A complete strategy for efficient and accurate multibody dynamics of flexible structures with large lap joints considering contact and friction
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Abstract
This paper deals with the dynamics of jointed flexible structures in multibody simulations. Joints are areas where the surfaces of substructures come into contact, for example, screwed or bolted joints. Depending on the spatial distribution of the joint, the overall dynamic behavior can be influenced significantly. Therefore, it is essential to consider the nonlinear contact and friction phenomena over the entire joint. In multibody dynamics, flexible bodies are often treated by the use of reduction methods, such as component mode synthesis (CMS). For jointed flexible structures, it is important to accurately compute the local deformations inside the joint in order to get a realistic representation of the nonlinear contact and friction forces. CMS alone is not suitable for the capture of these local nonlinearities and therefore is extended in this paper with problemoriented trial vectors. The computation of these trial vectors is based on trial vector derivatives of the CMS reduction base. This paper describes the application of this extended reduction method to general multibody systems, under consideration of the contact and friction forces in the vector of generalized forces and the Jacobian. To ensure accuracy and numerical efficiency, different contact and friction models are investigated and evaluated. The complete strategy is applied to a multibody system containing a multilayered flexible structure. The numerical results confirm that the method leads to accurate results with low computational effort.
Keywords
Flexible multibody dynamics Joint contact Dry fiction Model order reduction1 Introduction
Complex mechanical structures commonly consist of different substructures connected by joints. In this article, the term “joint” is used for the region where two substructures interact with each other. Inside the joint area, nonlinear contact and friction forces occur, which can strongly influence the overall dynamic behavior of such structures. This is especially true for structures that include joints with a large spacial distribution. In this work, only joints where the relative displacement of the two involved surfaces with respect to each other remains small are of interest. Such joints, for example, bolted joints, spot welded seams, and joints, are of technical relevance.
In engineering practice, structures that include nonlinear contact and friction forces are commonly analyzed with the direct finite element method (FEM). If the jointed structure needs to be considered in a multibody simulation where it can be connected to other rigid or flexible bodies and undergo a large rigid body motion, then a different strategy must be applied. Linear flexible bodies are often considered using component mode synthesis (CMS) [1, 2, 3, 4, 5] in combination with the floating frame of reference formulation (FFRF) [6, 7] in the multibody simulation (MBS). For local nonlinearities, like in jointed structures, the trial vectors provided by CMS are not suitable to capture small local deformations and, consequently, the local contact pressure inside the joints. Therefore, in [8] a problemoriented extension of the common reduction base is presented using special joint trial vectors (JTVs) that are based on trial vector derivatives (TVDs). These JTVs add local flexibility to the joint area to ensure an accurate computation of the gapping or penetrating areas inside the joint. TVDs have already been used in [9, 10, 11, 12] to extend the common reduction base in order to capture nonlinear effects. In [12] TVDs are also used in the context of MBS for an efficient reduction of structures showing geometric nonlinearities.
In addition to the JTVs, the contact and friction models are also of high importance for the efficient consideration of structures with lap joints in the MBS. In order to produce realistic results within acceptable computational time, the models must combine the properties of accuracy and numerical efficiency. Systematic research on jointed structures has been carried out by Gaul and his associates [13, 14, 15]. Their contributions are based on an experiment with an isolated generic joint, which has been systematically investigated. The tangential stiffness and damping properties of the joint were studied with respect to normal pressure, excitation frequency, and excitation amplitude. Based on the experimental results, proper mathematical models were deduced from literature and applied to certain problems. The conclusions drawn from these investigations are summarized in the review paper [16]. It was revealed that a Coulombtype friction model is able to describe the main characteristics of a lap joint with dry friction. This observation has been shared by other authors as well, and examples of variations of the Coulomb fiction models (e.g., Iwan model) are given [17, 18]. In [18] a jointed beam is compared to a monolithic structure with the same geometric dimensions, and the results underline the importance of the proper consideration of joints in dynamic simulations. For the local energy dissipation, the local contact pressure is of significant importance. Consequently, a varying normal pressure distribution throughout the joint due to varying deformations must be regarded for accurate dynamics [19].
The required flexibility inside the joint area of the reduced structure is achieved by extending the reduction base with special JTVs. Furthermore, an investigation into contact and friction models with a focus on numerical efficiency is performed to ensure a low overall computational time.
After a general problem description and introduction in Sect. 1, the paper continues in Sect. 2 with a short review of the properties of dry friction joints. Out of this review, the physical requirements on the friction model are defined. In Sect. 3 the theory of JTVs in the context of multibody simulation and the FFRF will be given. Section 4 discusses how nonlinear contact and friction forces inside a joint are considered in the vector of generalized forces together with their contribution to the Jacobian. Following that, a comparison of different contact models (Sect. 5.1) and friction models (Sect. 5.2) in terms of numerical efficiency and accuracy is given. Nonpenalty methods, for example, the Lagrange method or the augmented Lagrange method [20] are not included in this investigation. This section ends with a numerical investigation of the friction models on a simple multimass oscillator. A numerical example of a multibody system (car pendulum model with multilayer sheet metal structure) is presented in Sect. 6, where the suggested JTVs and contact models are investigated in more detail. Finally, a discussion and some conclusions are given in Sect. 7 and Sect. 8.
2 Brief review of lap joints with dry friction

