Lateral Bending Modes Research Articles

Modal characteristics analysis and vibration control of long-span cable-stayed-suspension hybrid bridge based on strain energy method

The research on the modal characteristic analysis of long-span bridges is crucial for vibration analysis and control, yet literature regarding the modal characteristics of long-span bridges remains limited. This study thus aims to introduce a strain energy-based damping model to estimate the modal damping of long-span cable-supported bridges. Two different systems of cable-supported bridges are analyzed, and the strain energy proportions and modal damping ratios of various substructures for each mode are calculated. Based on the analysis of modal strain energy results, a strategy to control the vibration of long-span bridges is proposed and validated using the finite element methods. The results indicate that the main girder exhibits the highest strain energy proportion in low-order lateral bending modes, while it is the main cable in low-order vertical bending modes. Different substructures in higher-order modes contribute to the strain energy in varying proportions. For a specific mode, increasing the material damping of the substructure with the highest strain energy proportion effectively suppresses the vibration amplitude of the main girder in that mode. Additionally, installing V-shaped damping cables to control the vibration of substructures with higher strain energy proportions effectively suppresses the vibration of the main girder.

Three-dimensional nonlinear flutter analysis of long-span bridges by multimode and full-mode approaches

Based on amplitude-dependent flutter derivatives (FDs), the present study established three-dimensional (3D) nonlinear flutter analysis methods, including a multimode and a full-mode approach, for long-span bridges. Then, by adopting the proposed methods, the influences of 3D effects, self-excited drag force, and multimode aerodynamic coupling effects on nonlinear flutter response were investigated. Besides, critical structural modes that have a great impact on nonlinear flutter were identified, their contributions were quantified and roles were clarified. The results show that the 3D effects of structural modes can reduce the system stability mainly by increasing the negative uncoupled aerodynamic damping. Thus, if the 3D effects are ignored, the stable amplitude of nonlinear flutter will be underestimated. For long-span bridges whose foundational symmetric torsional mode couples a minor lateral-bending mode shape, the added aerodynamic damping from self-excited drag force, especially the term of FD P1∗∗1, can improve the stability of the system. Thus, it will overestimate the nonlinear flutter response if the self-excited drag force is ignored. Meanwhile, the aerodynamic coupling effect among different structural modes should be considered, otherwise the nonlinear flutter responses of long-span bridges may be misestimated significantly. For the nonlinear flutter dominated by the fundamental symmetric torsional modal branch, the symmetric vertical bending modes with a frequency lower than the fundamental symmetric torsional mode should be considered. Besides, the other higher-order structural mode coupled with relatively significant fundamental symmetric torsional mode shape is also worth being taken into account. The selection of structural modes is an important and experience-based work in the multimode method. Thus, the full-mode method, which can consider the potential higher-order structural modes that may have contributions to the 3D nonlinear flutter, is also needed, especially when the amplitude is large enough. Otherwise, the nonlinear flutter response may be underestimated by the multimode method.

Shimmy analysis of light weight aircraft nose wheel landing gear

This paper presents mathematical modeling and analysis of shimmy oscillations for a light weight airplane single wheel nose landing gear. Shimmy is a self-excited oscillation which occurs usually on the nose wheel landing gear assembly during ground maneuvers which is governed by the dynamic characteristics of the landing gear and tires. Shimmy oscillation may lead to reduce the fatigue life of the landing gear and fuselage structure. So, the study of dynamic response and stability boundaries of landing gear plays crucial role while designing of airplanes. In earlier studies of vehicle shimmy only 3 degrees of freedom (DOF) considered such as torsional mode, lateral bending mode and tire lateral deformation. In this work along with above mentioned DOF, two more additional DOF introduced such as axial vibration of strut and tire in order to include the effect of vertical dynamics on shimmy model. Gyroscopic coupling effect also included in the model to study its influence on shimmy. Analysis carried out to determine critical velocity region for occurrence of shimmy and to investigate the effectiveness of ground unevenness on the landing gear system for two different runway conditions such as flat runway and random roughness runway. The results are more helpful to study significant interaction between the different parameters of landing gear and to represent stability boundaries.

