M I C H E L L N Y L U N D S I Z A R S H A M O O N

Transcription

1 M I C H E L L N Y L U N D S I Z A R S H A M O O N 4B1015 Examensarbete inom teknisk akustik Örebro 011

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3 Marcus Wallenberg Laboratoriet för Ljud- och Vibrationsforskning Master thesis on factors affecting the dynamic properties of rubber isolators Michell Nylund and Sizar Shamoon Godkännes Datum: Ansvarig handledare: ISP B1015 Examensarbete inom teknisk akustik

4 Abstract Rubber is a highly complicated material. Its properties are strongly dependent of temperature, preload, frequency, amplitude and filling materials. In order to have better knowledge about how rubber reacts for these different parameters, a number of different papers that treats these parameters have been reviewed. Main focus is on dynamic properties of carbon-black filled rubber. Mechanical models of rubber isolator with different kinds of possible solution have been suggested. A basic mathematical equation is derived to estimate the static and dynamic stiffness of a rubber isolator depending on the preload for a geometry that is commonly used by Atlas Copco Rock Drills AB. Tests of a rubber isolator with the same geometry is also conducted to validate the model. It is shown that the model corresponds very well to measurement data. i

5 Table of content Introduction... vii Part I: Literary Study Rubber properties Chemical properties of rubber Mechanical properties of rubber Frequency Temperature Amplitude Preload Natural frequency of rubber isolator Geometry dependence... 7 Rubber Modeling Modeling components Elastic part Viscous part Frictional part Existing models Generalized Maxwell model Implementation of the generalized Maxwell model Zener model Three way Maxwell model Implementation of the three way Maxwell model Elastic force model Implementation of the elastic force model Summary literary study... 1 Part II: Experimental Part... 1 Introduction... 3 Isolators and geometry Method Experimental Theory Dynamic measurement Uncertainty estimation of dynamic measurement Static measurement Uncertainty estimation of static measurement ii

6 3. Mathematical Isolator Stiffness with no preload Isolator Stiffness with preload Result and discussion Evaluation of mathematical model Conclusion Appendix A... A-A Appendix B... B-A Appendix C... C-A Appendix D... D-A iii

7 Introduction: D IL F Notation [db] Insertion loss F dist [N] Disturbing force after L F [db] Vibration level after isolation L before F [db] Vibration level before isolation c [Ns/m] Damping k [N/m] Spring stiffness m [kg] Mass u [m] Displacement u [m/s] Velocity u [m/s ] Acceleration Part I: A [m ] Cross sectional area A B [m ] Bulge area A L [m ] Load area C c [Ns/m] Critical damping level E [N/m ] Elasticity modulus E 0 [N/m ] E-modulus, Young s modulus E [N/m ] Bulk modulus E c [N/m ] Effective compression modulus E t [N/m ] Spring rate in tension F [N] Force F c [N] Force applied in compression direction F s [N] Force applied in shear direction G [N/m ] Shear modulus G [N/m ] Long term shear modulus K [N/m] Spring rate K c [N/m] Spring rate in compression K s [N/m] Spring rate in shear I b [kg m ] Moment of inertia of the cross sectional area M [kg] Mass of supporting load M f [kg] Total mass of vibration isolator S [-] Shape factor T [m] Thickness T ABS [-] Transmissibility Y [N/m ] Maximal frictional stress c [Ns/m] Damping d c [m] Compression displacement d s [m] Shear displacement f n [Hz] Natural frequency for undamped system f nd [Hz] Natural frequency for damped system l [m] Length r [-] Ratio of exciting vibration- to the system natural-frequency r g [m] Radius gyration of the cross-sectional area about neutral axis of bending t [s] Time iv

8 t r [s] Relaxation time u [m] Displacement u [m/s] Velocity β [-] Loss factor in vibration isolator ε [-] Strain δ [-] Damping factor ε [-] Loss factor κ [-] Shear strain κ y [-] Yield shear strain μ [-] Intensity of resonance caused by standing waves ζ [N/m ] Stress η y [-] Yield shear strain η [N/m ] Shear stress θ [-] Elastomers compression coefficient ω 0 [Hz] Fundamental frequency Part II: A [m ] Loaded area F [N] Force F screw [N] Force from the screw joint F load [N] Force from the applied load G [N/m ] Shear moduli H [m] Hight of undeformed isolator K [N/m] Stiffness K dyn [N/m] Dynamic stiffness of isolator, theory K dynmeas [N/m] Dynamic stiffness of isolator, measurement data K st [N/m] Static stiffness of isolator, theory K stmeas [N/m] Static stiffness of isolator, measurement data L kav [db] Level of frequency-band-averaged dynamic transfer stiffness R y [m] Outer radius of isolator, loaded area R i [m] Inner radius of isolator, average inner radius S [-] Shape factor h [m] Hight of deformed isolator Δl [m] Length difference u [m] Deformation u 0 [m] A given deformation u screw [m] Deformation from screw joint u over [m] Deformation on the isolator mounted over the support plate u load [m] Deformation on the isolator mounted under the support plate δ ins [db] Uncertainty caused by instrumentation δ rep [db] Uncertainty caused by installation repeatability of test element δ rig [db] Uncertainty caused by the test rig δ lin [db] Uncertainty caused by approximate linear behavior ζ ins [%] Estimate of standard deviation, instrumentation ζ meth [%] Estimate of standard deviation, measuring method ζ tot [%] Estimate of standard deviation, totally χ [-] Dynamic factor v

9 Appendix: C ij [] Represent linear dependence I i [] Strain invariant W [-] Energy strain function α i [N/m ] Material constant n i [-] Principal eigen vector tr(b) [-] Trace of (B) λ i [-] Principal eigen value μ i [N/m ] Shear modulus vi

