Influence Coefficient Method Research Articles

Extended influence coefficient method for meshing stiffness evaluation of spur gears considering rotor flexibility

Rotor-supported gears are applied in transmission systems, and the rotor flexiblility leads to the misalignment of the gear pair with multiple degrees of freedom, which ultimately leads to the decrease of meshing stiffness. To evaluate the meshing stiffness of rotor-supported gears, the extended influence coefficient method (EICM) is proposed in this paper. Firstly, the gear contact position adjustment under misalignment condition is carried out. Then, the extended influence coefficients of bending, shearing, axial compression, Hertzian contact and fillet foundation deformation of gears are derived to obtain the extended influence coefficient matrix. Afterward, the load-deformation equations under different contact states are obtained, and the meshing stiffness is calculated. Subsequently, a cantilever rotor-gear coupling dynamic model is established by the rigid body element-extended influence coefficient method (RBE-EICM), and the dynamic meshing stiffness considering rotor flexibility is obtained. The accuracy of EICM and RBE-EICM is verified by comparison with finite element method, literature methods and experimental data, respectively. Finally, the effects of supporting layout, input torque and rotor diameter on the dynamic characteristics of cantilever rotor-supported gears are discussed.

A new modeling method of assembly deviation of large flexible parts based on improved method of influence coefficient

A new modeling method of assembly deviation of large flexible parts based on improved method of influence coefficient

Development of Flexible Rotor Balancing Procedure Using Response Matching Technique

Abstract Purpose The two main methods under use for balancing of flexible rotor know as Influence Coefficient Method (ICM) and Modal Balancing Method (MBM). It needs few trials to arrive at predicting the unbalance mass. In presence of initial bow along with unbalance these methods will fail to give accurate predictions. It becomes further complex when rotors are mounted on flexible supports. To overcome the limitations of these two methods, a new method called response matching method is disused here for flexible rotor balancing. Method The objective of present work is to develop a balancing procedure for flexible rotor balancing, wherein unbalance response of Finite element model (FEM) is matched with experimental response using iterative technique. Unbalance masses, moments and shaft stiffness of FEM model was iterated to match experimental unbalance responses at three distinct speeds below its first flexural critical speed. Further, unbalances mass was estimated iteratively to cross the bending critical speed with lower residual amplitudes at critical speeds. Results Experimental test setup was developed to validate the proposed method. Results show that present method predicts the unbalance masses and moments to compensate the bow more accurately. Conclusion Results shows that with single run, rotor unbalance masses and movements are predicted accurately. Corrected masses are used in experimental test setup to check the rotor amplitudes during its bending critical speeds. It shows rotor passes first bending critical speed without excessive amplitudes.

Study on contact characteristics of double-disk arc-contact lapping for cylindrical rollers based on influence coefficient method

The double-disk arc-contact lapping (DDACL) method is a novel for precision machining of the rolling surface of cylindrical roller. The contact characteristics of DDACL process can affect the lapping quality of the bearing roller. However, there is a limited amount of study on the contact theory of DDACL. Therefore, the objective of this research is to create a semi-analytical modeling approach that can effectively and accurately forecast the contact characteristics of DDACL. In the initial step, the spatial coordinate transformation was employed to establish accurate mathematical models that represents the surfaces of the cylindrical roller and arc groove. Subsequently, the potential contact area was determined and subdivided into discrete elements based on the differential geometry principles. The accurate flexibility of the potential contact area is achieved by utilizing the interpolation theory and considering both the global and local deformations of the roller and groove. By employing the influence coefficient method and the force-deformation coordination relation, the deformation and contact force can be calculated. Finally, the accuracy and efficiency of this calculation is confirmed through a comparison between the finite element method (FEM) and the proposed method.

Continuous optimization method of micro geometry for unparallel beveloid gears

To enhance the bearing capacity and minimize the loaded transmission error for unparallel beveloid gears, a continuous optimization method of micro geometry is proposed. Considering that the conjugate characteristic of tooth surfaces is destroyed by misalignments and modification, an unload contact analysis model is first developed. This includes a contact tooth sequence determination model and an unloaded transmission error calculation model for multi-tooth contact. To determine contact due to deformation, a potential contact point matching model of non-contact tooth surfaces is introduced. Further, a numerical loaded contact analysis model based on the influence coefficient method for unparallel beveloid gear is developed. For multi-tooth contact of modified tooth surfaces, the transmission error compatibility condition is introduced. Based on the loaded contact model, a continuous optimization model for the comprehensive performance of contact pressures and loaded transmission error within a meshing cycle is established. To improve optimization efficiency, a calculation strategy for continuous optimization is developed. The feasibility of the proposed method is validated using a numerical example.

