Effective Vibro-Acoustical Modelling of Rubber Isolators

Transcription

1 Effective Vibro-Acoustical Modelling of Rubber Isolators by Michael Coja Stockholm 2005 Doctoral Thesis Royal Institute of Technology School of Engineering Sciences Department of Aeronautical and Vehicle Engineering The Marcus Wallenberg Laboratory for Sound and Vibration Research

2 Akademisk avhandling som med tillstånd av Kungliga Tekniska Högskolan i Stockholm framläggs till offentlig granskning för avläggande av teknologie doktorsexamen den 16 juni 2005, kl i D3, Lindstedtsvägen 5, KTH, Stockholm. ISSN TRITA-AVE 2005:25 ISBN c Michael Coja, June 2005

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4 Kookaburra Koo-ka bur-ra bor i ett Gum - mi trä - d Gum-mi trä-det va-jar han va - jar me - d Sjung Koo-ka - bur-ra Sjung Koo-ka - bur-ra Gum - mi trä - dets Kung!

5 Preface The work forming this thesis was carried out between June 2001 and June 2005 at the Marcus Wallenberg Laboratory for Sound and Vibration Research (MWL) part of the Department of Aeronautical and Vehicle Engineering at the Royal Institute of Technology in Stockholm. The main source of funding for my research activities coming from the European Project for Growth NORMA, I gratefully acknowledge the European Commission and the Marcus Wallenberg Laboratory for their financial support. During these years I encountered people who have helped me to accomplish this thesis and I would like to express my gratitude to them. First of all, I want to thank my supervisor Leif Kari for his constant availability and his excellent guidance in the subject and in the world of scientific research. I am also grateful to Anders Nilsson (former Head of the MWL) who first invited me to start working at the MWL; to Kent Lindgren and Danilo Prelevic for their technical savoir faire in the laboratory works; to my room colleagues for their conviviality; to Anthony Vinogradoff for his linguistic help and to all the MWL staff which forms such an interesting multi-cultural working environment. I would like to send special thanks to my family, Danielle, Henri, Agnès and Isabelle who have always been a great support to me despite the physically long distance separating us. Finally, to all the persons I mentioned and to those I forgot to mention I would like to offer this song, on the opposite page, since music occupies such an important part of my life and also might have been at the very origin of my orientation to acoustical matters. Michael Coja Stockholm in May 2005.

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7 Abstract This thesis, gathering four papers, concerns the enhancement in understanding and modelling of the audible dynamic stiffness of vibration rubber isolators including experimental measurements. Paper A studies the performances of three different types of vibration isolator using an indirect measurement technique to estimate the blocked dynamic transfer stiffness of each specimen. The measurements are performed over a wide audible frequency range of 200 to 1000 Hz in a specially designed test rig enabling the investigation of arbitrary preload influences. Paper B addresses the modelling of the audible-frequency stiffness of the rubber conical mount experimentally appraised in Paper A accounting for preload effects. The model is based on a simplified waveguide approach approximating the nonlinearities attributed to the predeformations by adopting shape factor considerations. The carbon black filled rubber is assumed incompressible, displaying a viscoelastic behavior based on a fractional derivative Kelvin Voigt model efficiently reducing the number of required material parameters. In Paper C the focus is on the axial dynamic stiffness modelling of an arbitrary long rubber bushing within the audible frequency range. The problems of simultaneously satisfying the locally non-mixed boundary conditions at the radial and end surfaces are solved by adopting a waveguide approach, using the dispersion relation for axially symmetric waves in thick-walled infinite plates, while fulfilling the radial boundary conditions by mode-matching. The results obtained are successfully compared with simplified models but display discrepancies when increasing the diameter-to-length ratios since the influence of higher order modes and dispersion augments. Paper D develops an effective waveguide model for a pre-compressed cylindrical vibration isolator within the audible frequency domain at arbitrary compressions. The original, mathematically arduous problem of simultaneously modelling the preload and frequency dependence is solved by applying a novel transformation of the prestrained isolator into a globally equivalent homogeneous and isotropic configuration enabling the straightforward application of a waveguide model to satisfy the boundary conditions. The results obtained present good agreement with the non-linear finite element results for a wide frequency range of 20 to 2000 Hz at different preloads. Key words: Rubber isolator, Bush mounting, Dynamic stiffness, Waveguide, Fractional derivatives, Mode-matching, Pre-compressed, Prestrain, Preload, Predeformation, Viscoelasticity, Dispersion

