Bushing Elements Research Articles

Research on the Impact of Non-Uniform and Frequency-Dependent Normal Contact Stiffness on the Vibrational Response of Plate Structures

Mechanical interfaces are prevalent in industries like aerospace and maritime, where the normal contact stiffness on these surfaces is a crucial component of the overall stiffness of mechanical structures. From the perspective of structural mechanics, normal contact stiffness significantly affects the dynamic response of mechanical structures and must be considered in mechanical design and simulation analysis. Essentially, the mechanical interface represents a typical type of nonlinear contact, characterized by both its non-uniform distribution and its frequency-dependent properties under external excitations. A plate structure was designed to study the distribution and frequency-dependent characteristics of normal contact stiffness on the mechanical interface. Experiments validated that the normal contact stiffness not only significantly increases the fundamental frequency of the plate but also alters its mode shapes. To replicate the experimental results in simulations, the BUSH elements were used to model the normal contact stiffness within the plate structure, arranging it non-uniformly and setting it to vary with frequency according to the plate’s mode shapes and frequency response. This method accurately replicated the plate’s mode shapes and response curves within the 0–1000 Hz range in simulations.

Quantification of soft tissue artifacts using CT registration and subject-specific multibody modeling

The potential use of gait analysis for quantitative preoperative planning in total hip arthroplasty (THA) has previously been demonstrated. However, the joint kinematic data measured through this process tend to be unreliable for surgical planning due to distortions caused by soft tissue artifacts (STAs). In this study, we developed a novel motion capture framework by combining computed tomography (CT)-based postural calibration and subject-specific multibody dynamics modeling to prevent the effect of STAs in measuring hip kinematics. Three subjects with femoroacetabular impingement syndrome were recruited, and CT data for each patient were collected by attaching marker clusters near the hip. A subject-specific multibody hip joint model was developed based on reconstructed CT data. Spring–dashpot network calculations were performed to minimize the distance between the anatomical landmark and its corresponding infrared reflective marker. The STAs of the thigh was described as six degrees of freedom viscoelastic bushing elements, and their parameter values were identified via smooth orthogonal decomposition. Least squares optimization was used to modify the pelvic rotations to compensate for the rigid components of STAs. The results showed that CT-assisted motion tracking enabled the successful identification of STA influences in gait and squat positions. Furthermore, STA effects were found to alter maximal pelvis tilt and hip rotations during a squat. Compared to other techniques, such as dual fluoroscopic imaging, the adopted framework does not require additional medical imaging for patients undergoing robot-assisted THA surgery and is thus a practical way of evaluating hip joint kinematics for preoperative surgical planning.

Modeling viscoelastic behavior in flexible multibody systems

Viscoelasticity plays an important role in the dynamic response of flexible multibody systems. First, single degree-of-freedom joints, such as revolute and prismatic joints, are often equipped with elastomeric components that require complex models to capture their nonlinear behavior under the expected large relative motions found at these joints. Second, flexible joints, often called force or bushing elements, present similar challenges and involve up to six degrees of freedom. Finally, flexible components such as beams, plates, and shells also exhibit viscoelastic behavior. This paper presents a number of viscoelastic models that are suitable for these three types of applications. For single degree-of-freedom joints, models that capture their nonlinear, frequency-dependent, and frequency-independent behavior are necessary. The generalized Maxwell model is a classical model of linear viscoelasticity that can be extended easily to flexible joints. This paper also shows how existing viscoelastic models can be applied to geometrically exact beams, based on a three-dimensional representation of the quasi-static strain field in those structures. The paper presents a number of numerical examples for three types of applications. The shortcomings of the Kelvin–Voigt model, which is often used for flexible multibody systems, are underlined.

