Nonlinear Stiffness Model Research Articles

Modified analytical solution of shield tunnel lining considering nonlinear bending stiffness of longitudinal joint

As discontinued segmental joint plays a great important role in the mechanical behavior of shield tunnel lining, design for the joint suffers much from the difficulty in selecting proper parameters i.e., reduction factor η, spring stiffness. The mechanical behavior of longitudinal joint is highly nonlinear and can be classified into four stages. A four stage analytical model for determining the joint bending stiffness is proposed considering the influence of bending moment and axial force. The proposed analytical model is verified by 3D FE model and experimental results of Shanghai Yangtze River Tunnel. An analytical full ring solution based on Lee’s model is proposed incorporating the nonlinear joint stiffness model. The modified analytical solution is compared with the Lee’s model and 3D FE model with a case study of metro tunnels. Compared to Lee’s model, the modified solution has a better prediction of bending moment distribution especially at tunnel invert while has a close prediction of axial force distribution. The proposed solution has a good agreement with FEM results in tunnel deformation including horizontal convergence and joint opening. The maximum prediction errors are 1.5 mm for horizontal convergence and 0.36 mm for joint opening compared to 3D FE model analysis.

On the characteristics of gas foil conical bearings considering misalignment

PurposeThe purpose of this paper is to study the characteristics of gas foil conical bearings (GFCBs) considering the misalignment, the static and dynamic performances with different misalignment cases were studied.Design/methodology/approachA test rig on the air compressor supported by GFCBs has been developed to measure the practicability. A nonlinear bump stiffness model and one-dimensional beam top foil stiffness model were used as a basis for the calculation of static and dynamic performance. The finite element method and finite difference method are adopted to solve the Reynolds equation and the film thickness equation coupled, in which different misalignment cases were considered by changing the film thickness.FindingsThe supporting performance of GFCB is excellent, and the film clearance plays a critical role. The misalignment effects depend on the assembled angle and the misalignment angle. The load capacity, friction torque, temperature of GFCB decrease when the misalignment assembled angle is between 120° and 240°, while the dynamic bearing stability is improved. The static and dynamic performances show the opposite law for the other assembled angles, and the misalignment effect is more dramatic when there is a larger misalignment angle. Moreover, the bearing and running parameters largely affect the bearing performance.Originality/valueThe present study focuses on the static and dynamic characteristics of GFCB and investigates the effects of misalignment on the bearing performance.Peer reviewThe peer review history for this article is available at: https://publons.com/publon/10.1108/ILT-04-2020-0117

Nonlinear Connection Stiffness Identification of Heritage Timber Buildings Using a Temperature-Driven Multi-Model Approach

‘Que-Ti’ is an important component in typical Tibetan heritage timber buildings and it performs similar to corbel brackets connecting beam and column in modern structures. It transfers shear, compression and bending moment by slippage and deformation of components as well as limited joint rotation. A rigorous analytical model of ‘Que-Ti’ is needed for predicting the behavior of a timber structure under extreme loadings. Few researches have been done on this topic, particularly with the parameters describing the performances of this connection subjected to external loads. In this paper, a new temperature-driven multimodel approach is proposed to identify the stiffness parameters of a ‘Que-Ti’ connection in its operating environment. Models with nonlinear compression and rotational springs have been developed to take into account the change of mechanical behavior of the ‘Que-Ti’ affected by the temperature variation in typical heritage Tibetan buildings. The column–beam connection is modeled as two nonlinear rotational springs and one nonlinear compressive spring. Ambient temperature variation is treated as a force function in the input (temperature)–output (local mechanical strains) relationship, and stiffness identification is conducted iteratively via correlating the calculated strain responses with measured data. The nonlinear model of the joint is reproduced with a number of linear local models in different deformation scenarios that are corresponding to different temperature ranges. The stiffness parameters can be identified using a multimodel approach. Numerical results show that the method is effective and reliable to identify the nonlinear connection stiffness of the ‘Que-Ti’ accurately with the temperature change even with 10% noise in measurements. The monitoring data from a long-term monitoring system installed in a typical heritage Tibetan building is used to further verify the method. The experimental results show that the identified stiffness by the proposed method with nonlinear connection stiffness model can get better results than that obtained from the linear connection stiffness model.

