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Modelling of frictional joints in dynamically loaded structures – a review HENRIK WENTZEL Dept. of Solid Mechanics Royal Institute of Technology (KTH) (SE-100 44 Stockholm) Sweden Email: [email protected] Telephone : ++46+8 553 807 39

Introduction The dynamic behaviour of joined structures is affected by the characteristics of the joints, in particular the force-displacement relation for tangential loading. Much research has been focused on this area triggered both by the automotive industry’s need for shortened development cycles and by the aeronautic/military industry’s need to reduce costly physical testing. It is a challenging subject where non-linearities arise from different physical phenomena such as friction, chatter and impact. Often friction is a desired property as it will damp vibration amplitudes in vehicle structures that otherwise would grow large and create fatigue damage and/or noise. For example heavy vehicles with leaf-springs suspension show desirable high damping characteristics compared to airspring suspended vehicles, which often makes the use of additional shock absorbers unnecessary in the former (Johansson, 2005). On occasion friction is unwanted in rotating machinery equipment as it increases energy losses and destabilizes the system (Mottershead, 1997) or adds noise and chatter (Ibrahim, 1994). Indeed, friction is an important factor for damping and may account for up to 90% of the total damping in joined structures (Padmanabhan, 1990). Goodman presented a review on interfacial slip damping of joined structures in 1959 (Goodman, 1959) and Ungar two more exhaustive studies on the subject in 1964 and 1973 (Ungar, 1964) and (Ungar, 1973). The surveys conclude that dry friction in joints only creates damping when the applied forces are shearing the joint. Indeed, an experimental set-up consisting of dozens of dry steel plates stacked on top of each other did not show any measurable damping when loaded with a force normal to the contact area. Shearing forces that are not sufficiently large to create global sliding will still create some energy dissipation due to micro-slip in the joint. Shearing forces create a difference in normal strains in the tangential direction in the plates of the joint that results in a relative tangential motion. The frictional

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forces oppose this motion and so energy dissipates. Figure 1 illustrates the phenomenon. In (Earles, 1966) it was found that the maximum damping capacity ratio for a riveted joint is obtained if all the shearing forces are transferred via friction such that the rivet carries no shearing force. The theories used in (Goodman, 1959), (Ungar, 1973), and (Earles, 1966) are clearly comparable to Johnson’s theory on contact and sliding between spherical objects (Johnson, 1990) and based on the assumption that Coulomb friction governs the tangential interaction in the contact area.

Figure 1. Micro-slip in joints, a reproduction from (Segalman, 2001).

Except for in laboratory conditions, with constant clamping pressure over a well defined area and uni-axial loading, the analytical theories fail to predict the quantity of the energy loss. Part of the explanation is that a homogenous contact pressure over a well-defined area is very rare in reality, particularly so for bolted or riveted joints (Cullimore, 1964). However, the analytical theories do yield valuable results as to how the energy dissipation ∆W varies with Young’s modulus E, clamping force Fclamp, coefficient of friction μ and amplitude the amplitude of the applied force Fapplied . Both analytical and experimental results show that the energy dissipation is influenced by these parameters in the following way:

(

amplitude ΔW = ΔW E , Fclamp , μ , Fapplied

∂ΔW < 0, ∂E

∂ΔW < 0, ∂Fclamp

)

∂ΔW < 0, ∂μ

∂ΔW >0 amplitude ∂Fapplied

Increasing the Young’s modulus makes all members of the joint stiffer, thereby reducing the magnitude of all displacements including the relative tangential slip responsible for energy loss. Increased clamping force and increased coefficient of friction also makes the joint stiffer thereby decreasing the amount of slip, but at the same time the slip that remains is restrained by larger frictional forces, something that increases the energy