Threshing cylinder unbalance detection using a signal extraction method based on parameter-adaptive variational mode decomposition

The threshing cylinder will wear and deform during the threshing process, causing dynamic balance problems. The combine harvester has multiple vibration excitation sources and a complex vibration environment, making it challenging to extract weak unbalanced signals from strong background noise. A novel three-step filtering framework is proposed in this paper. A zero phase filter was used as the pre-processing layer to filter out the high frequency components in the original signal and reduce the number of parameter-adaptive variational mode decompositions (PAVMD) needed. The PAVMD was used to decompose the non-stationary vibration signal before Adaptive Neuron Linear (Adaline) function was used to fit sinusoidal signal parameters. A measurement index, termed the correlation amplitude (CA) index, is constructed. The parameterisation of PAVMD was guided by the CA index, and the modal component of the unbalanced fault features were located. The simulation and real cylinder signals proved that the proposed method could effectively extract unbalanced signals under noise interference, and the unbalance was identified accurately by the influence coefficient method. Experiments on a threshing cylinder showed that the amplitude identification error was <24 g in single-sided unbalance identification, and the amplitude identification error was <27 g in double-sided unbalance identification. The proposed method had high robustness and small identification error, especially under short-time working conditions, compared with other similar approaches.

Dynamic balancing of rigid rotors by the influence coefficient method

Applied to rigid rotors, the influence coefficient method is the most adaptable method for rotor dynamic balancing because it uses only experimental information. The sole requirement for balancing rotors with this method is that there exists proportionality between mass unbalance located at correction planes and vibration measured at measuring planes. Hence, the rotors only need to be described by correction and measuring planes. The sensitivity of the measuring planes compared to the correction planes determines the response of the amount of unbalance for a specified rotation speed. The paper aims to describe in detail the experiment for dynamic balancing of rotors to verify the influence coefficient method. The direct calibration on two-plane rigid rotor using this method showed that a significant reduction of unbalances can be obtained. The results can be helpful for the design and safety assessment of dynamic systems equipped with rotors for marine engineering. Keywords: Rotor dynamic balancing, influence coefficient method, residual unbalances, Discrete Fourier Transform

Influence mechanism of tip clearance on flutter stability with different aerodynamic coupling effects for transonic compressor

Flutter is a highly destructive aeroelastic problem in modern compressors, which hinders the improvements of aero-engine performance and reliability. Because the aerodynamic work is mainly concentrated in the blade tip region, the tip clearance which is associated to the complex tip clearance flow has a considerable effect on the flutter stability. In this paper, in order to investigate the influence mechanism of the tip clearance on the flutter stability at different nodal diameters (ND), a series of compressor models were established with different tip clearances based on a transonic compressor rotor. The aerodynamic damping at each ND is obtained by influence coefficient method (ICM), while phase-shifted boundary method (PSB) is adopted to analyze the influence of the tip clearance on the flutter stability at different NDs. The results indicate the worst flutter stability for each tip clearance always appears at ND=1, where the aerodynamic damping exhibits a nonmonotonic trend of increasing first and decreasing thereafter along with the rising tip clearance. Two kinds of action are exerted by the tip clearance flow on the flow structures to alter the flutter stability: one is the tip clearance vortices which is generated on the suction side and impinge on the pressure side; the other is the interference of tip clearance vortices to the shock wave and to the flow separation. Besides, the influence of the changing aerodynamic coupling effect makes the above action bring about more drastic fluctuations to the unsteady pressure. At larger NDs, the unsteady pressure amplitude and phase fluctuate more sharply for larger tip clearances. Therefore, diverse change trends of the aerodynamic damping with the increasing tip clearance appear at different NDs, while the impingement of tip clearance vortices on the adjacent pressure side has a constantly stabilizing effect at different NDs.

