is defined in a specific way, in the form of complex probability of service conditions probability (load or stress spectrum) and failure probability (gear teeth wear, bearing failure,
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seal failure, etc.). Those probabilities are possible to obtain by extensive testing and experimental data processing. Testing and data processing methodology are presented. • Vibrations and noise of power transmission components are presented by specific and new models suitable to be the design constraints. Dimensions of gear transmission components are constrained by necessary reliability. Limited level of vibrations and noise have to be satisfied by variation of gear design parameters, bearing parameters and especially gearbox housing design parameters, without dimension variation which is already constrained by reliability. The analysis of vibration and noise generation is the wide area of experimental research, which provides a great support to robust design of PT components. Acknowledgement This work is a contribution to the Ministry of Science and Technological Development of Serbia funded projects TR 14052 and TR 14033.
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