Application of the MEBDFV solver to dynamic simulation of some specific multibody systems: an example of a cervical spine model
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Abstract
Application of the MEBDFV code in multibody dynamics is discussed. The solver is based on the modified extended backward differentiation formulae of Cash. It is especially suited for the solution of differentialalgebraic equations with time and statedependent mass matrix. Such a case can occur when motion of a multibody openchain system is described within the classical Lagrangian formalism, which still has some advantages. An outline of the numerical algorithm is given. As an example, a simplified mathematical model of the human headcervical spine system is presented. In a numerical experiment the model is used to analyze motion of the head–neck during rearend impact which may lead to whiplash injury. The obtained results are compared to literature data. Performance of the MEBDFV solver is examined in terms of algorithmic energy conservation. The test is based on an appropriately formulated ‘kinetic energy–work’ relation.
Keywords
Multibody dynamics Lagrangian formalism Discrete model Implicit ordinary differential equations Cervical spine Whiplash1 Introduction
Multibody dynamics is one of the most rapidly developing branches of computational mechanics. The specific area is strictly based on wellknown concepts of classical and analytical mechanics such as D’Alembert’s principle, Newton–Euler equations, Lagrangian formalism, etc. As an extension of these approaches some generalized methodologies for formulating dynamic equations have appeared over the past few decades. These modern techniques are especially suitable for computer implementation and can be applied to a vast class of multibody systems. In general there are two basic formulations which are widely used: the embedding techniques and the augmented formulation [1, 2]. Both of them have adavantages and disadvantages related to computational issues like number of equations, their nonlinearity and complexity. Usually the techniques involve applying the free body principle in order to obtain the equations of motion.
However, in some cases purely analytical methods seem to be preferable. The typical examples are conventional and inverted multiple pendula which over the last few decades have been the subject of theoretical and experimental studies mainly from the viewpoint of nonlinear dynamics and control theory (see e.g. [3, 4, 5, 6, 7, 8, 9, 10]). In general, multibody systems with a chain topology are especially likely to be modelled via the standard Lagrangian formulation. Basically, by the term ‘standard formulation’ the authors mean applying the Lagrange’s equations of the second kind, where all the generalized coordinates are independent and the dynamic equations do not involve constraint forces, e.g. the ones which act at the joints connecting rigid members. In such a case the standard approach can be regarded as an embedding technique. However, if some additional geometric constraints are imposed on the system (for instance, the other end of the chain is fixed), the former equations of motion are still useful. In the method of Lagrange multipliers, for example, the primary model can be extended by introducing constraints equations and undetermined multipliers which are related just to the new reaction forces [11, 12]. This procedure, in turn, corresponds to the augmented formulation.
There are some advantages of the standard approach to the openchain systems. Firstly, minimal or almost minimal (if the additional constraints exist) set of coordinates is used to describe a system. Moreover, the internal (connection) forces are not included in the dynamic equations. Both the factors reduce the problem dimensionality. Secondly, the equations of motion can be derived from a few scalar quantities specific for a certain system: the kinetic energy, potential energy and dissipation function. This feature is crucial especially when dealing with a system of bodies interconnected by many coupling elements like springs, dampers and actuators.
Although rarely described, among commonly available IVP solvers there are a few which are suited for solving IDEs [15]. In this paper the code MEBDFV designed by T.J. Abdulla and J.R. Cash is employed. We discuss its application to computer simulation of the openchain multibody systems. In particular, we consider a simplified mathematical model of the human headcervical spine system. The aim of this paper is to show that equations of motion in the implicit form can be solved satisfactorily and the standard Lagrangian approach, readily used in the field of theoretical mechanics, is not doomed to failure.
2 The MEBDFV code for multibody dynamics problems
2.1 General form of dynamic equations and a numerical model
2.2 The origins of the modified extended BDF
As can be seen, the system of IDEs (3) is a special case of DAEs (8). Nevertheless, the problems that involve both differential and algebraic equations are more demanding numerically. Similar to the case of stiff systems of ODEs, methods used for DAEs have to satisfy rigorous stability requirements. Other difficulties are related to error estimation and convergence, which enforces more sophisticated strategies in comparison to the methods intended for standard ODEs [16].
