Shaft Stiffness Research Articles

Optimization of two support spindle shaft on nonlinear elastic supports by rigidity characteristics

One of the main structural elements of metalworking machines is the spindle assembly (spindle), which is used to hold cutting tools or workpieces. The rigidity of the spindle assembly plays a decisive role in ensuring the accuracy and efficiency of the machine as a whole. The assessment of the spindle shaft stiffness is carried out on the basis of the analysis of the static bending of the spindle shaft, which made it possible to formulate and solve the problems of optimizing the spindle shaft according to the stiffness characteristics for two supporting structures on nonlinear elastic supports. To determine the stiffness of roller bearings, the work uses the dependence obtained on the basis of solving the problem of contact interaction of an elastic steel cylinder with curvilinear elastic steel half-spaces. For the considered design scheme, the optimization goals were chosen for the conditions of the smallest displacement of the end section of the spindle shaft console, the achievement of the minimum angle of rotation in this section or the minimum of their normalized superposition, which ensures maximum rigidity in the processing zone. Consideration has also been given to minimizing the swing angle at the front support to maximize bearing life. Mathematically, the problem is presented in the form of minimizing one of the 4 proposed objective functions by changing the variable parameters - the length of the cantilever and the value of the inter-support distance, represented as dimensionless quantities - the cantilever coefficient and the inter-support distance coefficient. Minimum and maximum values ​​of the cantilever length and shaft span were considered as constraints on the variable parameters. Varying the console coefficients and the inter-support distance was carried out by the method of sequential enumeration within the specified constraints, the solution of optimization problems is presented in a graphical form. The solution to the problem of shaft bending was carried out on the basis of the equation of the bent axis of the beam in the framework of the Euler - Bernoulli hypotheses and presented in an analytical form together with analytical dependencies for calculating the radial stiffness of a roller bearing as a function of the supporting force acting on it. The algorithm for solving optimization problems is implemented in the MatLAB package. Optimal solutions have shown that the minimum of the combined functions, consisting of the sum of the relative deflection values ​​at the end of the console and the angles of rotation at the end of the console and on the front support, is achieved at the same variable parameters as the minima of the angles of rotation at the end of the console and on the front support. The proposed approach to the design of the shafts of spindle units of metal-cutting machines, which are optimal in terms of rigidity characteristics, forms a tool for a reasonable choice of bearings and design parameters of spindle shafts.

Research on dynamic load sharing characteristics of double input face gear split-parallel transmission system

The face gear power-split system has huge superiorities over the traditional transmission form in the application of modern rotorcraft, and it has become the research trend of the industry in recent years. Thus this paper took the double input face gear split-parallel transmission system used in the rotorcraft as the research target, and established its dynamics model through the lumped parameter theory. Based on the Newtonian second law, the dynamics equations were built and solved to gain the meshing forces and load sharing coefficients of the transmission system. Simultaneously, the impacts of the eccentric errors, support stiffness, and torsional stiffness on the load sharing characteristics were studied. The results show that the meshing forces and load sharing coefficients of each gear pair have periodic changes; the eccentric errors of each drive stage gear have only a significant effect on the corresponding drive stage. Moreover, the changes in the support stiffness of the split-torque shafts and double gear shafts mainly affect the load distribution of the parallel stage, and the shaft torsional stiffness is less sensitively to maintain load balance. In addition, the increment of the shaft stiffness increases the load sharing coefficients of the corresponding gear pairs.

Effects of residual stresses on the bending stiffness of shafts strengthened by enveloping de-formation

