Influence Coefficient Method Research Articles

Experimental Subsonic Flutter of a Linear Turbine Blade Cascade With Various Mode Shapes and Chordwise Torsion Axis Locations

Abstract Axial flow turbomachinery is susceptible to self-excited vibrations known as flutter, primarily affecting high-aspect ratio rotor blades. Experimental studies of blade flutter are essential to understanding the phenomenon. The existing research has shown that the most critical factor in determining the aerodynamic stability of the blade is the mode shape. In this work, measurements of subsonic flutter of a linear turbine blade cascade with various mode shapes and torsion axis locations in a chordwise direction are carried out. The cascade consists of eight blades representing the reduced tip sections of the last stage blade of the steam turbine rotor, and central four flexibly mounted blades are forced to vibrate in pure bending and pure torsion modes and a combined motion. Both the traveling wave mode approach and the influence coefficient method are tested. The results show that the pure torsion mode is insensitive to increasing reduced frequency and is aerodynamically unstable for each positive angle of incidence tested, while pure bending and combined modes become less stable with the increased angle of incidence. In addition, high sensitivity of blade behavior to changes in torsion axis position and phase shift between bending and torsion motions is demonstrated.

Balancing of Motor Armature Based on LSTM-ZPF Signal Processing

Signal processing is important in the balancing of the motor armature, where the balancing accuracy depends on the extraction of the signal amplitude and phase from the raw vibration signal. In this study, a motor armature dynamic balancing method based on the long short-term memory network (LSTM) and zero-phase filter (ZPF) is proposed. This method mainly focuses on the extraction accuracy of amplitude and phase from unbalanced signals of the motor armature. The ZPF is used to accurately extract the phase, while the LSTM network is trained to extract the amplitude. The proposed method combines the advantages of both methods, whereby the problems of phase shift and amplitude loss when used alone are solved, and the motor armature unbalance signal is accurately obtained. The unbalanced mass and phase are calculated using the influence coefficient method. The effectiveness of the proposed method is proven using the simulated motor armature vibration signal, and an experimental investigation is undertaken to verify the dynamic balancing method. Two amplitude evaluation metrics and three phase evaluation metrics are proposed to judge the extraction accuracy of the amplitude and phase, whereas amplitude and frequency spectrum analysis are used to judge the dynamic balancing results. The results illustrate that the proposed method has higher dynamic balancing accuracy. Moreover, it has better extraction accuracy for the amplitude and phase of unbalanced signals compared with other methods, and it has good anti-noise performance. The determination coefficient of the amplitude is 0.9999, and the average absolute error of the phase is 2.4°. The proposed method considers both fidelity and denoising, which ensuring the accuracy of armature dynamic balancing.

Uncovering the Root Causes of Stall Flutter in a Wide Chord Fan Blisk

Flutter was encountered at part speeds in a scaled wide chord fan blisk designed for a civil aeroengine during a rig test when the fan bypass flow was throttled toward its stall boundary. Analysis of the blade tip timing measurement data revealed that the fan blades vibrated at the first flap (1F) mode with nodal diameters of two and three. To facilitate a further rig test and ultimately eliminate the flutter problem, a numerical campaign was launched to help understand the root causes of the flutter. Both the influence coefficient method (ICM) and the traveling wave method (TWM) were employed in the numerical investigation to analyze unsteady flows due to blade vibration, with the intention to corroborate different numerical results and take advantage of each method. To eliminate nonphysical reflections, a sponge layer with an inflated mesh size was used for the extended inlet and outlet regions. Steady flow field and unsteady flow field were examined to relate them to the blade flutter. The influences of vibration frequency, mass flow rate, shock, boundary layer separation and acoustic mode propagation behaviors on the fan flutter stability were also investigated. Particular attention was paid to the acoustic mode propagation behaviors.