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9 Dissertation The doctoral thesis consists of this summary and four appended papers listed below and referred to as Paper A to Paper D. Leif Kari has acted as a supervisor for this thesis. Paper A M. Coja, L. Kari 2003: Rubber versus steel vibration isolators the audible frequency contest. Submitted to Kautschuk Gummi Kunststoffe. Paper B M. Coja, L. Kari 2003: A simple engineering audible-frequency stiffness model for a preloaded conical rubber isolator. Submitted to Applied Acoustics. A modified version is accepted for presentation at the 4 th European Conference on Constitutive Models for Rubber. Paper C M. Coja, L. Kari 2005: Axial audio-frequency stiffness of a bush mounting the waveguide solution. Submitted to Applied Mathematical Modelling. A modified version is accepted for presentation at the 12 th International Congress on Sound and Vibration. Paper D M. Coja, L. Kari 2004: An effective waveguide model for pre-compressed isolators. Submitted to Acta Acustica united with Acustica. A modified version was accepted for presentation at the 17 th Nordic Seminar on Computational Mechanics and at the Nordic Vibration Research 2004.

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11 Contents 1 Introduction Background Rubber bushing Rubber mounting Dynamic stiffness measurement 5 3 Dynamic stiffness modelling Rubber material dynamic properties Simplified approach Waveguide solution Preload dependence Conclusions and future works 21 Bibliography 23 Paper A Paper B Paper C Paper D

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13 Chapter 1 Introduction The origin of the word rubber is attributed to the British chemist Joseph Priestley ( ) who noticed in 1770 the ability of the gummy substance imported from ancient Mesoamerica to erase or rub out pencil marks from paper. Natural rubber comes from the milky latex sap tapped from the rubber tree and it is believed that Mayan people used it for ritual games and handcrafts as long ago as 1600 BCE. Their word for the material was cahuchu meaning the weeping wood in quechua language. The rubber material was quite sensitive to the weather getting sticky during warm days and brittle when it got chilly making its use rather restrained until the accidental discovery of Charles Goodyear ( ) who dropped some lead and sulfur into hot rubber warming on a stove giving birth to a process known as vulcanization. The improvement of the mechanical and chemical properties of rubber broadcast its industrial production and utilization worldwide. Today the manufacturing process of rubber is extremely developed including many types of ingredients such as synthetic rubbers, carbon black fillers, oils, et c. and the indisputable place of the material in a large number of applications can somehow comfort the tears of its father. 1.1 Background Vibrations produced by industrial or private equipment are usually observed as undesirable disturbances for their environment as well as for the equipment itself. A classic situation consists of an exciting source transmitting structure-borne sound to a receiving structure eventually radiating noise to the vicinity. The attenuation of these unwanted vibrations can be achieved in a passive way either by creating an impedance mismatch at the interfaces or by introducing damping into the system. The former implies an increase in the mass of the system or the integration of a soft component as compared to the rest of the structure which will act as a mirror 1

14 reflecting the audio waves backwards. In most of the applications like for instance vehicle construction an overweigh is synonymous to a drawback as the induced power increase required to drive the system will subsequently lead to an augmentation of the fuel consumption being economically and environmentally unfriendly. Hence the second approach is usually preferable. Rubber components are particularly well suited to fulfill the task, providing an adequate softness as well as relevant mechanical and chemical properties for many applications. Furthermore, the inherent damping of the rubber material contributes to the reduction of structure-borne sound by partly dissipating the parasite motion into thermal energy. In order to control the effect of such devices, an accurate understanding of their dynamic behavior is necessary. In this thesis the modelling of the audible-frequency dynamic stiffness of different rubber vibration isolators using a waveguide approach is assigned. 1.2 Rubber bushing Rubber bushings are commonly utilized in the design of many types of constructions for their capacity to account for large axial and torsional deflections while providing stiffer radial and tilting responses. Their efficiency in attenuating audible frequency noise and vibration transmissions through the structure constitutes an advantage of growing interest for many industrial applications. Typically, a rubber bushing consists of two concentric shafts bonded to a tubular rubber bearing, finding applications in e.g. truck transmission axles, mountain bikes saddle suspensions. Paper C proposes a model of the axial dynamic stiffness of a rubber bushing, illustrated in Figure 1.1, accounting for any arbitrary dimensions and valid over a wide audible frequency range. 1.3 Rubber mounting Cylindrical rubber mountings are also widely used in many isolation applications like ship engine mounts, train bogie suspensions, bridge bearings, et c. They possess advantageous resilient characteristics, essential for the attenuation of undesirable vibrations, and support large axial loadings. A rubber mounting is traditionally composed of a rubber component firmly bonded to rigid metal parts, varying in geometry and types of rubber depending on the application. Paper B and D focus on the modelling of the audible frequency dynamic stiffness of two particular types of cylindrical rubber isolators, including preload and frequency dependence. The first mounting, displayed in Figure 1.2.a is composed of two conical rubber layers enrobing a central metal element and being firmly attached to an inner and outer metal sleeve. The second isolator, illustrated in Figure 1.2.c comprises a rubber cylinder firmly bonded at each end to a metal plate facilitating the connection to the rest of the equipment. 2