The aim was to study the effects of technological residual stresses on the bending stiffness of cylindrical parts of shafts and axes. Experiments were conducted for elongated cylindrical specimens made of steel grade 35 with a diameter of 30 mm using boring and turning methods. Specimens were annealed in a protective medium to remove initial residual stresses. Experiments were carried out using an Amsler laboratory hydraulic testing machine and VK8 grade hard-alloy matrices. The experiments showed that, for an extremely low degree of relative crimping of 0.1 to 0.5%, the size of the layer with tangential residual compression stresses gradually decreases. The stiffness of such cylindrical workpieces remains almost unchanged. An increase in relative crimping (from 0.5 to 1.2%) leads to a decrease in resi dual compression stresses on the part surface. The layer thickness with tangential residual compression stresses starts to increase. This leads to a decreased residual buckling and an increased bending stiffness. It was found that the degree of relative crimping has no effect on the variation of distribution depth of axial residual stresses. Optimal distribution of tangential residual compression stresses can be reached by increasing their depth. A linear relationship was found for relative crimping of 0.1 to 1.0%. The highest bending resistance was recorded for specimens strengthened by residual crimping of about 1.0%. By processing workpieces using enveloping deformation with crimping of 0.1% and loading them with a transverse force of 0.6 kN, bending distortion can be decreased and the strength of parts can be increased by 5 times. It was found that the bending stiffness of cylindrical shafts is greatly affected by residual compression stresses. The bedding depth of residual stresses has various effects on the stiffness of cylindrical parts. Thus, correct use of strengthening enveloping deformation can form a high-quality surface layer on parts with the pre-defined distribution of residual stresses.

Tooth surface modification of double-helical gears for compensation of shaft deflections

Abstract. Based on gear meshing theory, the tooth surface equation with tooth profile modification parameters is deduced, the tooth surfaces of unmodified and modified gears are constructed, the three-dimensional model of unmodified and modified double helical gear-shaft-bearing system is established and then the three-dimensional contact finite element model of double helical gear-shaft-bearing system is established and the load-bearing contact analysis of the tooth surface is carried out. The actual contact state of the tooth surfaces of double helical gears under different shaft stiffness and power transmission paths is investigated, and the influence of tooth modification parameters on the load distribution of the tooth surfaces of double helical gear pairs is studied. The results show that the tooth surface bearing the contact of the herringbone gear system has the phenomenon of partial load due to the supporting deformation, and the unmodified herringbone gear has obvious contact stress concentration. However, the phenomenon of partial load and stress concentration can be effectively improved by gear tooth modification.

Nonlinear vibration of the spiral bevel gear under periodic torque considering multiple elastic deformation evaluations due to different bearing supports

This paper investigates two parameters effect on vibrational responses of the spiral bevel gear. Changing the gear system overall stiffness (GSOS) considering elastic deformation and periodic torques are the two parameters which are represented as the main goals of this study. In order to investigate the effects of shaft stiffness and elastic deformation, two different cases with different support locations are considered. The first case is presented by locating the support close to the gear, and in the latter one, the distance between gear and support is increased. Besides, to study the effect of torque, two main types are considered: constant and periodic excitation torque. To illustrate the dynamic behavior, the governing differential equations are solved numerically according to the Runge–Kutta method. The equations are nonlinear due to backlash and time-varying coefficients as the results of GSOS variation. Vibrational phenomena are illustrated by means of bifurcation diagrams, RMS, and Poincaré maps. Particular vibrational behaviors such as “chaos” and “period-doubling” phenomena are illustrated with details. By investigating the effect of shaft stiffness, results show that when the support is far away from gear, the vibration response increased by 67.5%. Moreover, while the input torque is constant, the support movement does not cause undesirable responses such as chaotic or period-doubling responses. The periodic torque causes undesirable responses such as chaos and bifurcation and period-doubling responses.Article HighlightsWhat is done in the present paper can be mentioned in three main parts:The nonlinear dynamics of the spiral bevel gear pair under two different support situations is investigated in this paper.To scrutinize the dynamic behavior of the spiral bevel gear-pair in a nearly real situation, the input driver torque is periodically variable.The results show that the spiral bevel gear pair may comprise chaotic response with periodic torque excitation if the bearing supports locate far enough from the gears.

Input Torque Measurements for Wind Turbine Gearboxes Using Fiber Optical Strain Sensors