Diagnosing and Balancing Approaches of Bowed Rotating Systems: A Review

Driven/driving shafts are the most important portion of rotating devices. Misdiagnosis or late diagnosis of these components could result in severe vibrations, defects in other parts (particularly bearings), and ultimately catastrophic failures. A shaft bow is a common problem in heavy rotating systems equipped with such attachments as blades, discs, etc. Many factors can cause the shaft bending; this malfunction can be temporary, such as the bow resulting from a rotor gravitational sag, or can be permanent, such as shrink fitting. Since bending effects are similar to those induced by the classic eccentricity of the mass from the geometric center, i.e., unbalancing, distinguishing the differences in dynamic behaviors, as well as the symptoms, can be a labor-intensive and specialized task. This article represents a review of almost all the investigations and studies that have been carried out on the diagnosing and balancing of bowed rotating systems. The articles are categorized into two major classes, diagnosing and balancing/correcting approaches to bowed rotors. The former is divided into three subclasses, i.e., time-domain, frequency-domain, and time–frequency-domain analyses; the latter is divided into three other sub-sections that concern influence coefficient, modal balancing, and optimization method in correcting. Since the number of investigations in the time domain is relatively high, this category is subdivided into two groups: manual and smart inspection. Finally, a summary is provided, as well as some new research prospects.

Kinematics and dynamics analysis of a six-degree of freedom parallel manipulator

Modeling and analysis of inverse kinematics and dynamics for a novel parallel manipulator are established in this article. The manipulator is a spatial mechanism, which consists of six identical kinematic chains connecting to the moving platform. Firstly, screw theory is applied to compute the degree of freedom of this manipulator. Then the inverse position is achieved based on the homogeneous coordinate transformation principle while motion law of the moving platform is given. Furthermore, the first-order influence coefficient method is employed to obtain the Jacobian matrices of the considered manipulator and the links, so do the velocities. Afterward, the rigid-body dynamics model is derived from the Lagrange formulation. To obtain the integrated inverse dynamics model, an approach for simplified flexible dynamics analysis is proposed. Finally, simulations are conducted to compute the position and driving force of this considered manipulator, which validate the new method simultaneously.

Vibration analysis of a multi-disk bolted joint rotor-bearing system subjected to fixed-point rubbing fault

This study aims to investigate the nonlinear vibration properties of a rotor-bearing system carrying a multi-disk bolted joint with a fixed-point rubbing fault. For this purpose, a mathematical model of a multi-disk bolted joint rotor system is explored to determine the effect of rotor-stator rub-impact and rub-impact positions on the nonlinear dynamics of a certain type of aero-engine compressor. First, the dynamics of the rotor system with a rubbing fault are examined numerically. Then, the effect of the rub-impact position in the multi-disk bolted joint on the nonlinear dynamics is studied. The shaft is modeled using a massless shaft, where the stiffness matrix of the shaft is determined based on the flexibility influence coefficient method. The governing equation of the multi-disk bolted joint is deduced considering both the lateral and bending stiffness between the adjacent disks. The dynamic model of the overall system can then be obtained by combining the shaft and multi-disk bolted joint. The resultant governing equations of motion are solved using direct numerical integration. The numerical results obtained from the established model are compared with those of a similar structure in other literature to verify the accuracy of the method used in this study. Then, the dynamic behavior of the rotor system and the mechanical properties of the multi-disk bolted joint under a rubbing fault are presented. Finally, the chaotic response with the rub-impact force acting on different positions of the multi-disk bolted joint is investigated through numerical simulations.