Gaping: The involved surfaces are not in contact at the considered point. Consequently, there is no friction stress.

Sticking: The involved surfaces at the considered point are in contact, and the local friction stress is lower than a certain sticking stress limit. The sticking stress limit typically depends on the contact pressure and a friction coefficient. During sticking, no energy is dissipated, and the relative displacement of the two surfaces is zero or elastic (reversible).

Slipping: The involved surfaces at the considered point are in contact, and the local friction stress is higher than the sticking stress limit. Slipping leads to energy dissipation because the relative displacement of the involved surfaces is irreversible. The frequency dependency of the energy dissipation is very small and can be neglected. Note that also in the case of local sliding, the small displacement assumption still holds due to the construction of the joint.

The sticking condition is satisfied at each location throughout the joint.

The joint is partially slipping and sticking. In the literature, this state often constitutes microslip.

There is no sticking subarea, and the entire joint is slipping. This state is commonly referred to macroslip. Note that in such a case, the former restriction of small relative displacement of the joint surfaces with respect to each other no longer holds. For many types of joints, the appearance of macroslip indicates joint failure. Therefore, the macroslip state lies outside the scope of this paper.
3 Joint trial vectors based on trial vector derivatives
The computation of contact and friction forces is based on the deformation of the joint area. Classical trial vectors for model reduction, for example, CMS, do not describe the deformation inside the joint accurately enough. In this section a problemoriented extension of classical reduction bases with the purpose of getting sufficient accuracy in the joint deformation is discussed. Based on an accurate deformation of the joint, the contact and friction forces can be precisely computed.
Such an extension was introduced by Witteveen and the author [8] for the FEM and is based on the computation of TVDs. This approach is extended to the MBS in this and the following sections.
A different strategy, which also leads to a set of new trial vectors that do not contain rigid body content, is proposed in [26]. This strategy separates a set of arbitrary trial vectors into pseudofreesurface modes and rigid body modes.
It should be mentioned that the computation of TVDs and JTVs is not restricted to fixed interface trial vectors. In [8] a remark on the computation of TVDs for free interface methods (e.g., the Rubin method [2, 27]) can be found.
4 Contact and friction forces in the framework of MBS
Because the vector of nonlinear forces contributes to the vector of generalized forces \(\mathbf{Q} \), the derivatives \(( \boldsymbol{\Phi } ^{i\,\mathsf{T}} \mathbf{f}_{\mathrm{nl}}^{i} ) _{\mathbf{q}}\), \(( \boldsymbol{\Phi } ^{i\,\mathsf{T}} \mathbf{f}_{\mathrm{nl}}^{i} ) _{\dot{\mathbf{q}}}\) need to be computed for the Jacobian. Note that the superscript \(i \) marking the body index in the MBS is omitted from now on. As a consequence of the penalty formulation for contact forces and the claimed velocity independence of the friction model, the derivation with respect to the generalized velocities, \(( \boldsymbol{\Phi }^{\mathsf{T}} \mathbf{f}_{\mathrm{nl}} ) _{\dot{\mathbf{q}}} \) can be set to zero.
Because of the timeinvariant matrices \(\boldsymbol{\Phi }\) and \(\mathbf{P}\), the derivatives \(\frac{\partial \mathbf{p}_{N}}{\partial \mathbf{q}_{f}}\), \(\frac{ \partial \boldsymbol{\tau }_{F}}{\partial \mathbf{q}_{f}}\) can be computed for the derivatives needed in Eq. (18). The computation of these derivatives will be further discussed in the following subsections.