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dissipation. However both analytical and experimental investigations show that the overall result when increasing the clamping force or the coefficient of friction is a decrease in energy loss. Finally the dissipation is strongly influenced by the magnitude of the applied force in a positive way. Specifically one of the analytical findings is that the energy dissipation is proportional to the amplitude of the applied force raised to the power of 3. In physical testing of actual joints this exponent is found to be closer to 2.5 (Ibrahim, 2005), (Gregory, 1999), (Smallwood, 2000) but still fairly constant within some regions of the applied load amplitude. Extended analytical studies by Song et al. (Song, 2005) show that the exponent 3 is valid only for homogenous contact pressure but that other exponents apply to other contact pressure distributions. A number of analytical and experimental studies of bolted beam structures indicate that for a specific load case there exists an optimal clamping force which maximises the energy loss and hence the damping (Earles, 1966), (Beards, 1983), (Beards, 1985), (Ferri, 1995), (Ren, 1994), (Wentzel, 2005). Most presented studies treat simple shear joints subjected to unidirectional shear and have not investigated complicated load cases involving rotation of the joint, although this type of loading may create much larger energy dissipation (Beards, 1977), (Wentzel, 2005). Friction models The Coulomb friction model is a phenomenological description of friction and has experimentally been validated for global sliding between two rigid bodies. For a sliding velocity v and a normal force FN the frictional force Ff is: F f = − sgn(v) ⋅ μ ⋅ FN Coulomb friction is the most widely used model for sliding but also for micro-slip (Gaul, 2001), and several modified friction models have been derived from it. Stribeck investigated roll bearings and noted that the coefficient of friction decreases at the onset of sliding until a certain velocity before it increases again (Stribeck, 1902). This behaviour is called the Stribeck effect. A commonly used phenomenological model is the stiction model that behaves similar to the Coulomb model but has a higher value of friction at zero velocity (sticking) than in the sliding regime. In rubber tire applications experimentally validated models with a coefficient of friction that strongly depends on the velocity in a complex way are commonly used (Thorvald, 1998), (Gipser, 1990). These models are a mixture of friction and tire models because some of the velocity dependence is caused by the relative tangential deformations in the tire to ground contact and not by the local tangential interaction in the contact surface (Deur, 2004). For low frequency 3

excitation of joined structures the velocities are low so the Stribeck effect is rarely noticeable. Physical models of friction have been proposed relating to the field of tribology see for example (Hagman, 1993) or (Olofsson, 1995). The basic assumption is that the (spherical) asperities on the contact surfaces interact and deform elastically according to Hertz theory. The distribution and the radii of asperities determine the coefficient of friction. There exist several variants on this theory, notably the bristle-model (Haessing, 2001) where the tangential interaction is seen as an interaction between bristles populating the contact surfaces. It has been noted that the energy dissipation in joints with machined contact surfaces is influenced by the machined lay orientation (Rogers, 1975), (Murty, 1982). Asperities are subject to plastic deformation and wear and the influence of asperities is likely to change over time, this was measured in (Padmanabhan, 1991) where the energy dissipation per cycle stabilized after a couple of hundred load cycles. Numerical implementation of friction is commercially available in several Finite Element (FE) codes. ABAQUS (HKS, 1998) offers the possibility to enforce the Coulomb friction model by means of Lagrange multipliers, which sometimes is theoretically satisfactory but tends to create numerical problems so that the solution does not easily converge. The stiction model and other velocity dependent models are also widely available. Many FE codes (HKS, 1998), (LS-Dyna, 1997) also include an elastic-slip friction model where the surface interaction is elastic for small deformations. This avoids the numerical difficulties of Lagrange multipliers and speeds up the analysis considerably. Experimental methods and results Experimental studies of the dynamics of joined structures are usually performed with hydraulic or electric shakers, or with the aid of an impact hammer. The dynamic properties of structures are often visualised and evaluated with Frequency Response Functions (FRF), which is the equivalent of the systems transfer function. Commonly used FRFs are admittance, mobility, and acellerance (Ewins, 1994). They are all based on the Fourier transform of the response signal (u or u& or u&& ) divided by the Fourier transform of the exciting force. ℑ(u ) admittance: ℑ(F ) ℑ(u& ) mobility: ℑ(F ) ℑ(u&&) accelerance: ℑ(F )