A Balancing Method for Multi-Disc Flexible Rotors without Trial Weights

Rotor dynamic balancing is a classical problem. Traditional balancing methods such as the influence coefficient method and the modal balancing method, have low balancing efficiency because they need to run many times to add trial weights. Although the model-based balancing method improves the balancing efficiency, it cannot accurately identify the position, amplitude and phase of each unbalance fault for rotors with multi-disc structures, so it is difficult to apply it to actual balancing. To solve the above problems, based on the traditional modal balancing theory, this paper deduces that the continuous and isolated unbalance in the rotor-bearing system can be represented by isolated unbalance on several balancing planes approximately. The model-based method is used to identify the above-mentioned equivalent isolated unbalances, and then the corrected mass is added to the balancing planes so as to complete the balance of multiple flexible rotor without trial weights. Considering the practical situation, the proposed balancing method includes two steps: low-speed balancing and high-speed balancing. The proposed balancing method is verified using three and four-disc rotors. The simulation results show that the balancing method can effectively reduce the vibration of the flexible rotor after low-speed and high-speed balancing, and the amplitude at the measurement point is reduced by 79.74~97.60%, respectively.

Robust Dynamic Balancing of Dual Rotor-Active Magnetic Bearing System Through Virtual Trial Unbalances as Low and High Frequency Magnetic Excitation

Abstract This work focuses on in situ residual unbalance estimation of the dual rotor system with the implementation of active magnetic bearing (AMB) as a controller and exciter. The excessive vibration generated due to the presence of residual unbalances limits the operating speed of the system. The compact structure of the dual rotor system provides constraints to the conventional balancing procedure that requires manual addition of the trial unbalances for balancing. In order to overcome the difficulty in balancing of the dual rotor system, an identification algorithm based on the modified influence coefficient method (MICM) is developed for the simultaneous estimation of residual unbalances in both inner and outer rotors with the generation of virtual trial unbalances as magnetic excitation through AMB. The controlling action of AMB attenuates the vibrational response of the system within the required limit and allows the safe operation of the system in the presence of rotor faults and additional excitations. The vibrational responses of the system at the limited locations and the magnitude and phase of the virtual trial unbalances are only required in the MICM for the estimation of unbalances. To numerically illustrate the present methodology, the displacement response is obtained from the developed finite element model of the dual rotor system with discrete disk unbalances and randomly distributed shaft. The robustness of the algorithm in the estimation of residual unbalances is verified with the addition of a different percentage of measurement noises. After balancing, the dual rotor system is found to traverse its critical speed with a less vibrational response.

Fatigue damage estimation of icebreaker ARAON colliding with level ice

This paper presents a methodology for precisely calculating the fatigue damage of an arctic vessel colliding with level ice. For accurate prediction of local ice loads acting on the external surface of the hull, focusing on the bow area, numerical simulation using the finite element method (FEM) was conducted first. The existing Drucker-Prager plastic model was combined with damage mechanics to simulate the icebreaking phenomenon. Furthermore, element erosion technology was considered to simulate crack formation and propagation through level ice that collides with a moving structure. Both dynamic drag and static buoyancy were applied to consider the dynamic behavior of submerged ice fragments. With this model, the ice resistance acting on rigid structures and a ship hull passing through level ice at multiple forward speeds were numerically calculated and compared with the test results. An additional simulation was performed in which ice fragments are removed immediately after being broken, with an attempt to decompose the total ice resistance into icebreaking resistance and the resistance of submerged ice fragments. Afterward, the spatial distribution of extreme values and frequencies of loads from the forefront to the vicinity of the hull shoulder area was derived. Finally, the ice load applied to the icebreaker ARAON at full scale under level ice conditions was calculated and then converted into stress at the target point using the influence coefficient method. Based on the converted stress information, the cumulative fatigue damage according to the speed was calculated, and the tendency of the cumulative fatigue damage was analyzed.