Determination of the Fundamental Lateral Bending Mode Frequency of a Spacecraft from its Static Deflection

The primary structure of a spacecraft that involves the central cylinder is designed to meet a specific value of fundamental lateral mode frequency. Though this frequency can be theoretically estimated by using a finite element model, its experimental verification is only possible when the entire spacecraft is made ready. However, static load tests are done on the primary structure of the spacecraft as soon as the structure is realized. In this work, a methodology to determine the fundamental bending mode frequency of a spacecraft from the deflections measured when the structure alone is subjected to static load tests is developed. An expression relating the fundamental lateral bending mode frequency of a spacecraft, having several masses attached at different locations, and the deflection at its tip is derived. Two spacecraft, one of 3000kg class and the other of 6000kg class are taken as examples. The fundamental lateral frequencies are determined using this methodology. These frequencies are also obtained through resonance search tests. It is shown that the derived expression will be very useful in estimating the fundamental lateral mode frequency of a spacecraft with the help of the measured deflection.

Cross-disc coupling in a flexible shaft–disc assembly in presence of asymmetric axial–radial bearing supports

In a flexible shaft–disc assembly supported by linear bearings, the disc 1 Nodal Diameter (ND) modes are known to couple with the shaft lateral (bending) modes, whilst the 0ND modes can couple with the shaft axial modes. In addition to these well known coupling phenomena, a previous work by the authors has shown that, in presence of an asymmetric axial–radial bearing supporting structure, shaft axial and lateral modes can interact and lead to a coupling with a single flexible disc 0 and 1 ND modes simultaneously. Given that in most circumstances a shaft carries more than one disc, this work extends the previous findings to a shaft carrying two flexible discs and particularly investigates the mechanisms of cross disc coupling due to an asymmetric supporting structure. A full 3D FEM model of the assembly has been developed to model its dynamic behaviour. New classes of coupled modes involving the shaft and the two discs have been identified and a physical explanation will be provided, considering forces/moments applied at the interface amongst subcomponents and following the hypothesis that each disc acts like an independent dynamic absorber.A parametric study of the dual discs arrangement varying stiffness, thickness and position of one disc further highlighted the dynamic interaction of the subcomponents. Specific arrangements will allow an Engine Order forcing pattern applied to one disc to excite a different mode on the other disc, with the shaft and the supports acting as the vibration energy transmitter between the two discs. The industrial implications of such phenomena are also discussed throughout this work.

Vibration modes and wave propagation of the rail under fastening constraint

This paper investigates three-dimensional (3D) rail vibrations under fastening constraint up to 5000 Hz and provides insights into rail vibration control by fastening parameters. A methodology is proposed, including experimental investigation and numerical simulations of rail vibrations. Three steps are considered: 1) experimental investigation of rail vibrations under fastening constraint; 2) validation and analysis of 3D finite element (FE) modeling of rail-fastening systems; 3) rail vibration control by fastening parameters. In Step 1, operating deflection shape (ODS) and synchronized multiple-acceleration wavelet (SMAW) measurements are applied to identify rail vibration modes and measure wave propagation characteristics under fastening constraint. In Step 2, a 3D FE model capable of reproducing the dynamic behaviors of rail-fastening up to 5000 Hz is developed to analyze rail vibrations and validated using measurements from Step 1. In Step 3, insights into the control of rail vibrations are gained by sensitivity analysis of fastening parameters using the validated 3D FE model from Step 2. The results indicate that (1) under fastening constraint, ODS measurement identifies vertical bending modes, longitudinal compression modes, and lateral bending modes of the rail with shifted frequencies and significantly reduced vibration amplitude compared to that of free rail. (2) Vertical wave attenuation of rail-fastening is relatively small between 1800 and 3600 Hz, and lateral wave attenuation presents a dominant peak at about 3800 Hz. (3) Compared to the vertical and lateral directions, the fastening system constrains the longitudinal rail vibrations less strongly. (4) The change of fastening stiffness and damping can control rail mode frequencies and their vibration amplitude, and influence the wave propagation velocities and attenuation along the rail.

