A nonlinear two-node superelement is proposed for efficient modelling of arbitrary-shaped flexible members with two interfaces in a flexible multibody model. The formulation is based on a small rotation and displacement hypothesis in a local co-rotational frame. Component mode substructuring methods can then be used to determine the dynamical properties of the superelement from a linear finite element model. The key contribution of this paper is the inclusion of the so-called deformable-interface modes to model the deformability of the interface surfaces. This allows for a compliant connection to other superelements. With this capability, a component can be modelled with a number of superelements, and its dynamical properties can be accurately analysed even for large deflections provided that the deformations remain small with respect to the co-rotational frame. Three examples demonstrate the applicability of the method. In the first example, large deflections of a relative short sheet flexure are analysed. Next, the formulation is used to obtain a dynamically reduced model of a complex-shaped component. In the third example, the time-response of a compliant mechanism is considered that is composed of the components of the first two examples. For all three examples, eigenfrequency results are in good agreement with results obtained using a classical nonlinear finite element method.

Flexure hinges inherently lose stiffness in supporting directions when deflected. In this paper a method is presented for optimizing the geometry of flexure hinges, which aims at maximizing supporting stiffnesses. In addition, the new ∞-flexure hinge design is presented. The considered hinges are subjected to a load and deflected an angle of up to ±20 deg. The measure of performance is defined by the first unwanted natural frequency, which is closely related to the supporting stiffnesses. During the optimization, constraints are applied to the actuation moment and the maximum occurring stress. Evaluations of a curved hinge flexure, cross revolute hinge, butterfly flexure hinge, two cross flexure hinge types, and the new ∞-flexure hinge are presented. Each of these hinge types is described by a parameterized geometric model. A flexible multibody modeling approach is used for efficient modeling while it accounts for the nonlinear geometric behavior of the stiffnesses. The numerical efficiency of this model is very beneficial for the design optimization. The obtained optimal hinge designs are validated with a finite element model and show good agreement. The optimizations show that a significant increase in supporting stiffness, with respect to the conventional cross flexure hinge, can be achieved with the ∞-flexure hinge.

A nonlinear two-node superelement is proposed for the modeling of flexible complex-shaped links for use in multibody simulations. Assuming that the elastic deformations with respect to a corotational reference frame remain small, substructuring methods may be used to obtain reduced mass and stiffness matrices from a linear finite element model. These matrices are used in the derivation of potential and kinetic energy expressions of the nonlinear two-node superelement. By evaluating Lagrange’s equations, expressions for the internal and external forces acting on the superelement can be obtained. The inertia forces of the superelement are derived in terms of absolute nodal velocities and accelerations, which greatly simplifies the dynamic formulation. Three examples are included. The first two examples are used to validate the method by comparing the results with those obtained from nonlinear beam element solutions. We consider a benchmark simulation of the spin-up motion of a flexible beam with uniform cross-section and a similar simulation in which the beam is simultaneously excited in the out-of-plane direction. Results from both examples show good agreement with simulation results obtained using nonlinear finite beam elements. In a third example, the method is applied to an unbalanced rotating shaft, illustrating the potential of the proposed methodology for a more complex geometry.

A reduction method is proposed for efficient time-integration of compliant mechanism models that undergo large deflections. Of particular importance for the modelling of this class of mechanisms is the accurate description of geometric non-linearities, as stiffness characteristics can change significantly during deflection. A finite element-based flexible multibody approach is used to describe the compliant mechanism in terms of independent generalized coordinates. The modelling of large deflections requires a sufficient number of finite elements to ensure that deformations remain small in a co-rotational context. Increasing the number of elements, increases, besides the number of degrees of freedom, the largest eigenfrequency in the model. This reduces the allowable step size for explicit time-integrator methods. The proposed reduction method aims to suppress the high frequency vibrational modes which are not important for the desired simulation results, while retaining the geometric non-linearities in the reduced model. For this purpose we add constraint relations between the independent generalized coordinates. These constraint relations can be linear or non-linear. Both cases are investigated in this paper and are implemented as a fixed and an interpolated basis method, respectively. The effectiveness of the two methods is demonstrated by a simulation of a compliant straight guidance in a gravity field that undergoes large deflection. Both methods can yield accurate results with a significant increase in computational efficiency.