Generally, complexity of a problem increases with an index of DAEs, \(i_D\). For example, DAEs (6) are of index 3 for most constrained multibody systems; in the case of ODEs (\({\hat{\mathbf{M}}} = \mathbf{I}\)) and IDEs (\({\hat{\mathbf{M}}} \ne \mathbf{I}\) and \(\det ({\hat{\mathbf{M}}}) \ne 0\)), in turn, \(i_D = 0\). For more details on the concept of DAEs index the reader is referred to [17, 18].
Majority of the codes originated in the 1980’s and 1990’s have been restricted to \(i_D \le 3\) (e.g. DASSL, LSODI, GAMD). Therefore, many methods have been developed to reduce the problem index, e.g. the GGL formulation, the Baumgarte’s technique [18]. Today there are several solvers commonly used for the solution of DAEs of higher index. However, not all of them are suited to the DAE systems with nonconstant mass matrix (2) or the unstructured ones (e.g. BiMD or RADAU5) [15].
The MEBDFV solver is suitable for the numerical solution of DAEs with index \(i_D \le 3\), given in the form (2). The code is based on the modified extended backward differentition formulae (MEBDF) of Cash. In 80’s the researcher proposed a separate class of numerical methods, clearly different from both the standard backward differentition formulae (BDF) and the widely used Runge–Kutta (RK) type methods [19, 20]. The new approach resulted in very efficient schemes with extremely good stability properties. Thus, the MEBDF formulae can be regarded as a third path, aside (i) linear multistep methods, which are cheap to implement but, due to the famous Dahlquist’s barrier, at higher orders have pure stability, and (ii) implicit Runge–Kutta methods with potentially excellent stability but expensive implementation. The origins of the new approach were described by Cash [21].
Like other codes based on MEBDF, designed for initial value problems in ODEs and DAEs of various forms, the MEBDFV solver is available on the Internet [22]. As usual, the code contains a brief description of input and output parameters as well as the subroutines which must be supplied by a user. Needless to say, the information does not give a detailed insight into mechanisms of the numerical integration.
Treated as fundamentals of numerical methods for differential equations, the BDF and RK formulae have been discussed in many books, e.g. [13, 14, 23, 24, 25]. The MEBDF approach, in turn, has not become commonly known as yet. Moreover, issues related to IDEs or DAEs are covered in literature very rarely. Taking into account their importance to computational dynamics and capabilities of the MEBDFV solver, we feel that the MEBDF approach in application to DAEs of the form (2) merits further attention.
2.3 MEBDFV from the inside
This section does not contain a deep analysis of implementation details of the MEBDF. From practical point of view, however, it is important to generally understand computational ideas associated with the code MEBDFV, which prevents from using it in a blackbox manner.
Assume that approximate solutions \(\mathbf{X}_1,\, \mathbf{X}_2,\, \ldots ,\, \mathbf{X}_{i1}\) have been computed at the corresponding step points \(t_1, t_2,\, \ldots ,\, t_{i1}\). The key idea of the MEBDF approach is to find \(\mathbf{X}_{i}\) by using not only the previous values of \(\mathbf{X}\) and \(\dot{\mathbf{X}}\), but also their approximations at the superfuture point \(t_{i+1}\).
 1.
First BDF step
Given the predicted value \(\widetilde{\mathbf{X}}_{i}\), use a standard \(k\)step BDF to evaluate the derivative approximation \(\widetilde{\dot{\mathbf{X}}}_{i}\):Note that \(\widetilde{\dot{\mathbf{X}}}_{i}\) is expressed in terms of \(\widetilde{\mathbf{X}}_{i}\), thus, now (9) becomes a nonlinear algebraic system where \(\mathbf{X}_{i}\) is the only unknown. Find an approximate solution \(\mathbf{X}_{i}^{*}\) and use formula (11) to recompute the derivative, i.e. to obtain \(\dot{\mathbf{X}}_{i}^{*}\).$$\begin{aligned} \widetilde{\mathbf{X}}_{i} + \sum _{j=1}^{k} \alpha _j \mathbf{X}_{ij} = h \beta _0 \widetilde{\dot{\mathbf{X}}}_{i}\,. \end{aligned}$$(11)  2.