10 Introduction When mounting a vibrating device on a structure it is desired to isolate the vibrations from spreading on to the structure for different reasons. One of these reasons is that if vibrations are allowed to spread the structure will emit sound to the environment. Examples of this are shown for several application used in industries over the world where engines (vibrating devices) are mounted on chassis of different kind, see FigA 1. FigA 1 Engine mounted on a chassis with vibration isolators The most common components used to isolate the vibrations are called rubber or vibration isolators. They should not only be able to carry the load of the component which is to be isolated but also reflect and dissipate as much of the vibrations as possible. A suitable measure of vibration isolation is the insertion loss D F IL before F after F L L, [db] EqA 1 where L F before and L F after is the force level of a vibrating device acting on a foundation before and after vibration isolation is added, see FigA. vii

11 FigA Vibrating device mounted on foundation before and after isolation If the mechanical system is idealized as presented in FigA 4, the equation of motion from solid mechanics is mu cu ku, EqA F dist where m is the mass, c a viscous constant, u the displacement and k is the stiffness. They are shown in FigA 3. FigA 4 A simplified mechanical system of the dynamic system in FigA Knowledge about the spring stiffness and damping will provide information needed to obtain good vibration isolation. But rubber is an advanced material where stiffness and damping depend on frequency, amplitude, temperature and preload. The basis for this thesis is to explore the dynamic properties of rubber isolators. It is basically divided into two parts. The first part is a literary study about how rubber isolator s dynamic properties depend on different factors to give a better insight on the material. Later some basic engineering formulas and models of rubber are presented. The second part is based on experimental measurements which are used to evaluate a mathematical rubber isolator model. viii

12 Part I: Literary Study 1

13 1 Rubber properties Rubber is a special material with various features. 1.1 Chemical properties of rubber Rubber is a material which is widely used in engineering applications because of its unique features, and is the most common material for vibration and shock isolation between stiff structures and mounting joints. Rubber is a denotation used for a big group of materials with different chemical compositions but similar molecular structure and mechanical properties. Due to the high elasticity features, rubber is often called elastomers. Elastomers are polymers with very long molecular chains. They are made of a wide variety of organic substances for example from latex, which is the sap of a tropical rubber tree, especially Hevea brasiliensis [1]. It coagulates in thin sheets and bundles into bales. The important process of vulcanization, which was discovered by Charles Goodier (Charles Goodyear) in 1839, converts the plastic raw elastomeric material into a solid and elastic consistency. Vulcanization is a chemical process where the long molecular chains are linked together and thereby from a stable and more solid molecular structure. There are many kinds of rubber mixtures. The cross linking is enabled by a small amount of sulfur that is mixed with raw material. When the mixture is heated to about 170 C the vulcanization process starts and cross links are formed, connecting the molecular chains, see Fig 1-1. Fig 1-1 Microstructure of a Carbon-black filled rubber vulcanized Gray circles: Carbon particles. Solid lines: Polymer changes. Dash and zigzag-lines: Cross links []

14 Natural rubber used in some specific engineering applications needs to be reinforced by addition of carbon-black filling (very small particles of carbon 0μm-50 μm). These particles are added to natural rubber before vulcanization. The carbon-black filling increases the stiffness of the material and its wear resistance. The rubber phase forms a constituent network, and the filler material forms agglomerates inside the rubber network. The material is thus a two-phase material made from constituents with completely different mechanical properties []. 1. Mechanical properties of rubber The two basic and most discussed features of rubber are stiffness (modulus) and the ability to dissipate energy (loss factor). The loss factor is a quantity that measures the total mechanical energy losses in the rubber isolator. These losses are coupled to frictional and viscous effects that are generated in the rubber isolator when it is subjected to vibrating loads. Modulus and loss factor are dependent on frequency, amplitude, preload and temperature Frequency At low frequencies rubber is soft and gets stiffer as the frequency increases. The loss factor has its peak value where the change of the stiffness is at its maximum, see Fig 1-. A jump in the modulus can be witnessed when going from static to quasi-static frequencies. For filled rubber the phenomenon is more prominent [3]. Fig 1- Stiffness and loss factor dependence on frequency [4] 3

15 1.. Temperature At low temperature rubber is stiff and as the temperature rises it goes through a transition region before it reaches the area where the material acts like rubber. As with the frequency dependence the loss factor is at its highest when the absolute change of the stiffness is maximum, see Fig Amplitude Fig 1-3 Stiffness and loss factor dependent on temperature [4] The Payne effect is an amplitude dependence which is more pronounced for filled rubber isolators. When the excitation amplitude increases the dynamic modulus decreases. Both loss angle and amplitude are presented in Fig 1-4 [3, 4, 5]. The Mullin s effect is also a phenomenon coupled to the amplitude dependence, it is an effect that occurs when rubber is subjected to an initial load. There is a reduction in the peak stress values after the first oscillations. This is believed to be the result of rearrangements in the filler molecules. Mullin s effect is more pronounced in rubber with carbon-black filling [6, 4, 7, 8]. Fig 1-4 Stiffness and loss angle dependent on amplitude, Payne effect [4] 4

16 1..4 Preload For most rubber components the dynamic stiffness within the quasi-static frequency range gets higher as the preload increases. An example of a rubber isolator subjected to three different load cases is shown in Fig 1-5. At frequencies above 300Hz one can observe a change in the curve shape with applied preload. Fig 1-5 Measured results (b) transfer stiffness magnitude (c) and phase angle as a function of frequency and different preloads. 0N (solid thin), 100N (dashed line) and 3350N (dotted line) [9] 1.3 Natural frequency of rubber isolator At high frequencies there are resonances when the dimensions of the rubber isolator overlap with multiples of a half-wavelength. There are some simple assumptions made to roughly estimate these resonance frequencies. Ref. [10] assumes a rubber isolator as a mass less spring and plots the transmissibility vs. frequency from a cylindrical rubber isolator. It is found that the fundamental natural frequency is f n =30Hz and the first high-frequency peak is observed at f n =500Hz. The first natural frequency corresponds well to calculations but the assumption does not hold from the first high-frequency resonance. Another assumption is that the isolator can be designed as a cylindrical rod with uniformly distributed mass. The high-frequency wave resonance for the isolator occurs at i, i 1,,3..., Eq 1-1 i 0 where ω 0 is the fundamental natural frequency and γ is M / M f M / Al, Eq 1- where M is the mass of the object supported by the isolator and M f is the total mass of the isolation element, ρ is its density, A cross sectional area and l is its length [10]. It is also possible to calculate the high resonance frequency intensity with ' 0. M 0. f tan, Eq 1-3 M tan 5