A spatial closed-form nonlinear stiffness model for sheet flexures based on a mixed variational principle including third-order effects

The sheet flexure is commonly used to provide support stiffness in flexure mechanisms for precision applications. While the sheet flexure is often analyzed in a simplified form, e.g. by assuming planar deformation or linearized stiffness, the deformation in practice is spatial and sufficiently large that nonlinear effects due to the geometric stiffness are significant.This paper presents a compact analytical model for the nonlinear stiffness characteristics of spatially deforming sheet flexures under general 3-D load conditions at moderate deformations. This model provides closed-form expressions in a mixed stiffness and compliance matrix format that is tailored to flexure mechanism analysis. The effects of bending, shear, elongation, torsion and warping deformation are taken into account, so that the stiffness in all directions, including the in-plane lateral support direction, is modeled accurately. The model is verified numerically against beam and shell-based finite elements. The approach for deriving closed-form solutions in a nonlinear context is detailed in this paper. The Hellinger–Reissner variational principle with a specific physically motivated set of low-order interpolation functions is shown to be well-suited to the geometrically nonlinear analysis of flexures.An extension of the derivation approach to the nonlinear closed-form analysis of general flexure mechanisms consisting of multiple sheet flexures connected in parallel is presented. This is demonstrated with the case of a spatially deforming parallelogram flexure mechanism and a cross-hinge flexure mechanism.

Effect of taper error on the performance of gas foil conical bearing

Purpose This paper aims to study the bearing performance with different cone angle errors, to study the effect law of manufacturing taper error on the properties of gas foil conical bearing (GFCB). Design/methodology/approach For the GFCB supported by separated bump foil strips, a nonlinear structure stiffness model considering Coulomb friction and arch characteristics was proposed. The finite element method and finite difference method were used to solve the Reynolds equation and the film thickness equation by coupling, and the properties of the GFCB were obtained. The effect of foil and bearing structure parameters on the static and dynamic performance under different taper error cases was analyzed. Moreover, a test on the air compressor supported by GFCBs was conducted to verify the practicability. Findings The taper error has a largely adverse effect on the load capacity of GFCB. When the taper error is −0.03°, the radial load capacity Fr and axial load capacity Fz decrease by 37.5 and 58.3%, respectively. The taper error decreases the direct stiffness and cross-coupled damping of GFCB, which will weaken the bearing stability. Moreover, the performance of GFCB is closely related to the foil and bearing parameters. Originality/value The taper error adversely affects the static and dynamic characteristics of GFCB, which should be concerned by bearing designers, researchers and academicians. Peer review The peer review history for this article is available at: https://publons.com/publon/10.1108/ILT-03-2020-0089/

Study on the performance of novel gas foil conical bearings

To reduce the mass and size of low-power turbine machinery, a new type of gas foil conical bearing was proposed, and its static and dynamic performance was systematically studied. Based on a nonlinear bump stiffness model considering rounding and friction, the structural stiffness distribution, load capacity, dynamic stiffness, and damping coefficients of gas foil conical bearing were calculated, and the influence of bearing parameters on its static and dynamic characteristics was studied. In addition, a pair of gas foil conical bearings was used to replace the traditional radial-thrust foil bearing support scheme on an air compressor to explore the practicability of the novel bearing. The results show that the new gas foil conical bearing has an excellent supporting performance and broad prospects for application. The bump radius, rounding radius, friction coefficients, and foil thickness will significantly influence the bump foil stiffness. The bearing parameters such as structural stiffness, nominal clearance, cone angle, and eccentricity have a large effect on its static and dynamic performance, and we can obtain the desired bearing characteristics by tailoring these parameters.

Study on the performance of gas foil thrust bearings with stacked bump foils

Purpose This paper aims to improve the load capacity of gas foil thrust bearing (GFTB) and to introduce and study a novel bearing with stacked bump foils. Design/methodology/approach For the proposed novel GFTB supported by stacked foils, some bump-type gaskets with several partial arches are inserted below the regular bump foil, and the height of each arch can be made differently. These features make the bump foil thickness and height gradually increase, which can bring enhanced support stiffness and convergent film at the trailing edge. Based on a new nonlinear bump stiffness model considering bump rounding and friction force, the finite element and finite difference method were used to solve the coupling Reynolds equation, energy equation and foil deformation equation. Finally, the structural stiffness and static characteristics of the novel GFTB were gained and compared with the traditional bearing. Findings The novel GFTB has an additional convergence effect in the parallel section, which improves the static performance of bearing. The bearing capacity, friction moment, power loss and temperature rise of the novel GFTB are all higher than those of the traditional bearing, and the static characteristics are related to the parameters of stacked bump foils. Originality/value The stacked bump foils bring a fundamental enhancement on the load capacity of GFTB. The results are expected to be helpful to bearing designers, researchers and academicians concerned. Peer review The peer review history for this article is available at: https://publons.com/publon/10.1108/ILT-10-2019-0449/