15 z r l Inner shaft r 1 Outer shaft z r 2 r ϕ Figure 1.1: Rubber bushing schema. 3

16 b) a) Figure 1.2: Rubber mountings: a) Conical rubber isolator; b) Combined steel-rubber isolator; c) Cylindrical rubber isolator. c) 4

17 Chapter 2 Dynamic stiffness measurement In contrast with the well established dynamic properties of vibration isolators made of steel, those containing rubber components still require more work to acquire a thorough comprehension of their dynamic behavior. When the energy transmission through a resilient mount placed between an exciting source and a receiving structure is focused upon, the dynamic transfer stiffness is an essential quantity to estimate. The difficulty of the task residing essentially in the complexity of the rubber material, the components geometrical shape and the large pre-deformations under working conditions, often leads to a single possible modelling approach, although laborious to develop in some cases: the finite element (FE) analysis. Additionally, the changes in temperature, the fatigue and the oxidation of the material encountered during a lifetime in service can result in important dynamic stiffness alterations cumbrous to model. Already at its manufacturing process, a rubber compound including a wide variety of ingredients can contract anisotropy likely to be produced by molecular orientation and amplified by the adjunction of carbon black fillers, like in the case, for instance, of injection-molded rubber. An alternative to assess the dynamic stiffness of rubber components is by performing measurements on a test object, avoiding thereby the theoretical complications. For that purpose, two techniques are mainly employed, namely the direct and the indirect methods, an international standardized measurement procedure also referring to them. The direct method used by Snowdon (1979), Nadeau and Champoux (2000), consists in directly measuring the force at the blocked termination of the isolator using a force transducer while capturing the displacement with an accelerometer placed at the excited side, the main disadvantage being the rapid limitation in applicable preload since load cells are fragile instruments. In the indirect measurement technique 5

18 utilized by Thompson et al. (1998), Kari (2001), the accelerations at the top and bottom terminations of the isolator are recorded to derive the blocked transfer stiffness by use of Newton s second law applicable for a wide audible frequency range. Paper A Shakers Test object Figure 2.1: Pictures of the measurement test-rig. evaluates the performances of three different types of resilient mount through the indirect measurement of their dynamic transfer stiffness. The experiments are achieved in a specially designed test rig, see Figure 2.1 and 2.2, enabling application of preloads through a hydraulic piston. Two symmetrically positioned electro-dynamic vibration generators excite the distribution plate ensuring a vertical motion while auxiliary isolators reduce flank transmission to the frame. The three vibration isolators are under appraisal to equip the engine suspension system of new competitive European high speed ferries. The first one is a rubber conical mount including a reinforcing central conical steel sleeve, the vulcanized rubber layers being firmly bonded to an inner and outer metal skirt facilitating the connection to other components. The second resilient mount is entirely made of metal consisting of eight identical coil-steel elements filled with damping metallic foam. The last retained isolator combines a central helical steel spring with inclined natural rubber pads bonded to metal elements. The investigation produced over a wide audible frequency range of 200 to 1000 Hz at preloads varying from vanishing to a maximum of 60 kn, clearly reveals a superior vibro-acoustical isolation of the rubber conical mount presenting up to an 6