Second BDF step
Predict the superfuture value \(\widetilde{\mathbf{X}}_{i+1}\) and use the same BDF as before, but go one step further, i.e. evaluate \(\widetilde{\dot{\mathbf{X}}}_{i+1}\):Insert \(\widetilde{\mathbf{X}}_{i+1}\) and \(\widetilde{\dot{\mathbf{X}}}_{i+1}\) to (9) and find \(\mathbf{X}_{i+1}^{*}\). Use formula (12) again to compute a new value \(\dot{\mathbf{X}}_{i+1}^{*}\).$$\begin{aligned} \widetilde{\mathbf{X}}_{i+1} + \alpha _{1} \mathbf{X}_{i}^{*} + \sum _{j=1}^{k1} \alpha _j \mathbf{X}_{ij} = h \beta _0 \widetilde{\dot{\mathbf{X}}}_{i+1}\,. \end{aligned}$$(12)  3.
MEBDF step
Use both \(\mathbf{X}_{i}^{*}\), \(\dot{\mathbf{X}}_{i}^{*}\) and \(\mathbf{X}_{i+1}^{*}\), \(\dot{\mathbf{X}}_{i+1}^{*}\) to compute a corrected derivative \(\dot{\mathbf{X}}_{i}\) according to the modified extended BDF:One more time solve the system of nonlinear equations (9). The found solution \(\mathbf{X}_{i}\) is the final result of the time integration process.$$\begin{aligned}&\mathbf{X}_{i}^{*} + \sum _{j=1}^{k} \bar{\alpha }_j \mathbf{X}_{ij}\nonumber \\&\quad = h \left[ \bar{\beta }_1 \dot{\mathbf{X}}_{i+1}^{*} + \beta _0 \dot{\mathbf{X}}_{i} + (\bar{\beta }_0  \beta _0) \dot{\mathbf{X}}_{i}^{*} \right] \,. \end{aligned}$$(13)
It should be emphasized that \(\mathbf{J}\) is kept fixed for all \(N\) iterations. What is more, a specific construction of the method allows to use the same Jacobian matrix at each of the stages (1–3). It considerably reduces the computational effort related to evaluation and factorization of \(\mathbf{J}\). In addition, the first and third stages are usually provided with excellent approximations as mentioned before. Therefore, only a few iterations are needed to reach convergence [20, 26].
The discussed implementation of MEBDF goes even further. Indeed, Abdulla and Cash adopted the strategy of updating the Jacobian matrix only when it is indispensable. Thus, an old matrix \(\mathbf{J}\) is used for many time steps, as long as no ‘ill effects’ arise. If convergence is not achieved, for example, the Jacobian is reevaluated and Newton iterations are retried; if still computation fails, the stepsize is halved [20, 26].
Of course, the code MEBDFV employs numerous other techniques and strategies related to such issues as: selecting optimal stepsize, maintaining good convergence of the Newton scheme, estimating the convergence rate, controlling local errors, etc. Nevertheless, these aspects refer to detailed numerical analysis, which goes beyond the overview necessary to comprehend main concepts of the MEBDF implementation.
At the end of this section it is worth noting that, basically, the user supplies the routines which compute the matrix \({\hat{\mathbf{M}}}\) (banded or full), the righthand side vector \({\hat{\mathbf{F}}}\) and \(\mathbf{J}\) (banded or full). If the matrix of partial derivatives \(\mathbf{J}\) is to be evaluated numerically, a routine computing the residual delta \({\hat{\mathbf{F}}}\) should be specified instead.
3 Biomechanical system and equations of motion
3.1 Human cervical spine and its models
The cervical spine consists of seven vertebrae (denoted by C1–C7); each of them has different shape and geometric parameters. The main role of this spine column is to carry the weight of a head and ensure its appropriate mobility. For every vertebra (excluding C1), a vertebral body is the part responsible for carrying load resulting from muscles and the head. Between adjacent vertebral bodies (excluding C1–C2) there is an intervertebral disc which, thanks to its flexibility, stabilizes them and responds to compressive forces. Additionally, unproper mobility of the particular parts is restricted by ligaments which couple the adjacent vertebrae [27, 28].
Since it is difficult to precisely describe mechanical properties of bone tissue and soft tissues, one can find various mechanical models of certain segments of the spine. One of the earliest models of the head/neck system based on multibody dynamics was proposed by Deng and Goldsmith [29]. The authors modeled vertebrae as rigid bodies and included intervertebral connections, muscles and ligaments with nonlinear characteristics. They studied dynamics of the system during road accidents. Dauvilliersa et al. [30] proposed a model with linear charactristics of ligaments and elastic, isotrophic intervertebral discs. De Jager et al. [31, 32] developed the model of Deng and Goldsmith by assumption that the head and vertebrae are rigid members but discs are represented through viscoelastic joints with nonlinear characteristics determined experimentally. Yang et al. [33] analyzed motion of the cervical spine in sagittal plane leading to spinal injuries. In this model both elastic and viscoelastic nature of the ligaments was considered. Van der Horst [34] extended the de Jager’s model by describing the discs and ligaments as viscoelastic nonlinear elements. Van Lopik and Acar [35] presented a threedimensional multibody model of the head and neck for the analysis of whiplash motion. Apart from the discs and ligaments they took into account facet joints too.