17 where β is the loss factor of the vibration isolator. It is obvious from previous formulas that both the positioning and the intensity of high resonance peaks is depending on the ratio of the mass for the isolator and mass supported by the isolator. The conclusion that the high-frequency resonance peaks will shift for different loads can be made. The natural frequency of an undamped spring-mass system is 1 K f n, Eq 1-4 M where K is the spring rate of the carrying component. One can also derive a corresponding equation for the damped system. The derivation of this equation is based on the equation of motion for the undamped system. The solution is based on that the transmissibility. The transmissibility depends on both system characteristics and the vibrations see T ABS 1 (1 r ), Eq 1-5 where r is the ratio between the excitation frequency and the system natural frequency and f r. Eq 1-6 f n The natural frequency for almost all real systems varies slightly with amount of damping in the system. This is the reason for introducing a variable ζ which is the damping factor. The damping factor is approximately half of loss factor /. Eq 1-7 The natural frequency for a damped system is f nd fn 1. Eq 1-8 It is obvious from the equation that the natural frequency for the damped system is related to the undamped system. The damping factor is C, Eq 1-9 C c where the critical damping level C c for the damped vibratory system is Cc KM, Eq 1-10 and the equation for absolute transmissibility of a damped system is 6

18 T ABS (r) 1 (r) 1 r. Eq 1-11 Fig 1-6 Transmissibility for a damped and undamped system [11] The Fig 1-6 above shows the transmissibility for a damped and undamped system. It can be recognized from the figure that the damping reduces the transmissibility in regions around r=1. It can also be noticed that the damping reduces the amount of protection in region where r> [11]. In real life, the isolator systems do not follow the above named model perfectly due to some non-linearity, especially for elastomers which are very sensitive to vibration level, frequency, temperature and more. 1.4 Geometry dependence As mentioned before the mass/spring system is the most basic model when finding the fundamental frequencies in rubber isolators. The key design parameter for this model is K, which define the spring stiffness (sometimes called spring rate). K is F K, Eq 1-1 d where F is the applied force and d is the deflection. K varies with load deflection (shear, compression, tension) AG Ks, Shear stiffness Eq 1-13 T AEc Kc Compression stiffness Eq 1-14 T 7

19 and AE t K t, Tension stiffness Eq 1-15 T where A L is effective load area, T is the thickness of the undeformed elastomer. The parameters G, E c and E t stand for the shear, compression and tension modules of the elastomer. For the shear load Eq 1-13 could be used for simple shear calculations of sandwich models (rubber bonded with metal). The equation is only valid when deformation due to bending is not taken into account. When the ratio of length to thickness of the element exceeds 0.5, the argument about bending shear is not valid anymore and the stiffness could be calculated according to F s AG 1 K s, Eq 1-16 d s T 1 T /36rg where r, Eq / g ( I b / A) is the radius of gyration of the cross-sectional area about the neutral axis of bending, and I b is the moment of inertia of the cross-sectional area A about the neutral axis of bending [7]. For the compression load case named above, K c is calculated according to Eq To get a reliable result for this calculation, E c (effective compression modulus) must be known. E c is a function of both material properties and component geometry. To calculate the spring rate in compression K c, there are different analytical techniques used. For instance, E c for a simple sandwich block could be calculated as E or c E (1 S ) For a bidirectional strain Eq E c 1.33E0 (1 S ), For one-dimensional strain Eq 1-19 (long, thin compression strip) where E 0 is young s modulus θ is elastomers compression coefficient (see TableC 1 in Appendix C) and S is the shape factor A A L S, Eq 1-0 B where A L is the load area and A B is the bulge area. 8

20 The formulas named above can be used to estimate the spring rate roughly. For more accurate result, some more advanced analytical techniques or finite element analysis are required. Rubber Modeling Many different rubber models have been developed with the approach of determining attributes of interest e.g. dynamic stiffness, damping and loss factor. The models are built up of almost the same elements, viscous dampers, elastic and friction elements which are coupled together in different ways. An example is the common three way Maxwell model seen in Fig -1. Fig -1 Rubber model existing of elastic, viscous and frictional elements Not all models can give a good physical representation. Some models are better at representing amplitude dependence and some models are better at predicting the frequency dependence. There are other variables that affect the properties of rubber namely temperature, filling type and preload. It is difficult to include all these parameter in one single model. Consequently the researchers often try to test one single parameter at a time..1 Modeling components The different elements that build up some of the basic models used in scientific literature will now be presented in a short introduction. Physical representation and derivation of the fundamental equations that describes the model components will be presented as well. 9

21 .1.1 Elastic part The single linear spring is one of the basic elements that a model could consist of, see Fig -. Fig - Basic elements for rheological models [5] It represents the linear elastic part of the model. The linear elastic part is not capable to describe a complete model. There are some models which are more complicated and include some non-linearity by using non-linear components, see Fig -3. And the force depends non-linear on the displacement..1. Viscous part Fig -3 Non-linear elastic element [4] The viscous part can also be modeled by various elements. One possible model is a single linear dashpot, see Fig -4. Fig -4 Basic dash-pot element for rheological models [5] The force over the dashpot can be written as F v c u Eq -1 and it is proportional to the velocity. The dashpot element causes a time delay when cyclic load is applied i.e. the force and displacement will be out of phase. If a plot is made with force versus displacement a hysteresis loop is displayed, see Fig -5. This is obviously valid for all dynamic systems [8]. 10

22 Fig -5 Force and displacement [4] The dynamic behavior of a viscoelastic material is analogical with a corresponding spring, dashpot combination. The rheological models represent the elastic behavior of spring and viscous behavior of the dashpot. Another possible model of viscous element called spring-pot is presented in Fig -6. Fig -6 The spring-pot element [4] The spring-pot model treats the constitutive equation with fractional time derivatives, while the dash-pot model uses the standard integer order derivative. 11

