Elastic Support Research Articles

Effects of boundary conditions on phase shift of a dual-tube conveying fluid in the Coriolis mass flowmeter

This paper investigates the effects of boundary conditions on the performance of a Coriolis mass flowmeter with double straight tubes conveying fluid. The non-self-adjoint governing equations of motion are derived in dimensionless form, and the Galerkin method is used to discretize them into a set of ordinary differential equations. The free vibration analysis of the structure, including extraction of the natural frequencies and the symmetric and antisymmetric mode shapes of double tubes, is presented. In addition, the effects of the stiffness of the elastic supports on the phase shift induced between specific points of the tubes are investigated. The greater the phase shift between two special points, the easier it will be to measure by pickoffs; consequently, and the Coriolis flowmeter will be more accurate. The results revealed that the phase shift approaches zero for small translational stiffness but approaches a limited non-zero value when the rotational stiffness approaches zero. The maximum phase shift is obtained for a specific translational and rotational stiffness. The effects of the springs’ stiffness on the stability of the tubes are also investigated.

Vibration of prestressed beams: Experimental and finite-element analysis of post–tensioned thin-walled box-girders

In published works, very different explanations are given for the effect of prestressing on the beam dynamics, especially on the relationship between the vibrational effects caused by an external compressive force and those due to a prestressing force. To solve this conflict, free transverse vibrations of a post–tensioned thin-walled steel box-girder were investigated. Its fundamental frequencies were then measured for different values of prestressing. Subsequently, the experimental data were compared with a formula considering shear effects. The experimental data were also compared with two high-fidelity finite-element models assuming geometric nonlinearities. According to the obtained findings, and comparisons with tests reported in literature, the beam dynamics strongly depends on the contact of the cables with the surrounding section. If the tendons are not in contact with the surrounding cross-section, the beam dynamics due to an external compressive force is practically coincident to that caused by a prestressing force for low values of prestressing. In this case, the dynamics of prestressed beams are initially ruled by the compression-softening effect. For high values of prestressing, the beam would behave as a shallow arc with horizontal elastic supports. In this case, an increase of fundamental frequency is observed, but the yield stress would be exceeded. Conversely, in thin-walled steel girder-bridges with tendons in contact with the cross-section or draped with a deviator, the fundamental frequency is practically unaffected by prestressing. Thus, the fundamental frequency cannot be used as indicator for identifying long-term phenomena of prestressing losses, such as anchorage slips, tendon friction and relaxation.

Linear stability and nonlinear rotor responses of the gas foil bearing with multiple sliding beams

This article reports the stability and dynamic characteristics of a passive gas foil bearing (MSGFB) in which multiple sliding beams provide the elastic support and Coulomb damping. The linear and nonlinear dynamic models of rotor-MSGFB system are established, both considering the complex contact constraints inside foil structures. The instability mechanism of rotor-foil bearing system is revealed from the equivalent dynamic stiffness and the critical whirl speed. The half speed whirl and the frequency locked vibrations are both observed in the simulations, indicating the nonlinearities of gas film and foil structure respectively. The results also show that larger rotor preload of MSGFB help suppress the half speed whirl vibrations and assist improving high speed rotor stability especially under the friction effect.

Vibration and stability analysis of a spinning shaft with arbitrary boundaries subjected to partial load

This paper deals with the flexural vibration and dynamic stability of a spinning shaft subjected to partial load. Based on the theorem of moment of momentum and the theorem of motion of center of mass, the governing equation of motion for the spinning shaft system is obtained. Four kinds of boundary conditions including pinned-pinned, clamped-clamped, free-free, and elastic supports are considered. The exact whirl frequency equation is derived analytically for different boundary conditions. The obtained model is confirmed by comparing with the results reported in the available literature, and excellent agreement is presented. A numerical investigation is carried out to examine the effects of support stiffness, mass ratio, eccentric distance, and slenderness ratio on the whirl characteristics and stability of the system in terms of the obtained model. The results show that the whirl frequency and critical spinning speed of the system are dependent on these parameters, and the stability is mainly governed by the mass ratio and eccentric distance.