Theoretical and experimental applications of a rotor balancing technique without using trial weights based on augmented Kalman filter

Unbalance is the most common fault of rotating machines and efforts devoted to developing balancing techniques are the goal of several researchers around the world. The most common balancing approach is the well-known influence coefficient method, which requires the use of trial weights to determine the relationship between unbalance forces and vibration responses of the rotor; consequently, it is time-consuming regarding industrial applications. The present contribution investigates the augmented Kalman filter (AKF) technique to balance a flexible rotor without using trial weights. There are few studies in this area and the majority of the previous research works are dedicated to theoretical studies using simple or reduced rotor models. This paper considers a finite element (FE) model of a realistic rotor system and investigates the effectiveness of the AKF as a balancing technique by considering several unbalance conditions. Therefore, the novelty of the present contribution is characterized by a complete numerical and experimental AKF based balancing procedure that is performed by using a realistic rotor system. As mentioned above, limited experimental contributions are found in the literature involving real-world applications. Additionally, a sensitivity analysis on the force covariance matrix is proposed to investigate the most favorable conditions to apply the AKF technique in the context of rotor balancing. The methodology proved to be effective to deal with a variety of rotor balancing applications.

Optimal design of strain sensor placement for distributed static load determination

Abstract In many applications it is desirable to inverse-calculate the distributed loading on a structure using a limited number of sensors. Yet, the calculated loads can be extremely sensitive to the placement of these sensors. In the case of predicting point loading applied at a known location, best results are typically achieved when one sensor is collocated with the force. However, the extension of this rule to distributed loading remains uncertain, and even simple sensor system design scenarios often require the designer to directly optimize the sensor placements using a numerical model. In an effort to provide designers with guidance, we identify optimal sensor configurations for predicting static distributed loads on beams with classical boundary conditions. An influence coefficient method, wherein the strain is related linearly to the static load, is used to estimate the applied forces. The loading distribution on the structure is assumed to be either a piece-wise linearly-distributed load or a uniformly-distributed load, allowing for distributed loads to be estimated using the magnitudes of a small number of control points. Given the simplicity of the beam structure, the equations of the influence coefficient method are derived analytically, which allows for the sensor placement to be specified using continuous optimization methods. The condition number of the influence coefficient matrix is used as a surrogate for error during optimization. ‘Rules of thumb’ for sensor placement are presented based on the optimization results. Results show that the optimal and rule-of-thumb sensor configurations are more resistant to input noise than naïve configurations, with the rule-of-thumb configurations yielding similar force predictions relative to the optimal configurations. We expect the rules of thumb to be useful guidelines for engineers designing tests on beam-like structures such as aircraft wings or marine propellers where the inverse calculation of distributed loads is of interest.

Design, kinematic and fluid-structure interaction analysis of a morphing wing

The application of morphing wing enables vehicles to show better aerodynamic performance in various flight conditions. This article proposes a modular bilateral triangular pyramid (BTP) unit-based morphing wing skeleton (MWS) mechanism that may achieve continuous variable span-wise bend, span-wise twist, and sweep. A lockable morphing (LM) unit with the BTP unit is designed. Based on the Denavit–Hartenberg coordinate method and motion influence coefficient method, the kinematics of MWS mechanism is conducted. The theoretical models of the kinematic characteristics of the MWS mechanism during the three morphing patterns are obtained. The kinematic simulation models of three morphing configurations of the MWS mechanism are performed. The driving speed ratio of the linear actuator of span-wise bend and twist is analyzed during the twisting phase, and the analysis shows that there be a speed ratio minimizes the coupling bend angle. By comparing the results between the theoretical and simulation models, the accuracies of kinematic theoretical models under different morphing configurations are verified. At last, this article conducted a two-way fluid-structure interaction (FSI) analysis of morphing wing to obtain the influence of aerodynamic load on the flexible skin and skeleton mechanism, as well as the influence of structural deformation on the aerodynamic characteristics. Results show that the initial and swept states have the greatest impact on the out of plane bearing capacity of the flexible skin. In addition, the out-of-plane bulging deformation located at the trailing edge of the flexible skin is beneficial for improving aerodynamic performance, while the protrusion located at the leading edge increases air resistance. These findings provide important theoretical basis for the application of the MWS mechanism.