23 .1.3 Frictional part The frictional part is usually represented by two different elements. An element with smooth characteristic is presented as a frictional part shown in Fig -7. Fig -7 Smooth frictional component [4] This element takes account for the amplitude dependence (Payne effect) during the simulation[4]. Another element is the frictional solid element. It consists of a spring and frictional blocks see Fig -8. Fig -8 Frictional solid element [5] An advantage of modeling with smooth frictional component is that you need only one element, which gives two parameters to determine to get a good representation. Not like the frictional solid element shown in Fig -8 where two or three elements are needed which gives four to six parameters for good model fitting to experimental data.. Existing models Rubber has been used in the industry for over a hundred years. Many models have been developed during this time to describe the characteristic. From the beginning only simple models which consisted of dashpots and springs built up different structures that could simulate the behavior of rubber e.g. Kelvin-voigt model, which is a dashpot in parallel with a spring but it does not give a realistic physical behavior []...1 Generalized Maxwell model The Maxwell element is shown in Fig -9. Fig -9 Maxwell element [5] The generalized Maxwell model contains a spring which is parallel with Maxwell elements, see Fig

24 Fig -10 The generalized Maxwell model [5] The accuracy of the model depends on the amount of Maxwell elements. The total stress... 1 n, Eq - where n denotes the stress over n:th element. The Maxwell elements are considered as viscoelastic fluid model. The relaxation behavior can be generalized as E r (t) which is a time dependent fundamental function that describes the behavior of viscoelastic material. For a single Maxwell element with a spring and dashpot serial coupled, the requirement is that the total strain is dashpot spring. Eq -3 The time derivative are dashpot spring, Eq -4 dashpot Eq -5 and spring. Eq -6 E 13

25 This gives E E, Eq -7 which is the differential equation that describes the strain-stress relation for a Maxwell model. The normalized relaxation behavior derives by solving this equation for a step strain. When t>0 the time derivative for strain is 0, inserting this in Eq -7 gives E 0 at t > 0. Eq -8 When the step strain is applied, the dashpot acts like a rigid part due to the infinity strain at t=0, see Fig -11. The initial stress is given only by the elastic spring and the initial condition will be ζ(0)=eε 0. Using this to solve Eq -7 yields E t ( t) E e. Eq -9 0 The relaxation modulus for the model can now be defined as E R ( t) t tr Ee, Eq -10 where the relaxation time is t r =ε/e. The relaxation behavior of this element is illustrated in Fig -11. Fig -11 Maxwell model s relaxation behavior [5] 14

26 ..1.1 Implementation of the generalized Maxwell model The generalized Maxwell model gives a good representation of the frequency dependence of rubber components. The representation gets better with a growing amount of Maxwell elements implemented. However this results in a larger amount of parameters to be determined. In FE-codes the viscoelastic part are modeled with different strain energy functions e.g. Neo-Hookean or Mooney-Rivlin. All the physical properties have to be determined by experimental measurements. A step by step method of implementing frequency dependence and the amplitude dependence in commercial FE-code is presented in ref. [3]. It is done in two ways. The first way is that a single generalized Maxwell model is used to model the whole rubber isolator. The second way is to let several generalized Maxwell models build up the isolator representing different parts of the component depending on different strain levels, this way of modeling is proven to give a better representation but is harder to implement. The step by step method is schematically showed in Fig -1. If amplitude dependence is ignored (e.g. non-filled or soft rubber) the shear and bulk modules are extracted from experimental measurements and is inserted in the FE-code, the dynamic properties can be obtained from applying a dynamic step. If amplitude dependence is included (e.g. filled or hard rubber) the calculations get more complicated as an equivalent strain needs to be calculated from quasi-static measurements at the specific amplitude range of interest. Fig -1 Practical procedure of the simplified methodology [3] 15

27 .. Zener model Zener model is very similar to the generalized Maxwell. It consists of only two elements, one single spring and one Maxwell element which are parallel coupled, see Fig -13. Fig -13 Zener model [5] The Zener model is a simple viscoelastic model which acts like a broad frequency range. It has reasonably good relaxation behavior but limited creep behavior. If we now look at the stress in the different elements in this model and call the stress over spring to ζ and the stress over the Maxwell element to ζ M, the total stress shown in Fig -13 will be. Eq -11 M Rewriting this equation gives M Eq -1 and the time derivative is M. Eq -13 Inserting this relation to the differential equation derived for the Maxwell model follows in E, Eq -14 E E E Eq -15 and E EE ( E E ). Eq -16 The last equation is the differential equation for the Zener model. It defines the relations between stress and strain. This equation could be solved for a strain step history. For further details please check ref. [5]. The relaxation behavior of a Zener model is illustrated in Fig

28 ..3 Three way Maxwell model Fig -14 Relaxation behavior of Zener model [5] The three way Maxwell model is also called the generalized viscoelastic elastoplastic model, see Fig -15. Fig -15 Generalized viscoelastic elastoplastic model [5] There are two quite similar options of this model. The first option can be observed in Fig -15, where the total stress is. Eq -17 e v f The second option has an elastic part consisting of a non linear spring. The first term in Eq -17 is different in the non-linear viscoelastic elastoplastic model. The total stress in this model is E (. Eq-18 e ) v f The previous expression describes the stress for only three elements. The total stress in the whole model can be expressed as v f i Ee ( ) i i i, Eq

29 where i=1,, 3 and subscripts v and f denotes the viscoelastic elements and frictional elements Implementation of the three way Maxwell model Ref. [6] has a good suggestion for implementing such a model in ABAQUS. The model is shown in Fig -16. Fig -16 One dimensional generalized viscoelastic elastoplastic model [6] As mentioned before in this case shear quantities are used M i1 N e ve ep e ve ep, Eq -0 i j1 j where is the shear stress and the subscripts e, ve and ep stand for elastic, viscoelastic and elastoplastic elements. The total stress can be obtained of the summation of the distribution of single elements in the model. The amplitude dependence and the rate dependence can be considered to be independent. It gives rather good results for some kind of rubber materials, especially rubber with filling. For the three dimensional model the total stress is achieved for the total distribution of single stresses. For the finite element model the summation is obtained by an overlay of finite element meshes, see Fig