РАЗРАБОТКА ОПОРНО-ОСЕВОГО ПРИВОДА ПАССАЖИРСКИХ ЭЛЕКТРОПОЕЗДОВ С АСИНХРОННЫМИ ТЯГОВЫМИ ДВИГАТЕЛЯМИ

The problem of import-substituting components for the support-axial traction drive of an electric train with a wheel diameter of 950 mm is considered. As a result of the analysis, the disadvantages of the drive using foreign components are revealed: a high unsprung mass, which worsens the impact of the crew on the track, the difficulty of providing the required dynamic properties with elastic support of the motor-gear unit on the axle, the complexity of manufacturing and assembling the drive. The inexpediency of copying foreign solutions to create a domestic analogue of a traction drive is established. Variants of technically possible traction drive adapted for domestic production and easier to manufacture are considered. It is proposed to use a traction drive with a high-torque support - frame traction engine, with an increased outer diameter and the same circumferential speed of the rotor and an axial gearbox. It is proved that in this case, due to a decrease in the speed of the drive shaft, the design of the gearbox is simplified, which can be made single-reduction, and it becomes possible to use the developed and studied domestic flat rubber-cord couplings. A new design of the transverse rubber-cord coupling is proposed. For the case of using a domestic high-speed traction electric motor in the drive, it is proposed to use an aggregate-type traction drive. Two utility model patents are obtained for the proposed technical solutions and a patent application is filed.

Mathematical modeling and vibration analysis of rotating functionally graded porous spacecraft systems reinforced by graphene nanoplatelets

This paper investigates the theoretical modeling and coupled free vibration behaviors of a rotating double‐bladed shaft assembly resting on elastic supports in a spacecraft system. According to the Kirchhoff plate theory and the Euler–Bernoulli beam theory, the theoretical model is established. The presented rotor is considered to be made of porous foam metal matrix and graphene nanoplatelet (GPL) reinforcement. Nonuniform distributions of porosity and GPLs are taken into account, which leads to the functionally graded (FG) structure. The effective material properties of the double‐bladed shaft assembly are varying along the radius and thickness directions of the shaft and blade, respectively. Moreover, the rule of mixture, the Halpin–Tsai model, and the open‐cell scheme are employed to determine its material properties. Considering the gyroscopic effect, the Lagrange equation is utilized to derive the coupled equations of motion. Then the traveling wave frequencies of the double‐bladed shaft assembly are obtained by applying the assumed modes method and substructure modal synthesis method. A detailed parametric analysis is conducted to examine the effects of the rotating speed, GPL weight fraction, GPL distribution pattern, GPL length‐to‐thickness ratio, GPL length‐to‐width ratio, porosity coefficient, porosity distribution pattern, shaft length‐to‐radius ratio, blade length‐to‐thickness ratio, support stiffness and support location on the free vibration behaviors of the double‐bladed shaft assembly.

Effect of wear rings on buckling load capacity of Two-Stage hydraulic cylinder

In a two-stage hydraulic cylinder, the rod is a critical component for stability against buckling failure. A small disturbance in the rod may increase the instability that may lead to catastrophic failure. A small support against the instability may increase the system's flexural strength to make it more stable. The wear rings provide an elastic support against the rod and cylinder's instability and increase the flexural strength of the connections. In this paper, a theoretical model for the buckling load of a pin-pin-mounted two-stage hydraulic cylinder is proposed. The buckling load calculation by the proposed approach is in close match (6.96% deviation) with the experimental results reported in the literature for the case of single-stage hydraulic cylinders. The effect of bending stiffness on stability is considered by considering the stiffness of the rod and cylinder along with the wear rings. A hyperelastic model is used to determine the bending stiffness of wear rings in three different cases: first when all wear rings are present, second when three wear rings are present, and third when only two wear rings are present. The stiffness of the rod and piston head is determined with the help of the conjugate beam method. Eigenvalue buckling analysis (finite element method) is carried out to verify the buckling load of a two-stage hydraulic cylinder with two, three, and four wear rings. Theoretical results are observed to be in good agreement with the numerical simulations (FEA) results, with a percentage deviation of 0.22 % − 9.22 % for different cases of wear rings positions. It is observed that when all four wear rings are present, the buckling load capacity increases by 1.14 times, whereas the removal of wear rings on the gland side decreases it by 0.96 times.