Estimation of ice loads on offshore structures using simulations of level ice-structure collisions with an influence coefficient method

During their service, ice-class vessels are exposed to ice collision loads. Precise estimation of such ice loads is essential to the design of ice-class vessels. Due to difficulties encountered in real measurement of ice loads, a simple yet logically straightforward method, namely the influence coefficient method (ICM), has been widely employed to convert strain data to ice pressures. However, the method's accuracy and reliability depend on how strain gauges are arranged and load cells are divided accordingly. This study investigated those aspects in order to enhance the accuracy and reliability of the ICM in estimating ice loads incurred in collisions of ice with offshore structures. It began by developing a numerical model of level ice-structure interactions. For this, an ice-material model applying the Drucker-Prager (DP) plasticity model combined with an element erosion technique's simplified damage mechanics was developed. The developed numerical model was verified by comparing its results with the relevant model test data and corresponding numerical results available in the open literature, and good agreement was achieved. Next, the present study performed, based on the numerical simulation strategy thus validated, simulations of level ice collisions with actual offshore structures showing inflexible and deformable behaviors. Subsequently, a finite element (FE) model of the structures’ instrumented area, which is the same as the scantling used in the ice collision simulation, was developed to construct an influence coefficient matrix for the ICM. The structural responses, including stress/strain data and ice-contact pressure observed numerically, were then applied to an investigation of the ICM capability. The effects of the strain gage arrangements and corresponding load cells on the estimation of the local ice pressures acting on actual offshore structures were demonstrated. The proposed procedure is expected to provide insights into how the ICM can be improved in reliably and cost-effectively estimating ice loads on ice-class offshore structures.

Intentional Mistuning Effect on the Blisk Vibration with Aerodynamic Damping

Intentional mistuning is a well-known factor affecting the vibration of the blisk. This paper mainly focuses on the intentional mistuning effect on the aerodynamic damping and forced response of an axial compressor blisk with random mistuning, and the intentional mistuning types involved in this paper include alternating mistuning, single mistuning, and sector mistuning. The aerodynamic damping of the blisk and its composition are calculated based on the fundamental model of mistuning and influence coefficient method. Under the influence of intentional mistuning, the aerodynamic damping and weight coefficient of each blade in the blisk vary with the modal amplitude and phase. Then, the forced response of the blisk is calculated by the modal superposition method, and the results indicate that intentional mistuning can reduce the increase of forced response caused by random mistuning in two ways: reducing the sensitivity of the blisk to random mistuning and improving the aerodynamic damping of the blisk. Finally, it is found that sector mistuning is better in reducing the sensitivity of the blisk to random mistuning, whereas alternating mistuning is better in improving the aerodynamic damping of the blisk.

탄성지지된 티모센코 호의 면외 자유진동 해석

Arcs have widely been used as structural elements in many mechanical, aerospace, and civil engineering applications. The authors previously analyzed the in-plane free vibration of Timoshenko arcs with elastic supports by using the transfer influence coefficient method. The basic concept of this method is based on the transfer of influence coefficient. In this study, the transfer influence coefficient method is applied to the out-of-plane vibration of Timoshenko arcs. The authors formulate a computational algorithm to analyze the out-of-plane free vibration of the arcs with elastic supports by using the transfer influence coefficient method. The reliability of the transfer influence coefficient method is confirmed by comparing both the numerical results of the transfer influence coefficient method and those of other researchers. The authors verify that the transfer influence coefficient method is more effective than the transfer matrix method in analyzing an arc with a hinged point at its intermediate.

Research on a novel robot mooring system based on dual-parallel elastic under-actuated mechanisms

Parallel mechanisms have the advantages of a high carrying capacity, large stiffness and compact structure. To develop the theory and key technology of robot mooring systems, in this paper, focusing on a 266,000 m3 LNG ship model with a scale of 1:60, a novel robot mooring system based on dual-parallel elastic under-actuated mechanisms is proposed. The configuration, structure parameters and workspace of the system are determined by experiments and simulations of the ship model with conventional mooring arrangements. Based on the influence coefficient method, the global stiffness model of a parallel-robot mooring system (PRMS) is presented, yielding a stiffness matrix in tensor form. The boundary conditions for the PRMS-moored ship in an experimental basin are introduced and a dynamic equation is given considering the two work modes of the PRMS. Numerical simulations and experiments are carried out to obtain the motions of the PRMS-moored ship when encountering unidirectional irregular waves, and the results indicate that the system can reduce the frequency drift motions of ships in surge, sway and yaw induced by the second order wave drift force.