Model Updating of Multiple Nominally Identical Car Components

A method for estimation of rubber bushing stiffness parameters is presented. Four individual rubber bushings, mounted in a car rear subframe are considered. A traditional model of the bushing elements using a generalised spring model, known as a CBUSH element in Nastran, is compared to a geometrically more realistic approach where the bushing is modelled with solid elements and a linear elastic material model. Each bushing is mass loaded to better reveal the bushing’s dynamic behaviour in a lower frequency range of interest. In an initial step, the overall subframe model is updated towards test data. In a second step, the bushing parameters are updated. Three nominally identical components are used to investigate the spread between the identified parameters. The model updating procedure is based on frequency responses and equalised damping. The undamped behaviour at frequencies below 300 Hz are considered. To quantify the parameter uncertainty, with respect to measurement noise for each individual, an uncertainty quantification procedure is proposed, using a linear-in-parameters surrogate model with bootstrapping.

A new methodology to determine the design sensitivity of critical automotive body joints for basic design cycle

As a result of more stringent requirements for improved fuel economy and emissions, there has been an increasing research activity to make vehicles lighter weight under some predetermined structural performance targets such as the stiffness of the vehicle body. The vehicle body structure is one of the most significant contributors to the weight of an automotive. Therefore, understanding the automotive joint properties on vehicle body performance is of significant importance as they are closely linked to structural integrity and weight of the vehicle body. In this paper, we develop a new methodology to quantify the sensitivity of critical joints of an automotive on the key performance indices. Torsional stiffness is chosen as static key performance index, while vehicle body modes are selected as dynamic key performance indices. Lower and upper sections of the A-pillar, B-pillar, C-pillar, and D-pillar of an automotive body are replaced by bushing elements having appropriate stiffness properties in the simplified model. Stiffness of bushing elements is tuned by minimizing the error between the original and simplified models on the aforementioned key performance indices. Once a satisfactory correlation is achieved between the simple model and the original model, bushing stiffness for each section is varied to determine the sensitivity of each joint. The proposed approach is demonstrated on a finite element model of 2010 Toyota Yaris. Finally, a design study is presented to improve the body key performance indices using the sensitivity results. The simulation results show that the methodology has a potential for the basic design cycle, where the targets for section properties need to be defined and at later design cycles, where the joints can be realized in design using the sensitivity of joints resulting in more efficient body structure considering the trade-offs between structural integrity and weight.

A Jacobian formulation for efficient simulation of multibody chain dynamics

This research proposes an efficient analysis method with an implicit integration method for multibody chain dynamics. Absolute Cartesian coordinates which are related to the body reference frame are used as generalized coordinates. Contact between a sprocket and chain links is formulated as a circle to circle contact. Chain links are connected by bushing elements. Newton-Raphson method is used to solve the combined equations of motion, constraints, and implicit integration formulas. Jacobian matrix must be calculated and LU decomposed with the implicit integration method in Newton-Raphson method. Since Cartesian coordinates with respect to the body reference frame are used as the generalized coordinates and the contact and bushing elements exist between two bodies, the Jacobian matrix for these elements depended on only the relative motion of two adjacent bodies. Therefore, the Jacobian matrix hardly changes unless the relative motion changes significantly. This research investigates the Jacobian matrix to find small-changing and large-changing submatrices. Investigation showed that 65.4 % and 34.6 % of the Jacobian matrix belong to the small changing and the large changing submatrices, respectively. The small changing submatrices are calculated and LU decomposed only once at the initial time and reused at every time step afterwards. The large changing submatrices are updated and LU decomposed at every time step. As the sprocket rotates, the generalized coordinate sequence of the Jacobian matrix for the links is shifted one by one to keep the Jacobian matrix change of the small change submatrices small until the end of simulation time. The proposed method is implemented and its results are compared to those obtained from the original method which updates and LU decomposes the whole Jacobian matrix at every time step. Numerical experiments showed that the proposed method is about 11 times faster and yields almost identical results to the original method.

