Effect Of Support Stiffness Research Articles

Nonlinear dynamics of viscoelastic fluid-conveying pipe installed within uniform external cross flow by pipe clamps

Nonlinear dynamic response of a viscoelastic fluid-conveying pipe excited by uniform external cross flow is investigated, which is installed by pipe clamps. A pipe clamp theoretical model is proposed for the actual pipe clamp instead of the commonly used simply supported or clamped boundary conditions. Based on the model, equivalent support stiffness and torsional stiffness are obtained from structure and material parameters of the pipe clamp. The vortex-induced lift force and the mean drag force are described by van der Pol oscillator model. The nonlinear equations of motion with the boundary conditions of different constraint stiffness are discretized by Differential Quadrature Method (DQM) and then solved by Runge-Kutta algorithm. The results are verified by Galerkin method for the pinned-pinned pipe, and by ANSYS for the pipe with two arbitrary constraint stiffnesses. The dependence of critical external fluid velocity upon constraint stiffness and pipe length is investigated. Moreover, the effects of support stiffness, torsional stiffness, internal fluid velocity, and viscoelastic coefficient on the amplitude response of the pipe are studied in detail. The influence of constraint stiffness on characteristics of lock-in region is elucidated. Consequently, pipe clamp spacing, material and structure parameters of the rubber mat can be designed to ensure the requirement of economy and safety, which are helpful to guide the layout and design of pipe clamps.

Non-Linear Dynamic Analysis of Drill String System with Fluid-Structure Interaction

In this research, the non-linear dynamics of the drill string system model, considering the influence of fluid—structure coupling and the effect of support stiffness, are investigated. Using Galerkin’s method, the equation of motion is discretized into a second-order differential equation. On the basis of an improved mathematical model, numerical simulation is carried out using the Runge—Kutta integration method. The effects of parameters, such as forcing frequency, perturbation amplitude, mass ratio and flow velocity, on the dynamic characteristics of the drill string system are studied under different support stiffness coefficients, in which bifurcation diagrams, waveforms, phase diagrams and Poincaré maps of the system are provided. The results indicate that there are various dynamic model behaviors for different parameter excitations, such as periodic, quasi-periodic, chaotic motion and jump discontinuity. The system changes from chaotic motion to periodic motion through inverse period-doubling bifurcation, and the support stiffness has a significant influence on the dynamic response of the drill string system. Through in-depth study of this problem, the dynamic characteristics of the drill string can be better understood theoretically, so as to provide a necessary theoretical reference for prevention measures and a reduction in the number of drilling accidents, while facilitating the optimization of the drilling process, and provide basis for understanding the rich and complex nonlinear dynamic characteristics of the deep-hole drill string system. The study can provide further understanding of the vibration characteristics of the drill string system.

Study on Parametric Resonance of Support Parameters to Flow Pipeline

The influence of support parameters on the parametric resonance of fluid conveying pipeline under pulsating flow excitation is studied. Firstly, the mathematical model of fluid conveying pipeline and support system is established, Then the Galerkin method is used for discretization, and the incremental harmonic balance method is used to solve the problem. The effects of support stiffness, support damping, support position, support number and support spacing on the boundary of parametric resonance region and the parametric resonance response are discussed. The results show that under the fixed pulsating amplitude, the pulsating frequency range of system instability becomes smaller with the increase of support stiffness, support damping and the number of support, while the support position and support spacing have less influence on it. The change of support stiffness and support position has little influence on the pulsation frequency that the parametric resonance lasts to the end, but the increase of brace damping makes its value decrease obviously. The research results can provide a theoretical basis for the design and installation of pipeline support system.

Drilling Vibration in a Micro-Drilling Process Using a Gas Bearing Spindle

In industry, high-density packaging technology is an unavoidable requirement. Therefore, the drilling hole of printed circuit boards, PCB, requires being much smaller, even down to 0.1 mm or less. Drill fractures are frequently found in the micro drilling process due to the micro scale, hole-location errors, reaming. In all micro drilling failure cases, there existed a large vibration or instability is frequently found because of the insufficient rigidity of supports for a system with a super-high spinning speed. To improve the drilling quality and avoid drill breakage, the effects of support stiffness and high rotational speed of the vibration in the micro drilling process must be studied. Most investigations on the vibration of micro drilling are focused on only drill self-structure. However, in an actual engineering application, the micro drill must be attached in a bearing spindle system. This study considers the vibration of a micro drill with a gas bearing spindle. Hence, it includes the effects of the rotation speed, air pressure, and clearance of gas bearing on the vibration in a micro drilling process. After constructing the governing equations of the system, the numerical analysis by the Fortran programming is performed to solve for the frequency and amplitude response of the spindle system over time. The results indicated that the spindle with the gas bearings effect increased the vibration in the micro-drilling process.