30 Fig -17 Overlay method Finite element meshes [6] The main idea of the finite element model is to achieve the total stress tensor from the parallel contribution of finite element meshes. This method is called the Overlay method and the total stress tensor is obtained from the separate finite elements which is included in the model. The modeling of the elastic and viscoelastic parts is obtained by implementing a hyperelastic model (Yeoh or Neo-Hooke), which is already existing in ABAQUS. The elastoplastic part however could not be modeled directly because of the lack of a piece wise kinematic hardening model in ABAQUS/explicit, instead an elastoplastic model based on a hypoelastic description was used. The stress summation is achieved by assembling each layer of elements into one set of nodes. This approach yields a model able to represent the combined rate and amplitude dependence. The main drawback with this solution is the amount of parameters to decide. To avoid this problem a general step by step method is suggested to extract the model parameters using an error function and the steps are presented bellow. Firstly a rough guess of the material parameters is made. Both yield strains and relaxation times are given a logarithmic distribution over the measured amplitudes and frequencies respectively. Secondly, the shear moduli of the elastic and the elastoplastic elements are fitted to only the lowest frequency for which the influence of the viscoelastic elements may be neglected. The yield strains for the elastoplastic contributions are unchanged at this stage. Thirdly is to fit all of the shear moduli to all test data. After this step the model should be fairly accurate. Fourthly is to fit all material parameters to all test data, resulting in a minor adjustment of the material model. These four steps together with an analytical approach give material parameters that correspond well with measurement results. 19

31 ..4 Elastic force model Fig -18 The elastic force model [4] The elastic force model consists of three components, much like the three way Maxwell model with the difference that nonlinear effects are taken into account. It gives a good representation of a rubber component with only five parameters to determine. It is a model relating component compression with component force and accounts for both frequency and amplitude dependence, while neglecting temperature and preload. It is described with three elements, from the left in Fig -18. The first element is an elastic spring showing nonlinear behavior to simulate the geometrical and material nonlinearities that are associated with a real rubber component. The second element is used to describe the frequency dependence through a fractional derivative, spring pot. The third element is associated with amplitude dependence through a smooth frictional component Implementation of the elastic force model The elastic stiffness at different compressions is K elast 3 D L 4 ( L x) D 1, Eq -1 8L( L x) this gives the compression force in F elast 3 D x D (L x) 1, Eq - 4 L x 16L ( L x) where x is the compression ( 0 x L ), D is the diameter, L is the original height and μ is the static shear modulus. The second part is associated with the frequency dependence. It is modeled by a part named spring-pot. This model relates the force to displacement through a noninteger time derivative. The relation is shown in Ffreq bd x, Eq -3 0

32 where α (0< α<1) and b>0 are model constants. The relation can be rewritten through a Grünwald definition as F freq ( t ) bd n x n t b ( ) n1 j0 ( j ) x ( j ) n j, Eq -4 the Γ is the Gamma function. This model is dependent on the displacement history this suits rubber isolators very well because of the memory that is displayed by rubber. The third part accounts for the amplitude dependence. The force of the friction is defined in two different directions when x is increasing F frict F fs x xs x (1 ) ( x x s ( F ) fmax F ), Eq -5 fs holds and when x is decreasing F frict F fs x xs ( F x (1 ) ( x x ) s fmax F ), Eq -6 fs is accurate, where F fmax and the displacement parameter x (the rate of friction force development) are defined by the user. The total force response is a super positioning of the three contributions according to F tot( tn freq n elast n frict ) F ( x( t )) F ( x( t )) F ( x( t )). Eq -7 Later a model that describes frequency, preload and temperature dependence is presented, for more insight reading of ref. [4] is recommended. n 3 Summary literary study The aim of this part of the master thesis has been to give further knowledge about rubber and to study different models of rubber isolators and decide the accuracy of these models and eventually suggest some appropriate ways to calculate the dynamic behavior of rubber isolators. Rubber is an advanced material. The selection of rubber model depends on the application of interest. There are mainly two models that consider both frequency and amplitude dependence that are presented. The models are named the three way Maxwell model and the elastic force model. Both give good representation with an advantage for the elastic force model because of less parameters to determine. On the other hand the three way Maxwell model presented in ref. [6] shows a step by step method implemented with existing FE-code which may lead to faster establishment for industry use. 1

33 Part II: Experimental Part

34 1 Introduction Previous studies show that the static and dynamic properties given by the subsuppliers of rubber isolators are not always correct. Measurements of a rubber isolator that is commonly used in the industry have been conducted. Later a model is derived that gives the static stiffness with regards to preload and geometry. The dynamic stiffness can be obtained by introducing a dynamic factor. 3

35 Isolators and geometry The geometry of the investigated isolator, CB05-, is shown in FigII -1 and Table -1. It should be mentioned that the mount consists of two pieces. Illustration of the mounted isolator can be seen in FigII -. FigII -1 Geometry of the investigated rubber isolator [1] Table -1 The dimensions [mm] of the isolator that is investigated [1] Series Number A B C D E F G CB ,1 85,9 64,8 31,8 37,8 FigII - Mounted rubber isolator 4

36 3 Method Now follows a description of the experimental theory and a derivation of the staticand dynamic-stiffness with geometry and preload dependence. 3.1 Experimental Rubber is an advanced material and many precautions have to be made before conducting measurements. In the upcoming sections some information about the theory behind the measurement and also how the experiment, both static and dynamic, has been carried out will be presented Theory One applies a force and measures the displacement when calculating the stiffness, see FigII 3-1. The stiffness is K F, EqII 1-1 l where F is the applied force and Δl is the change in thickness. FigII 3-1 Illustration of basic stiffness A cylindrical metal plate is mounted in between the two pieces of the vibration isolator. A cylindrical metal tube with one closed and one opened end is mounted on top of the isolator resulting in that the excitation force can be led through to the support plate, see FigII 3-. 5

37 FigII 3- A exploded view of the vibration isolator assemble An excitation force is applied to the cylindrical metal tube when calculating the dynamic stiffness. The movement of the support plate relative to the underlying surface is measured, see FigII 3-3. FigII 3-3 Force and displacement positions The force and displacement are measured in the time domain and then Fourier transformed to the frequency domain before the operation resulting in the dynamic stiffness in K dyn F( ). EqII 1- l( ) During the experiment (both static and dynamic) the isolator is prestrained with listed torque value of 500Nm that is used during assemble. 6