In many classical models of the human spine, different approaches to modeling of bone tissue and soft tissue were used. In study of the cervical spine dynamics usually vertebrae are treated as rigid bodies. Where it comes to ligaments, one can find numerous models. The simpliest way to reflect the ligaments character is to assume that they are two [36] or threedimensional [37] linear elements. In more advanced approaches the ligaments were considered to be linear elastic elements with the same value of Poisson’s ratio and diverse stiffness coefficient [38]. Dauvillers et al. [30], in turn, modeled the ligaments as linearly elastic elements with damping and used identical stiffness constant for all of them. Some authors proposed nonlinear models [39, 40, 41]. Due to common use of the finite element method in numerical simulations, shell and beam models [33] as well as the spring and axial (link) ones [42] have become popular.
Similar to the ligaments, the intervertebral discs are described in diverse manner. In problems related to cervical spine biomechanics, solid models are widely applied. It is assumed that a disc is a homogenuous elastic body with known Young’s modulus and Poisson’s ratio [38, 39, 43]. In order to take into account the acting loads, the discs were divided into three parts: annulus, annulus fibrosus and nucleus pulposus. In [33, 37] authors proposed discs comprised of two bodies representing the annulus and nucleus. Goel et al. [44] additionally applied a beam model for the annulus fibrosus. In turn, de Jager et al. [40] used viscoelastic linear elements to represent the annulus and nucleus. Kumeresan et al. [42] employed a solid model of the annulus, beam model of the annulus fibrosus; the nucleus was treated as incompressible liquid. In the model of van Lopik and Acar [35] the discs were represented by nonlinear viscoelastic ‘bushing’ constraints.
3.2 Multibody model of cervical spine
It is not the purpose of this section to present a novel and complete (in some measure) model of the cervical spine. Actually, the multibody system discussed below has a twodimensional, simplified nature. However, despite its limited complexity, the model is a good example of application of the standard Lagrangian approach leading to a system of IDEs.

For the sake of simplicity, analysis is restricted to plane motion of the system, i.e. its motion in the midsagittal plane.

The head (H) and seven vertebrae (C1–C7) are treated as rigid bodies connected by joints.

The torso (with the Th1 vertebra of the thoracic spine) plays a role of a base for the system.

The ligaments are represented by elasticdissipative elements of known characteristics.

Effect of musculature on the system motion is neglected.

Geometric and inertial properties of the members as well as attachement points of the elasticdissipative elements are given.
 \(U_i^G\) is the gravitational potential energy of the \(i\)th member:$$\begin{aligned} U_i^G = m_i g y_{Si} \end{aligned}$$(29)
 \(U_{Ai}^S\) is the potential energy of the spring that connects points \(A''_{i1}\) and \(A'_i\):where \(k_{Ai}\) denotes the stiffness coefficient of the spring and \(\xi _{Ai}\) is the distance between \(A''_{i1}\) and \(A'_i\):$$\begin{aligned} U_{Ai}^S = \frac{1}{2} k_{Ai} \xi _{Ai}^2\,, \end{aligned}$$(30)$$\begin{aligned} \xi _{Ai} = \sqrt{(x_{A'i}  x_{A''i1})^2 + (y_{A'i}  y_{A''i1})^2} \end{aligned}$$(31)
 \(U_{Bi}^S\) is the potential energy of the spring that connects points \(B''_{i1}\) and \(B'_i\):where \(k_{Bi}\) denotes the stiffness coefficient of the spring and \(\xi _{Bi}\) is the distance between \(B''_{i1}\) and \(B'_i\):$$\begin{aligned} U_{Bi}^S = \frac{1}{2} k_{Bi} \xi _{Bi}^2\,, \end{aligned}$$(32)$$\begin{aligned} \xi _{Bi} = \sqrt{(x_{B'i}  x_{B''i1})^2 + (y_{B'i}  y_{B''i1})^2} \end{aligned}$$(33)