Seismic responses of hyperbolic cooling towers under horizontal and vertical earthquake

Following the dynamic property analysis and elaboration, linear response spectrum analysis (RSA) and response history analysis (RHA) were conducted on a representative hyperbolic cooling towers (HCT) in present study. The seismic responses in tower shell were illustrated in detail, including the internal force amplitude, modal contribution, influence from damping ratio, comparison of results got from RSA and RHA and especially the latitude distributions of internal forces. The results show that the eigenmodes could be classified in a new method into four types according to their mode shapes and only the lateral bending modes and vertical stretching modes are meaningful for horizontal and vertical earthquake correspondingly. The bending modes and seismic deformation display the same feature which is global lateral bending accompanied by minute circular flow displacement of section. This feature also decides the latitude distributions of internal forces as sine or cosine. Moreover, the following method is also proposed for approximate estimation of internal force amplitudes without time-consuming response history analysis: getting the response spectrums of the selected ground accelerations and then comparing values of response spectrums at the natural period of first lateral bending mode because it is always prime dominant for horizontal seismic responses.

Biomechanical Study of Cervical Disc Arthroplasty Devices Using Finite Element Modeling.

Many artificial discs for have been introduced to overcome the disadvantages of conventional anterior discectomy and fusion. The purpose of this study was to evaluate the performance of different U.S. Food and Drug Administration (FDA)-approved cervical disc arthroplasty (CDA) on the range of motion (ROM), intradiscal pressure, and facet force variables under physiological loading. A validated three-dimensional finite element model of the human intact cervical spine (C2-T1) was used. The intact spine was modified to simulate CDAs at C5-C6. Hybrid loading with a follower load of 75 N and moments under flexion, extension, and lateral bending of 2 N·m each were applied to intact and CDA spines. From this work, it was found that at the index level, all CDAs except the Bryan disc increased ROM, and at the adjacent levels, motion decreased in all modes. The largest increase occurred under the lateral bending mode. The Bryan disc had compensatory motion increases at the adjacent levels. Intradiscal pressure reduced at the adjacent levels with Mobi-C and Secure-C. Facet force increased at the index level in all CDAs, with the highest force with the Mobi-C. The force generally decreased at the adjacent levels, except for the Bryan disc and Prestige LP in lateral bending. This study demonstrates the influence of different CDA designs on the anterior and posterior loading patterns at the index and adjacent levels with head supported mass type loadings. The study validates key clinical observations: CDA procedure is contraindicated in cases of facet arthroplasty and may be protective against adjacent segment degeneration.

Free vibration of simply supported steel I-girders with trapezoidal web corrugations

Natural frequency is essential information required to perform the dynamic analysis. In this study, the straight I-girder element with trapezoidal web corrugations and singly symmetric cross-section is derived based on Kang and Yoo’s thin-walled curved beam with doubly symmetric cross-section theory. Each node of the element has seven degrees of freedom including one warping degree of freedom. Using Hamilton’s principle, governing linear dynamic differential equations of equilibrium, elastic element stiffness matrix, and consistent element mass matrix are established. Matlab code is employed for natural frequency analysis of simply supported steel I-girders with trapezoidal web corrugations. The results are compared with the natural frequencies obtained with ABAQUS. Finally, it is found that the proposed equations provide a good prediction of natural frequency for the first three lateral bending modes and the first five torsional modes.

Analytical and finite element solutions of free and forced vibration of unrestrained and braced thin-walled beams

In this article, vibration of thin-walled beams with arbitrary open cross-section shape is investigated. Based on the beam element model accounting for warping and flexural–torsional coupling, analytical solutions for different boundary conditions are derived for higher free vibration modes in bending, torsion and flexural–torsional coupled modes. In the model, the effects of rotational inertial kinematic terms are considered. The finite element approach of the model is also investigated. Three-dimensional beams with seven degrees of freedom per node are adopted in the mesh process. Free vibration and forced vibration analyses are possible. In forced vibration, the behaviour of the beams is studied in the frequency domain using the steady-state method (modal analysis). Damping is considered using the Rayleigh model. The model is validated by comparing the results to benchmark solutions found in the literature and to other recent numerical and experimental results. Additional finite element simulations are performed by means of commercial softwares (Abaqus and Adina). In slender unrestrained beams, the vibration behaviour is predominated by torsion and lateral bending modes. In design, recourse to braces is a good compromise. This solution is discussed, and improvement of the vibration behaviour in the presence of intermediate braces is confirmed. Application of higher vibration modes in building and bridge design is outlined. The effects of the number and distribution of the intermediate braces to improve structural stability against vibration behaviour is outlined.

