Link-spring Model Research Articles

Theoretical Evaluation of Lubrication Performance of Thrust-Type Foil Bearings in Liquid Nitrogen

The development of reusable liquid rocket turbopumps has gradually highlighted the disadvantages of rolling bearings, particularly the contradiction between long service life and high rotational speed. It is critical to explore a feasible bearing scheme offering a long wear life and high stability to replace the existing rolling bearings. In this study, liquid nitrogen is adopted to simulate the ultra-low temperature environment of liquid rocket turbopumps, and theoretical evaluations of the lubrication performance of thrust-type foil bearings in liquid nitrogen are conducted. A link-spring model for the bump foil structure and a thin-plate finite element model for the top foil structure are established. The static and dynamic characteristics of the bearings are analyzed using methods including the finite difference method, the Newton–Raphson iteration method, and the finite element method. Detailed analysis includes the effects of factors such as rotational speed, fluid film thickness, thrust disk tilt angle, and the friction coefficient of the bump foil interface on the static and dynamic characteristics of thrust-type foil bearings. The research results indicate that thrust-type foil bearings have a good load-carrying capacity and low frictional power consumption. The adaptive deformation of the foil structure increases the fluid film thickness, preventing dry friction due to direct contact between the rotor journal and the bearing surface. When faced with thrust disk tilt, the direct translational stiffness and damping coefficient of the bearing do not undergo significant changes, ensuring system stability. Based on the results of this study, the exceptional performance characteristics of thrust-type foil bearings make them a promising alternative to rolling bearings for the development of reusable liquid rocket turbopumps.

Experimental and theoretical investigation of rotordynamic characteristics of a rigid rotor supported by an active bump-type foil bearing

This study aims to investigate the rotordynamic characteristics of a rigid rotor supported by an active bump-type foil bearing (ABFB) with real-time controllable mechanical preload. ABFB is composed of a traditional bump-type foil bearing with controlling substructures that consist of flexure hinges, lever amplifiers and piezoelectric stack actuators (PZTs). The structure realises the adjustment of bearing's mechanical preloads by actively controlling the DC voltage on the PZTs (U). Experimental investigations based on rotordynamic test rig are performed to measure the effects of ABFB on rotordynamic responses of rotor-ABFB system. Combined with the dynamics of the active control structures and modified link-spring model, linear and nonlinear mechanical models of the rotor-ABFB system are established, and the rotordynamic responses of the rotor are predicted. Both the experimental and theoretical results verify the possibility of building active rotor-gas foil bearing systems. Moreover, the effects of rotor unbalance on the rotordynamic responses of the rotor-ABFB system are investigated.

Study of Complex Looping With Five Kinks in Thermosonic Wire Bonding by Using Variable-Length Link-Spring Model

Complex looping with five kinks has been commonly used in the stacked die 3-D packaging and simulation models were desired to help control the loop profile in modern thermosonic wire bonding. The traditional finite-element model was time consuming and cost intensive, and the simple link-spring analysis model cannot provide precision simulation of the complex looping as it ignores the stage when the capillary is moving upward from the first bond to the highest point. In this paper, a variable-length link-spring model is proposed to deal with the above problems. In this model, the wire segments and moment equilibrium equations at the opening end of the wire are dynamically added during the capillary upward movement stage, to simulate the wire feeding and loop formation process. The experiment was carried and the complete wire loop profile in the model agree well with the experimental results. Based on this verified model, the effects of various parameters on loop profiles with five kinks were studied. It seems that the first kink height has little effect on the final loop profiles, while the second kink height can change the loop profiles and sweep resistance obviously.

Buckling and vibration analysis of nonlocal axially functionally graded nanobeams based on Hencky-bar chain model

• We introduce Hencky bar net model for inhomogenous nanobeams for the first time. • We show the formal analogy between Hencky bar net model and Finite Difference Model. • We demonstrate the convergence to continuum plate results by increasing the grid size for any proposed example. • The model deals with both stepped and tapered microbeams. Buckling and free vibration analyses of nonlocal axially functionally graded Euler nanobeams is the main objective of this paper. Due to its simplicity, the Eringen's differential constitutive model is adopted for describing the nonlocal size dependency of nanostructure beam. The nonlocal equilibrium equation is derived using the principle of the minimum potential energy principle, and discretized by using the link-spring model known in literature as Hencky bar-chain model. The general applicability of the proposed approach allows analyses of functional graded microbeams without any restriction on variability, boundary and loading conditions. A comparison with results available in the literature shows the reliability of the method.

Characterisation and calculation of nonlinear vibrations in gas foil bearing systems–An experimental and numerical investigation

This paper states a unique classification to understand the source of the subharmonic vibrations of gas foil bearing (GFB) systems, which will experimentally and numerically tested. The classification is based on two cases, where an isolated system is assumed: Case 1 considers a poorly balance rotor, which results in increased displacement during operation and interacts with the nonlinear progressive structure. It is comparable to a Duffing-Oscillator. In contrast, for case 2 a well/perfectly balanced rotor is assumed. Hence, the only source of nonlinear subharmonic whirling results from the fluid film self-excitation. Experimental tests with different unbalance levels and GFB modifications confirm these assumptions.Furthermore, simulations are able to predict the self-excitations and synchronous and subharmonic resonances of the experimental test. The numerical model is based on a linearised eigenvalue problem. The GFB system uses linearised stiffness and damping parameters by applying a perturbation method on the Reynolds Equation. The nonlinear bump structure is simplified by a link-spring model. It includes Coulomb friction effects inside the elastic corrugated structure and captures the interaction between single bumps.