Dynamical modeling and experimental validation for squeeze film damper in bevel gears

Squeeze film damper (SFD) effectively reduces rotor motion amplitude while crossing critical speeds for vibration and noise reduction in a wide application of aero-engine rotor system. The squirrel cage elastic support (SCES) as an essential component of SFD, accurate modelling of SCES using component modal synthesis (CMS) to account for its flexibility, and calculation of nonlinear oil film forces for SFD using the finite difference method. The CMS method is also used to model the bevel gear wheel body and to establish a finite element nodal dynamics model of the gear-rotor-bearing-SFD system considering time-varying mesh stiffness, tooth clearance, gyroscopic effects, transmission error excitation and SFD oil film forces. The modal frequency of the gear shaft in aero-engine power transmission is exercised to verify the established bevel gear model. Moreover, a bevel gear test platform with SFD is established to study the vibration reduction effect of SFD. It is found that SFD can effectively improve the transmission of the bevel gear system, and the flexibility of the SCES makes the critical speed shift. Actually, the smaller the stiffness of SCES, the more the critical speed is moved to the left. By using CMS to determine the flexibility of bevel gear and SCES, it can show the complete phenomenon of the dynamic behaviors of the bevel gear system with SFD.

Weak-End and Frequency Detection of Elastically Supported Bridges by Contact Residual Response of Two-Axle Test Vehicle in a Round Trip

The two support bearings of an elastically supported (ES) bridge may be unequal in stiffness due to aging or other factors. This paper proposed an effective technique of using the contact residual response (CRR) generated by a two-axle test vehicle in its round-trip movement to detect the weak end and frequencies of the bridge. First, the round-trip CRR is presented in closed form. It is featured by three facts: (1) the CRR is given in terms of the vehicle–bridge “contact” response and thus is free of the annoying vehicle’s frequency; (2) being created as the “residual” of the responses of the two axles, the CRR is immune to the disturbance of the surface roughness; and (3) by letting the vehicle move in “round trip,” the weak end of the bridge can be detected, while all the bridge frequencies are enhanced. A procedure is presented for calculating the CRR considering the discrete nature of field measured data recorded by the test vehicle. In the numerical study, the closed-form solution is validated using the finite-element method (FEM). The results from a realistic example demonstrated that weak-end amplification can be clearly recognized for the CRR (both temporal and spectral) for the vehicle moving in a round trip. Such a feature enables the weak end of the bridge to be easily detected. The other advantage is that more bridge frequencies of higher orders can be identified using the CRR generated by the vehicle moving from the weak end.

Capped Asymmetric Elastic Net Support Vector Machine for Robust Binary Classification

Recently, there are lots of literature on improving the robustness of SVM by constructing nonconvex functions, but they seldom theoretically study the robust property of the constructed functions. In this paper, based on our recent work, we present a novel capped asymmetric elastic net (CaEN) loss and equip it with the SVM as CaENSVM. We derive the influence function of the estimators of the CaENSVM to theoretically explain the robustness of the proposed method. Our results can be easily extended to other similar nonconvex loss functions. We further show that the influence function of the CaENSVM is bounded, so that the robustness of the CaENSVM can be theoretically explained. Other theoretical analysis demonstrates that the CaENSVM satisfies the Bayes rule and the corresponding generalization error bound based on Rademacher complexity guarantees its good generalization capability. Since CaEN loss is concave, we implement an efficient DC procedure based on the stochastic gradient descent algorithm (Pegasos) to solve the optimization problem. A host of experiments are conducted to verify the effectiveness of our proposed CaENSVM model.

Flexural Wave Bandgaps in a Prestressed Multisupported Timoshenko Beam with Periodic Inerter-Based Dynamic Vibration Absorbers

Locally resonant (LR) metamaterial structures possess bandgaps in which wave propagation is significantly attenuated. In this paper, we discuss flexural wave bandgaps in an LR beam subjected to a global axial force and multiple vertical elastic supports. An array of inerter-based dynamic vibration absorbers (IDVAs) was periodically attached to the LR beam. The flexural wave band structure of this prestressed multisupported LR beam was first derived using the transfer matrix method (TMM) and then explicitly illustrated through a numerical example. Four bandgaps were identified: a bandgap located in the low-frequency zone, a Bragg band generated by Bragg scattering, and two LR bands generated by the local resonance of the IDVAs. The effects of the IDVA parameters, axial force, and vertical elastic support on the properties of the bandgaps were evaluated. In particular, the bandgaps merged accompanied by an exchange of their edge frequencies. The bandwidth of the merged bandgap was nearly equal to the sum of the bandwidths of the bandgaps involved, indicating a method for controlling broadband flexural vibration through the bandgap splicing mechanism.