Performance of Thrust Foil Bearings with Nonlinear Foil Stiffness Model

Performance of Thrust Foil Bearings with Nonlinear Foil Stiffness Model

A Method to Solve the Stiffness of Double-Row Tapered Roller Bearing

By analyzing the roller force, the nonlinear stiffness model of the double-row tapered roller hub bearing is derived, and the method of solving the hub bearing stiffness matrix is summarized: if the displacement between the inner and outer rings is known, the stiffness of the hub bearing can be directly calculated. If only the external load of the hub bearing is known, the displacement of the hub bearing needs to be solved by numerical method, and then the stiffness of the hub bearing can be calculated. The improved Newton-Raphson method is used to solve the stiffness matrix of the hub bearing. Three-dimensional FE model of DRTRBs is presented and validated the proposed the stiffness matrix of the hub bearing. It is found that the radial stiffness of the hub bearing is greater than the axial stiffness. The stiffness of the hub bearing is greatly affected by the vertical force of the ground and the wheel driving torque, showing obvious nonlinearity. The smaller the vertical ground load and wheel driving torque, the greater the influence of vertical ground load and wheel driving torque on the hub bearing stiffness.

Study on seismic performance of shield tunnels under combined effect of axial force and bending moment in the longitudinal direction

Shield tunnels are usually subject to combined axial force and bending moment in the longitudinal direction during earthquake. In traditional analytical models, the combined effects of axial force and bending moment are considered separately. In this paper, the modified longitudinal nonlinear stiffness model considering the combined effects of axial load and bending moment together is proposed. The structural deformation of the shield tunnel is divided into five patterns based on the elastic-plastic state of joint bolts and the contact state of circumferential joints. For each pattern, the longitudinal equivalent bending stiffness and tensile stiffness are obtained individually. In order to verify the accuracy of the proposed model, a case study based on a real tunnel structure in Tianjin in China is carried out. The longitudinal seismic response of the tunnel is investigated. The study shows that the elastic-plastic state of longitudinal bolts has significant influence on the structural deformation when the seismic intensity is large. The deformation of the modified longitudinal nonlinear stiffness model is 20.79% smaller than that of the longitudinal continuous model. The research results also show that the opening of the circumferential joint and the internal force of the longitudinal bolt both reach the maximum value when the incident angle of the seismic wave reaches about 45°.

Nonlinear behavior of rail fastening system on slab track at railway bridge ends: FEA and experimental study

Rail fastening systems on a slab track at railway bridge ends are often damaged, which increases the maintenance costs. The serviceability evaluation of a rail fastening system on a slab track at railway bridge ends is based on the assumption that the stiffness of the rail fastening system is linear. The behavior of the rail fastening system on slab track at the railway bridge ends is studied in detail in this study. In the experimental model, the abutment and pier of the railway bridge were considered to be directly connected to two H-beams through a rail and rail fastening systems. A stiffness model of the rail fastening system for the finite-element analysis was established by performing a clamping-force test. In the finite-element analysis, the rail fastening system was considered as both linear and nonlinear stiffness models. The experimental and numerical results were very similar when the rail fastening system was considered as the nonlinear stiffness model. With the same uplift force acting on the rail fastening system, the displacement results of the nonlinear stiffness model were larger than those of the linear stiffness model. Therefore, it is necessary to employ the nonlinear behavior of the rail fastening system and investigate a displacement-based design method of the rail fastening system when evaluating the serviceability of the rail fastening system on the slab track at railway bridge ends.