Analytical calculation of mesh stiffness for spiral bevel gears with an improved global tooth deformation model

The paper proposes an efficient and accurate analytical calculation model of time-varying mesh stiffness (TVMS) for the spiral bevel gear. Firstly, an enhanced numerical loaded tooth contact analysis (NLTCA) model that considers both the global tooth and local contact deformations is developed. Based on Tredgold's approximation, spiral bevel gears are sliced into a series of spur gears. An improved global tooth deformation model is developed by accounting for axial tooth deformation and the effect of non-contacting tooth slices. The influence coefficient method is utilized to determine the local contact deformation. Subsequently, based on the results of the NLTCA model, a detailed calculation procedure for determining the TVMS of spiral bevel gears is described. Finally, some numerical examples are provided to illustrate the efficiency and accuracy of the proposed model through comparisons to finite element analysis (FEA) results. The proposed model can achieve satisfactory accuracy without FEA modification. Furthermore, ignoring the effects of the non-contacting tooth slices will lead to excessive elastic deformation when the length of the contact area across the tooth width is significantly shorter than the tooth width.

Thermal-elastohydrodynamic lubrication study of misaligned journal bearing in wind turbine gearbox

Owing to the simple structure, large load-carrying capacity and long life, journal bearings supporting planet gears (JBSP) are increasingly used in the wind turbine gearbox. The moment caused by the asymmetric meshing leads to the misalignment of the JBSP and further deteriorates its lubrication behaviors. A thermal-elastohydrodynamic (TEHD) model is developed for the misaligned JBSP, where the finite element method (FEM) and influence coefficient method are used to compute the temperature and deformation of this bearing, respectively. Besides, the above model is also experimentally verified. It is found that the misalignment of the JBSP can be suppressed through reducing its modification amount, helix angle and radial clearance, which have the optimal values to improve the TEHD behavior of the JBSP.

Investigation of loaded contact characteristics of planetary roller screw mechanism based on influence coefficient method and machine learning

The performance of the planetary roller screw mechanism (PRSM) is directly determined by the mechanical behavior of the loaded contact. The traditional method based on a finite element modeling to describe the loaded contact characteristics (LCCs) is computationally expensive, while the use of Hertzian contact theory and the thread connection model results in a loss of computational accuracy. As a solution, this study aims to develop a semi-analytic modeling technique that can predict the LCCs of PRSM accurately and efficiently. To achieve this, the study first establishes a mathematical model using the spatial coordinate transformation theory to accurately describe the thread surface. The potential contact area is discretely divided using the differential geometry principle. Then, the linear deformation and nonlinear deformation are matched using node interpolation technique to obtain the accurate compliance of the potential contact area. Finally, the contact forces for all activated contact points are derived by a nonlinear iterative algorithm based on the influence coefficient method. The performance of the proposed model is verified by numerical examples. Further, the sensitivity weights of the geometric parameters to the LCCs are identified utilizing machine learning, and a preferred zone is also constructed to optimize these parameters. This study provides a valuable tool for evaluating and improving the LCCs of PRSM in engineering practice.

Multivariable optimization of pyramidal compound substrates for cooling of power-electronics in modern hybrid and electric propulsion systems

We present a method for the optimization of the thermal cooling of heat sources mounted on top of layered composites and pyramidal substrates, that are widely used in the power electronics of hybrid-electric propulsion systems. The analytical solution of the Laplace’s heat equation is approximated via Fourier expansion series and it is coupled to the Influence Coefficient Method (ICM) to provide a functional of the overall thermal stress to minimize. A multivariable optimization method is derived by coupling the equations for the heat transfer with the Sequential Least-Square Quadratic Programming (SLSQP), or the Bounded Limited-Memory BFGS (L-BFGS-B) algorithm. Code validation is performed against three-dimensional simulations and experimental data available from the literature. It is shown that an optimal component relocation and apportionment of the overall thickness of the multilayer substrate promotes a sensible reduction of the thermal stress.