Abstract. Accurate knowledge of the input torque in wind turbine gearboxes is key to improving their reliability. Traditionally, rotor torque is measured using strain gauges bonded to the shaft. Transferring the resulting signal from the rotating shaft to a stationary data acquisition system while powering the sensing devices is complex and costly. The magnitude of the torques involved in wind turbine gearboxes and the high stiffness of the input shaft pose additional difficulties. This paper presents a new alternative method to measure the input torque in wind turbine gearboxes based on deformation measurements of the static first stage ring gear. We have measured deformation using fiber optic strain sensors based on fiber Bragg gratings because of their advantages compared to conventional electrical strain gauges. The present study was conducted on a Siemens Gamesa Renewable Energy gearbox with a rated power of 6MW, in which a total of 54 fiber optic strain sensors were installed on the outer surface of the first stage ring gear. The gear mesh forces between the planets and the ring gear cause measurable deformations on the outer surface of the stationary ring gear. The measured strains exhibit a dynamic behavior. The strain values change depending on the relative position of the strain sensors to the planet gears, the instantaneous variations of the input torque, and the way load is shared between planets. A satisfactory correlation has been found between the strain signals measured on the static ring gear and torque. Two signal processing strategies are presented in this paper. The first procedure is based on the peak-to-peak strain values computed for the gear mesh events, and therefore, torque can only be estimated when a gear mesh event is detected. The second signal processing procedure combines the strain signals from different sensors using a Coleman coordinate transformation and tracks the magnitude of the fifth harmonic component. With this second procedure, it is possible to estimate torque whenever strain data of all sensors is available, leading to an improved frequency resolution up to the sampling frequency used to acquire strain data. The method presented in this paper could make measuring gearbox torque more cost-effective, which would facilitate its adoption in serial wind turbines and enable novel data-driven control strategies, as well as a more accurate assessment of the consumed fatigue life of the gearboxes throughout their operation.

Nonlinear characteristic investigation of tribo-crack-dynamic rotor system based on full spectrum analysis

The tribo-crack-dynamic model and experimental method to research dynamic mechanism are proposed based on the full spectrum in this paper. In view of the nonlinear characteristics from rub-impact forces, oil-film forces and time-varying stiffness of crack shaft, the tribo-crack-dynamic rotor model is established to serve the analysis of dynamic mechanism and characteristics. The numerical simulation is developed to obtain the precession signal and bifurcation diagram for the tribo-crack-dynamic rotor system. The results show that the negative frequency components are more abundant and their amplitudes increase in the full spectrum, showing obvious reverse precession characteristics when cracks and rub-impact faults occur at the same time. Experiment investigation on nonlinear dynamics is designed, which verifies the tribo-crack-dynamic model and the positive and negative precession characteristics in multi-fault rotor system.

The influence of shaft stiffness on joint kinematics and kinetics during hiking

Hiking boots provide an interface for walking in challenging environments, typically equipped with a shaft to provide ankle joint stability in rough terrains. Currently it is unclear if the ankle joint is stabilized to an extent that protects against ankle injuries, and if so, to what degree this added ankle stability sacrifices ankle mobility and hence decreases efficient gait propulsion. The aim of the present study was to compare the effect of shaft construction and stiffness on lower extremity kinematics and kinetics during level and step-down walking to simulate hiking conditions. Thirteen healthy males walked in one low-cut and three shafted commercially available hiking shoes with varying shaft stiffness. Lower extremity kinematics and ground reaction forces were recorded simultaneously. During level walking, ankle plantar-dorsiflexion range of motion was significantly reduced for the stiffest shaft hiking shoe compared to the low-cut shoe. A reduction in the muscle contribution to ankle joint work was found for all shafted shoes compared to the low-cut shoe. The reduced ankle joint work for the shafted shoes conversely increased eccentric knee joint work. Kinematic and kinetic differences between shoes diminished during box step-down walking. The present study shows that shaft height and stiffness can influence ankle joint range of motion, and ankle and knee joint work, with the high-shaft shoes redistributing load from the ankle to the knee joint. This may have implications for gait efficiency and increase the risk of knee joint loading or injuries.

Nonlinear forced vibration analysis of the composite shaft-disk system combined the reduced-order model with the IHB method

In this paper, the internal resonance phenomena of a composite shaft-disk system with multi-degrees-of-freedom are analyzed. The force caused by the unbalanced mass of the disk is considered as an external excitation force. The shaft is simply supported. Shear deformation and gyroscopic effects are considered. The strain–displacement relationship of the shaft element is expressed using the Timoshenko beam theory. Each node has 5 degrees of freedom. SHBT (simplified homogenized beam theory) is applied to calculate the stiffness of the composite shaft. WQEM (weak form quadrature element method) is used to construct the element matrices, and the system matrices are established using the element matrix assembly rule of the FEM (finite element method). The reduced-order model is applied to reduce the calculation time. IHB (incremental harmonic balance) method is utilized to solve the nonlinear equations of motion of the composite shaft-disk system. The nonlinear vibration characteristics of the Jeffcott rotor are analyzed using the proposed method and compared with the results of previous researches, and the results are very similar. Based on these considerations, the nonlinear vibration phenomena of the composite shaft-disk system with multi-degrees-of-freedom are considered at the several resonance points.