A unified formulation for mechanical joints with and without clearances/bushings and/or stops in the framework of multibody systems

Virtually all machines and mechanisms use mechanical joints that are not perfect from the kinematic point of view and for which tolerances, in the fitting of their components, are specified. Together with such controlled clearances, mechanical joints may require the use of bushing elements, such as those used in vehicle suspensions. Furthermore, in many situations the joints exhibit limits (stops) in their translational or rotational motion that have to be taken into account when modeling them. The dynamic response of the mechanical systems that use such realistic mechanical joints is largely dependent on their characteristic dimensions and material properties of the compliant elements, implying that correct models of these systems must include realistic models of the bushing/clearance joints and of the joint stops. Several works addressed the modeling of imperfect joints to account for the existence of clearances and bushings, generally independently of the formulation of the perfect kinematic joints. This work proposes a formulation in which both perfect and clearance/bushing joints share the same kinematic information making their modeling data similar and enabling their easy permutation in the context of multibody systems modeling. The proposed methodology is suitable for the most common mechanical joints and easily extended to many other joint types benefiting the exploration of a wide number of modeling applications, including the representation of cut-joints required for some formulations in multibody dynamics. The formulation presented in this work is applied to several demonstrative examples of spatial mechanisms to show the need to consider the type of imperfect joints and/or joints with stops modeling in practical applications.

Experimental Validation of the EBF3PanelOpt Vibroacoustic Analysis of Stiffened Panels

Vibroacoustic experiments are performed at the Structural Acoustics Loads and Transmission facility of NASA Langley Research Center to measure the sound radiation from and transmission loss of two panels with straight and curvilinear stiffeners, respectively. The focus of the work addressed in this paper is to validate the vibroacoustic analysis methods implemented in a multi-objective design optimization framework, EBF3PanelOpt. The stiffened test panels with six stiffeners have been designed using EBF3PanelOpt for turbulent boundary-layer excitation. To build the confidence in finite-element models, free-vibration tests of the manufactured stiffened panels are also performed. The excited experimental mode shape and eigenfrequencies match quite well with the finite element analysis results for the free-vibration test. A medium-density fiberboard and aluminum frame is used to clamp the stiffened panels in the Structural Acoustics Loads and Transmission loss window. Because of strong coupling of the stiffened test panels with the frame, a finite-element model of the stiffened panels with frame is developed. To capture the effect of the bolted joints in the test setup, bush elements with optimal values of rotational and translational stiffnesses are used. Measurements are also performed to estimate the damping in the stiffened panels with the frame. The numerical prediction of sound radiation from and transmission loss of stiffened panels is in fair agreement with the measured values for both the panels.

Numerical method on load sharing problem of thick laminate joints

Accurately and efficiently predicting the load sharing of multi-bolt thick laminate joints is necessary to quicken the optimization of the large-scale structures over various design variables, and a two-dimensional (2D) finite element method (FEM) is introduced to meet such a demand. The deformation contributions of the joint zone are analyzed and calculated separately, including the shearing deformation of the fasteners shank, the bending deformation of the fasteners shank, and the bearing deformation of the fasteners and joint plates. These deformations are all transferred and incorporated into the components of the fastener’s flexibility. In the 2D finite element model, the flexibilities of the beam elements and bush elements are used to simulate different components of the fastener’s flexibility. The parameters of the beam elements which include the bending moment of inertia and intersection area, and the parameters of the bush elements which include the stiffness in different directions, are all obtained through equalizing the fasteners flexibilities. In addition, the secondary bending effect introduced by the single-lap joints is also taken into account to verify the flexibilities of the fasteners in practical application. The proposed FEM is testified to be more accurate than the traditional 2D FEMs and more efficient than the three-dimensional (3D) FEM in solving load sharing problem of multi-bolt single-lap thick laminate joints. With the increase of joint plates’ thickness, the advantages of the proposed method tend to be more obvious. The proposed 2D FEM is an effective tool for designing bolted joints in large-scale composite structures.