19 11 db lower mean stiffness magnitude compared with the two other devices. In addition the steel and combined steel-rubber designs show highly resonant comportments over the examined frequency range, as disclosed by the transfer stiffness peaks, deteriorating the isolation, while the rubber mount dynamic stiffness varies much more smoothly with frequency. Besides the strong preload dependence of the rubber isolator dynamic transfer stiffness, resulting in neat frequency shifts of the anti-resonance peak and low-frequency stiffening, the frequency dependence demonstrates a ratio of maximum to minimum stiffness magnitude of approximately 24(!), notably modifying the definition of spring constant. In the light of these experimental observations, the complex behavior recorded of the rubber vibration isolator implies the use of modelling approaches capable of accounting for the non-linearities attributed to the finite deformations caused by preload and include the strong frequency dependence. Charge Amplifiers Frequency Analyzer Amplifier Personal Computer Figure 2.2: Schema of the test-rig and measurement set-up: 1. Test object; 2. Force distribution plate; 3. Blocking mass; 4. Accelerometers; 5. Strain gauge; 6. Electro dynamic vibration generators; 7. Upper auxiliary isolators; 8. Lower auxiliary isolators; 9. Heavy and rigid body; 10. Strong columns; 11. Crosshead. 7

20 Chapter 3 Dynamic stiffness modelling The problem of modelling the dynamic stiffness is decomposed into two assignments: one considering the influence of the isolator geometry, or boundary conditions while the other accounts for the rubber material frequency and prestrain dependent structureborne sound properties. To simplify the task, assumptions of homogeneity and isotropy of the rubber material are made. Although the manufacturing process of the rubber compound can engender anisotropy like for instance by injection-molding, a traditional compression molding allied with a proper blending of the ingredients is likely to produce negligible non-homogeneous and anisotropic characteristics. Furthermore, the rubber material is assumed nearly incompressible, as usually admitted as long as the hydrostatic stress does not become too large, being elastic in dilatation while presenting a linear viscoelasticity in shear deformation. The stiffness dynamic amplitude dependence is disregarded referring to the study by Payne and Whittaker (1971) who reported the absence of low dynamic strain influence on rubber properties in an audible frequency range. Concerning the frequency dependence of rubber material properties, three characteristic zones are distinguished, namely the rubber region, the transition region and the glassy region, where the shear modulus and the loss factor display singularly different tendencies as shown in Figure 3.1 displaying results obtained by Kari et al. (2001). In the present work, the focus on a frequency range spanning in the audible domain, typically 20 to 20 khz, confines the material behavior to the rubber-like zone, while the considered ambient temperature conditions prevent low temperature crystallization and glass hardening to occur. 3.1 Rubber material dynamic properties Within a vibrating system, the decay of the propagating waves and the moderation of the resonance amplitudes disregarding attenuation processes due to discontinuity at 8

21 10 9 a) Loss factor Shear Modulus [N/m 2 ] Shear Modulus [N/m2] Loss factor 10 8 Rubber Region Transition Region Glassy Region Rubber Region Rubber Region Transition Region b) Transition Transition Region Glassy Region Glassy Region Glassy Region Frequency [Hz] Frequency [Hz] Figure 3.1: a) Shear modulus and; b) loss factor versus frequency; Measurement (cross) and fitting function (solid line). structure junctions (Cremer et al., 1988) is attributed to the phenomenon of damping describing a conversion of mechanical energy into thermal energy or heat throughout the material composing the considered medium. For linear materials, a common definition of the damping, in the frequency domain, is through the dynamic modulus, e.g. the loss factor of the shear modulus taking the ratio of the shear modulus imaginary part also interpreted as the loss modulus (Christensen, 1982) over its real part or storage modulus. A simple way, commonly used for engineering materials, accounting for this dissipative attribute is to apply a hysteretic model with a frequency-independent complex dynamic modulus. However this approach fails to fulfill the causality requirement for transient loads as enunciated by Crandall (1970). Another commonly used approach is to apply a viscoelastic Kelvin Voigt model assuming a frequency directly proportional dependence of the loss modulus. Yet, the dynamic behavior of 9