Characteristics analysis of rotor model with support–foundation looseness fault considering elastic–plastic contact

In this paper, the nonlinear response characteristics of a single degree of freedom lumped mass model with a new support looseness fault are analyzed. In this model, a new nonlinear contact stiffness model between the support and foundation is considered, which considers the elastic–plastic deformation of the contact parts. The nonlinear response of the system is obtained by numerical integration methods. The nonlinear results are compared with an asymmetric model and a piece-wise contact model. By comparing three different looseness fault models, the mechanism of generating multi-frequencies and fractional frequencies is analyzed. The result shows that plastic deformation makes the displacement amplitude increase rapidly; odd harmonics appear in symmetric stiffness model due to the modulation of symmetric shock forces; however, harmonic response appears in asymmetric stiffness model; when the contact number is odd, the frequency components contain odd or even order fractional frequencies at supercritical speeds due to the incompatibility of contacts on the upper and lower surfaces; when the contact number is even, multiple frequency components appear at subcritical frequencies.

Nonlinear trapping stiffness of mid-air single-axis acoustic levitators

We describe and experimentally explore a nonlinear stiffness model of the trapping of a solid particle in a single-axis acoustic levitator. In contrast to the commonly employed linear stiffness assumption, our nonlinear model accurately predicts the response of the system. Our nonlinear model approximates the acoustic field in the vicinity of the trap as a one-dimensional sinusoid and solves the resulting dynamics using numerical continuation. In particular, we predict a softening of stiffness with amplitude as well as period-doubling bifurcations, even for small excitation amplitudes of ≈2% of the wavelength. These nonlinear dynamic features are observed experimentally in a single-axis levitator operating at 40 kHz and trapping millimetre-scale expanded polystyrene spheres. Excellent agreement between the observed and predicted behaviour is obtained suggesting that this relatively simple model captures the relevant physical phenomena. This new model enables the dynamic instabilities of trapped particles to be accurately predicted, thereby benefiting contactless transportation and manipulation applications.

A novel nonlinear contact stiffness model of concrete–steel joint based on the fractal contact theory

The heavy-duty machine tool is usually assumed in the concrete foundation, in which the machine tool-foundation joints have a critical effect on the working accuracy and life of heavy-duty machine tool. This paper proposed a novel contact stiffness model of concrete–steel joint based on the fractal theory. The topography of contact surface exist in concrete–steel joint has a fractal feature and can be described by fractal parameters. Asperities are considered as elastic, plastic deformation in micro-scale. However, the asperities of concrete surface will be crushed when the stress is larger than their yield limit. Then, the force balance of contact surfaces will be broken. Here, an iteration model is proposed to describe the contact state of concrete–steel joint. Because the contact asperities cover a very small proportion (less than 1%), the load on crushed asperities is assumed carried by other no contact asperities. This process will be repeated again and again until the crushed asperities are not being produced under external load. After that, the real contact area, contact stiffness of the concrete–steel joint can be calculated by integrating the asperities of contact surfaces. Nonlinear relationships between contact stiffness and load, fractal roughness parameter G, fractal dimension D can be revealed based on the presented model. An experimental setup with concrete–steel test-specimens is designed to validate the proposed model. Results indicate that the theoretical vibration mode shapes agree well with the experimental variation mode shapes. The errors between theoretical and experimental natural frequencies are less than 4.18%. The presented model can be used to predict the contact stiffness of concrete–steel joint, which will provide a theoretical basis for optimizing the connection characteristic of machine tool-concrete foundation.

A Composite Bolted Joints Non-Linear Stiffness Model and its Application

The paper presents a nonlinear stiffness model on the bearing damage and bolt-load distribution rule of composite bolted joints. Based on the experimental stress-strain curves of composite single-bolted joints, the bearing damage process of structures can be divided into:linear non-damage stage; nonlinear damage propagation stage and linear degeneration stage. In this paper, the composite bolted joints are represented by a system of springs in which the static friction effects between laminates are considered in the linear stage, the non-linear interlaminate shear behavior and the nonlinear damage function for composite are defined and failure behavior based on the energy abortion principle is simulated in the linear degeneration stage. At last, the multi-fasters spring model is analyzed by importing the single-faster spring model.Comparing with the experimental data of multi-fasters shows the numerical result is in good agreement with experimental curves and has little error on the bolt-load distribution, which satisfies to the requirements in the engineering.