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A commonly used technique is to estimate the damping by using the halfpower points of the FRF. The damping as fraction of critical damping, ζ is defined as the ratio between the dissipated energy per cycle and the peak kinetic energy during vibration (times 4π). For a single Degree Of Freedom (DOF) system with viscous damping it may be extracted from the mobility plot as:

ζ =

Δf , 2 fn

where Δf is the distance between the half-power points around the peak at the natural frequency fn. This method also provides a good estimation of the equivalent viscous damping for other types of damping and for multiple DOF systems if the damping is small (ζ < 10 %) and the natural frequencies are well isolated (Ewins, 1994). Modern methods to determine the parameters of visco-elastic system models usually involve the minimization of the error between a model FRF and a measured FRF, (Balmes, 2005):

J = FRFmeasured (ω ) − FRFmodel (ω ) , 2

where the following three parameter model FRF may be used for SDOF systems, A FRFmodel (ω ) = 2 . 2 ωn − ω + 2iζωω n For multi-DOF visco-elastic models this technique may be generalised. The FRF of a multi DOF model may be written in the polynomial Laplace form as

FRFmodel (s ) =

A(s ) with B (s )

J = FRFmeasured (s )B(s ) − A(s )

2

being a reasonable choice of the error function J. The frequency response function may also be used to estimate joint model parameters instead of global system parameters. Ren and Beards assumed a linear visco-elastic joint model described by mass, stiffness and damping matrices and developed a general methodology to extract these matrices from experimental FRF data (Ren, 1995). Ratcliffe and Lieven assumed the same linear joint model and enhanced the parameter identifying methodology (Ratcliffe, 2000). Inamura has proposed a method for joint model parameter identification that not only uses the FRF but also the mode shapes (Inamura,

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1979). There are numerous other published studies on this approach, see for example the review paper by Ibrahim (Ibrahim, 2003). For linear systems the FRF is a system property and the same regardless of which time history force is applied to the system. However, because of the non-linear behaviour of joints and the increase in energy dissipation as function of applied load amplitude, joined systems have signal dependent FRF’s, i.e. for different input signals different FRF’s are obtained see for example (Ewins, 2000). Experimentally this is well illustrated in (Gaul, 1993) where measurements were performed on a system consisting of two blocks of steel connected with a single bolt lap joint. The entire system was suspended with weak springs and excited with a sine-sweep signal by means of an electric shaker. In that particular study different amplitudes of the excitation signal resulted in different Frequency Response Functions, Figure 2.

Figure 2. Frequency Response Functions for a system with a bolted joint for different levels of harmonic excitation (Gaul, 1993).

It may be seen from Figure 2 that as the amplitude of excitation and response increases so does the equivalent viscous damping. The figure also reveals that the stiffness decreases as the amplitude increases, because the deformation in the joint increases with the amplitude. This is probably an inherent property of systems with frictional joints. For a treatise on nonlinear systems in general and their FRF’s refer to (Thompson, 1993). Hartwigsen used a 3-piece beam structure forming a bolted double lap joint, which was suspended in weak elastic cords in the experimental study 6

(Hartwigsen, 2004). The structure was excited with an impact hammer and the FRF was measured for different levels of excitation. The equivalent viscous damping was extracted from the FRF for several modes of vibration and it was presented in an amplitude-of-vibration to damping diagram, Figure 3.

Figure 3. Amplitude-of-vibration to damping relation for mode no. 1 in Hartwigsen’s experimental setup (Hartwigsen, 2004).