For simpler reading, only a nodetonode contact is considered, and the derivatives are only computed for a single contact pair. A general extension to the vector containing all contact pairs is quite straightforward.
4.1 Computation of the contact pressure
In Sect. 5 the derivative \(\frac{\partial p _{N,i}}{\partial g_{N,i}}\) for some selected penalty models is computed.
4.2 Computation of the friction stress
5 Selection of contact and friction models based on efficiency criteria
The reduction base presented in Sect. 3 shows an efficient way of getting an accurate representation of the deformation inside a joint during dynamic simulations. To ensure low computational effort for the entire computation process, it is also important that the contact and friction models are numerically efficient. In addition to the numerical efficiency, it is also important that the models capture the physical characteristics of the joint. Based on these requirements, in the following sections, different contact and friction models are evaluated, and a recommendation is given.
5.1 Investigation of contact models
Contact models have to ensure that the contacting surfaces do not penetrate (or at least not perceptibly). Furthermore, the model should give a realistic representation of the contact mechanics and the contact pressure with (few) physically meaningful parameters. In [32] a review and comparison of different contact force models for contact and impact in multibody dynamics can be found, but these do not consider lap joints inside flexible structures or numerical efficiency. In the following subsections, different penalty contact models are investigated. For readability reasons, the derivative \(\frac{\partial p_{N,i}}{\partial g_{N,i}}\) for different models is given in Appendix A.
5.1.1 Linear penalty model
5.1.2 Multistage linear penalty model
5.1.3 Powerfunctionbased nonlinear penalty model
5.1.4 Combined quadraticlinear penalty model
5.1.5 Exponential penalty model
5.1.6 Jointadapted exponential penalty model
5.1.7 Comparison of contact models
Comparison of contact models
(A)  (B)  (C)  (D)  (E)  (F)  

simplicity of mathematical description  +  ∘  +  −  ∘  ∘ 
number of parameters  1  ≥3  2  2  3  3 
physical interpretation of parameters  +  +  −  ∘  ∘  ∘ 
contact mechanical interpretation  −  −  +  +  ∘  + 
continuity of slope  −  −  +  +  +  + 
The exponential and the power function based pressuregap relationship can be associated with the Hertzian contact theory by using statistical models for the roughness of the surfaces. For all other models, such a contact mechanical interpretation is not possible.
A study investigating the numerical efficiency of the different models follows in Sect. 6.2.
5.2 Investigation of friction models
In the sections that follow, different models for calculating the magnitude of the friction stress \(\tau_{F,i}\) found in the literature are reviewed and compared to the characteristics of dry fiction mentioned in Sect. 2. A good starting point for a literature review on friction models is given by [16, 44, 45]. An investigation of alternatives to the Coulomb friction model can also be found in [46], where especially the continuity of the models is investigated. The focus in this paper is on friction models for dry friction inside joints that can capture the microslip range mentioned in Sect. 2. Therefore, friction models describing sliding friction (LuGre friction model [47], signumfriction models, Karnopp model, etc.) are not further discussed here. The interested reader is referred to [44, 45, 48, 49].

Two different nonzero slopes for the stick and slip regime.

No energy dissipation in case of sticking.

Frequencyindependent energy dissipation.