3-D analytical model for bending stiffness of thick-walled composite hollow shaft

In this paper, the Layer-wise theory was improve based on the alternate characteristics of layer angle of composite hollow shaft. The ±φi laminate was considered as the minimum mechanical analysis element, which can effectively simplify the stress coupling in mechanical analytical method. On this foundation, a 3-D analytical method of bending stiffness < EI> for thick-walled composite hollow shaft was derived. The CFRP hollow shaft specimens made of same dimensions but different laminate were used for bending test. Then, the bending stiffness calculated by theory of beams, 2-D analytical method, 3-D analytical model and experiment were compared. All the results show that the 3-D analytical model predicts the bending stiffness of thick-walled composite hollow shaft more precisely than the theory of beams and analytical method, and agree well with experimental results. And the 3-D analytical model can reflect the influence of stacking sequence on the bending stiffness of the composite hollow shaft.

Study on shaft alignment of propulsion shafting system depending on single reaction force supporting position of aft stern tube bearing

Trends of large-scale ships have seen propulsion shaft and propeller sizes increase. This has enabled shafts to have greater stiffness, yet has caused flexibility to lessen and induce bearing failure at the aft stern tube bearing. In general, shaft alignment is calculated and evaluated in accordance with classification societies’ rule requirements. Especially, positioning reaction support and stiffness of aft stern tube bearing are based on the practical experience of shaft alignment. Therefore, in this study, to evaluate the feasibility of the reaction force supporting position and stiffness of the aft stern tube bearing as recommended by classification societies in shaft alignment, theoretical reaction force supporting positions for various ship propulsion shafting systems were examined and the differences were evaluated. The reaction force supporting position of the aft stern tube bearing provided by classification societies was evaluated and it was found that a propeller shaft diameter less than 600 mm is within the provided range. However, in the evaluated shafting system, a propeller shaft diameter of more than 600 mm tends to cause the ship to move to the forward side due to increased shaft stiffness.

Design optimization analysis of an anti-backlash geared servo system using a mechanical resonance simulation and experiment

Abstract. In many mechatronic systems, gear transmission chains are often used to transmit motion and power between motors and loads, especially for light, small but large torque output systems. Gear transmission chains will inevitably bring backlash as well as elasticity of shafts and meshing teeth. All of these nonlinear factors will affect the performance of mechatronic systems. Anti-backlash gear systems can reduce the transmission error, but elasticity has to be considered too. The aim of this paper is to find the key parameters affecting the resonance and anti-resonance frequencies of anti-backlash gear systems and then to give the design optimization methods of improving performance, both from element parameters and mechanical designing. The anti-backlash geared servo system is modeled using a two-inertia approximate model; a method of computing the equivalent stiffness of anti-backlash gear train is proposed, which comprehensively considers the total backlash of transmission chain, gear mesh stiffness, gear shaft stiffness and torsional spring stiffness. With the s-domain block diagram model of the anti-backlash geared servo system, the influences of four main factors on the resonance and anti-resonance frequencies of system are analyzed by simulation according to the frequency response, and the simulation analysis results dependent on torsional spring stiffness of anti-backlash gear pair and load moment of inertia variation are verified by the experiment. The errors between simulation and experimental results are less than 10 Hz. With these simulation and experiment results, the design optimization methods of improving the resonance and anti-resonance frequencies such as designing the center distance adjusting mechanism to reduce the initial total backlash, increasing the stiffness of torsional spring and lightweight design of load are proposed in engineering applications.

Experimental study of loop shape using 0.025-inch ERCP guidewires (with videos).