 \(U_{Oi}^S\) is the potential energy of the torsional spring located in the pivot \(O_{i}\). Much information on determining stiffness of the intervertebral joints of the upper and lower cervical spine was presented by de Jager [32], who derived strongly nonlinear loaddisplacement curves from experimental data. Therefore, instead of defining the energy \(U_{Oi}^S\), we postulate the generalized elastic forcesin the following form [8]:$$\begin{aligned} Q_{Oi}^S =  \frac{\partial U_{Oi}^S}{\partial \varphi _i} \end{aligned}$$where$$\begin{aligned} Q_{Oi}^S = \left\{ \begin{array}{l@{\quad }l@{\quad }l}  M_{Oi} + M_{Oi+1} &{} \quad \text {if}\;\; &{} i = 1,\, 2,\, \ldots ,\, n1 \\  M_{Oi} &{} \quad \text {if}\;\; &{} i = n \end{array} \right. \end{aligned}$$(34)is the elastic moment arising in \(i\)th joint; \(k_{Oi}\) denotes the stiffness coefficient of the spring and \(\theta _i\) is the relative angular coordinate given by$$\begin{aligned} M_{Oi} = k_{Oi} \frac{\tan (\theta _i/2)}{\cos (\theta _i/2)} \end{aligned}$$(35)The resulting characteristics of the intervertebral joints is shown in Fig. 4.$$\begin{aligned} \theta _i = \left\{ \begin{array}{l@{\quad }l@{\quad }l} \varphi _i &{} \quad \text {if}\;\; &{} i=1 \\ \varphi _i\varphi _{i1} &{} \quad \text {if}\;\; &{} i=2,\, 3,\, \ldots ,\, n \end{array} \right. \end{aligned}$$(36)
 the generalized potential force \(Q_i^{p}\) given bywhich has the two basic components:$$\begin{aligned} Q_i^{p} =  \frac{\partial U}{\partial \varphi _i}\,, \end{aligned}$$(40)where \(Q_i^{G}\) results from gravity and \(Q_i^{S}\) is the generalized elastic force including the term (34)$$\begin{aligned} Q_i^{p} = Q_i^{G} + Q_i^{S}\,, \end{aligned}$$(41)
 the generalized nonpotential force \(Q_i^{np}\) which is a sum of the two components: the dissipative forceand a generalized external force \(Q_i^{\text {ext}}\) applied to the member$$\begin{aligned} Q_i^{R} =  \frac{\partial R}{\partial \dot{\varphi }_i} \end{aligned}$$(42)
4 Numerical experiment
In order to present numerical perfomance of the MEBDFV code in dynamic simulations based on the discussed model, whiplash motion of the head–neck system is analyzed. The term ‘whiplash’ is related to one of the most frequent cervical spine injuries which occur mainly in automobile accidents and lead to significaant societal costs. Such an injury results from a sudden, excessive movement of the head with respect to torso, which produces soft tissue damage in the neck. Despite lots of various hypotheses, there is no definitive explanation of the mechanisms leading to the whiplash trauma. Dynamics of the head–neck system during impacts is still studied both experimentally and computationally [35, 45].
Although whiplash motion can be generated in all impact configurations, usually it is a consequence of rearend collisions. Panjabi and coworkers [45, 46] used isolated cervical spine specimens and a benchtop sled apparatus to perform rearimpact simulation. The base of a specimen was subject to horizontal acceleration of 3.5, 5, 6.5 and 8\(g\). The phenomenon was analyzed numerically by van Lopik and Acar [35] who studied peak accelerations of 2.5, 4.5, 6.5 and 8.5\(g\).
Stiffness and damping coefficients of the springdamper elements for \(i=1,\,2,\,\ldots ,\,n\)
Quantity  Unit  Value 

Translational stiffness, \(k_{Ai}\)  N/m  \(10^4\) 
Translational stiffness, \(k_{Bi}\)  N/m  \(3\times 10^4\) 
Rotational stiffness, \(k_{Oi}\)  Nm/rad  \(600.0\) 
Translational damping, \(c_{Ai}\)  Ns/m  \(10.0\) 
Translational damping, \(c_{Bi}\)  Ns/m  \(10.0\) 
Rotational damping, \(c_{Oi}\)  Nm\(\cdot \)s/rad  \(1.0\) 
Initial configuration of the system \(\mathbf{q}(0)\) is specified according to the radiograph. Generalized velocities, in turn, are set to zero: \(\dot{\mathbf{q}}(0) = \mathbf{0}\).