22 rubber materials present substantial deviations with these models, requiring a more adjustable approach to capture its rheological comportment, being likely to find itself between elasticity and viscosity. The usual manner to improve the spring-dashpot model is by including higher order derivatives of the stress and strain yielding a series of spring-dashpot elements also known as the generalized Kelvin Voigt model. Lesieutre and Mingori (1990) propose an augmenting thermal field model later extended to a three-dimensional anelastic displacement model by Lesieutre and Bianchini (1995). McTavish and Hughes (1993) uses a mini-oscillator approach and Dovstam (1995) presents an augmented Hooke s law model. Unfortunately, these approaches augment rapidly and substantially the number of needed parameters to properly match the true tendency of the material. An attractive alternative which gained high interest in recent decades in the description of elastomers stress-strain relation, is the introduction of fractional derivatives into the viscoelastic theory efficiently reducing the amount of material parameters. Since experimental investigations made by Nutting (1921) and Gemant (1936) leading their authors to suggest respectively a fractional power time dependence of the stress relaxation and a dynamic modulus varying with frequency raised to fractional power, numbers of models applying the fractional calculus theory (Oldham and Spanier, 1974) to derive material constitutive laws appeared. Bagley and Torvik proposed a general fractional derivative model using three to five empirical constants to describe the dynamic behavior of elastomers reviewing related contributions in Bagley and Torvik (1983) as well as establishing a theoretical basis of the fractional calculus approach by molecular theory considerations also recorded by Gaylord et al. (1986) comforting the physical fundament of the approach. Koeller (1984) presents an interesting connection between fractional calculus and the theory of Abel s integral equation for material with memory and also proposes an expression of the relaxation and creep process in terms of Mittag Leffler functions. More recently Enelund and Lesieutre (1999) combined fractional calculus with an approach based on anelastic displacement fields to model weak frequency dependence damping characteristics. Pritz (2003) applied a five-parameter fractional derivative model to experimental data from different types of elastomers. Schmidt and Gaul (2002) implement a FE formulation with three-dimensional fractional constitutive equations to perform a time stepping analysis of a viscoelastic structure. Non-linear viscoelasticity focusing on the modelling of the large strain dependence of polymers dynamic modulus is studied by Adolfsson and Enelund (2003), Lion and Kardelky (2004) and Ramrakhyani et al. (2004). In Paper B, a four-parameter fractional derivative model is employed to match the structure-borne sound properties of the vulcanized rubber material containing carbon black fillers assumed linear, non-aging, isotropic, homogeneous, viscoelastic and incompressible. Expressed in the frequency domain the fractional derivative 10

23 Kelvin Voigt model used to describe the dynamic shear modulus takes the form of a simple, first degree polynome ordered to fractional power efficiently capturing the material behavior over the considered audible frequency range. The fractional order being fixed to 1/2 as recommended by Bagley and Torvik (1983) and the static shear modulus being drawn from Tables, the remaining material parameters are optimized in a least square sense to fit the stiffness measurement curve at vanishing preload. The shear modulus and the loss factor obtained with the fractional derivative Kelvin Voigt model are plotted in Figure 3.2 over a frequency range spanning from 200 to 1100 Hz. Magnitude (N/m 2 ) 3 x Loss Factor Frequency (Hz) Figure 3.2: Shear modulus and loss factor versus frequency obtained with a fractional derivative Kelvin-Voigt model. A fractional derivative standard linear solid model embodying a Mittag Leffler relaxation kernel is used in Paper C and D to describe the unfilled, sulfur cured natural rubber material dynamic, viscoelastic shear behavior while the spherical part of the constitutive relation is based on pure elasticity motivating the latter by the near incompressibility of the rubber. The optimized material parameters are drawn from the investigation made by Kari et al. (2001), Figure 3.1 displaying the corresponding shear modulus and loss factor over a wide frequency range. 11

24 3.2 Simplified approach The analytical estimation of the dynamic stiffness can rapidly become an arduous task necessitating to some extent the use of numerical methods like FE. However under judicious and physically plausible assumptions simplified models can approximate the reality with acceptable results. u r π/2 α U Z u z U R a) R 02 w 02 R R 01 w 01 π/2 α α Z r r 01 h 02 z b) h 01 R 2 w 2 R R 1 w 1 π/2 α α Z r r 1 h 2 z c) h 1 Figure 3.3: Schematic drawings of the vibration isolator geometrical configurations: a) Unloaded isolator configuration; b) Simplified unloaded isolator configuration; c) Equivalent volume preserving preloaded isolator configuration. Paper B proposes an efficiently simplified model of the rubber isolator measured in Paper A. The complex geometry of the sandwiched conical mount is simplified by flattening the stress-free curved surfaces of the rubber layers, as indicated in Figure 3.3. In addition, under a finite deformation the idealized rectangular cross sections are assumed to remain rectangular, simply increasing in width and decreasing in 12