Evaluation of gravity effects on the vibration of fluid-conveying pipes

Gravity effects are generally ignored in the study on the pipe vibration. However, in this paper, the effect of gravity on the nonlinear vibration of a pipe conveying fluid has been evaluated. The nonlinear vibration of the pipe with an intermediate elastic support is investigated, and the equilibrium configuration of the pipe caused by gravity is calculated. The governing equation of the nonlinear vibration near the equilibrium configuration is established. Then, the natural frequencies and modes of the transverse vibration of the pipe are obtained. The vibration characteristics of the pipe around the equilibrium configuration are studied. In addition, the effects of gravity on the vibration characteristics of the pipe are illustrated by comparing the same system without gravity effects considered. The influence of pipe parameters on the gravity effects is also investigated. The results show that there are some different trends in the natural frequencies with and without considering gravity effects. The lower-order natural frequencies are more affected by gravity. Moreover, the contribution of this paper is to propose an impact factor of gravity to evaluate the relative difference caused by gravity. Furthermore, the critical impact factors of the gravity of three metal pipes are proposed. In short, this paper evaluates the effects of gravity on the vibration of the pipe conveying fluid and provides an effective strategy for engineering design.

Numerical Simulation and Validation of Flow-Induced Vibration of the Specific Rod under Elastic Supports using One-Way Fluid-Solid Interaction

The vibration induced by the cooling fluid flow around the fuel rods in the fuel Assembly of nuclear reactors causes the rods to be destroyed and eventually leak due to the fretting wear in the place of contact with their supports for a long time. In this paper, the vibration caused by axial fluid flow around a specific fuel rod under elastic supports is numerically simulated. In this study, the fluid flow is modeled using the Large Eddy Simulation (LES) turbulence model in the FLUENT software. The fluid-structure interaction is also modeled using the ANSYS coupling system. To validate the implemented numerical model, the test results of the reported brass rod vibration similar to the studied problem in this research are used. Due to the long execution time of the two-way fluid-structure interaction simulations with a high grid number, the one-way fluid-structure interaction method is proposed. The results of simulations show that the one-way fluid-structure interaction method can be used in cases where the vibration amplitude of the structure is less than the height of the viscous sub-layer. Also, this method reduces the simulation time by 80%. Finally, the results of the flow-induced vibration simulation of the fuel rod show that the vibration range of the fuel rod will increase by 20 times if the contact of the elastic supports with the rod is lost, which will lead to the intensification of the wear caused by the rod oscillation. Also, the main natural frequency of the rod decreases when the rod loses contact with the supports and falls within the range of the reactor excitation frequency, i.e. 0 to 50 Hz, which should be avoided.

Stochastic Higher-order Finite Element Model for the Free Vibration of a Continuous Beam resting on Elastic Support with Uncertain Elastic Modulus

This paper deals with a continuous beam resting on elastic support with elastic modulus derived from a random process. Governing equations of the stochastic higher-order finite element method of the free vibration of the continuous beam were derived from Hamilton's principle. The random process of elastic modulus was discretized by averaging random variables in each element. A solution for the stochastic eigenvalue problem for the free vibration of the continuous beam was obtained by using the perturbation technique, in conjunction with the finite element method. Spectral representation was used to generate a random process and employ the Monte Carlo simulation. A good agreement was obtained between the results of the first-order perturbation technique and the Monte Carlo simulation.

An Improved Static Membrane Deflection Analysis for Collapse Mode CMUT Fabricated by Sacrificial Release Method

To improve the accuracy of the static membrane deflection calculation for the collapse mode capacitive micromachined ultrasonic transducer (CMUT) fabricated by sacrificial release (SR) method. The beam theory was used and the substrate supporting force can be modeled by a coefficient to the load on the membrane. The surrounding supporting post bending can be modeled by applying the elastic support boundary conditions (ESBCs) for the peripheral part of the membrane. CMUT devices have been fabricated using the SR process to validate this analytical analysis. Young’s modulus of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\text {Si}_{{3}} \text {N}_{{4}}$ </tex-math></inline-formula> was derived as 200 GPa by a method using center deflections of several different radii single-layer conventional devices under 1 atm. The analytical deflection results and contact radius results for the collapse mode CMUT membranes were testified by experimental results under 1 atm and were found matching the experimental results. This showed that the analysis can provide improved results for SR fabricated CMUT devices by including the substrate supporting effect and the post bending effect.