Locally in the joint the rate of energy dissipation increases rapidly with increased amplitude of motion and globally this is observed as an increase of the damping. As the length of the slip region in the joint increases the stiffness of the joint decreases (refer to Figure 1). For the global system this is seen as a reduction of the resonance frequency for higher amplitudes. The response surface methodology used in (Padmanabhan, 1992) provides a method to estimate the reliability of the damping measurements and predictions. Padmanabhan performed quasi-static measurements on a machine joint and varied the normal pre-load and the level of excitation of the joint. The energy dissipation is described as a polynomial function of the level of excitation and the clamping force, Figure 4. From the surface gradient it is apparent how the energy dissipation varies with the clamping force and the applied load amplitude.

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Figure 4. Energy dissipation per cycle as a surface function of clamping force, P, and amplitude of excitation, Tm (Padmanabhan, 1992).

The non-linear properties clearly visualized in (Padmanabhan, 1992), (Gaul, 1993) and (Hartwigsen, 2004) are imperative to include in a joint model. Joint modeling

A good joint model should be as simple as possible and still capture all the important physical properties of the actual joint. The important properties considered here are the force-displacement behaviour, the resulting hysteresis loop i.e. the damping, and the influence of velocity. The wear aspect will not be considered at this stage nor will the possibility to create failure criteria for the joint be investigated. Six different joint modelling techniques from the scientific literature are presented here together with a brief description: Linear Visco-Elastic model, a damping matrix C is chosen for the global system such that the viscous energy dissipation equals the expected dynamic frictional dissipation in the joints. This is by far the most widely used technique. Linear Complex stiffness model, a complex stiffness matrix K is chosen for the global system such that the energy dissipation equals the expected dynamic frictional dissipation in the joints. This technique has with some

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success been used to model high-frequency vibration and acoustics in the frequency plane. Iwan networks model, a network of springs and sliders replaces the actual joint. The network is calibrated to produce the desired quasi-static or dynamic force-displacement characteristics (including the dissipation) of the actual joint. Valanis model, a first order differential equation from Valanis plasticity model replaces the actual joint. The parameters in the equation are calibrated to produce the desired quasi-static or dynamic force-displacement characteristics of the actual joint. Bouc-Wen model, an equation containing time derivates and sign functions of the state variables replaces the actual joint. The parameters are chosen to produce the desired quasi-static or dynamic force displacement characteristics of the actual joint. Detailed FE model, a detailed FE model is used to simulate the micro-slip in the joint and calibrated to produce the quasi-static force displacement characteristics of the actual joint. Each of the above mentioned techniques are examined closer in the following. Linear visco-elastic joint model

The most common technique for modelling dynamics of systems with frictional joints is to use a linear elastic joint model for the stiffness in combination with viscous Rayleigh-damping. The value of Rayleigh damping is chosen such that approximately the same amount of energy is dissipated viscously as the frictional losses in the joints would amount to. Thus the dissipation is not localized in space but evenly distributed in the structure; see (Pavic, 2005) for a theoretical discussion on the validity of this approach. The joint parameters are the linearized stiffness for each joint and the global systems modal damping. The linearized stiffness is often computed with FEM while the global systems modal damping is estimated with rules of thumb (Chang, 1964) or measured in physical testing see for example (Ellison, 1972) or (Fischer, 2000). This modelling technique is unable to capture neither the variations in damping nor the changes of stiffness in the joint region discussed in the previous section. It is uses a damping force that is proportional to the velocity in order to model the frictional forces which are in general only marginally dependent of velocity. Despite the method’s theoretical limitations it is very popular in the automotive industry and has also been used successfully in the aerospace industry see for example (Chang, 1969), (Ellison, 1972).