Physically reasonable parameters (as few as possible).
5.2.1 Threeparameter Coulombtype friction model
The threeparameter Coulombtype friction model mentioned in Sect. 2 is used as a reference model for the computation of dry friction inside joints. The parameters of this model are the sticking stress limit \(R_{G} \) and two stiffness parameters \(c_{1}\), \(c_{2}\) for the slope of the sticking and slipping motion.
The reference model is implemented using a trial frictional stress \(\tau_{F}^{\mathrm{tr}} \) and decomposing the tangential displacement into elastic (reversible) and plastic (irreversible) components \(s = s ^{e} + s^{p} \) as discussed in [50].
5.2.2 Adapted Dahl friction model
The Dahl friction model [51] is a simple friction model, which was developed in 1968. This model implies that the friction stress is only a function of the tangential displacement. The displacement dependency of the friction stress \(\frac{d \tau_{D}}{d s}\) was described in further studies [52] by Eq. (B.1), where \(\sigma_{0}\) is a stiffness parameter, and \(\alpha \) is a model parameter.
In this form the Dahl friction model is well documented, for example, in [44, 45]. In order to get the two required stiffness regimes for sticking and slipping, it is necessary to extend the Dahl friction model by a parallel linear spring in the form of Eq. (B.3).
The Dahl friction model has three parameters \(\left( \sigma_{0}, c _{2},\alpha \right) \) in the discussed form. The two stiffness parameters can be easily related to the reference model as \(\sigma_{0} = c_{1}; c_{2} = c_{2}\). It must be mentioned that, for \(R_{G} \rightarrow 0\), numerical problems and a division by zero can occur. Therefore, precautions must be applied to capture this case.
5.2.3 Valanis friction model
The Valanis model, originally known from plasticity theory, is used as a friction model in [13, 15, 16]. In [15] a detailed derivation of the original plasticity model to the Valanis friction model can be found. The final equation for the friction stress is given by the differential equation (B.4) with four model parameters \(E_{0}\), \(E_{t}\), \(\kappa \), \(\lambda \).
A physical interpretation of the parameters for joints is possible. A detailed investigation of the parameters in [15] results in the conclusion that \(E_{0} = c_{1} + c_{2}\) represents the sticking stiffness and \(E_{t} = c_{2}\) the sliding stiffness analogous with the reference model. The parameter \(\kappa \) influences the transition between sticking and sliding and therefore the shape of the hysteresis curve. For a physical meaningful hysteresis \(( E_{0}>E_{t} ) \), the parameter has to be chosen between \(0<\kappa <1\). The best fit of the hysteresis curve compared to the reference model can be achieved with high values of \(\kappa \). The parameter \(\lambda \) can be set into context to the other parameters by Eq. (B.6).
5.2.4 Bouc–Wen friction model
5.2.5 Viscous damping models
5.2.6 Comparison of the friction models
Comparison of friction parameters
Reference  Dahl  Valanis  Bouc–Wen  Viscous  

Physical parameters  \(R_{G}\), \(c_{1}\), \(c_{2}\)  \(R_{G}\), \(\sigma _{0}\), \(c_{2}\)  \(E_{t}\), \(E_{0}\), \(\lambda _{0}\)  A  – 
Model parameters  –  α  κ  B, γ, n  d, \(c_{1}\), \(c_{2}\) 
Comparison of friction models
Reference  Dahl  Valanis  Bouc–Wen  Viscous  

simplicity of mathematical description  ∘  ∘  –  –  + 
sticking/sliding  +  ∘  +  +  – 
no energy dissipation during sticking  +  –  ∘  ∘  – 
frequency independent  +  +  +  +  – 
few physical parameters  +  ∘  ∘  –  – 
5.2.7 Evaluation of friction models on a multimass oscillator
Cases multimass oscillator
Gaping  Varying \(R_{G} \)  Sticking/sliding  Pure sticking 