In modelling flexure based mechanisms, generally flexures are modelled perfectly aligned and nominal values are assumed for the dimensions. To test the validity of these assumptions for a two Degrees Of Freedom (DOF) large stroke compliant mechanism, eigenfrequency and mode shape measurements are compared to results obtained with a flexible multibody model. The mechanism consists of eleven cross flexures and seven interconnecting bodies. From the measurements 30% lower eigenfrequencies are observed than those obtained with the model. With a simplified model, it is demonstrated that these differences can be attributed to wrongly assumed leaf spring thickness and misalignment of the leaf springs in the cross flexures. These manufacturing tolerances thus significantly affect the behaviour of the two DOF mechanism, even though it was designed using the exact constraint design principle. This design principle avoids overconstraints to limit internal stresses due to manufacturing tolerances, yet this paper shows clearly that manufacturing imperfections can still result in significantly different dynamic behaviour.

In overconstrained mechanisms inherent alignment errors cause self-stress. The level of stress can be reduced by inserting flexure releases making the mechanism exactly constrained. The location and orientation of releases can be optimized for a combination of minimum self-stress and maximum stiffness. We compare two methods for optimization in a case study of a four-bar mechanism with three overconstraints.The first method analyzes the kinematics of a mechanism using a multibody modeling approach and a singular value decomposition. The second method lumps the compliance of the mechanism to the joints. It is shown that in order to obtain a mechanism with large stiffness and small self-stress over a range of motion the mechanism must be exactly constrained in special poses. In a special pose an exactly constrained mechanism can become statically and kinematically indeterminate due to an alignment of releases.The singular value decomposition method is a powerful tool to find the special poses and to design exactly constrained configurations. It provides sufficient insight to assist the selection of release locations, without requiring all the stiffness properties of the mechanism. (C) 2013 Elsevier Ltd. All rights reserved.

Numerical simulations are essential to determine the characteristics, performance and structural integrity of mechanisms and robots. With increasingly higher demands on the specifications of such devices, the demands on the accuracy of the numerical models increases as well. Increasing the complexity of the models, inherently increases the computational time of the simulations. Model reduction techniques can offer a reduction of the simulation time, while maintaining sufficient accuracy. In this work, the modelling of flexible multibody systems is considered. The starting point is a numerical approach based on non-linear finite elements such as beams, trusses and hinges. In particular the spatial beam element has proven to offer a numerical efficient analysis of mechanisms that consist of beam-like components. However, for systems with complex-shaped parts, or for systems composed of a rather large number of beams, the dimensionality of the model increases. Hence, two types of model reduction techniques are investigated for the efficient and accurate modelling of such flexible multibody systems. The first type deals with the efficient and accurate modelling of individual components that may be flexible and have a complex geometry. This is referred to as component model reduction. The second type attempts to reduce a complete non-linear multibody system and can therefore be referred to as a system model reduction approach. Both methods are demonstrated by modelling a largestroke two degree of freedom compliant positioning mechanism.

Flexure hinges inherently lose stiffness in supporting directions when deflected. In this paper a method is presented for optimizing the geometry of flexure hinges, while supporting stiffnesses are retained. These hinges are subjected to a load and deflected an angle of up to ±20°. The measure of performance is defined by the first unwanted eigenfrequency, which is closely related to the supporting stiffnesses. During the optimization, constraints are applied to the actuation moment and the maximum occurring stress. Evaluations of three cross flexure hinge types and a butterfly flexure hinge are presented. A flexible multibody modeling approach is used for efficient modeling. Each of these hinge types is described by a parameterized geometric model. The obtained optimal hinge designs are validated with a finite element model and show good agreement. The optimal solution of the butterfly flexure hinge shows the least decrease in the supporting stiffnesses of the evaluated hinges.

Flexure based stages are particularly important for vacuum applications because they combine low hysteresis, no wear and no contamination with a high supporting stiffness. However, flexure hinges inherently lose stiffness in supporting directions when deflected. Therefore the workspace to footprint ratio is limited. In this article we present the design and modeling of a two degrees of freedom cross flexure based stage that combines a large workspace to footprint ratio with high vibration mode frequencies. Because the mechanism is an assembly of optimized components, the stage is designed according to the exact constraint principle to avoid build-up of internal stresses due to misalignment. FEM results have been validated by measurements on an experimental test setup. The test setup has a workspace-area to footprint ratio of 1/32. The lowest measured natural frequency with locked actuators over a 60 × 60mm workspace was 80Hz.