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In order to estimate the energy loss during vibration and hence the damping Ellison and Jones used substructure testing of satellites transported by the Saturn rocket (Ellison, 1972). Modal damping for mode r is defined as

ζr =

ΔWr , 4π ⋅ Trpeak

where Trpeak is the peak kinetic energy contained in mode r and ΔWr is the energy loss per cycle of mode r. Each joint in the structure is characterised by a force to energy dissipation relation. The energy dissipations in the individual joints are measured experimentally for typical load cases of typical amplitudes. In this way a relation between the energy dissipation and the applied load amplitude F is obtained for each joint i

ΔW i = ΔW i ( F i ) . Each mode shape r contribute with a force on each joint i proportional to the amplitude of that mode qr as

F i = ∑ a ri q r , r

where the coefficient ari depends on the geometry and the mode shape. Thus when the structure vibrates at a single mode r the energy loss is given by the expression

ΔWr = ∑ ΔW i = ∑ ΔW i (a ri q r ) . i

i

Recognising that for vibration at the resonance frequency the peak kinetic energy is well determined if the amplitude of motion is known, Trpeak =

1 1 u&&2 dm = qr2ωr2 ∫ φr2 dm , ∫ 2 mass 2 mass

where the last integral is unity if the mode shapes are normalized with respect to the modal mass. Since the energy loss for each mode is known as a function of the modal coordinate the energy equivalent modal damping is obtained with the relation

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∑ ΔW (a q ) i

ζr =

i r

i

2πq r2ω r2

r

.

The method of using substructure testing and then computing the modal forces may work on systems with low damping. However, Bowden showed that for highly damped systems this approach leads to erroneous results, (Bowden, 1988). For lightly damped systems the method does produce a damping that is energy equivalent to the dissipation in the joints when the structure vibrates at a single mode at specific amplitude. It is not calibrated for and is not likely to produce correct damping for vibrations at several simultaneous modes or for dynamic processes of varying amplitude. An alternative to Rayleigh damping is to model only the actual joint as a visco-elastic element. In this way a simple frequency analysis of the structure gives not only the natural frequencies and the mode shapes but also a corresponding modal damping. The damping is largest for the modes that create the largest relative velocities in the joint region. For highly damped systems this model may be used in large-displacement analyses in the actual coordinates (without transformation to the modal space). The joint parameters are the stiffness and the viscosity of the joint material model or the spring dashpot. In (Baraco, 1981) joints of sheet metal were experimentally characterised by the number of bolts, clamping force and the way the loading was applied. Different local energy equivalent viscous damping coefficients were computed for different joints and load cases. Coefficients that later mat be used for linear dynamic FE simulations. A similar approach is used in (Dubigeon, 1982) for bending of bolted joints, resulting in very simple viscous models that are treated analytically. In (Bowden, 1988) the joints are modelled with a parallel spring and dashpot. For any modal vibration a joint participation factor is computed which is a measure on how much the joints are exercised during vibration. Although the usage of viscous damping models seem to be somewhat out of date in the scientific literature it remains the most common tool in the industry. One of the reasons for its popularity is that dynamic analyses of large structures are generally performed in modal coordinates for which linear viscous damping is very easily incorporated. Linear complex stiffness joint model

A variant of the linear visco-elastic joint model is the use of complex Young’s modulus in the material model of the joints. Complex eigenvalue solvers are commercially available and permit the extraction of complex eigenvalues (Neumark, 1969). The approach permits the computation of a deformation dependent modal damping, where the modes that create the

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largest strain in the joint region are damped the most. In the frequency domain the FRF of a complex stiffness problem may be formulated as:

FRF (ω ) =

1 − ω M + (1 + iγ sgn (ω ))K 2

It should be remarked that when transformed to the time domain this equation does automatically fulfil causality, see (Inaudi, 1995), and complex stiffness models are indeed generally used exclusively in the frequency domain for acoustic applications. The method is not restricted to usage of an imaginary stiffness that is proportional to the real stiffness although this guarantees that the modal vectors are unchanged. Also for this approach it remains to define the magnitude of the complex stiffness such that the energy dissipation corresponds to the actual dissipation in the joints, a task equally challenging as that of defining a viscous damping. The energy equivalent viscous damping for a specific mode of vibration excited at the natural frequency is related to that mode's corresponding complex eigenvalue with the relation:

ζr =

Re(λ ) Im(λ )

This modelling technique is unable to describe the variations in damping as the amplitude of motion increases, but it has with some success been used for high frequency analysis of engine components (Fischer, 2000), and for analysis of break squeal (Lou, 2004). Joint models with Iwan networks

Many joint models are derived from friction or material models that include plasticity. Iwan’s material model (Iwan, 1967) consisting of a network of spring and slider elements is currently the basis for many joint models. These models are simplifications of the joint but permit some liberty in the design of the force-displacement characteristics. A typical Iwan network is depicted in Figure 5. The sliders are non-linear elements that implement the Coulomb friction model with a predetermined normal force and coefficient of friction resulting in a break-free force fi.

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Figure 5. Schematics of parallel Iwan networks.

For a finite number of spring-slider units, n, the force displacement relationship is given by: n

F (u ) = ∑ k i (u − xi ) , i =1

where xi is the current displacement of slider i. The above relation results in the hysteresis curve depicted in Figure 6, where n = 4.

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Figure 6. Force displacement hysteresis loop of a discrete parallel Iwan network of four spring-slider units with stiffness 1 and break-free forces 1,2,3, and 4 respectively.

Of particular interest are parallel Iwan networks with an infinite number of spring-slider units where all the springs have the same stiffness k. The computation is performed using a population distribution ρ(f) of the breakfree force of the slider elements. For such a network the force displacement relationship is written (Iwan, 1967): ∞

F (u ) = ∫ ρ ( f )k [u − x( f )]df , 0

where x(f) is the current displacement of all sliders with break-free force f. Segalman proposes the use of the moments of distribution (Segalman, 2001) of ρ(f) to facilitate the computation. The moments of distribution are defined as: θ

Λ n (θ ) = ∫ ρ ( f ) f n df 0

In (Segalman, 2001) it is demonstrated that for distributions of the type

ρ ( f ) = Rf χ , the energy dissipation during harmonic loading with a force of constant amplitude F depends as: 14

ΔW ∝ F 3+ χ Refer to Gregory and Smallwood et al. (Gregory, 1999), (Smallwood, 2000) where it was experimentally shown that this exponent, ( 3 + χ ) may be somewhere between 2.5 and 3 indicating a negative value of χ. Song et al. (Song, 2004) developed a finite element based on Iwan networks labelled the Adjusted Iwan Beam Element (AIBE) that was used for dynamic simulation of a joined structure. The Iwan network consisted of an infinite number of spring-slider units all with the spring stiffness k. The break-free forces of the spring-slider units are defined by the population distribution:

β

[

]

H ( f − (1 − β ) ⋅ f y ) − H ( f − (1 + β ) ⋅ f y ) 2 fy In addition, the network is adjusted by adding an extra spring with stiffness kextra = k α without a slider in series (or, equivalently, a slider with breakfree force f = ∞). Thus, the initial stiffness of the network is k(1+α). The AIBE element consists of two such networks, Figure 7, and can transfer both shear forces and bending moment. It is an eight parameters model with the parameters:

ρ( f ) =

{k , α , f 1

1

y1

, β1 , k 2 , α 2 , f y 2 , β 2 }

Figure 7. Schematic of the AIBE element, (Song, 2004).