25 thickness when submitted to compression. The alteration in dimension subsequent to an arbitrary predeformation is derived by expressing the volume conservation of each rubber layer during the finite geometrical transformation, being motivated by the rubber s incompressible property. A straight-forward one-dimensional dispersive waveguide model accounting for the surface area variation can then be separately applied in the radial and axial directions to each rubber layer. Considering the rubber Magnitude (N/m) Preload 0kN 10 9 Model Measurement Phase (Degree) Magnitude (N/m) Frequency (Hz) Preload 60 kn 10 9 Model Measurement Phase (Degree) Frequency (Hz) Figure 3.4: Transfer stiffness magnitude and phase versus frequency at vanishing and 60 kn preload. Model (solid line) and measurement (dashed). layers as infinitely long strips, the radial and axial layer transfer stiffness are derived before obtaining the layer total axial stiffness by properly expressing the total axial 13

26 displacement and force fields in terms of their radial and axial components. In addition, the influence of the service preload is accounted for by inserting shape factor formulations expressed in terms of the predeformation dependent dimensions of the rubber layers. The Newton s second law provides the global isolator axial frequency dependent stiffness, assuming the metallic conical elements as a rigid body compared to the rubber parts. Finally, a correction factor given in Tables (Gent, 1992) is included to account for the amount of carbon black fillers in the vulcanized rubber. Figure 3.4 displays the good agreement obtained between the proposed model and measurements in the case of vanishing and maximum preloads. 3.3 Waveguide solution The description of wave propagation within a medium of finite dimensions is not a straightforward matter. Although some simple well known cases allow to correctly derive the wave propagation pattern within a given bounded medium, practical problems usually cause difficulties hindering a completely theoretical treatment. Regarding the modelling of isolator dynamic stiffness, numbers of simplified theoretical approaches have been extensively used as in the case of cylindrical geometries, the long rod model assuming lateral, plane cross sections remain plane and lateral as well as stress is distributed uni-axially and uniformly over the section, while neglecting radial expansion and contraction. The Love model extends the long rod including the effects of radial displacement, however those simplified models produce too inaccurate results as presented by Kari (2003) especially in more realistic applications where the width-to-length ratio of the cylindrical isolator is increased. The waveguide solution approach presented in Paper C and D, proposes to simultaneously satisfy the isolator boundary conditions by adopting a subregion matching mode technique to fulfill the conditions at the bonded surfaces using the dispersion relation of the infinite cylinder, in D, and that of the thick-walled infinite plate, in C, applying traction free conditions at the remaining boundaries. In this way, the influence of higher order modes and dispersion phenomena can be thoroughly investigated. In addition it can be noticed that the application of locally mixed boundary conditions is usually more favorable as the locally non-mixed conditions tend to obstruct the resolution of the modelling, grouping it into a non-separable category as indicated by Miklowitz (1984). In Paper D a waveguide approach is proposed to model the preload and frequency dependence of the dynamic axial stiffness for a pre-compressed cylindrical vibration isolator made of rubber. The well known Pochhammer Chree dispersion relation, (Pochhammer, 1876; Chree, 1889) for an infinitely long circular cylinder is used to derive the axial wave numbers, satisfying the traction free condition on the radial surface. Paper C provides a new formulation of the obtention of the dispersion 14

27 relation for the axially symmetric non-torsional waves in a tube happening to be in its geometrical limits, the tube being asymptotically extended to a thick-walled plate, the same relation as the Rayleigh Lamb frequency equation for straight-crested waves in a plate. Since these dispersion relations were derived, lots of related works were published but a problem was persisting: the number of real and imaginary wave numbers was finite hindering arbitary conditions to be set at the boundary. Adem (1954) was the first to draw the conclusion that an infinite set of wave numbers was in fact achievable when including the complex solutions. The cumbersome task of Frequency (Hz) Imaginary wave number (1/m) Real wave number (1/m) 400 Figure 3.5: Three dimensional frequency spectrum from Paper C. calculating the wave numbers is done by using a Newton Raphson method where the initial values are obtained by a winding integral method using an implemented version of the algorithm proposed by Invansson and Karasalo (1993). This evaluation is based on the argument principle stating that 2π times the difference between the zeros and the poles of a function is given by the total variation of its argument around a close single path surrounding a complex domain. The dispersion relations discussed above are however not ideally suited for this treatment since the presence of the square root operators leads to branch points, namely the longitudinal and the transversal wave 15