38 3.1. Dynamic measurement The measuring chain for the dynamic measurement can be seen in FigII 3-4. A total of six accelerometers where mounted, three above and three under the isolator. The result from the three displacements was then averaged to get a final outcome. The isolators are also preloaded with a load that is 80% of the maximum recommended load listed in Table 3-5 before the dynamic tests are conducted. A random burst signal with the frequency span of 0 to 5000Hz and executed every two seconds with a pause of two seconds in between every burst. Isolators consisting of blends of natural rubber, carbon black, sulfur and other chemical agents where investigated. Thus the Mullin s effect (see heading 1..3 in Part I for details) should be more pronounced. To make sure that the Mullin s hade dissipated some excitations where conducted before the measurement commenced. The frequency range of interest is 0-00Hz. FigII 3-4 Measuring chain of the dynamic test setup The acceleration is measured by six piezoelectric accelerometers where each pair is symmetrically mounted. The used equipment when conducting the dynamic measurement is shown in Table 3-1. Table 3-1 The equipment used during measurements Accelerometers S/N Charge Amplifiers S/N Data Collection S/N Brüel & Kjaer Brüel & Kjaer Agilent 8408A - Brüel & Kjaer Brüel & Kjaer Agilent 8491B - Brüel & Kjaer Brüel & Kjaer Agilent E143A - Brüel & Kjaer Brüel & Kjaer Brüel & Kjaer Brüel & Kjaer Brüel & Kjaer Brüel & Kjaer Force Transducer S/N Brüel & Kjaer Kistler

39 Uncertainty estimation of dynamic measurement The uncertainty estimation of the dynamic measurements is based on ref. [14]. The test method described is similar to the one conducted in this thesis. The frequency-band-averaged dynamic transfer stiffness is dyn 0 K Lkav 10lg, EqII 1-3 k where k 0 =1[N/m]. With the introduction of error quantities, becomes L Lk, [db] EqII 1-4 k av av ins rep rig lin where δ ins is the error caused by instrumentation, δ rep an error caused by installation repeatability of test element, δ rig the error caused by the test rig and δ lin is the error caused by approximate linear behavior. In this thesis the uncertainties are shown in Table 3-. The sensitivity coefficients are estimates. Quantity Transfer stiffness level Table 3- Uncertainty budget for dynamic measurements [14] Estimate db Standard uncertainty u i db Probability distribution Sensitivity coefficient c i Uncertainty contribution c i u i db L k av 0.3 Normal δ ins Normal δ rep 0 p / 3 Rectangular 0 0 δ rig Rectangular δ lin Rectangular The error is defined for a coverage probability of 95% and assumed to be a normal distribution, the value of the expanded uncertainty is given by 5 U ( c u i i). EqII 1-5 i1 And in this case the expanded uncertainty is: U=1.11 [db] 8

40 3.1.3 Static measurement The measurement chain and the equipment used are shown in FigII 3-5 and Table 3-3. FigII 3-5 Measuring chain of the static test setup Table 3-3 Equipment for statical measurement Data Collection S/N Position Sensor S/N Agilent 8408A - HBM 1-WA150MM-L Agilent 8491B - HBM 1-WA150MM-L Agilent E143A - Low Pass Filter S/N Force Transducer S/N HBM ML55B - Instron Model803 - HBM ABA - At the start of the measurement a load of 110N is applied and the position sensors are zeroed. The load is increased a minimum of 300N and left under pressure a time of at least 30 minutes to let the viscous effects of the rubber dissipate. The force and displacement signal is sampled for 60 seconds, an average of this is the force at a specific displacement. 9

41 Force [N] Uncertainty estimation of static measurement The uncertainty estimation of the static measurement is based on a summation of the uncertainty of the instrumentation (force transducer, positioning sensors etc), measuring method and other factors (viscous effects, fatigue etc). These estimations are shown in Table 3-4. Table 3-4 Estimation of standard deviation in percent Standard Estimate Basis for estimation deviation [%] ζ ins 0.8 Calculated from uncertainty given by the suppliers of the measuring equipment. ζ meth 1 The test method gives a low uncertainty due to the long relaxation and sampling time. ζ other Other factors that could affect the uncertainty. ζ tot 3.8 Summation of all standard deviations 6000 Force Statical measurement of CB WRONG measurement RIGHT Measurement Extrapolation Deflection [m] x 10-3 FigII 3-6 Load against deformation of 05- isolator. Blue line: Measurement with WRONG method Crosses: Measurement with RIGHT method Red line: Extrapolation of RIGHT measurement points If the viscous effects during the static measurements are not taken into account the results are not correct. An example of this is illustrated in FigII 3-6. In this case a load was applied with a step increase of 0.005mm/s (blue line), instead of waiting for the viscous effects to disappear (red line).the slope of the line is the static stiffness. In this case the isolator has a larger stiffness for a smaller load. 30

42 3. Mathematical This model of rubber isolator is purposed to fit some isolators tested in KTH on behalf of Atlas Copco Rock Drills AB, Rocktech Division. The model is specific for a cylindrical rubber isolator with an opening in the middle of the specimen FigII 3-7. FigII 3-7 Rubber investigated rubber isolator [1] Stiffness of the isolator showed above will be discussed for two different cases, with and without preload. Simple formulas for estimating the static and dynamic stiffness for this isolator geometry in different load cases are derived. The investigated isolator is produced by Lord Corporation is tabulated according to Table 3-5. Table 3-5 Data of isolators [1] (*all elastomers are blends of Natural Rubber, Carbon Black, Sulfur and other chemical agents) T-Thick support Plate (recommended) Part Elastomer Elastomer Elastomer Thickness T Load/ Number code nominal ShoreA [mm] Deflection [N/mm] CB-04-3 NR A079P at.3 CB-04-5 NR A111P at.3 CB-05- NR A060P at.3 CB-05-3 NR A079P at.3 CB-05-4 NR A091P at.3 CB-05-5 NR A111P at.3 31