Semi-analytical modeling and experimental verification of a flexible varying section disk–blades system with elastic supports

A semi-analytical model of a flexible varying section disk–blades (FVSDBs) system with elastic supports is proposed, where the spin softening, centrifugal stiffening, and Coriolis force effects of the system are considered. Timoshenko beams with a stagger angle are used to simulate blades, and a varying section wheel based on plate theory is used to construct the disk. Hamilton’s principle and Galerkin method are utilized to derive the differential governing equations of the FVSDBs system. The proposed semi-analytical model accounts for the lateral vibration of the disk, and the longitudinal and lateral vibration of the blades. Besides, the validity of the proposed model has been verified by comparing the natural frequencies, mode shapes, and vibration responses obtained from the proposed model, finite element model (FEM), and experiment. The results indicate that the proposed semi-analytical model has high precision and computational efficiency, which can provide an alternative scheme for predicting the vibration characteristics of varying section disk–blades and overcoming the high computational cost of finite element models.

Dynamic Green Function Solution of Beams Under Moving Loads With Elastically Supports

The dynamic analysis of bridges simulated as Euler-Bernoulli beam models with elastic supports subjected to mobile loads are analyzed by conventional methods to obtain a new solution for displacement. Generally, these beam supports can be characterized by springs with a given stiffness, which considerably influence the structure's dynamic behavior and even attenuate the dynamic amplification. The solutions proposed until now are defined only on span but not supports. In this paper, we used Green's function, considering boundary and continuity conditions and shear force to study the global behavior of the beam. A new displacement formula is proposed for the beam to support a span according to the velocity of the mobile load, the beam rigidity, and the stiffness of supports. A further study leads to the present two new formulas, which directly give displacements at the level of supports according only to the beam rigidity and supports stiffness and to the load value at any time. The result of this analysis shows that several combined factors influence the vibratory behavior of the beam when it is supported on elastically supports, namely the stiffness of the supports, the rigidity of the beam, its length, the value of the mobile load, and its velocity. The evolution of support stiffness leads to classical boundary conditions. A study of coupling between the beam and supports is presented, with the study of the comportment in function to the ratio between the beam rigidity and spring stiffness.

Forced rotation enhances cylinder flow-induced vibrations at subcritical Reynolds number

When a cylinder is mounted on an elastic support within a current, vortex-induced vibrations (VIV) may occur down to a Reynolds number (Re) close to $20$ , based on the body diameter ( $D$ ) and inflow velocity ( $U$ ), i.e. below the critical value of $47$ reported for the onset of flow unsteadiness when the body is fixed. The impact of a forced rotation of the elastically mounted cylinder on the system behaviour is explored numerically for $Re \leqslant 30$ , over wide ranges of values of the rotation rate (ratio between body surface velocity and $U$ , $\alpha \in [0,5]$ ) and reduced velocity (inverse of the oscillator natural frequency non-dimensionalized by $D$ and $U$ , $U^\star \in [2,30]$ ). The influence of the rotation is not monotonic, but the most prominent effect uncovered in this work is a substantial enhancement of the subcritical-Re, flow-induced vibrations beyond $\alpha =2$ . This enhancement is twofold. First, the rotation results in a considerable expansion of the vibration/flow unsteadiness region in the $({Re},U^\star )$ domain, down to $Re=4$ . Second, the elliptical orbits described by the rotating body are subjected to a major amplification, with a transition from VIV to responses whose magnitude tends to increase unboundedly with $U^\star$ , even though still synchronized with flow unsteadiness. The emergence of such galloping-like oscillations close to the onset of vibrations disrupts the scenario of gradual vibration growth with Re, as amplitudes larger than $10$ body diameters may be observed at $Re=10$ .

Dynamic stiffness characteristics of aero-engine elastic support structure and its effects on rotor systems: mechanism and numerical and experimental studies

Dynamic stiffness characteristics of aero-engine elastic support structure and its effects on rotor systems: mechanism and numerical and experimental studies

The transverse vibration of Mindlin rectangular plates with internal elastic supports and arbitrary boundary supports

The transverse vibration of Mindlin rectangular plates with internal elastic supports and arbitrary boundary supports