Obviously, better agreement between the simulation and the results presented by other authors requires more complete head–neck model and its validation. Nevertheless, the aim goes beyond the scope of this paper.
From the computational point of view, performance of the solver is essential. It can be examined in many ways, not only by measuring CPU time related to numerical integration of dynamical equations. In case of mechanical systems, the test criterion can be based naturally on the socalled algorithmic energy conservation. It is commonly known that in simulation the given system may artificially gain or loose energy due to numerical inaccuracy. Very often the energy decreases which is referred to as ‘numerical dissipation’. The problem appears in dynamics of rigid bodies and multibody systems modeled via various approaches, even if a set of DAEs with a constant mass matrix is obtained. Hence, many researchers apply certain formulations and/or design specific time integration schemes, which leads to good conservation properties of the resulting algorithm [49, 50, 51, 52].
5 Conclusions
In this paper we have discussed application of the MEBDFV solver to dynamic simulation of multibody systems. The code is particularly useful for openchain systems modeled via the classical Lagrangian formulation, since such an approach leads to a set of implicit differential equations or DAEs with nonconstant massmatrix. The most important concepts of the numerical algorithm have been outlined. As it has been shown, the solver does not use the explicit form of DAEs, thus, when the imposed constraints require the use of Lagrange multipliers, singularity of the mass matrix does not constitute any numerical problem.
As an example, a twodimensional multibody model of the cervical spine has been considered. The proposed system includes bone tissue, intervertebral discs and selected ligaments. The equations of motion have been derived in the framework of the Lagrangian formalism. This approach allows to take into account various models of interactions between spinal vertebrae and skull. Also kinematical excitation of the system may be considered. Consequently, one can deal with some important biomechanical problems like the whiplash trauma. In the numerical experiment we have especially focused on the ligaments forces and their values at failure. Although the presented model is incomplete and its validation has not been conducted, the obtained results are qualitatively similar to the ones reported in other works.
Moreover, the MEBDFV solver performance has been evaluated in terms of the algorithmic energy conservation. Since the system is subjected to rheonomic constraints, the ‘kinetic energy–work’ relation has been applied in an appropriate form, including the relative kinetic energy and the transport inertia forces. The numerical experiment indicates that the solver provides satisfactory results, with low energy inconsistency. Furthermore, it can be easily checked that in case of different test problems the computation process is timeefficient compared to wellknown, twodimensional dynamic simulation environments such as Working Model 2D.
Notes
Acknowledgments
This work has been supported by 21387/2011 DSMK and 21407/2012 DSMK Grants. It is an extension of an earlier paper presented during 12th Conference on Dynamical Systems: Theory and Applications (DSTA 2013).
References
 1.Shabana A (2010) Computational Dynamics, 3rd edn. Wiley, New YorkCrossRefzbMATHGoogle Scholar
 2.Shabana A (2005) Dynamics of Multibody Systems, 3rd edn. Cambridge University, CambridgeCrossRefzbMATHGoogle Scholar
 3.Awrejcewicz J, Kudra G, Lamarque C (2004) Investigation of triple pendulum with impacts using fundamental solution matrices. Int J Bifurcat Chaos 14(12):4191–4213CrossRefzbMATHMathSciNetGoogle Scholar
 4.Awrejcewicz J, Kudra G (2005) Stability analysis and Lyapunov exponents of a multibody mechanical system with rigid unilateral constraints. Nonlinear Anal 63:e909–e918Google Scholar
 5.Lobas L (2005) Generalized mathematical model of an inverted multilink pendulum with follower force. Int Appl Mech 41(5): 566–572Google Scholar