Vibration control of loosely supported cross-flow heat exchanger tube undergoing fluid elastic instability

In this theoretical work, the effect of support stiffness in controlling the nonlinear vibration of loosely supported cross-flow heat exchanger tube and support-impact forces is studied. The bending of slender heat exchanger tube is modelled using Euler-Bernoulli beam theory. The cross-flow induced forces on tube are modelled using quasi-steady flow model whereas the loose support is modelled with trilinear spring model. The Galerkin discretisation and time integration are used to solve the nonlinear governing differential equation of motion. From the results, it is observed that with the reduction of support stiffness, the chaotic motion of the tube gradually changes to periodic motion, the drastic changes in the acceleration of the tube during its motion reduces, and also the magnitude of the contact forces greatly reduces. Thus, if the support is made of low stiffness material or placing a low stiffer buffer layer over stiff support, the vibration and impact forces can be significantly reduced..

Dynamic Characteristics Analysis of Rotor-Bearing System and Experimental Validation

As for the vibration problem of rotor system for large rotating machinery such as aero engines, a finite element model for dynamic similarity experiment system of rotor is established based on finite element theory. The effect of support stiffness and support position on the dynamic characteristics of the rotor system have been emphatically analyzed. Moreover, the effect of support stiffness on the dynamic characteristics of rotor system have been verified on the designed test rig. The results show that, under different support stiffness intervals, each order natural frequency of the rotor system has a different variation tendency with the support stiffness or position. In addition, the variation of the mode shapes with the support stiffness or position are closely related to the variation of the natural frequencies with the support stiffness and position. Combining with the practice, the natural frequencies of the rotor system can be reasonably designed and the harmful mode shapes can be avoided by adjusting the supporting characteristics of the rotor system.

Coupling vibration behaviors of drum-disk-shaft structures with elastic connection

The drum-disk-shaft system is a significant part of the aero-engine rotor structure and directly influences the rotor dynamics. The dynamic model of a drum-disk-shaft structure was established considering the elastic support, flexible connection, and the Coriolis and centrifugal effects on the drum due to the rotation. The effects of support stiffness, coupling stiffness and rotating speed on vibration behaviors of the system were investigated. Some results were obtained. It is found that the disk's vibration and the drum's translational motion are coupled whatever the circumferential number (n) is, and the local deformation of the drum is coupled with the disk's vibration only under n = 1. When n is equal to 1, the more veering phenomena can be found during the process of the variation of the coupling stiffness, and support stiffness and the mode shapes are plotted to show the phenomena. At last, the effect of rotating speed on the frequencies of the coupling system was investigated. It is shown that the forward and backward traveling waves are generated in all the modes of the system under n = 1 due to the rotating speed and coupling effects, but only in the modes of the drum as n is not equal to 1. Also, the coupling stiffness and support stiffness can significantly affect the critical speed of the system.

Effect of Support Stiffness on Bogie Motion of Railway Vehicle

Effect of Support Stiffness on Bogie Motion of Railway Vehicle

Development of arching action in longitudinally-restrained reinforced concrete beams

Membrane action of reinforced concrete (RC) beams is one of the primary mechanisms that enhances progressive collapse resistance of frames and influences the robustness of a structure under an unforeseen overload event. Compressive membrane action increases both the capacity and stiffness of longitudinally-restrained reinforced concrete members. In this study, six scaled reinforced concrete beam assemblages with longitudinal end restraints were tested under a monotonically increasing displacement applied at the mid-span. The longitudinal reinforcement ratio and configuration of stirrups were the main variables investigated in the experimental programme. The effect of support stiffness and strain penetration on arching behaviour is studied using a numerical model calibrated against the experimentally measured rotation of the end supports. It is concluded that, for RC beams, the longitudinal reinforcing ratio and stirrup configuration has only a minor influence on arching action, whereas compressive strength of the concrete and strain penetration have pronounced effects on the peak as well as post-peak response.

A generic model for investigation of arching action in reinforced concrete members

A generic nonlinear compound frame model is developed and applied for investigation of arching action in reinforced concrete beams. This model takes account of geometrical and material nonlinearities including cracking and crushing of the concrete and yielding of reinforcing steel. Further, a non-local model is adopted to resolve the numerical sensitivity associated with compressive softening of concrete. The effect of support stiffness is incorporated into the formulation by a set of nodal rotational and translational springs that, respectively, represent the flexural and axial stiffness of the supports. The developed model is verified by available experimental data and advanced continuum-based FE models and it is shown that the generic model can predict the ultimate loading capacity and load–displacement response of reinforced concrete beams and one-way slabs with reasonable accuracy. Lastly, a parametric study is undertaken and effects of end supports stiffness and concrete compressive strength on development of arching action is investigated.

Effects of support settlement on continuous deep beams and STM modeling

This paper investigates the effects of differential support settlement on the strength and behavior of two-span Continuous Deep Beams (CDBs). An experimental program consisting of six specimens has been carried out with spring supports to investigate the effects of differential settlement and the web reinforcement on the beam behavior. Test results including crack patterns, load–deflection curves, strains in steel, serviceability loads and ultimate loads are presented and discussed with the effects of support stiffness and web reinforcement. It is shown that settlement of the middle support significantly affects the serviceability load, crack pattern and failure mode. However, for ultimate strength, the beams exhibit considerable tolerance in accommodating differential support settlement, particularly for those with web reinforcement. A direct Strut-and-Tie Model (STM) is proposed for beams subjected to support settlement. Good agreement is found between the test results and STM predictions on support reactions and ultimate loads. The method can be easily implemented through an excel spreadsheet, thus making it an attractive tool for engineers.