A novel stiffness model for a wall-climbing hexapod robot based on nonlinear variable stiffness

In this article, the most contribution is to propose a novel general stiffness model to analyze the stiffness of a wall-climbing hexapod robot. First, we propose a new general stiffness model of serial mechanism, which includes the linear and nonlinear stiffness models. By comparison, the nonlinear stiffness model is a variable stiffness model which introduces the external load force as a variable, obtaining that the nonlinear stiffness model can greatly improve the accuracy of stiffness model than linear stiffness model. Then, the stiffness model of one leg of the robot and the overall stiffness model of the robot are derived based on the general stiffness model. Next, to improve the stiffness of the robot, a new minimum and maximum stiffness are introduced, which provide with effective reference for the selection and optimization of the structural parameters of the robot. Finally, we develop a new wall-climbing hexapod robot based on selection and optimization of the structural parameters, then the experiments are used to show that the selection of structure parameters of the robot effectively improve the stiffness of the robot.

A Non-Linear Stiffness Model for Serial and Parallel Manipulators

The paper presents a methodology to enhance the stiffness analysis of serial and parallel manipulators with passive joints. It directly takes into account the loading influence on the manipulator configuration and, consequently, on its Jacobians and Hessians. The main contributions of this paper are the introduction of a non-linear stiffness model for the manipulators with passive joints, a relevant numerical technique for its linearization and computing of the Cartesian stiffness matrix which allows rank-deficiency. Within the developed technique, the manipulator elements are presented as pseudo-rigid bodies separated by multidimensional virtual springs and perfect passive joints. Simulation examples are presented that deal with parallel manipulators of the Ortholide family and demonstrate the ability of the developed methodology to describe non-linear behaviour of the manipulator structure such as a sudden change of the elastic instability properties (buckling).

A new fatigue failure theory for multidirectional fiber-reinforced composite laminates with arbitrary stacking sequence

Fatigue failure is one of the most important failure types of fiber-reinforced composites. In this paper, a new fatigue failure theory for multidirectional fiber-reinforced composite laminates with an arbitrary stacking sequence is developed, by combining nonlinear residual strength and residual stiffness models with the recently improved Puck’s failure theory which includes the in situ strength effect. This fatigue theory can predict the fatigue life, residual strength and residual failure envelope of fiber-reinforced composite laminates under multidirectional loadings. For these predictions it is necessary to recalculate the fatigue lives of laminae after each cycle since the stresses in the laminae change due to stiffness degradation. It is also necessary to account for the nonlinear accumulation of damage at the new stress level in the laminae resulting from stiffness degradation. This is achieved by using the concept of equivalent cycle. The theoretical predictions are in good agreement with available experimental results.

Nonlinear stiffness models of a net twine to describe mesh resistance to opening of flexible net structures

Numerical simulation of marine flexible net structures allows predicting the behavior of fishing gears and aquaculture cages. In recent years, the tendency toward the use of thicker and stronger twines in netting materials has made its resistance to opening a key factor in the performance of such structures. To accurately describe the mesh resistance to opening, a net twine was modeled as a double-clamped beam and its force–displacement response was calculated by finite element analysis. Fitting techniques were used to develop three different dimensionless stiffness models that express elastic forces in the twine as an explicit nonlinear function of its deformation: (1) a polynomial fitting of the force, (2) a spline fitting of the potential energy, and (3) a spring-based model able to deal with large axial deformations. Each model has different characteristics and advantages. Numerical and experimental tests were used to assess and compare them with previous models described in the literature. The results show that the presented models have very good accuracy and high computational efficiency. They will allow introducing accurate simulation of mesh resistance to opening in numerical simulations of marine netting structures without a high impact in the computational performance.

Development of a higher-order model for a roll simulator

A four degree-of-freedom model of a roll simulator is developed. The roll simulator consists of a sled-platform assembly, where a recreational off-highway vehicle (ROV) is mounted to the platform which has the freedom to rotate up to 90 degrees. The purpose of the roll simulator is to re-create rollover manoeuvres ascertained from in-field vehicle dynamic experiments. Previous studies developed a two degree-of-freedom model; one for translational and one for rotational motion. This study expands on that model and increases the number of degrees-of-freedom to more closely emulate the roll simulator. The model incorporates non-linear stiffness models of the wire ropes and enables the model to have three translational degrees-of-freedom and one rotational degree- of-freedom. The equations of motion (EOM) are numerically simulated using Matlab’s Simulink. Comparisons are made with experimental data derived from the roll simulator to validate the model.