Experimental and numerical analysis of the dynamic behaviour of a foil bearing structure affected by metal shims

This paper gives a detailed overview of a shimmed first generation gas foil bearing (GFB). Three metal shims are inserted inside the elastic GFB structure between the bumps and the bearing sleeve. Dynamic shaker tests with a harmonic excitation are used for evaluating dynamic stiffness and damping of the elastic structure. In addition, a numerical structural bump model, based on a link-spring model, is applied to confirm the experimental results, where a dynamic friction model with pure Coulomb friction is used. It neglects Stribeck’s effects and avoids numerical problems in time domain analysis. Good agreement between experiments and simulations is observed.

Experiments and predictions on the performance of double-bump foil bearings: Effects of bearing loads and height difference between bumps

The concept of multilayer bump foils was introduced in the design of bump foil bearings to produce a double-bump foil bearing, which can provide increased load capacity and damping by adding another bump foil in the underlying structure. The height difference between the upper and lower bumps is a crucial parameter in the design and application of such structure. In this study, two double-bump foil bearings with various height differences between bumps are designed and fabricated to compare with an ordinary bump foil bearing. Three bearings are examined via static and dynamic load tests to estimate the structural stiffness and equivalent viscous damping. Test results indicate that lower bumps can enhance both the structural stiffness and equivalent viscous damping. A theoretical link-spring model, which exhibits good agreement with the data obtained from the static load test, is adopted to analyze the effect of height difference between bumps on gas film thickness and gas pressure of double-bump foil bearings. Results show that lower bumps of the double-bump foil bearing with a smaller height difference become active more easily and are more likely to form a stable double-bump supporting structure.

Semi-analytical solutions for optimal design of columns based on Hencky bar-chain model

This paper is concerned with the shape optimization problem of columns for a given volume and length against buckling by using the discrete link-spring model or the so-called Hencky bar-chain model (HBM). This discrete beam model comprises a finite number of rigid segments connected by frictionless hinges and rotational springs. In particular, the rotational spring stiffness of HBM is a function of the square of cross-sectional area of columns with regular polygonal or circular cross-sectional shape. Therefore, the design of optimal rotational spring stiffnesses of a HBM allows one to obtain the optimal shape of a column provided that the assumed number of springs is sufficiently large. The present formulation of HBM for column optimization is prompted by some discrepancies in the volume calculations and the specification of the spring stiffness at the clamped end in Krishna and Ram (2007) discrete link-spring model formulation. By using the correct formulation and the semi-analytical method proposed by Krishna and Ram (2007), we determine the optimal shape of clamped-free, pinned-pinned, clamped-spring-supported columns. In addition, we extend the semi-analytical method to optimize the shape of clamped-free columns under distributed loads. Also presented herein are exact buckling solutions for the uniform HBM under axial load and selfweight as well as the non-uniform HBM under axial load with a specific class of spring stiffnesses.

A valid method of gas foil bearing parameter estimation: A model anchored on experimental data

Gas foil bearings are a smart green technology and suitable for the next generation of small turbo machinery e.g. turbochargers, micro gas turbines, range extenders and compressors of fuel cells. A combination of low power loss, high speed operation and the omission of an oil system are the major advantages. To enable access to this technology, it is essential to evaluate critical speeds and onset speeds of subharmonic vibration of the rotor system in the first design stage. Hence, robust and valid models are necessary, which correctly describe the fluid structure interaction between the lubrication film and the elastic bearing structure. In the past three decades several experimental and numerical investigations of bearing parameters have been published. But the number of sophisticated models is small and there is still a lack of validation towards experimental works. To make it easy for designers dealing with this issue, the bearing parameters are often linearised about certain operating points. In this paper a method for calculating linearised bearing parameters (stiffness and damping) of gas foil bearing is presented. Experimental data are used for validation of the model. The linearised stiffness and damping values are calculated using a perturbation method. The pressure field is coupled with a two-dimensional plate model, while the non-linear bump structure is simplified by a link-spring model. It includes Coulomb friction effects inside the elastic corrugated structure and captures the interaction between the single bumps. For solving the separated perturbed Reynolds equation a static stiffness is used for the 0. order equation (stationary case) and a dynamic stiffness is applied for 1. order equation (dynamic case). Therefore, an additional dynamic structural model is applied to calculate the dynamic stiffness. The results depend on the load level and friction state of each bump. Different case studies including the impact of clearance, frictional contacts and the comparison of a linear and non-linear structure are carried out for infinitesimal perturbations. The results show, that the linear structure underestimates main and cross-coupling effects. The impact of the clearance is notable, while the impact of the overall frictional contacts is small due to relatively small loadings. The infinitely small perturbation model is adapted to the experimental setup by using a superposition of two resulting bearing parameters identifications of two total loadings including shaker forces. Due to this adaptation a good correlation with the experimental results of the bearing parameters is achieved.