A comparative study of joint formulations: application to multibody system tracked vehicles

This paper is focused on the dynamic formulation of mechanical joints using different approaches that lead to different models with different numbers of degrees of freedom. Some of these formulations allow for capturing the joint deformations using a discrete elastic model while the others are continuum-based and capture joint deformation modes that cannot be captured using the discrete elastic joint models. Specifically, three types of joint formulations are considered in this investigation; the ideal, compliant discrete element, and compliant continuum-based joint models. The ideal joint formulation, which does not allow for deformation degrees of freedom in the case of rigid body or small deformation analysis, requires introducing a set of algebraic constraint equations that can be handled in computational multibody system (MBS) algorithms using two fundamentally different approaches: constrained dynamics approach and penalty method. When the constrained dynamics approach is used, the constraint equations must be satisfied at the position, velocity, and acceleration levels. The penalty method, on the other hand, ensures that the algebraic equations are satisfied at the position level only. In the compliant discrete element joint formulation, no constraint conditions are used; instead the connectivity conditions between bodies are enforced using forces that can be defined in their most general form in MBS algorithms using bushing elements that allow for the definition of general nonlinear forces and moments. The new compliant continuum-based joint formulation, which is based on the finite element (FE) absolute nodal coordinate formulation (ANCF), has several advantages: (1) It captures modes of joint deformations that cannot be captured using the compliant discrete joint models; (2) It leads to linear connectivity conditions, thereby allowing for the elimination of the dependent variables at a preprocessing stage; (3) It leads to a constant inertia matrix in the case of chain like structure; and (4) It automatically captures the deformation of the bodies using distributed inertia and elasticity. The formulations of these three different joint models are compared in order to shed light on the fundamental differences between them. Numerical results of a detailed tracked vehicle model are presented in order to demonstrate the implementation of some of the formulations discussed in this investigation.

Flexible joints in structural and multibody dynamics

Abstract. Flexible joints, sometimes called bushing elements or force elements, are found in all structural and multibody dynamics codes. In their simplest form, flexible joints simply consist of sets of three linear and three torsional springs placed between two nodes of the model. For infinitesimal deformations, the selection of the lumped spring constants is an easy task, which can be based on a numerical simulation of the joint or on experimental measurements. If the joint undergoes finite deformations, identification of its stiffness characteristics is not so simple, specially if the joint is itself a complex system. When finite deformations occur, the definition of deformation measures becomes a critical issue. This paper proposes a family of tensorial deformation measures suitable for elastic bodies of finite dimension. These families are generated by two parameters that can be used to modify the constitutive behavior of the joint, while maintaining the tensorial nature of the deformation measures. Numerical results demonstrate the objectivity of the deformations measures, a feature that is not shared by the deformations measures presently used in the literature. The impact of the choice of the two parameters on the constitutive behavior of the flexible joint is also investigated.

Robust design optimization of the dynamic responses of a tracked vehicle system

This paper presents the robust design optimization of the dynamic responses of a heavy military tracked vehicle system. The tracked vehicle model addressed in this study has 954 degrees of freedom and consists of 189 bodies in total: 37 bodies for the chassis, such as sprockets, road wheels, road arms, etc.; 76 track link bodies for each track subsystem; 36 revolute joints; and 152 bushing elements. The design objectives were to minimize the maximum vertical acceleration of the hull and its variance while satisfying the wheel travel constraints for torsion bars and the hydro-pneumatic suspension units within ±1σ ranges. To avoid the difficulty of the design sensitivity analysis and to overcome the numerical noise, a progressive meta-model technique was employed in the optimization process. First, space-filling methods were used to determine the minimum number of sample points. Second, the simultaneous kriging method was used to construct the initial meta-models, and the augmented Lagrange multiplier (ALM) method was then used to solve the robust design problems of the meta-models. Third, the new design results were added to the analysis results for the initial sample points, and the meta-models were updated automatically. Next, the optimizer resolved the robust design problems of the updated meta-models. These processes were repeated until the convergence tolerances were satisfied. The robust design optimization of the tracked vehicle system, with 11 random design variables, was solved in only 26 analyses, including 12 analyses for the initial meta-models and 14 analyses added during the iterative optimization process.