Whereas the use of compliant mechanisms is favorable for high precision applications, the constraints must be dealt with carefully. In an overconstrained design the actual natural frequencies and stiffnesses can differ considerably from their intended values. For this reason the awareness and possibly the avoidance of an overconstrained condition is important. We have developed a kinematic analysis with which under-constraints and overconstraints can be detected. A finite element based multibody approach is applied which offers a flexible beam element that is particularly suited to model the wire and sheet flexures frequently encountered in compliant mechanisms. For each element a fixed number of independent discrete deformations are defined that are invariant under arbitrary rigid body motions of the element. In the kinematic analysis only deformations associated with low stiffnesses are allowed, whereas the remaining deformations are prescribed zero. A singular value decomposition is used to determine the rank of the Jacobian matrix associated with the dependent nodal coordinates. Column and row rank deficiency indicate an underconstrained and overconstrained system, respectively. For an overconstrained system a statically indeterminate stress distribution can be derived from the left singular matrix. In this way the overconstraints can be visualized clearly as is illustrated with examples of compliant straight guidance mechanisms. The possible solutions to eliminate the overconstraints are found easily from the visualization.

A flexure which retains its support stiffness characteristics for large deflections, is optimized with respect to maximum allowable stress, low actuation stiffness and high support stiffnesses. Such an optimization requires an efficient model which accurately describes the stiffness characteristics and stress distribution of flexures. For this purpose a multibody modelling approach based on a non-linear finite element description is investigated and extended to include the computation of the stress distribution in the deformed configuration. It is shown that the accuracy of the maximum occurring stress is comparable with those obtained from a classical non-linear finite element analysis. An optimized shape of the flexure is found and for deflection angles larger than 7.4, it is preferable over a single leaf-spring flexure.

Generalized-α time integration methods were originally designed for large scale problems in structural mechanics [1] but may be used as well for constrained systems of moderate dimension that are typical of multibody dynamics [2]. For constrained systems, the Newmark like update formula qn+1 = qn + hvn + (0.5− β)han + βhan+1 , (1a) vn+1 = vn + (0.5− γ)han + γhan+1 (1b) for position coordinates q and velocity coordinates v and the generalized-α update scheme (1− αm)an+1 + αman = (1− αf )q̈n+1 + αf q̈n (1c) for the auxiliary vectors an are coupled to equilibrium equations Mq̈ = f −B>λ and constraints Φ = 0 at t = tn+1: M(qn+1)q̈n+1 = f(tn+1,qn+1,vn+1)−B(qn+1)λn+1 , (1d) 0 = Φ(qn+1) . (1e) In (1), the method parameters αf , αm, β, γ are chosen to satisfy the order condition γ = 1 2 + αf − αm, see [1], and the time step size h is for the moment considered to be constant for all time steps tn → tn+1 := tn + h. Following the principles of classical mechanics, the nΦ constraints (1e) are coupled to the dynamical equations (1d) by constrained forces −B>(q)λ with B(q) := (∂Φ/∂q)(q) and Lagrange multipliers λ(t) ∈ RnΦ . All other forces and moments of the system are summarized in vector f = f(t,q,v). The mass matrix M(q) is assumed to be non-singular and symmetric, positive definite. With an appropriate scaling of the corrector equations [3], the fixed step size implementation of (1) works well for reasonable time step sizes h > 0. The convergence analysis for h→ 0 proves stability and second order convergence if the stability conditions αm < αf < 1/2 and β > 1/4 + (αf − αm)/2 are satisfied, see [4] and the extension to the generalized-α Lie group integrator of Brüls and Cardona [5] in [6]. Because of numerical damping, the generalized-α method (1) shows a favourable long-time behaviour. But in a short transient phase, the method may suffer from large errors that are damped out rapidly, see Fig. 1. In a recent joint work with O. Brüls (Liège) and A. Cardona (Santa Fe), this order reduction was studied in detail for a Lie group integrator [7]. The problem is fixed by perturbed initial values v0. ∗Email: martin.arnold@mathematik.uni-halle.de