 6.Galán J, Fraserb W, Achesonc D, Champneysd A (2005) The parametrically excited upsidedown rod: an elastic jointed pendulum model. J Sound Vibr 280:359–377CrossRefzbMATHGoogle Scholar
 7.Fritzkowski P, Kamiński H (2009) Dynamics of a rope modeled as a discrete system with extensible members. Comput Mech 44(4):473–480CrossRefzbMATHMathSciNetGoogle Scholar
 8.Fritzkowski P, Kamiński H (2011) A discrete model of a rope with bending stiffness or viscous damping. Acta Mech Sin 27(1):108–113Google Scholar
 9.Bendersky S, Sandler B (2006) Investigation of a spatial double pendulum: an engineering approach. Discrete Dyn Nat Soc 2006: 1–22Google Scholar
 10.Plaut R, Virgin L (2013) Pendulum models of ponytail motion during walking and running. J Sound Vibr 332:3768–3780CrossRefGoogle Scholar
 11.Hand LN, Finch JD (1998) Analytical Mechanics. Cambridge University, CambridgeCrossRefGoogle Scholar
 12.Thornton S, Marion J (2004) Classical Dynamics of Particles and Systems, 5th edn. Brooks/Cole, New YorkGoogle Scholar
 13.Press W, Teukolsky S, Vetterling W, Flannery B (2007) Numerical Recipes: The Art of Scientific Computing, 3rd edn. Cambridge University, CambridgeGoogle Scholar
 14.Hoffman J (2001) Numerical Methods for Engineers and Scientists, 2nd edn. Marcel Dekker Inc., New YorkzbMATHGoogle Scholar
 15.Mazzia F, Magherini C (2008) Test set for initial value problem solvers. Tech. Rep. 4/2008, Department of Mathematics, University of Bari, Italy. http://www.dm.uniba.it/~testset/testsetivpsolvers/
 16.Petzold L (1982) Differential/Algebraic Equations are not ODE’s. SIAM J Sci Stat Comput 3(3):367–384CrossRefzbMATHMathSciNetGoogle Scholar
 17.Riaza R (2008) DifferentialAlgebraic Systems: Analytical Aspects and Circuit Applications. World Scientific, SingaporeCrossRefGoogle Scholar
 18.CostaCastelló R, Griñó R, Basañez L (1998) DAE methods in constrained robotics system simulation. Computatión y Sistemas 1(3):145–160Google Scholar
 19.Cash J (1980) On the integration of stiff systems of ODEs using extended backward differentiation formulae. Numer Math 34(3):235–246CrossRefzbMATHMathSciNetGoogle Scholar
 20.Cash J (1983) The integration of stiff initial value problems in ODEs using modified extended backward differentiation formulae. Comp Math Appl 9(5):645–657Google Scholar
 21.Cash J (2000) Modifed extended backward differentiation formulae for the numerical solution of stiff initial value problems in ODEs and DAEs. J Comput Appl Math 125:117–130CrossRefzbMATHMathSciNetGoogle Scholar
 22.Cash J (2000) Software for initial value problems. http://www2.imperial.ac.uk/~jcash/IVP_software/readme.php
 23.Conte S, de Boor C (1980) Elementary Numerical Analysis: An Algorithmic Approach, 3rd edn. McGrawHill, New YorkzbMATHGoogle Scholar
 24.Stoer J, Bulirsch R (1991) Introduction to Numerical Analysis, 2nd edn. Springer, New YorkGoogle Scholar
 25.Shampine L, Allen RC (1997) Fundamentals of Numerical Computing. Wiley, New YorkzbMATHGoogle Scholar
 26.Cash J, Considine S (1992) An MEBDF code for stiff initial value problems. ACM T Math Softw 18(2):142–155CrossRefzbMATHMathSciNetGoogle Scholar
 27.Middleditch A, Oliver J (2002) Functional Anatomy of the Spine. Butterworth–Heinemann, OxfordGoogle Scholar
 28.de Jager M (1993) Mathematical modelling of the human cervical spine: A survey of the literature. Tech. Rep. WFW93.027, Faculty of Mechanical Engineering Eindhoven University of Technology, The NetherlandsGoogle Scholar
 29.Deng Y, Goldsmith W (1987) Response of a human head/ neck/uppertorso replica to dynamic loading  II. Analytical/numerical model. J Biomech 20:487–491Google Scholar
 30.Dauvilliers F, Tarriere C, Lavaste F, Weiss M, Bendjellal F (1994) Development of finite element method of the neck. In: Proceedings of 38th Stapp Car Crash Conference, pp 77–91Google Scholar
 31.de Jager M, Sauren A, Thunnissen J, Wismans J (1996) A global and detailed mathematical model for head–neck dynamics. In: Proceedings of 40th Stapp Car Crash Conference, pp 269–281Google Scholar