Effects of environmental and operational actions on the modal frequency variations of a sea-crossing bridge: A periodicity perspective

Abstract The modal frequency variations of bridges associated with environmental and operational actions are among the primary research topics in structural health monitoring (SHM). Based on the periodic variations of the monitoring data obtained over 6 years for the Donghai Bridge (DHB), we investigated the effects of environmental and operational factors on the modal frequency variations and identified the main periods and influencing factors of the DHB frequency changes. Partial correlation coefficients and cyclic averaging were applied to compare the contributions of various actions to the frequency changes, and the underlying mechanisms of the frequency changes were preliminarily studied. The modal frequencies of the DHB were found to vary with periods such as 1 year, 1 week, 1 day, and 12.42 h in a manner corresponding to inherent variation cycles of structural temperature, traffic loading, wind loading, and sea level. Structural temperature and traffic loading were the primary factors influencing the DHB frequency changes, and their relative degrees of impact on the frequency changes were different for different cycles. All modal frequencies of the DHB decreased with increasing temperature due to variation in the temperature-dependent elastic moduli of the structural materials. However, as traffic loading increased, the fundamental frequencies of the vertical bending, lateral bending, and torsional modes of the girder decreased, while some higher-order frequencies increased. The influence of traffic on the modal frequencies could be qualitatively explained by employing an equivalent spring–mass model of the traffic flow on the bridge. The results of this study will deepen the understanding of the frequency changes occurring in bridges during operation and can serve as a reference for vibration-based SHM.

The Mechanical Role of the Radial Fiber Network Within the Annulus Fibrosus of the Lumbar Intervertebral Disc: A Finite Elements Study.

The annulus fibrosus (AF) of the intervertebral disc (IVD) consists of a set of concentric layers composed of a primary circumferential collagen fibers arranged in an alternating oblique orientation. Moreover, there exists an additional secondary set of radial translamellar collagen fibers which connects the concentric layers, creating an interconnected fiber network. The aim of this study was to investigate the mechanical role of the radial fiber network. Toward that goal, a three-dimensional (3D) finite element model of the L3-L4 spinal segment was generated and calibrated to axial compression and pure moment loading. The AF model explicitly recognizes the two heterogeneous networks of fibers. The presence of radial fibers demonstrated a pronounced effect on the local disc responses under lateral bending, flexion, and extension modes. In these modes, the radial fibers were in a tensile state in the disc region that subjected to compression. In addition, the circumferential fibers, on the opposite side of the IVD, were also under tension. The local stress in the matrix was decreased in up to 9% in the radial fibers presence. This implies an active fiber network acting collectively to reduce the stresses and strains in the AF lamellae. Moreover, a reduction of 26.6% in the matrix sideways expansion was seen in the presence of the radial fibers near the neutral bending axis of the disc. The proposed biomechanical model provided a new insight into the mechanical role of the radial collagen fibers in the AF structure. This model can assist in the design of future IVD substitutes.

Biomechanical Analysis of a Long-Segment Fusion in a Lumbar Spine-A Finite Element Model Study.

Examine the biomechanical effect of material properties, geometric variables, and anchoring arrangements in a segmental pedicle screw with connecting rods spanning the entire lumbar spine using finite element models (FEMs). The objectives of this study are (1) to understand how different variables associated with posterior instrumentation affect the lumbar spine kinematics and stresses in instrumentation, (2) to compare the multidirectional stability of the spinal instrumentation, and (3) to determine how these variables contribute to the rigidity of the long-segment fusion in a lumbar spine. A lumbar spine FEM was used to analyze the biomechanical effects of different materials used for spinal rods (TNTZ or Ti or CoCr), varying diameters of the screws and rods (5 mm and 6 mm), and different fixation techniques (multilevel or intermittent). The results based on the range of motion and stress distribution in the rods and screws revealed that differences in properties and variations in geometry of the screw-rod moderately affect the biomechanics of the spine. Further, the spinal screw-rod system was least stable under the lateral bending mode. Stress analyzes of the screws and rods revealed that the caudal section of the posterior spinal instrumentation was more susceptible to high stresses and hence possible failure. Although CoCr screws and rods provided the greatest spinal stabilization, these constructs were susceptible to fatigue failure. The findings of the present study suggest that a posterior instrumentation system with a 5-mm screw-rod diameter made of Ti or TNTZ is advantageous over CoCr instrumentation system.