\(R_{G}^{*} = \frac{R_{G0}}{2} \cos ( \frac{3 \pi }{2} t ) \) \(R_{G}= \begin{cases} R_{G}^{*} &\mbox{if } R_{G}^{*} >0 \\ 0 &\mbox{if } R_{G}^{*} \leq 0 \\ \end{cases} \)  \(R_{G}= \frac{R_{G0}}{2} \cos ( \frac{3 \pi }{2} t ) + R_{G0} \)  \(R_{G}= R_{G0} \)  \(R_{G}= R_{G0} \) 
\(R_{G0} =0.15~\mbox{N} \)  \(R_{G0} =0.15~\mbox{N} \)  \(R_{G0} = 0.15~\mbox{N} \)  \(R_{G0} = 5~\mbox{N} \) 
None of the investigated cases shows a qualitatively different result in terms of numerical efficiency than the averaged results shown in Fig. 13. As a result of the findings summarized in Table 3 and the numerical investigations on the multimass oscillator, the threeparameter Coloumbtype friction model (reference model) seems to be the best choice for the modeling of dry friction inside jointed structures although it is not continuous.
6 Numerical example—putting it all together
Model parameters: case FORCE
\(m_{c} = 0.95~\mbox{kg}\)  \(m_{pm} = 0.2~\mbox{kg}\) 
\(m_{p} = 0.157~\mbox{kg}\)  \(k = 1000~\mbox{N}/\mbox{m}\) 
\(f_{0} = 5~\mbox{N}\)  \(g = 0~\mbox{kg}\,\mbox{m}/\mbox{s}^{2}\) 
Model parameters: case GRAVITY
\(m_{c} = 0.95~\mbox{kg}\)  \(m_{pm} = 0.2~\mbox{kg}\) 
\(m_{p} = 0.157~\mbox{kg}\)  \(k = 1000~\mbox{N}/\mbox{m}\) 
\(f_{0} = 0~\mbox{N}\)  \(g = 9.81~\mbox{kg}\,\mbox{m}/\mbox{s}^{2}\) 
The differential algebraic equations describing the multibody system as defined by Eq. (12) and Eq. (13) are solved with a modified HHT solver [30, 31] written in Scilab [54]. The algorithm is very similar to the solver presented in [31].
6.1 Evaluation of the reduction base
Without the use of additional JTVs, the contact forces are distinctly overestimated. The contact forces computed with 60 JTVs are very close to the converged solution, and, therefore, all further computations are executed with 60 JTVs. For the practical application, also 20 JTVs would suffice, as the results in Sect. 6.3 show. Comparing the convergence of JTVs to the use of additional normal modes shows that with 20 JTVs the results are much closer to the converged solution than the results with 20 additional NMs. These results are in agreement with [8], where also a comparison to a full nonlinear computation is given.
In terms of computational time, it can be reported that the simulation with 60 JTVs needs only 0.2 % of the CPU time required for the simulation where all 903 contact pairs are considered separately.
6.2 Numerical investigation on contact models
Penalty parameters
(A)  \(\varepsilon _{N} = 400~\mbox{N}/\mbox{mm}^{3} \)  
(B)  \(\varepsilon _{N,1} = 667~\mbox{N}/\mbox{mm}^{3}\) \(\varepsilon _{N,0} = 133~\mbox{N}/\mbox{mm}^{3}\)  \(g_{0} = 0~\upmu\mbox{m}\)  \(g_{1} = 1.25~\upmu\mbox{m}\) 
(C1)  \(\varepsilon _{N}= 0.16~\mbox{N}/\mbox{mm}^{2}(\upmu\mbox{m})^{2}\)  m = 2  
(C2)  \(\varepsilon _{N}=4.86 \mathrm{e}^{2}~\mbox{N}/\mbox{mm}^{2}(\upmu\mbox{m})^{3.3}\)  m = 3.3  
(D)  \(p_{0}=1.23 \mathrm{e}^{4}~\mbox{N}/\mbox{mm}^{2}\)  \(g_{0} = 0.4~\upmu\mbox{m}\)  
(E)  \(\varepsilon _{N}=5.7 \mathrm{e}^{1}~\mbox{N}/\mbox{mm}^{2}(\upmu\mbox{m})\)  \(g_{0} = 0~\upmu\mbox{m}\)  \(g_{1} = 1.5~\upmu\mbox{m}\) 
(F)  \(\varepsilon _{N}=617~\mbox{N}/\mbox{mm}^{3}\)  \(g_{0} = 0~\upmu\mbox{m}\)  \(\lambda _{N} = 200~\mbox{mm}^{1}\) 
Considering the averaged results (Fig. 18a) of the numerical investigations, it seems that all models that have a smooth transition in the pressuregap relationship tend to have lower computational effort. Together with the comparison of Table 1, the jointadapted exponential model seems to be the best choice for computing the contact pressure inside jointed structures.
6.3 Car pendulum with friction
The figures clearly show the damping influence of the dry friction. In both figures, the results for using additional 20 NMs, 20 JTVs, and a converged solution with 60 JTVs are compared. It can be seen that the overestimation of the contact forces by the use of additional NMs leads to a different dynamic behavior of the flexible structure and the car. The solution with 20 NMs deviates from the converged solution in the amplitude and in the phase of the motion. Even in this configuration, where the point mass is directly excited by the external force, the accurate consideration of the nonlinear joint forces influences the displacement significantly. Note that this effect might be even more observable in the threedimensional case.
7 Discussion