If the initial stiffness is known in both directions the model is reduced to six parameters, and if in addition β = 1, which implies that micro-slip occurs for

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infinitesimal small loading the model is further reduced to four parameters. Song et al. trained a neural network to identify the four parameters from hammer excitation loading. Pettit used the AIBE element to model variability in joints (Pettit, 2004) and developed a method to identify the parameters from harmonic loading (Pettit, 2005). Comparison of simulated data to experimental results showed good agreement, particularly for the envelopes of the signals. Parallel Iwan networks show capability to model complex energy dissipation behaviour and can interpolate any energy dissipation that increases convexly with the applied force (Schindler, 2005). It is a static model where the energy dissipation and stiffness are independent of frequency and velocity. Furthermore Iwan networks are relatively easy to handle numerically when the number of spring-slider units is small or described by a few parameter population distribution. Joint models based on Valanis’ plasticity model

Valanis model of plasticity (Valanis, 1980) derives from a theory originally intended to unify the kinematic and the isotropic hardening models. Lenz and Gaul used this model to reconstruct measured force-displacement characteristics of a dynamically excited lap-joint (Lenz, 1995), (Gaul, 1997). The governing equation of the model they used is the first order differential equation: ⎡ ⎤ λ E 0 x& ⎢1 + sgn ( x& ) (Et x − F )⎥ E0 ⎣ ⎦, F& = λ 1 + κ sgn ( x& ) (Et x − F ) E0 where E0, Et, λ, and κ are constants and model parameters.

Just like the Iwan model the Valanis model fulfils what is sometimes referred to as the Masing hypothesis (Segalman, 2006): the forcedisplacement characteristics during cyclic loading may be obtained from reflection, translation and scaling of the force-displacement characteristics during monotonic loading. The Valanis model seems able to capture some of the non-linear phenomena in the joint transfer behaviour and is clearly easily implemented in existing FE-codes.

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The Bouc-Wen model

Wen has developed a model to describe the restoring force in a system with hysteresis (Wen, 1976), (Wen, 1980) based on a model first introduced in (Bouc, 1967). According to the model the following relation describes the total restoring force in a hysteretic system:

F ( x, x& ) = g ( x, x& ) + z ( x(τ ),τ = ] − ∞,τ ] ) where g is the non-hysteretic component and z is the hysteretic component. The value of z is governed by a first-order differential equation:

z& = Ax& − α ⋅ x& ⋅ z

n −1

n

− β ⋅ x& ⋅ z , with z(0)=0

where A, α, β, and n are parameters to be determined. The parameter A corresponds to the initial stiffness of the joint and the other parameters govern the shape of the hysteresis loop. Wen concluded that this mathematical model may be used to describe a very wide range of hysteretic behaviours. Oldsfield studied a particular case, rotation of an isolated joint, and used this model to describe hysteretic behaviour (Oldsfield, 2003). Detailed finite element modeling of joints

As the modelling capabilities of available FE software have increased, several attempts have been made to model the physics in joints with detailed FE-models. Notably the contact pressure distribution and the stick-slip state in the contact surface during loading may be studied with detailed FE models. In theory this permits computation of the frictional dissipation. In the 1990’s several two dimensional FE-models appeared in various publications. Generally these models correlate well with the analytical theories of two dimensional frictional problems and some of the experimentally investigated non-linearities may be predicted. See for example Lobitz numerical study (Lobitx, 2001) of the experimental investigations conducted by (Gregory, 1999) and (Smallwood, 2000) or (Gaul, 1993), where slip regions and power-law dissipation for a simple jointed resonator are correctly predicted. The two dimensional models are however of limited practical interest since so few real joints exhibit the plane stress or strain state and furthermore it is required that the load is applied in particular directions. The two dimensional models generally fail to quantify the dissipation even for very simple joints.

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Three dimensional models including friction have been used in order to investigate the ultimate failure of open bolted joints (Bursi, 1997) and regular lap joints (Chung, 2000). These authors found acceptable agreement in force-displacement characteristics under monotonic loading up to the point of maximum force. Pratt studied a conical-head bolted lap joint with three dimensional finite element models and compared force-displacement characteristics with experiments during cyclic loading (Pratt, 2002) using a rather coarse mesh but still obtaining seemingly good results. The three authors use three different solvers with slightly different numerical implementation of the frictional interaction and they all stress the importance of correctly choosing the analysis parameters. This is clearly an area of ongoing research.