Bridging the gap between design and manufacturing specifications for non-rigid parts using the influence coefficient method

The manufacturing process may lead non-rigid parts to endure large deformations which could be reduced during assembly. The manufacturing specifications of the single parts should refer to their free state or “as manufactured” state; the functional specifications should instead address the “as assembled” state. Therefore, a functional geometrical inspection requires dedicated fixtures to bring the parts in “as assembled” state. In this paper, through a linearized model that considers fixturing and elastic spring-back, we aim to correlate the functional specification to the manufacturing specifications. The model suggests a hybrid approach called “restricted skin model” that allows to reduce the degrees of freedom considering the form error when relevant. Firstly, the framework is verified in a mono-dimensional test case. Subsequently, it is verified including FEM simulation and actual measurement for two simple assemblies. The results show that the model can correctly interpret the theoretical assembly behaviour for actual applications. The use of the “restricted skin model” approach shows a negligible difference when compared to full FEM simulation with differences of 2.1 · 10−7 mm for traslations and 6.0 · 10−3 deg for rotations. The comparison with actual measurement values showed an error of ±0.2 mm at the assembly features. Furthermore, the linearized model allows a possible real-time application during production that enables to adjust manufacturing specification limits in case of process drifting.

Lubrication analysis of stern bearing during ship severe turning maneuver

Stern shaft deformation caused by ship maneuvering will lead to stern bearing lubrication failure and wear. Based on the displacement superposition method, the deflection equation of the stern shaft under the action of the interface force and the propeller load was established. The fluid-solid coupling model of the stern bearing was established by using the averaged Reynolds equation, the Greenwood-Tripp model, and the influence coefficient method. The model was experimentally validated. The influence of the shaft bending deformation caused by the propeller load change on the lubrication characteristics is discussed. The results show that the shaft bending deformation leads to the changes in lubrication characteristics along the axial and circumferential directions. Compared with the vertical load alone, the existence of horizontal propeller load will significantly reduce the minimum nominal oil film thickness and friction coefficient. Under hydrodynamic lubrication, the direction of horizontal load has little effect on friction coefficient.

Aerodynamic Damping of the Tubed Vortex Reducer in an Axial Compressor Disk Cavity

The tubed vortex reducer in the axial compressor can destroy the vortex in the disk cavity, by a thin-walled tube, to reduce the total pressure loss. The tube may suffer from vibration problems, such as flutter and forced vibration, which are closely related to aerodynamic damping. In this paper, the energy method and the influence coefficient method are used to study the aerodynamic damping of the tube. Based on the modal characteristics, steady and unsteady flow characteristics of the tube, the first and second modes with lower frequencies and greater vibration risk are selected as the analysis objects. The energy method is used to calculate the aerodynamic damping of the tube with different amplitudes, which shows that the results approximately meet the linear assumption and are accurate. The results obtained by the influence coefficient method show that the aerodynamic influence between adjacent tubes is very small, and the effect of inter-tube phase angle on the aerodynamic damping can be ignored. Finally, it is found that the effect of structural coupling caused by the support ring on the aerodynamic damping of the tube is mainly reflected in two aspects: frequency reduction and tube mode coupling.

Unbalanced variation after assembly and double-speed influence coefficient method in the threshing drum

Unbalanced variation after assembly and double-speed influence coefficient method in the threshing drum

Loaded contact pressure distribution prediction for spiral bevel gear

Contact pressure and load distribution are key indices used in the performance evaluation of spiral bevel gears, and their calculation accuracy and efficiency continue to be the focus of active research. An accurate and efficient numerical approach for predicting the contact pressure and load distribution of a spiral bevel gear is presented based on multi-tooth deformation compatibility. First, an improved discrete tooth surface contact analysis (DTCA) method is developed to obtain the contact characteristics of the gear set under the no-load condition. This method is a geometric approach that avoids numerical instability. A local triangular mesh refinement algorithm is proposed to improve the accuracy and efficiency of the DTCA method. Subsequently, the deformation compatibility relationships of a spiral bevel gear set are derived for single-tooth and multi-tooth meshing. By combining the deformation compatibility relationship and the influence coefficient method, a multi-tooth loaded contact analysis (MLTCA) model of the spiral bevel gear set is established. Moreover, an iterative algorithm based on the conjugate gradient and fast Fourier transform method is proposed to determine the mesh force and contact pressure distribution. Finally, the proposed method is tested using numerical examples to demonstrate its accuracy and efficiency by comparison with finite element simulations and experiment results.