Background and study aims Duct penetration by the guidewire sometimes occurs during endoscopic retrograde cholangiography, which might lead to adverse events such as acute pancreatitis. To prevent duct penetration, making a loop shape with the guidewire might provide a useful technique. The aim of this experimental study was thus to evaluate which types of guidewire can most easily form a loop shape. Methods This experimental study evaluated six guidewires (0.025-inch, angle type): MICHISUJI; VisiGlide 2; Jagwire; Pathcorse; RevoWave-α UltraHard 2; and M-through. Flexibility of the tip, shaft stiffness, and the ability to form a loop were evaluated for each type in an ex vivo model. Deformation behavior was also recorded on video, and factors suitable for making a loop shape in each guidewire were evaluated. Results Flexibility and stiffness of each guidewire differed significantly. During an experimental study regrading deformation behavior before forming a loop shape, maximum load was lower for MICHISUJI (6.8 g) than for other guidewires (Jagwire [11.3 g], M-through [12.9 g], VisiGlide 2 [12.9 g], Revowave [21 g], and Pathcorse [25.4 g]). Mean time required to achieve a loop shape was as follows: MICHISUJI, 6.2 seconds; M-through, 8.7 seconds; VisiGlide 2, 11.0 seconds; and Revowave, 7.1 seconds. Conclusion In conclusion, characteristics of flexibility and stiffness among guidewires were significantly different in the ex-vivo study. In the experimental study regrading deformation behavior until achieving a loop shape, maximum load also differed. To evaluate whether guidewires easily form a loop shape, clinical study is needed.

Nonlinear mechanical model of the shaft of a roll forming mill and parameter identification

Roll forming is a continuous process in which a moving metal sheet passes through numerous pairs of opposing forming rolls. The shafts of the roll forming mill are equipped with these rolls and must be set up and aligned to achieve the required final profile of the sheet. The practically relevant task of predicting the profile geometry of this incremental rolling process with varying characteristics of the metal sheet entering the mill requires an accurate description of the stiffness behavior of the shaft with rolls, which is the most compliant part of the roll forming mill. In this paper, the measured force-deflection characteristic of the shaft without rolls is compared with predictions of various theoretical models, followed by the adoption of the shear deformable beam model of the shaft with nonlinear elastic supports in the bearings. The coefficients of the cubic stiffness characteristics of the rotational springs as well as the effective length between the supports are identified based on the experimental data for the deflections, measured along the shaft for various loading levels. The theoretical predictions are obtained via the nonlinear finite element model of the shaft. The model thus provided shows high accuracy compared with the measurements. The paper’s results serve as a foundation for models to predict the stiffness of shafts with rolls.

Analysis of Factors Affecting Vibration During Wet Clutch Engagement

The shift characteristics of a wet clutch mainly depends on the variation characteristics of the friction coefficient during clutch engagement. Therefore, by analyzing the dynamic friction transmission mechanism of wet clutch engagement, a numerical simulation of the shift engagement of a certain type of vehicle clutch is performed based on a dynamic model of the power transmission system of a vehicle with four degrees of freedom. We discuss the effect of the damping coefficient of the wet clutch transmission system, the moment of inertia of the driving disc, the moment of inertia of the driven disc, the stiffness of the transmission shaft, and the variation law of the dynamic friction coefficient on vibration and its effect on shift engagement. The results show that under the condition of a fixed speed difference, with a greater gradient change in the dynamic friction coefficient over time, shift engagement vibration occurs and increases. By appropriately increasing the damping coefficient, the moment of inertia of the driving and driven discs, and the rigidity of the drive shaft, the vibration amplitude of clutch shift engagement can be effectively reduced. The research of dynamic friction coefficient can accurately analyze the vibration of the clutch engagement, which provides a reference for the optimization design of automatic transmission.

The dynamic loads of mobile continuous earthmoving machine during the working body lock

working body is locked significant dynamic loads appear that influence the drive shaft and might lead to the machine breakdown. Preliminary approximate calculation of these loads is made according to oneor two-mass dynamic loads. For more detailed and precise analysis of dynamic load in drive shaft elements of the earth moving machines it is necessary to review the multimass model machine element movement. Goal. The aim of research is definition of the dynamic load and working out methodology that appears in the continuous earth digging machine elements during working body lock. Methodology. The discreet dynamic system dynamic and mathematical model construction is performed by means of the programming language Modelica. The model relevance is assessed via a two-mass model analytical solution. Results. Drive shaft element loading is assessed via the dynamic coefficient, which depends on kinematic, flexible and inertial machine parameters. Originality For the first time the continuous earth digging machine dynamic model that enables the automobile chassis drive shaft to interact with the continuous digging chain is constructed. It was determined that the drive shaft dynamic loading parameter correlation change upon the earth digging machine working body meaningful parameters. With designed model usage, the input data for transmission element engineering calculation was defined and the automobile basic chassis construction, equipped with the continuous digging chain. Practical value. The obtained dynamic factor value and developed methodology can be applied during continuous earth moving machine designing and calculation. Dynamic load decrease is possible through shaft stiffness and rotating mass inertia adjustment, and also by introducing elastic protecting frictional coupling with a controlling device.