 32.de Jager M (1996) Mathematical head–neck models for acceleration impacts. Ph.D. thesis, University of EindhovenGoogle Scholar
 33.Yang K, Zhu F, Luan F, Zhao L, Begeman P (1998) Development of a finite element model of the human neck. In: Proceedings of 42nd Stapp Car Crash Conference, pp 195–205Google Scholar
 34.Horst MVD (2002) Human head neck response in frontal, lateral and rear end impact loading: modelling and validation. Ph.D. Thesis, Eindhoven University of TechnologyGoogle Scholar
 35.van Lopik D, Acar M (2004) A computational model of the human head and neck system for the analysis of whiplash motion. Int J Crashworthiness 9(1):465–473CrossRefGoogle Scholar
 36.Saito T, Yamamuro T, Shikata J, Oka M, Tsutsumi S (1991) Analysis and prevention of spinal column deformity following cervical laminectomy, pathogenetic analysis of post laminectomy deformities. Spine 16(5):494–502CrossRefGoogle Scholar
 37.Yoganandan N, Kumaresan S, Voo L, Pintar F (1996) Finite element application in human cervical spine modeling. Spine 21(15): 1824–1834Google Scholar
 38.Kleinberger M (1993) Application of finite element techniques to the study of cervical spine mechanisms. In: Proceedings of 37th Stapp Car Crash Conference, pp 261–272Google Scholar
 39.Williams J, Bellytschko T (1983) A threedimensional model of the human cervical spine for impact simulations. J Biomech Eng 105(4):321–331CrossRefGoogle Scholar
 40.de Jager M, Sauren A, Thunnissen J, Wismans J (1994) A three dimensional headneck model: validation for frontal and lateral impacts. In: Proceedings of 38th Stapp Car Crash Conference, pp 93–109Google Scholar
 41.Stemper B, Yoganandan N, Pintar F (2004) Validation of a headneck computer model for whiplash simulation. Med Biol Eng Comput 42(3):333–338Google Scholar
 42.Kumaresan S, Yoganandan N, Pintar F (1997) Pediatric neck modeling using finite element analysis. Int J Crashworthiness 2(4): 367–376 Google Scholar
 43.Nitsche S, Krabbel G, Appel H, Haug E (1996) Validation of a finite element model of the human neck. In: Proceedings of International Conference of Biomechanics of Impact, pp 107–108Google Scholar
 44.Goel V, Scifert J, Grosland N Finite element investigation of anterior plate and bone graft load sharing in the cervical spine. Tech. rep., Departments of Biomechanical Engineering and Orthopaedic Surgery, University of Iowa, Iowa CityGoogle Scholar
 45.Panjabi M, Pearson A, Ito S, Ivancic P, Wang J (2004) Cervical spine curvature during simulated whiplash. Clin Biomech 19(1): 1–9Google Scholar
 46.Ivancic P, Panjabi M, Ito S (2006) Cervical spine loads and intervertebral motions during whiplash. Traffic Inj Prev 7(4):389–399CrossRefGoogle Scholar
 47.van Lopik D, Acar M (2007) Development of a multibody computational model of human head and neck. Proc Inst Mech Eng 221:175–197Google Scholar
 48.Myklebush J, Pintar F, Yoganandan N, Cusick J, Maiman D, Myers T, Sances A (1988) Tensile strength of spinal ligaments. Spine 13(5):526–531Google Scholar
 49.Cohen D, Hairer E, Lubich C (2005) Numerical energy conservation for multifrequency oscillatory differential equations. BIT Numer Math 45(2):287–305CrossRefzbMATHMathSciNetGoogle Scholar
 50.Betsch P, Leyendecker S (2006) The discrete null space method for the energy consistent integration of constrained mechanical systems. Int J Numer Meth Eng 67:499–552CrossRefzbMATHMathSciNetGoogle Scholar
 51.Uhlar S, Betsch T (2009) Energyconsistent integration of multibody systems with friction. J Mech Sci Tech 23(4):901–909CrossRefGoogle Scholar
 52.Nielsen M, Krenk S (2012) Conservative integration of rigid body motion by quaternion parameters with implicit constraints. Int J Numer Meth Eng 92(5):734–752CrossRefMathSciNetGoogle Scholar
 53.Goldstein H, Poole C, Safko J (2001) Classical Mechanics, 3rd edn. Addison Wesley, San FranciscoGoogle Scholar
 54.Gürgöze M (2006) On the formulation of Lagrange’s equations with respect to moving coordinate systems: application to a point mass vibrating on a rotating base. Int J Mech Eng Edu 34(3): 263–272Google Scholar
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