43 3..1 Isolator Stiffness with no preload To make this derivation possible and easy, one has to define a simple geometry which approximately describes the real specimen, see FigII 3-8. FigII 3-8 Geometry approximation for rubber isolator with metal cap on top The static stiffness of the rubber isolator shown in FigII 3-8 without any preload can be written mathematically with equation K 3GA H st 1 S, EqII -1 where G is the shear modulus, H the height of the isolator (see TableC 1 in Appendix C) and A is the load area which is A R y R, EqII - i where R y and R i (see Table -1) is the outer and inner radius of the undeformed rubber isolator. S is the shape factor defined as the loaded area divided by the surface area Ry Ri S. EqII -3 R H y Because the load area is different on the support plate side and the support bracket side an average of these two inner radiuses is calculated. The static stiffness could now be used to calculate the dynamic stiffness K, EqII -4 dyn K st where χ is a dynamic factor given by the rubber producer or measured in a laboratory. It is recommended that this factor is measured for each special case because of its wide variety (1.3-5). 3

44 Isolator Stiffness with preload In this section, a similar case as in the previous section will be discussed but with respect to preload. When a load is applied upon the rubber isolator it will bulge as shown in FigII 3-9 (a) but mathematically it is easier to approximate the deformation as is shown in FigII 3-9 (b). The approximation takes account for nonlinearity in geometry, however the material is still treated linearly neglecting possible nonlinearity. FigII 3-9 (a) Real deformation of a rubber isolator (b) Mathematical approximation of deformation EqII -1 to EqII -3 is reused, the difference is that now there is deformation dependence and the stiffness of the isolator will now be ) ( 1 ) ( ) ( 3 ) ( st u S u h GA u u K, EqII -5 where the load area and height is i y ) ( R r u A EqII -6 and u H u h ) (. EqII -7 And the shape factor becomes u H r R R h r R R u S y i y y i y ) (. EqII -8 Assuming incompressibility yields i i y y i y i y R h H R R r h R r H R R. EqII -9 Eq -5 to Eq -9 gives

45 3 ( R y Ri ) ( Ry Ri ) Kst( u) 3GH. EqII ( H u) [( Ry Ri ) H Ri ( H u)]( H u) The dynamic stiffness is K dyn u) ( K EqII -11 st and the static stiffness is df Kst( u) ( u). EqII -1 du At a given deformation u 0 the total force is u u 0 0 df F ( u0 ) du Kst( u) du. EqII -13 du 0 0 FigII 3-10 Isolator with screw force and applied load The force acting on the rubber specimen from the screw joint will be the same for the isolator mounted over and under the support plate, see FigII This specific force will give a displacement on each isolator given by u screw H T D. EqII -14 When an external load is applied the total displacement of the isolators u over u u EqII -15 screw load 34

46 and u under u u. EqII -16 screw load The total force acting on the support plate can now be defined as u over u under F K ( u) du K ( u) du. EqII -17 st 0 st 0 st 35

47 Force F[N] 4 Result and discussion The last part of this thesis consists of analyzing the measurement result that has been obtained for an isolator of model 05- produced by LORD. To verify that the mathematical model in section 3. is valid, static- and dynamic test data from isolator CB05- is compared to the calculations. The equipment used during measurements is listed in Table 3-1 and Table 3-3. A personal computer provided with MATLAB (R010a) is used to perform the analysis. 4.1 Evaluation of mathematical model The dimensions of the rubber isolators are defined in Table -1 and the shear-moduli corresponding to shore value is presented in TableC 1 in Appendix C. Calculations of three different shore-values are plotted together with measurement data of isolator CB05-. The force is illustrated in FigII 4-1 and the stiffness is illustrated in FigII 4-. The displacement in the plots are defined from F load i.e. the screw joint is mounted and the displacement is zero at this point. Measurement and mathematical FORCE of Math Model G=370000Shore=35 Math Model G=450000Shore=40 Math Model G=540000Shore= extrapolation 1000 Given by LORD cataloge 00-5 measurement Displacement u [m] x 10-3 FigII 4-1 Comparison of static measurement force and mathematical model As the above figure shows the model corresponds well to the real isolator. The measurement values are illustrated by the crosses (error bars based on the standard deviation estimation are also shown) and the line is an extrapolation of the measurement results. Values given by LORD and a value given from the wrongfully 36

48 Static Stiffness K[N/m] conducted measurement is also marked, see section and FigII 3-6. The measurements seem to match with what has been seen in literature, namely that the isolator gets stiffer with higher preload as is expected. 3.8 x 106 Measurement and mathematical STIFFNESS of Math Model G=370000Shore=35 Math Model G=450000Shore=40 Math Model G=540000Shore= measurement Given by LORD catalog WRONG measurement Displacement u [m] x 10-3 FigII 4- Comparison of measurement stiffness, mathematical stiffness and stiffness calculated from LORD catalogue It should also be noted that the shore value of 05- is 43 but this value has a deviation of 10-15% because of the accuracy in the production batches. While the line in FigII 4- which follows the measurement data the best has a listed shore value of 40. As mentioned before it can be due to the deviation in batches during production. Table 4-1 Comparison of load and static stiffness from mathematical, measurements and LORD catalogue given data 05- Load[N] at.3[mm] Static Stiffness[N/mm^] at.3[mm] Math. Model 703 3,130 Measurements ,033 LORD 5340,3 WRONG measurement 600,364 In Table 4-1 the load and static stiffness at.3mm deflection is given. When comparing the LORD results to the measurement conducted there is a difference of roughly 77% both between load and static stiffness. One interesting observation is that the wrongfully conducted experiment results are closer to the LORD catalog given values. 37