Nonlinear Dynamic Analysis and Experiment Verification of Rotor-Ball Bearings-Support-Stator Coupling System for Aeroengine With Rubbing Coupling Faults

In this paper, a new rotor-ball bearings-support-stator coupling system dynamic model with rubbing coupling faults is established for practical aeroengine. In the model, the rubbing fault is modeled, the stator motion is considered, the flexible support and squeeze film damper are established, and the nonlinear factors of ball bearing, such as the clearance of the bearing, the nonlinear Hertzian contact force between balls and races, and the varying compliance vibration because of the periodical variety of the contact position between balls and races, are modeled. The numerical integral method is used to obtain the system responses, the effect of support stiffness on rotor responses is studied using a vibration amplitude-rotating speed plot, and the characteristics of the rubbing fault is analyzed using a 3D cascade plot. An aeroengine tester with a stator is established to carry out the rubbing fault experiments, the simulation results from the rotor-ball bearings-support-stator coupling model are compared with the experimental results, and the consistency of the results show fully the effectiveness of the new rotor-ball bearings-support-stator coupling model with rubbing fault.

Dynamic Responses of Beams with a Flexible Support Under a Constant Speed Moving Load

ABSTRACTThis paper deals with the linear dynamic responses of beams with a flexible support under a moving load with a constant speed. The entire system is modeled as a two-span beam and each span of the continuous beams is assumed to obey the Euler-Bernoulli beam theory. Considering the compatibility requirements on the flexible constraint, the relationships between two segments can be obtained. By using a transfer matrix method, the characteristic equation of the entire system can then be determined. The forced responses of the system under a moving load can then be obtained through modal expansion theory. Some numerical results are presented to the effects of support stiffness and the different speeds of the moving load.

Effect of Support Stiffness on the Stability of a Rotor System with Blade Flexibility

A simple mathematical model describing the motion of a rotating flexible blade, rigid-disk, and flexible shaft assembly is derived. The model provides a fast parametric study for the dynamic behaviour of the proposed rotor system. The Lagrangian approach combined with the assumed mode method is used to derive the governing equations of motion for this system. The model consists of asymmetrical continuous shaft mounted on two isotropic supports; an end support, represented by two rotational and two translational springs, and an intermediate support, represented by two translational springs only. A rigid disk, with a number of flexible blades of uniformly distributed stiffness and mass, cantilevered to the disk, is attached to the free end of the shaft. The effect of support flexibility on the stability of the system is studied. Instability regions are presented for different system parameters. Coupling between flexible blades and shaft coordinates showed very little effect on the instability regions. The mathematical model results are similar to those found in the literature.

Analysis of plate resting on elastic supports and elastic foundation by finite strip method

A procedure incorporating the finite strip method together with spring systems is proposed in this paper for treating plates on elastic supports. The spring systems can simulate different elastic supports, such as elastic foundation, line and point elastic supports, and also mixed boundary conditions. To illustrate the application of this procedure, two numerical examples are presented. A three-span simply supported plate is first considered and the effects of support stiffness on the static and free vibration responses and on the critical buckling stress are discussed. A plate resting on Winkler elastic foundation is next studied and the effects of dimension ratio on the static and free vibration responses are discussed. Numerical results show that the spring system can successfully simulate different kinds of elastic supports.

TRANSVERSE MODE LOCALIZATION IN A DUAL-SPAN ROTATING SHAFT

In this paper, transverse mode localization in a dual-span rotating shaft is studied. The shaft is elastically supported by transverse and rotational springs at an intermediate point and at both ends. A Timoshenko beam model including the effects of axial loads is employed, and a small disorder in the span length is introduced in the study of the mode localization. The exact eigensolutions of the systems are obtained by the phase closure principle. Based on actual values of typical bearing stiffness applied in turbo machinery, the effects of support stiffness, rotation speed and axial load on the mode localization are examined. These results are compared with those obtained from the Rayleigh shaft model to assess the effects of shear deformation. It is shown that mode localization depends strongly on the modelling of the bearing support.

Stability Analysis of Flexibly Supported Rough Submerged Oil Journal Bearings

A theoretical analysis is presented to study the effect of support stiffness and damping on the transient response of flexibly supported rotor-bearing systems, considering surface roughness. The transient response of rotor is plotted in the form of trajectories. The average Reynolds equation, which uses the Patir and Cheng flow simulation model of rough surfaces, is solved by the finite difference method with a successive over-relaxation scheme (Gauss-Seidel), while the equations of motion of both the rotor and bearing are solved by the fourth-order Runge-Kutta method. It has been found that the surface roughness considerably affects the stability of a rotor-bearing system and the stability may be improved greatly by a proper design of bearing supports.