Robust geometric parameter optimization of a crossed beveloid gear pair with approximate line contact

For a crossed beveloid gear pair with approximate line contact, the macroscopic geometric parameter is a dominant factor that affects meshing characteristics. However, geometric parameter design and reliability under misalignment are of great difficulty. To deal with these issues, an enhanced design method for a beveloid gear with approximate line contact is firstly proposed. Based on this, a robust optimization model of geometric parameters for crossed beveloid gear pairs with approximate line contact is developed. Three evaluation items are considered, namely the maximum contact pressure, contact trace offset distance and contact ratio. To reduce the calculation time, the influence coefficient method is applied to calculate the contact pressure. Considering the characteristics of installation deviation uncertainty and distribution, the mean maximum contact pressure, contact trace offset distance and ratio are considered as objective functions, formulated as a robust design optimization problem. Finally, robust optimization of a crossed beveloid gear pair based on a fast elitist nondominated sorting genetic algorithm (NSGA-II) is applied to solve the model to obtain the Pareto front of the macroscopic geometric parameters. A numerical example is provided to demonstrate the effectiveness of the proposed method.

Effect of Shock Wave Behavior on Unsteady Aerodynamic Characteristics of Oscillating Transonic Compressor Cascade

Abstract The unsteady behavior of the shock wave was studied in an oscillating transonic compressor cascade. The experimental measurement and corresponding numerical simulation were conducted on the cascade with different shock patterns based on influence coefficient method. The unsteady pressure distribution on blade surface was measured with fast-response pressure-sensitive paint (PSP) to capture the unsteady aerodynamic force as well as the shock wave movement. It was found that the movement of shock waves in the neighboring flow passages of the oscillating blade was almost antiphase between the two shock patterns, namely, the double shocks pattern and the merged shock pattern. It was also found that the amplitude of the unsteady pressure caused by the passage shock wave was very large under the merged shock pattern compared with the double shocks pattern. The stability of blade vibration was also analyzed for both shock patterns including three-dimensional flow effect. These findings were thought to shed light on the fundamental understanding of the unsteady aerodynamic characteristics of oscillating cascade caused by the shock wave behavior.

Online unbalance compensation of a maglev rotor with two active magnetic bearings based on the LMS algorithm and the influence coefficient method

Traditional rotor dynamic balancing requires equipment shutdown to add or subtract a counterweight mass so that a rotor can reach a certain level of dynamic balancing. However, a rotor may still have some residual unbalance. As the unbalance condition changes due to factors such as load changes and rust, the original dynamic balancing will be compromised. To solve this problem, an online unbalance compensation algorithm with active magnetic bearings (AMBs) based on the least mean squares (LMS) method and the influence coefficient method (ICM) was proposed. This algorithm first applies the LMS to extract the rotor unbalance vibration signal in real time. By implementing the AMB, the trial signal with the same frequency and phase with the rotor unbalance vibration signal is added to each AMB of the balancing plane, instead of mass addition or subtraction in the traditional dynamic balancing process. An eddy current displacement sensor is used to measure the rotor displacement response in each measuring plane, and the rotor unbalance vibration compensation current can be calculated to realize the online unbalance compensation during the normal operation. The online unbalance compensation converged through the iterative algorithm. The principle of the algorithm is simple and does not depend on the model of the controller. The effectiveness of the algorithm was verified by the online unbalance compensation experiment of a maglev rotor. The experimental results revealed that the vibration amplitude of the rotor decreased by 83.6%, 84.5%, 68.6%, and 84.3% after three iterations of the algorithm in each direction, and the amplitude of the main harmonic declined by 94.2%, 96.4%, 90%, and 96.1%. The vibration amplitude of the main harmonic in the Y direction at the left AMB specifically decreased by 28.8 dB. Moreover, the adaptability of the proposed algorithm to different unbalance masses and the phase of the rotor was verified through experiment validation.