49 Dymanic stiffness [N/m] 14 x 106 Dynamic stiffness of X: Y: 3.739e Frequency [Hz] FigII 4-3 Dynamic stiffness of isolator 05- When looking at FigII 4-3 one can observe peaks at 110Hz and 175Hz. The conclusion can be made that these peaks are associated with resonances in the rig and not the rubber isolator. Because the peaks are at the same frequencies for all rubber isolators independent of their stiffness, the conclusion could be made from earlier conducted measurements in the same test rig. The measurement data was not stable under 10Hz, for this reason it was hard to see the dynamic stiffness value of interest. To solve this problem an extrapolation was made from 10Hz to 100Hz and is a black line displayed in FigII 4-3. In FigII 4-3 the dynamic stiffness of 05- is plotted. The dynamic stiffness is evaluated at 10Hz and 80% of the force 5340N (Table 3-5) and gives the dynamic factor: Kdynmeas 3.739e6 1.4 [-] K 3.04e6 stmeas This leads to, from Eq-4: K dyn e6 3.79e6 [N/m^] This calculated dynamic stiffness can be used when calculating the first natural frequency of the 1-Dof system before selecting which isolator to use in an assembly. 38

50 5 Conclusion It is shown that the model presented is valid for an isolator in the CB05 series. The conclusion can be drawn that the model probably also will be valid for all isolators in both CB04- and CB05-series because of the strikingly good correspondence between test data and model in this evaluation. A comparison of all isolators against the mathematical model would have been favorable, but because tests of the other isolators had been delayed this could not be done at the time of writing this thesis. The MATLAB code with all calculations included and is found in Appendix D. 39

51 References 1. P.B. Lindley. Engineering design with natural rubber. The Malaysian rubber producer s research association, P.E. Austrell. Modeling of elasticity and damping for filled elastomers. Division of structural mechanics LTH, Report TVSM-1009, N. Gil-Negrete, J. Vinolas, L. Kari. A simplified methodology to predict the dynamic stiffness of carbon-black filled rubber isolators using a finite element code. Journal of sound and vibration 96 (006) , M. Sjöberg. On dynamic properties of rubber isolators. Department of vehicle engineering, KTH, ISSN: X, F. Karlsson and A. Persson. Modeling non-linear dynamics of rubber bushings parameter identification and validation. Division of structural mechanics, LTH, ISSN: , A. K. Olsson. Finite element procedures in modeling the dynamic properties of rubber. Division of structural mechanics, LTH. ISSN: , A. N. Gent. Engineering with rubber, nd edition. ISBN: , P. E. Austrell. Konstruktionsberäkningar för gummikomponenter. Division of structural mechanics, LTH. ISSN: , L. Kari. On the dynamic stiffness of preloaded vibration isolators in the audible frequency range: Modeling and experiments. MWL/Department of Vehicle Engineering, KTH, E. I. Rivin. Passive vibration isolation. ISBN: , Lord Corporation, Lord aerospace product design catalogue PC6116, Erie, PA, Lord Corporation, Lord industrial catalogue, Sveriges Plast- och gummitekniska institutet, Konstruera med gummi, ISBN , British Standard, Acoustics and vibration Laboratory measurement of vibroacoustic transfer properties of resilient elements. Part, BS EN ISO :008,

52 Appendix A A.1 Search methodic Prior to the writing of this thesis a literary search was conducted. The search logic, search engine and results are presented in TableA 1. To narrow the search down terms were added to the search logic until a manageable amount of titles were found. Only titles with relevance to rubber modeling and behavior where chosen for the basis of the literary study. TableA 1 Search results and search logic Search engine: Compendex/Inspec ( ): Search logic: [(((("vibration control" ) WN CV) OR ((NVH or "vibration isolation" or "vibration isolator" or "rubber mount" or "rubber mounting" or "rubber mounts") WN All fields)) AND (("dynamic stiffness") WN All fields)), English only, ]; [ vibration isolation AND rubber isolator AND dynamic stiffness ] Scopus Number of hits: 17hits (77hits/8hits) of 57hits; which hits were doubles; Interesting hits: 18hits 1hits [( vibration isolation OR vibration control OR rubber mount ) AND dynamic stiffness ] A-A

53 Appendix B B.1 FEM implementation In Part I a short introduction of different kind of elements that could build up a potential model is presented. The computer based solutions are the most efficient and FEM. There are many FE-software that enables modeling of isolator parts. The elastic and viscous parts are more common to model with FE-software. The frictional part is thought more difficult to model e.g. there is no existing model for this part so the researcher has to develop individual code. B.1.1 Energy strain functions For hyperelastic materials the stress could be expressed with strain energy functions. These functions are presented in terms of principal stretches and invariants [5]. W denotes the energy strain function and is assumed to depend on all strain components W W B) W(,,, n, n, ). Eq B 1 ( n3 If the tested material is assumed to be isotropic there is no direction dependence i.e. W will only depend on the principal stretch W,, ). Eq B ( 1 3 To determine the roots of this equation it is easier to express Eq B in terms of strain invariant instead of principal stretch this is done with the relations I I I 1 tr( B) (tr( B) tr( B) ) , Eq B 3 det( B) which gives W I, I, ). Eq B 4 ( 1 I3 For the incompressible material, there is no I 3 dependence because it represents the volume change, for rubber the bulk modulus is many times larger than the E-modulus that is why incompressibility could be assumed [1]. Thus, the energy strain function only depends on two invariants W W I 1, I ). Eq B 5 ( This equation seems to indicate the possibility for all values I 1, I. But in reality, these values are limited by the condition of incompressibility, see Fig B 1. B-A

54 Fig B 1 Limitation due to incompressibility [] To eliminate one of the three principal stretches consider principal directions and use the condition I 3 =1. This condition will give 1 3 Eq B 6 1 and I I Eq B This will make it possible to choose independent values of I 1, I. B.1. Energy strain functions format There are different formats of energy strain functions which have been investigated by scientists e.g. Mooney Rivlin, Ogden and Yeoh. All the existing models will not be presented in detail in this thesis, references to articles will be shown. The polynomial form is one of most implemented forms in FE-software. The energy strain functions for this model are expressed in forms of a series expansion i j W Cij ( I1 3) ( I 3) i0, j0, Eq B 8 where C ij is constant [5]. For the undeformed case, W is supposed to be zero and that is why the equation for W is expressed in terms of (I 1-3) and (I -3). In spite of that the sum above is written to infinity. Only the first few terms are usually used B-B