Figure 8. a) Details of the finite element model used by Bursi, (Bursi, 1997). b) Details of the finite element model used by Chung, (Chung, 2000). c) Details of the finite element model used by Pratt, (Pratt, 2002).

Although the detailed three dimensional FE models seem to be able capture many of the important physical phenomena in the joints the extensive demand of computational resources is a restraint, particularly so for analysis of dynamic processes. To overcome this problem studies have been made to use the detailed FEmodels in quasi-static simulation to compute simplified joint model parameters. Oldsfield studied a detailed FE-model of an isolated joint and used the results from static FE-simulations to design a parallel Iwannetwork and a Bouc-Wen model with similar properties, (Oldsfield, 2003). Wentzel used a detailed FE-model to compute the energy dissipation in joints during loading and computed an equivalent modal viscous damping for the global structure (Wentzel, 2005). Dynamics of joined structures and future development

The primary aim of simplified joint models is to permit simulation of dynamic processes. Distinction is made between cases where only the steady-state solution to a particular harmonic loading is sought for and cases where the actual time history of a long dynamic process is to be determined.

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Steady state solutions can be computed analytically for systems with a very limited number of DOF's and Coulomb friction (Nosonovsky, 2004). This is of limited interest in vehicle systems where steady state but rarely is reached and the models considered often are complex with many degrees of freedom. In transient response analysis a common technique is to use some kind of reduction method. One reduction technique is Gyuan reduction (Gyuan, 1965) where the mass of the internal points is neglected and only the external points of the structure are retained. A more common variant is Component Mode Synthesis (CMS), (Bampton, 1968), (Hurty, 1971). The CMS method retains a few internal DOFs such that the lowest internal mode shapes are retained, thus not completely neglecting the mass of the internal points. The method makes use of a transformation matrix composed by the local eigenvectors. The constituting parts of the global system that are approximately linear are reduced so that only the external (and a few internal if CMS is used) DOFs are retained. A linear model (constant stiffness and mass matrices) denoted linear substructure is replacing the reduced parts. The different linear substructures may be interconnected to each other via non-linear joint models such as the Iwan-, Valanis-, or BoucWen-model. It has also been envisaged to connect linear substructures to a detailed (not simplified) joint model (Gaul, 1993). However, not much is gained with this approach since it is the computation of the dynamic frictional interaction that is computationally expensive. In the foreseeable future it remains interesting to model the complex joint mechanics with simplified models. This is particularly important for applications where the actual joint is not the primary aim of the study, but where the forcedisplacement characteristics are needed in order to compute the structural response. This may for example be time-domain dynamic analyses of large joined structures. Of particular interest in the future are applications where multiple modes are excited simultaneously. Multiple mode vibration complicates joint mechanics even more and has scarcely been treated at all. Another area where the industry demands improvement is in the parameter estimation of the simplified models. Most of the authors in the scientific literature on joint models use experimental data to find suitable model parameters. Some attempts have been made to extract parameters for simplified models from detailed FE-models, (Oldsfield, 2003), (Wentzel, 2005). This approach will, if proven successful, have substantial impact in the industry.

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Concluding remarks

The force displacement characteristics of frictional joints play an important role in the dynamic behaviour of joined structures. Depending on the available test resources and knowledge of the system different joint models are available. In general, for simulations where the primary interest is to model the dynamics far from the joint, a phenomenological description of friction or damping is generally apt to provide satisfactory results. However it is necessary to further investigate the behaviour of these simplified models in systems with multiple simultaneous modes of vibrations. Detailed FE-models may in many cases capture the important characteristics during static or quasi-static loading. If these detailed models may be used to estimate parameters for simplified joint models much is won. The ongoing research on simplified joint models has so far provided several interesting alternatives of which Iwan networks is perhaps the most promising. The need for predictive joint models in the industrial product development phase and simplified joint models for dynamic simulations will continue to drive the development for years to come. References

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