28 numbers. A modification is hence made by partitioning the domain into adapted branch cut free sub-domains. A three dimensional plot derived in Paper C for an infinite rubber plate of L = 50 mm; displaying the imaginary and real parts of the first 16 radial wave numbers versus a frequency range of 20 to 2000 Hz is shown in Figure 3.5. The fulfillment of the boundary conditions of the different test objects is done by applying a mode-matching technique dividing the whole domain into subregions where integrations of the reduced boundary condition modal series can be performed. This approach enables to include the desired order of modes and improves the convergence of the results by over-determining the matrix system employing more equations than unknown modal coefficients. In the case of the cylindrical isolator studied in Paper D, a convergent prediction of the dynamic stiffness is obtained when a number of modes superior to 64 is included and a single-fold or three-fold over-determined equation system is used. Figure 3.6 displays the convergence of the transfer and driving Magnitude (N/m) Transfer stiffness 50 mm isolator 10 7 three fold one fold exactly 10 6 Driving point stiffness Magnitude (N/m) Frequency (Hz) Figure 3.6: Transfer and driving point stiffness magnitude versus frequency for H = 50 mm isolator at maximum pre-compression u 0 = 5 mm. Exactly determined (P = 64), single-fold (P = 128) and three-fold (P = 256) over-determined equation system with number of modes N = 128. point axial dynamic stiffness with respect to the degree of over-determination of the equation system for a cylindrical vibration isolator of diameter D = 100 mm and height H = 50 mm when the number of modes is set to 128. The sensitivity of the model to the number of modes is presented in Figure 3.7 where the transfer 16

29 Transfer stiffness 25 mm isolator Magnitude (N/m) modes 64 modes 32 modes 16 modes Driving point stiffness Magnitude (N/m) Frequency (Hz) Figure 3.7: Transfer and driving point stiffness magnitude versus frequency for H = 25 mm isolator at maximum pre-compression u 0 = 10 mm. Number of modes N = 16, 32, 64, 128, using a three-fold over-determined equation system (P = 2N). and driving point axial dynamic stiffness for an isolator of diameter D = 100 mm and height H = 25 mm is plotted, fixing the over-determination to three-fold and increasing the amount of modes. Physically these high-order modes contribute to the fulfillment of the displacement and stress boundary conditions at the cylinder ends. In the case of the rubber bushing modelled in Paper C a fewer number of modes is required to obtained a converging solution. Indeed, a relatively small number of higher order modes (N = 16) is sufficient to satisfy the boundary conditions at the inner and outer radial surfaces. 3.4 Preload dependence Under service conditions a vibration isolator is usually submitted to finite pre-straining being the result from the application of the vibrating source s weight. In Paper B and D the preload dependence of the audible frequency dynamic stiffness is studied, investigating the influence of several arbitrary static preloads on the isolator dynamic behavior. The imposition of a static load modifies the original stress-free configuration into a previously unknown shape rendering uncertain the application of the boundary conditions. The complex pre-strained configuration, clearly distinct from the reference configuration at rest, is obtained through a finite transformation 17

30 h 0 d 0 Top edge surface h 0 Bottom edge surface D Figure 3.8: Globally equivalent prestrained configuration. involving geometrical non-linearities. In addition, the rubber isolator response to a superimposed dynamic excitation displays seemingly anisotropic and non-homogeneous features in the finitely transformed state interpretable as material non-linearities. A suitable way to treat the problem is by using the single integral viscoelastic constitutive theories as presented by Zdunek (1992), Kari (2003). However, these approaches usually involving complex mathematical difficulties can render their applications arduous and laborious. Moreover, the use of non-linear finite element method packages does not generally facilitate the physical interpretations of the results as no-closed form is achieved. In Paper B a simplified approach is proposed to model the audible dynamic preload dependence of a rubber conical mount. Once approximating the geometrically complex rubber layers by equivalent infinite strips of rectangular cross-section, the rubber elements are assumed to respond in an incompressible manner to finite predeformation, the cross-section conserving their rectangular forms. Subsequently, shape factors, defined as in Lindley (1992) as the ratio of the loaded area to the free area, are derived and incorporated in a closed-form expression of the preload dependent dynamic stiffness. Despite the good agreement obtained with the measurements performed on the isolator, validating the method as a powerful tool for engineering applications, some refinements could be sought to ameliorate the rubber 18