A strong adaptive piecewise model order reduction method for large-scale dynamical systems with viscoelastic damping

This paper develops a novel and strong adaptive piecewise model order reduction (PMOR) method for large-scale dynamical systems with viscoelastic damping. Based on polynomial least squares approximations, the system is piecewise approximated by several kth-order (k≥2) dynamical systems in a wide target frequency band, in which the orders are adaptively determined with a curvature-based method. Then the convergent reduced-order models (ROMs) of the approximate systems are obtained gradually. In the above process, for each approximate system, the orthonormal basis is constructed iteratively via the kth-order Arnoldi method to span a projection subspace. To accelerate the convergence, an influence coefficient method and an order-dependent method are proposed to automatically determine the initial order of the ROM and the order increments, respectively. More importantly, a proposed error estimation strategy can predict all forms of estimated relative errors. According to the study of these forms, a comparison-selection method is presented to determine the final ROM for the whole target band interval by interval. Four examples comprehensively validate the strong adaptive ability, high efficiency and wide applicability of the PMOR method.

Fault identification in cracked rotor-AMB system using magnetic excitations based on multi harmonic influence coefficient method

On-site estimation of multiple fault parameters has been performed in a rotor integrated with active-magnetic bearing (AMB) with a cracked shaft supported on flexible conventional bearings. In addition to external viscous damping (at flexible bearings), internal damping (at transverse crack) is considered to show its influence on the dynamics of the cracked system. The instability generated in the rotor system due to the effect of internal damping at high speed are reduced with the control action of the AMB. A Multiple Harmonic Influence Coefficient Method (MHICM) has been proposed that requires the full spectrum amplitude and phases of the rotor responses and excitation forces to obtain the fault parameters of the rotor. The additive crack stiffness, residual unbalances, and internal damping can be estimated in the operating condition with less information on the rotor system. The intensity of the fault parameters is required to be monitored periodically to identify the safe operating speed of the system. As the AMB is an integral part of the rotor system, the multi-harmonic excitation can be initiated without changing the physical condition of the system. To check the robustness of the algorithm, different percentages of random noise are added to the rotor responses.

A Calculation Method of Surface Wear of Plastic Line Gear Pair Under Dry Friction Conditions

Abstract Parallel line gear pair (PLGP) can achieve pure rolling meshing along its theoretical contact curves under the ideal condition. However, the actual meshing positions may deviate from the theoretical ones under the actual operating conditions, which may result in the alternation of contact pressure distribution and cause relative sliding of the meshing surfaces. Contact pressure and relative sliding are two main factors causing surface wear. A calculation method of surface wear of plastic line gear (LG) pair under dry friction conditions was studied theoretically and experimentally, taking a polyoxymethylene (POM) PLGP as an example. Based on the geometric model of PLGP considering the misalignments under the actual operating conditions, this method employs the three-element model and influence coefficient method to compute the contact pressure with the consideration of the viscoelasticity of the gear material. Archard’s wear equation is applied to calculate the surface wear depth. Results of the wear experiment of the POM PLGP specimens validated the feasibility of the calculation method. This study provides a necessary basis for the engineering application of plastic LG pairs.

Review of Rotor Balancing Methods

This review is dedicated to balancing methods that are used to solve the rotor-balancing problem. To ensure a stable operation over an operating speed range, it is necessary to balance a rotor. The traditional methods, including the influence coefficient method (ICM) and the modal balancing method (MBM) are introduced, and the research progress, operation steps, advantages and disadvantages of these methods are elaborated. The classification of new balancing methods is reviewed. Readers are expected to obtain an overview of the research progress of existing balancing methods and the directions for future studies.