This paper describes the use of JTVs based on TVDs to extend the classical reduction base for model reduction. In context with FEM, this method is compared to other possibilities for model reduction of nonlinear mechanical systems in [8], whereas the focus of this discussion is on flexible multibody dynamics. An alternative approach for an extension of the reduction base for jointed structures is given in [24]. The advantages of JTVs based on TVDs compared to the trial vectors found in [24] are faster convergence in terms of additional trial vectors (especially for multilayered complex structures) and easier implementation of the algorithm. TVDs are also used in [12] for the model reduction of flexible multibody systems with geometric nonlinearities. In [12] the TVDs are directly used for the extension of the reduction base, which is a major difference to the JTVs, which are computed via POD out of all TVDs. Furthermore, the TVDs for geometric nonlinearities are symmetric, whereas for contact problems, they are not \((\frac{\partial \boldsymbol{\varphi}_{i}}{\partial q_{f,j}} \neq \frac{\partial \boldsymbol{\varphi}_{j}}{\partial q_{f,i}} ) \).
Different contact models can be found in the literature as mentioned in Sect. 5.1. The contact model preferred in this paper is an adaption of an exponential model and based on the usage of the exponential distribution describing the asperity of a metallic surface. In [38] a normal distribution is supposed for the height distribution of the summits a rough surface. This leads to a pressuregap relationship, which is similar to the power function based model in Sect. 5.1.3. The idea of using statistical models for describing asperity heights is also used in [55]. In this paper it is also mentioned that the height distribution of the asperity summits tends to be rather a normal distribution than an exponential distribution, but the latter is sufficient to describe the uppermost 25 % of the asperities of most surfaces.
The work of Gaul and his associates [13, 15] has focused on the investigation of dry friction and finding alternatives to the Coulomb friction model. In their contributions, however, the continuity and numerical efficiency of the models was not investigated. In [46] the continuity of different models is studied in detail, but the computational effort of the different model was not considered. The difficulties with the discontinuity of the Coulomb model in dynamic simulations are discussed in [56]. Nevertheless, the results of the numerical investigations showed that the continuity of the used friction model is of minor importance when the overall numerical efficiency is evaluated.
For preloaded structures (e.g., clamped joints), the deformation and stress state can be dominated by the preload force even through a dynamic simulation. For such structures, it seems to be a promising strategy to consider the preload state for the computation of the reduction base to achieve further decrease of the number of trial vectors. The Taylor series expansion of statedependent trail vectors leads in an intuitive way to the consideration of the preload state in the TVDs. Therefore, the use of JTVs based on TVDs for considering preloaded structures will be investigated in future work.
8 Conclusions

Accurate representation of the joint deformation with JTVs

Efficient computation of the contact forces

Efficient modeling of the dry friction characteristics
The numerical example in Sect. 6 confirms that the extension of the reduction base with JTVs based on TVDs leads to an accurate description of the joint deformation. Furthermore, the study shows that the computational time for a simulation with the number of additional JTVs required for a converged solution is significantly lower than for a simulation considering all contact pairs in the joint area.
The joint adapted exponential contact model presented in Sect. 5.1.6 is recommended for computing contact forces. The model turned out to be numerically efficient, and it has the advantage that it can be derived from some contact physical background.
Different friction models that capture the characteristics of dry friction were evaluated. Although the threeparameter Coulombtype friction model is not continuous, it turned out to be numerically very efficient and also fulfills all other defined criteria for a dry friction model.
Notes
Acknowledgements
Open access funding provided by University of Applied Sciences Upper Austria. The author was supported by a researcher development program financed by the government of Upper Austria.
Furthermore, this project was supported by the program “Regionale Wettbewerbsfähigkeit OÖ 2010–2013”, which is financed by the European Regional Development Fund and the government of Upper Austria.
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