Experimental and Computational Modal Analyses for Launch Vehicle Models considering Liquid Propellant and Flange Joints

In this research, modal tests and analyses are performed for a simplified and scaled first-stage model of a space launch vehicle using liquid propellant. This study aims to establish finite element modeling techniques for computational modal analyses by considering the liquid propellant and flange joints of launch vehicles. The modal tests measure the natural frequencies and mode shapes in the first and second lateral bending modes. As the liquid filling ratio increases, the measured frequencies decrease. In addition, as the number of flange joints increases, the measured natural frequencies increase. Computational modal analyses using the finite element method are conducted. The liquid is modeled by the virtual mass method, and the flange joints are modeled using one-dimensional spring elements along with the node-to-node connection. Comparison of the modal test results and predicted natural frequencies shows good or moderate agreement. The correlation between the modal tests and analyses establishes finite element modeling techniques for modeling the liquid propellant and flange joints of space launch vehicles.

Classical Adaptive Augmentation Control for a Typical Second Generation Launch Vehicle

Abstract Classical control technique is used for attitude control of launch vehicles worldwide, because of its established history of success. Usual method is to model launch vehicle dynamics by linear techniques to achieve adequate stability and tracking performance. Common type of feedback control system for launch vehicles is the Proportional-Integral (PI) controller with appropriate filters to stabilize the lateral bending modes and slosh modes and also ensure sufficient robustness margins for rigid body. This paper presents a Classical Adaptive Augmentation Control (CAAC) Algorithm for forward loop gain augmentation in real time, to cater to large dispersion in vehicle parameters beyond the capability of classical control system. The idea is to provide augmentation to a classical control designed autopilot when performance enhancement is required to tackle off-nominal conditions arising out of modeling errors and large dispersion in estimated vehicle parameters (thrust, inertia, slosh, aerodynamics, lateral bending modes). There is high chance that such large dispersion can arise during initial design of a new generation launch vehicle before actual flight. Finally, validation of the Adaptive Augmentation Control design for several credible launch vehicle failure scenarios show that the adaptive controller consistently and predictably improves performance and robustness, and achieves stability during extreme off-nominal situations.

Coupled flutter analysis of long-span bridges using full set of flutter derivatives

A convenient and effective finite element-based method for coupled flutter analysis of long-span bridges is presented. The exact formulation of the aerodynamic self-excited forces with eighteen flutter derivatives utilized by complex notation is proposed. The predictions of the flutter wind speed and the critical frequency are compared with those either given by existing methods or the wind tunnel test showing the effectiveness and accuracy of the present approach. Numerical flutter analysis for an asymmetric bridge is the application for engineering practice, and its obtained results highlight the important role of the first lateral bending and torsional mode in generating the coupled flutter. Multi-mode analyses that are based on only the symmetrical modes can predict accurately the bridge flutter onset. The consistent self-excited aerodynamic force formulations produce the flutter velocity that is closer to the experimental one of full-bridge model in the wind tunnel.

Influence of structural system measures on the dynamic characteristics of a multi-span cable-stayed bridge

A three-dimensional finite element model for the Jiashao Bridge, the longest multi-span cable-stayed bridge in the world, is established using the commercial software package ANSYS. Dynamic characteristics of the bridge are analyzed and the effects of structural system measures including the rigid hinge, auxiliary piers and longitudinal constraints between the girders and side towers on the dynamic properties including modal frequency, mode shape and effective mass are studied by referring to the Jiashao Bridge. The analysis results reveal that: (i) the installation of the rigid hinge significantly reduces the modal frequency of the first symmetric lateral bending mode of bridge deck. Moreover, the rigid hinge significantly changes the mode shape and effective mass of the first symmetric torsional mode of bridge deck; (ii) the layout of the auxiliary piers in the side-spans has a limited effect on changing the modal frequencies, mode shapes and effective masses of global vibration modes; (iii) the employment of the longitudinal constraints significantly increases the modal frequencies of the vertical bending modes and lateral bending modes of bridge deck and have significant effects on changing the mode shapes of vertical bending modes and lateral bending modes of bridge deck. Moreover, the effective mass of the first anti-symmetric vertical bending of bridge deck in the longitudinal direction of the fully floating system is significantly larger than that of the partially constrained system and fully constrained system. The results obtained indicate that the structural system measures of the multi-span cable-stayed bridge have a great effect on the dynamic properties, which deserves special attention for seismic design and wind-resistant design of the multi-span cable-stayed bridge.