31 layer waveguide model. Developing a similar approach to create a globally equivalent homogeneous and isotropic prestrained configuration by use of the rubber incompressibility, as illustrated in Figure 3.8, Paper D enlightens the influence of the boundary conditions applied on the lateral end surfaces of the cylindrical isolator, the model being based on an exact waveguide solution as described in Section 3.3. To that purpose four different types of boundary conditions are envisaged for the top and bottom edge surfaces (see Figure 3.8), namely the free where the shear and axial stress vanish, the fixed that is imposing an infinitesimal axial displacement and vanishing radial displacement on the top surface while restraining the axial and radial displacements at the bottom, the slip imposing an infinitesimal axial displacement and a vanishing shear stress at the top surface, while the shear and axial stress vanish at the bottom surface, and finally the no-edge assuming a preserved radius of the isolator during the precompression. Not surprisingly the fixed and slip boundary conditions, constraining more the motion, result in an over-estimated low-frequency stiffness magnitude while the no-edge leads to a softer isolator, under-estimating the static stiffness magnitude. Clearly, the best prediction is attained with the free condition. Indeed, the proposed Transfer stiffness 25 mm isolator Magnitude (N/m) Phase (degrees) mm waveguide 5 mm FEM 0 mm waveguide 0 mm FEM Frequency (Hz) Figure 3.9: Transfer stiffness magnitude and phase versus frequency for H = 25 mm isolator at vanishing and maximum pre-compressions. Effective waveguide and nonlinear finite element solution. method results in an excellent agreement with the correct non-linear FE solution over the whole examined frequency range of 20 to 2000 Hz at each considered preload, enabling, in addition, a study of higher order modes influence and dispersion aspects in the propagating media while remaining relatively simple and physically intuitive in its development and implementation. Considering a cylindrical isolator of diameter 19

32 D = 100 mm and height H = 25 mm, Figure 3.9 presents the results obtained with the proposed waveguide approach, being compared to those computed with FE for pre-compressions of 0 and 5 mm. In Figure 3.10, the pre-compression dependence of the transfer stiffness magnitude and phase versus frequency is indicated. Transfer stiffness 25 mm isolator Magnitude (N/m) Phase (degrees) Frequency (Hz) Figure 3.10: Transfer stiffness magnitude and phase versus frequency for H = 25 mm isolator at pre-compressions u 0 = 0, 1, 2, 3, 4, 5 mm. Direction for increased compression is marked. 20

33 Chapter 4 Conclusions and future works The main contributions proposed in the present work can be summarized as follows: The measurements of the audible axial dynamic transfer stiffness performed on three different types of engine mounts indicate a clearly superior isolating behavior of the rubber conical mounting compared with the steel coiled-spring system and the combined steel-rubber mount. In addition to be a performant comparison tool, the indirect measurement technique employed here also provides an accurate estimate of the dynamic stiffness extremely useful to check the validity of analytical models. A simple audible-frequency stiffness model for a preloaded conical rubber isolator agrees very well with the measurement results performed in a specially designed test rig. The complex mathematical difficulties mainly attributed to the complicated geometrical shape of the isolator, the rubber structure-borne sound properties and the non-linearities introduced by the finite precompression are successfully assigned by using a series of relevant approximations. An axial, dynamic stiffness model of an arbitrary long rubber bush mounting, being based on a waveguide approach, is developed within the audible frequency range. The rubber is assumed nearly incompressible, presenting dilatation elasticity and deviatoric viscoelasticity based on a fractional derivative, standard linear solid embodying Mittag Leffler relaxation kernel, thereby efficiently reducing the required number of material parameters. It is shown that only a few number of higher order modes (N = 16) is needed to fulfill the radial boundary conditions and yields a convergent solution. A novel, effective waveguide model for pre-compressed vibration isolators within an audible frequency range is presented displaying excellent agreement with 21

34 the non-linear FE solution. The originally complex, mathematical problem is simplified by transforming the pre-compressed isolator into a globally equivalent homogeneous and isotropic form, thereby enabling a waveguide model to be applied, where the boundary conditions are satisfied by mode-matching. The present study is believed to enlighten several new aspects of analytical modelling and experimental investigation of rubber vibration isolators. However, some interesting extensions of the work are worth mention as: Enhancing the rubber layer element model in Paper B. Including temperature and dynamic amplitude dependence both experimentally and analytically in the approach. Applying the waveguide modelling approach to other isolator s geometries like hollow cylindrical or rectangular shapes. Implementing the proposed models to designs made of magneto-rheological rubber material finding more applications. 22

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