On the modeling of the intervertebral joint in multibody models for the spine

The need to develop feasible computational musculoskeletal models of the spine has led to the development of several multibody models. Central features in these works are models for the ligaments, muscles, and intervertebral joint. The purpose of the present paper is to show how experimental measurements of joint stiffnesses can be properly incorporated using a bushing element. The required refinements to existing bushing force functions in musculoskeletal software platforms are discussed and further implemented using a SpineBushing element specific to the intervertebral joint. Four simple lumbar spine models are then used to illustrate the accompanying improvements. Electronic supplemental material for this article includes a complementary review of formulations of stiffness matrices for the intervertebral joint.

On the relative rotation of rigid parts and the visco-elastic torsion bushing element

In this paper the interaction between two rigid parts of a multibody, connected by an ideal spherical joint equipped with a visco-elastic torsion bushing element, is derived. The model allows for arbitrary relative rotations of the parts and involves a non-linear torsion stiffness of the bushing. An expression for the interaction between the parts is derived and a specialization to the isotropic bushing element is presented.

Homogenisation method to capture the non‐linear behaviour of intervertebral discs in multi‐body systems

AbstractThe present contribution is motivated by the desire to compute physiological loads on the intervertebral discs (IVD) of a human lumbar spine during activities like standing, bending and falling. Following this, a mechanical multi‐body system (MBS) is utilised to capture the overall mechanical behaviour of a human, whereas an inhomogeneous, anisotropic, multi‐phasic finite‐element model (FEM) is applied to resolve the resulting field quantities inside an IVD. In order to couple the FEM of the IVD with the numerically diverse MBS, a homogenisation procedure has to be applied such that field quantities can be converted into discrete quantities. In particular, the MBS captures the mechanical behaviour of an IVD using a bushing element, which provides discrete force‐displacement and moment‐rotation relations.The goal of this contribution is to present a homogenisation method for the IVD as well as a possibility to include the homogenised results in the MBS without the need for embedded FE computations in the MBS. Instead, certain deformation modes of the IVD are pre‐computed and represented using a non‐linear constitutive equations. This task becomes even more challenging, as the resulting discrete DOF of a motion segment appear in a coupled fashion due to the structure of the IVD, i. e., a rotation in the sagittal plane triggers a resulting moment and a resulting force. (© 2011 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Tensorial Deformation Measures for Flexible Joints

Flexible joints, sometimes called bushing elements or force elements, are found in all multibody dynamics codes. In their simplest form, flexible joints simply consist of sets of three linear and three torsional springs placed between two nodes of a multibody system. For infinitesimal deformations, the selection of the lumped spring constants is an easy task, which can be based on a numerical simulation of the joint or on experimental measurements. If the joint undergoes finite deformations, the identification of its stiffness characteristics is not so simple, especially if the joint itself is a complex system. When finite deformations occur, the definition of deformation measures becomes a critical issue. Indeed, for finite deformation, the observed nonlinear behavior of materials is partly due to material characteristics and partly due to kinematics. This paper focuses on the determination of the proper finite deformation measures for elastic bodies of finite dimension. In contrast, classical strain measures, such as the Green–Lagrange or Almansi strains, among many others, characterize finite deformations of infinitesimal elements of a body. It is argued that proper finite deformation measures must be of a tensorial nature, i.e., must present specific invariance characteristics. This requirement is satisfied if and only if the deformation measures are parallel to the eigenvector of the motion tensor.

Improved bushing models for general multibody systems and vehicle dynamics

The development and computational implementation, on a multibody dynamics environment, of a constitutive relation to model bushing elements associated with mechanical joints used in the models of road and rail vehicles is presented here. These elements are used to eliminate vibrations in vehicles, due to road irregularities, to allow small misalignment of axes, to reduce noise from the transmission, or to decrease wear of the mechanical joints. Bushings are made of a special rubber, used generally in energy dissipation, which presents a nonlinear viscoelastic relationship between the forces and moments and their corresponding displacements and rotations. In the methodology proposed here a finite element model of the bushing is developed in the framework of the finite element code ABAQUS to obtain the constitutive relations of displacement/rotation versus force/moment for different loading cases. The bushing is modeled in a multibody code as a nonlinear restrain that relates the relative displacements between the bodies connected with the joint reaction forces, and it is represented by a matrix constitutive relation. The basic ingredients of the multibody model are the same vectors and points relations used to define kinematic constraints in any multibody formulation. One particular, and relevant, characteristic of the formulation now presented is its ability to represent standard kinematic joints, clearance, and bushing joints just by defining appropriate constitutive relations. Spherical, revolution, cylindrical, and translational bushing joints are modeled, implemented, and demonstrated through the simulation of two multibody models of a road vehicle, one with perfect kinematic joints for the suspension sub-systems, and other with bushing joints. The tests conducted include an obstacle avoidance maneuver and a vehicle riding over bumps. It is shown that the bushing models for vehicle multibody models proposed here are accurate and computationally efficient so that they can be included in the vehicle models leading reliable simulations.