Effect of Torsional Damping on Aircraft Nose Landing-Gear Shimmy

The bifurcation analysis method is applied in dealing with shimmy oscillations of the aircraft nose landing gear. Algorithms and details of the procedure in bifurcation analysis are described. Torsional damping of the antiswing damper is studied based on the model consisting of torsional and lateral bending modes of the landing gear. It is concluded that more linear torsional damping means more stability from torsional shimmy, although excessive damping will result in lateral shimmy at the starting speed and low vertical force. The torsional shimmy occurs prior in the area of low speed and midvertical force without sufficient linear damping. And there exists a so-called ideal damping varying with forward speed, with which no shimmy happens in the global scope below a critical vertical force. Squared torsional damping has little influence on the area of shimmy unless Bautin bifurcation appears in the dynamics system, and it is effective in controlling the amplitude of shimmy oscillations. The torsional clearance increases the unstable area of torsional shimmy obviously, but the value of the clearance does not affect the shimmy area.

Four lateral mass screw fixation techniques in lower cervical spine following laminectomy: a finite element analysis study of stress distribution.

BackgroundLateral mass screw fixation (LSF) techniques have been widely used for reconstructing and stabilizing the cervical spine; however, complications may result depending on the choice of surgeon. There are only a few reports related to LSF applications, even though fracture fixation has become a severe complication. This study establishes the three-dimensional finite element model of the lower cervical spine, and compares the stress distribution of the four LSF techniques (Magerl, Roy-Camille, Anderson, and An), following laminectomy -- to explore the risks of rupture after fixation.MethodCT scans were performed on a healthy adult female volunteer, and Digital imaging and communication in medicine (Dicom) data was obtained. Mimics 10.01, Geomagic Studio 12.0, Solidworks 2012, HyperMesh 10.1 and Abaqus 6.12 software programs were used to establish the intact model of the lower cervical spines (C3-C7), a postoperative model after laminectomy, and a reconstructive model after applying the LSF techniques. A compressive preload of 74 N combined with a pure moment of 1.8 Nm was applied to the intact and reconstructive model, simulating normal flexion, extension, lateral bending, and axial rotation. The stress distribution of the four LSF techniques was compared by analyzing the maximum von Mises stress.ResultThe three-dimensional finite element model of the intact C3-C7 vertebrae was successfully established. This model consists of 503,911 elements and 93,390 nodes. During flexion, extension, lateral bending, and axial rotation modes, the intact model’s angular intersegmental range of motion was in good agreement with the results reported from the literature. The postoperative model after the three-segment laminectomy and the reconstructive model after applying the four LSF techniques were established based on the validated intact model. The stress distribution for the Magerl and Roy-Camille groups were more dispersive, and the maximum von Mises stress levels were lower than the other two groups in various conditions.ConclusionThe LSF techniques of Magerl and Roy-Camille are safer methods for stabilizing the lower cervical spine. Therefore, these methods potentially have a lower risk of fixation fracture.

Acute longitudinal ligament rupture following acute spinal trauma

The authors present a rare case of anterior longitudinal ligament (ALL) rupture in a 47- year-old gentleman following a bicycle accident. The ALL is a continuous band of a variable thickness that acts as a primary spinal stabiliser. Stress, strain or rupture of the ALL usually occurs as a result of hyperextension, with the primary perpetrator being whiplash injuries. Such injuries have been shown to result in cervical spine instability during extension, axial rotation, and lateral bending modes. Spine radiographs of such patients may be routinely assessed as normal, therefore this specific type of injury does not lend itself to identification by traditional imaging methods. This account demonstrates the importance of having a high index of suspicion of a ligamentous neck injury in the setting of normal plain radiographs but abnormal clinical examination.