Effects of misalignment on the structure characteristics of bump-type foil bearings: a comparison between experimental and theoretical results

Purpose – The purpose of this paper is to investigate the effects of misalignment on the static and dynamics characteristics of bump-type foil bearings (BFBs). High-speed and high-temperature oil-free turbomachinery can be realized with the use of gas foil bearings (GFBs). GFBs have a flexible supporting structure; thus, they can tolerate a higher degree of misalignment compared with rolling element bearings. Design/methodology/approach – A test rig for GFBs has been developed to measure the effects of misalignment on the structure characteristics of bump-type foil bearings. The link-spring model, which is the foil structure model presented previously by the authors, is used as a basis in the present study to predict the static and dynamic performances of the foil structure. In general, predictions of the dynamic characteristics exhibit good agreement with the measurements acquired from the dynamic load tests. Findings – Results from the static tests show that GFBs develop high stiffness when the misalignment angle increases. Moreover, the dynamic characteristics of GFBs are identified by considering the test bearing supported by a non-rotating shaft as a one-degree-of-freedom system. The results indicate that the dynamic characteristics of GFBs strongly depend on excitation frequency and excitation amplitude because of the variation in the dynamic friction force within the foil structure. The structural stiffness and equivalent viscous damping increase with an increase in the misalignment angle. Originality/value – The present study focuses on the misalignment of GFBs and investigates experimentally the effects of misalignment on the structure characteristics of GFBs.

Variable-Length Link-Spring Model for Kink Formation During Wire Bonding

Kinks have a strong influence on the structural performance of wire bonds. In this study, a variable-length link-spring model has been developed to better understand kink formation. In this model, a gold wire was decomposed into segments that were represented by a link and a torsional spring. One end of the gold wire was fixed, and the other end was free. The friction and air tension forces at the wire ends were considered a function of the capillary position, and the wire segments and moment balance equations were added at the free end as the wire length increased. By using this model, the wire profile, moment, and curvature diagrams at the reverse motion stage were obtained to study the dynamical kink formation and wire-length increasing processes. The analysis result was verified experimentally. Good agreement is obtained between the analytical and the experimental wire profiles. This study indicates that a moment magnitude of several hundred mN·μm is required to form a kink, and the wire profile is the result of the residual curvature and the instantaneous bending moment.

Prediction of Dynamic Coefficients of Bump-Type Foil Bearings with Bumps Considered as Link-Spring Structures

Bump-type foil bearings have demonstrated excellent performance in service experiences in recent 30 years. Due to their special structures, the performance prediction models must couple the air pressure with the elastic deflection of the compliant foil structure. The solution of air pressure has been well settled in the hydrodynamic lubrication theory, while the deflection of foils now still cannot be accurately solved due to the mechanical complexity of the corrugated bump structure. A theoretical model, link-spring model, was developed to predict the structure characteristics of foil strips in a previous work. Effects of four factors, i.e., the elasticity of the bump foil, the interaction forces between bumps, the friction forces at contact surfaces, and the local deflection in the top foil, had been taken into consideration. In this investigation, based on the link-spring model, a perturbation approach is used to determine the dynamic characteristics of bump-type foil bearings. The calculations have been performed with different meshes. And the accordant results show that the calculation is independent on the mesh. The effects on the dynamic coefficients of the friction forces and radial clearance are also evaluated using the perturbation method.

Parametric Studies on Static Performance and Nonlinear Instability of Bump-Type Foil Bearings

A theoretical analysis model is presented to investigate the nonlinear dynamic behavior of bump-type foil bearings, i.e., the instability and unbalance response. In the developed model, the foil structure of bump-type foil bearings was simulated using the link-spring model, which was presented and validated in a previous study. During the calculation, the Reynolds equation and the foil structure model were coupled though the pressure and film thickness. An iteration solution method with the equation of shaft motion was applied for the shaft orbit. Parametric studies of design parameters, such as the bump number, the length ratio of the segment between two bumps and a bump, the foil thickness, the bump height, and the Young's Modulus, were presented upon the analysis of static performance and the instability of bump-type foil bearings. It is noted that more bumps or lager length ratio lead to larger load capacity. But the bearing load capacity does not change with different bump heights. Moreover, the threshold speed of instability decreases significantly with the rise of the bump number, especially when the number of bumps is low. However, it almost does not affect by other four parameters.

546 バンプ型フォイル軸受の静特性と安定性に対するフォイル設計変数の影響

In this study, the simulation model presented and validated in a previous study, Link-Spring model, is used to describe the foil structure of bump-type foil bearing. A theoretical analysis is presented to investigate the nonlinear dynamic behavior, i.e., the stability and unbalance response of bump-type foil bearings. The trajectory of shaft is simulated with the iteration calculation of shaft motion equation at each time step, which is coupled with the Reynolds' equation and the foil structure model. Parametric studies of number of bump, size of bump, thickness of foil and material of foil are presented based on the bearing load prediction and the nonlinear orbit simulations.