Real-time multibody vehicle model with bush compliance effect using quasi-static analysis for HILS

Vehicle handling simulations such as J-turns, slaloms, double lane changes, and fish hook maneuvering are essential to evaluate the control subsystems of the intelligent vehicle in the HILS (Hardware-in-the loop simulation). It is well known that the compliance effects from bushes in the suspension subsystem are important to have reliable solutions for the handling simulations. In the multibody vehicle dynamics model, the bush elements in the suspension subsystem are usually modeled as high-stiffness springs. However, due to this high stiffness of the bush model, it requires small step-sizes in the numerical integration. With this kind of bush model, it is almost impossible to achieve real-time simulations for HILS. In this paper, a quasi-static analysis method is presented in order to consider bush compliance effects for a real-time multibody vehicle model. To validate the proposed method, the quarter car simulations of the McPherson strut and the Multilink suspension subsystems have been performed, comparing with the ADAMS simulations. Full car bump run simulations and fish hook handling test simulations have also been carried out to see the effectiveness of the proposed model. The CPU time measured shows the real-time capabilities of the proposed method.

Sensitivity of a vehicle ride to the suspension bushing characteristics

The sensitivity of the ride characteristics of a road vehicle to the mechanical characteristics of the bushings used in its suspension is discussed here. First, the development and computational implementation, on a multibody dynamics environment, of a constitutive relation to model bushing elements associated with mechanical joints is presented. Bushings are made of a rubber type of material, which presents a nonlinear and viscoelastic relationship between the forces and moments and their corresponding displacements and rotations. Suitable bushing models for vehicle multibody models must be accurate and computationally efficient, leading to more reliable models. The bushing is modeled in a multibody code as an arrangement of springs that penalize the motion between the bodies connected. In the methodology proposed here, a finite element model of the bushing is developed in the framework of a finite element (FE) code to obtain the curves of displacement/rotation versus force/moment for different loading cases. The basic ingredients of the multibody model are the same vectors and points relations used to define kinematic constraints in any multibody formulation. Spherical, cylindrical and revolute bushing joints are developed and implemented in this work, since the methodology is demonstrated through the ride over bumps, at different speeds, of two multibody models of a road vehicle: one with perfect kinematic joints, for the suspension sub-systems; the other with bushing joints, riding. Then, sensitivities of different vehicle kinematic responses to the characteristics of the bushings used in the suspension are evaluated, by using numerical sensitivities. Based on the sensitivity analysis, indications on how to modify the vehicle response by modifying the bushing characteristics are drawn.

실시간 다물체 차량 해석을 위한 준정적법의 컴플라이언스 효과 모델링

Compliance effect consideration method for real-time multibody vehicle dynamics is proposed using quasi-static analysis. The multibody vehicle model without bush elements is used based on the subsystem synthesis method which provides real-time computation on the multibody vehicle model. Reaction forces are computed in the suspension subsystem. According to deformation from the quasi-static analysis using reaction forces and bush stiffness, suspension hardpoint locations and suspension linkage orientation are changed. To validate the proposed method, quarter car simulations of McPherson strut and multilink suspension subsystems are performed. Full car bump run simulations and fish hook handling test simulations are also carried out comparing with the ADAMS vehicle model with bush elements. CPU times are also measured to see the real-time capabilities of the proposed method.