A system for simulating the kinematics and measuring the impact force from riding whips used in Thoroughbred horseracing

Thirty horse racing whips of four different designs were tested to measure dynamic impact force and compared using a specially designed mechanical testing device to simulate the whipping action of a jockey during racing. The whips tested included designs used in Thoroughbred horse racing in North America, which meet the design criteria established by the Association of Racing Commissioners International (ARCI) model rules, as well as the most common whip used in British horse racing. The objective of the device was to allow comparisons to be made between peak impact loads resulting from different whip designs. A high peak dynamic force on a horse’s shoulder or hind quarter may result in injuries, such as welts. The testing device contains a planar three-bar, open mechanical linkage loaded by torsion springs to model the arm motion of a jockey. The whip strikes a flat plate covered by an elastomeric pad. The energy input is replicated during the simulated impact. A single axis dynamic load cell under the loading plate and three single-turn precision potentiometers located at each joint of the three-arm mechanical system measure impact forces and relative angular positions, respectively. Force measurements are compared from the face of each whip and the edge or seam where applicable. In addition to the flap design, other physical differences between whip designs included mass, shaft length, shaft stiffness, flap cushion thickness/compression factor, flap surface area, and flap seam area. Statistically significant impact force differences were found between flap face and flap seam impact orientations, with higher impact forces delivered by the flap face. Significant differences were also found in impact forces between the three whip styles with seams.

MODELING AND IDENTIFICATION OF BALL SCREW FEED DRIVE SYSTEMS

In this paper, the ball screw feed drive system is simulated and it?s frequency response is studied. Various parameters effect on the dynamic behavior of the ball screw system have been investigated. Ball screw feed drive system is used in high speed machine tools due to their high efficiency. Estimation of the dynamic behavior of ball screw feed drive mechanism is very important in the industrial processes in order to get high demand for precision and accuracy in machine tools. Optimizing the drive operation can provide significant cost savings. A four degree of freedom system, lumped parameter model is used for modeling a single axis ball screw feed drive system and use it to study and analyze vibrations in this model. The mathematical modeling provides an important information about frequency response when applying different levels of table mass, stiffness of the nut, axial stiffness of the ball screw shaft and torsional stiffness of the ball screw on the system, to describe the effects of these parameters on the system dynamic behavior. The study of dynamic response of ball screw feed drive system provides a better performance control and better understanding of ball screw dynamics.

Impact of shear forces on variable cross section shaft stiffness

Influence of shear forces is presented in this paper for static analysis of a shaft with variable cross-section. A stiffness matrix method is used to evaluate displacement and reaction forces on the shaft in regard to the slope deflection equations. The Timoshenko beam theory is addressed according to the shear coefficient and the shear correction factor. Bending moments are calculated.

Modular modeling and dynamic response analysis of a driveline system during start-up process

This study is aimed at exploring the effect of the structural parameters of a driveline system on its dynamic response, during the start-up process. For this purpose, a modular modeling method was proposed, based on the power flow and mechanical connections in the driveline. Moreover, a dynamic model of the driveline for vehicle start-up was established, taking into consideration the time-varying mesh stiffness, shaft stiffness, damping, and the variation of the wet clutch friction coefficient with the relative slipping speed. Modal analysis of the driveline system was performed, and the influence of time-varying mesh stiffness of the gear set on the natural frequency was analyzed. Furthermore, an engine constant start-up control strategy was adopted, and the start-up process of the vehicle was simulated. The simulation model was verified with the experiment data. A new evaluation indicator was proposed taking into consideration the phase difference and amplitude change, induced by the time-varying stiffness and the self-excited vibration in the system. Lastly, the influence of the structural parameters on the dynamic response of the driveline system, during start-up, was investigated.