Nodal Diameter Research Articles

Exploration of an unconventional validation tool to investigate aero engine transonic fan flutter signature

Abstract Flutter, an aeroelastic blade vibration phenomena, experienced by the fan of an developmental aero gas turbine engine, result in blade failure. Hence, suitable flutter detection instrumentation is required during engine testing. Flutter signature capture from revolving blades is a challenging task that necessitates either a complicated strain gauge-based rotating instrumentation or a noncontact tip timing system. Authors investigated a unique way for identifying, measuring, and validating flutter signature by assessing wall static pressure pulsations produced during blade tip transit across a casing mounted high bandwidth sensor during this research. The authors devised a mathematical model to explain signal spectrum components that feature both amplitude and angle modulation properties at the same time. The theory was tested using first-stage fan rotor blades that were fluttering in the first flexural mode (1F) and forming the second nodal diameter (2ND). The approach’s estimated blade deflection was compared to measurements taken using a traditional tip timing method up to 7 mm and determined to be within 1% inaccuracy. This research provides a low-cost, easy alternative technique for measuring flutter during engine development testing.

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

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

Structural Optimization of Vented Brake Rotors with a Fully Parameterized Model

Vented brake rotors used in an automobile behave similarly to centrifugal fans, drawing cool air from the inboard side, passing it through the disc vents, and exhausting it from the periphery. A vented brake rotor with a better heat dispersing ability is often superior to a solid rotor, in both thermal performance and brake efficiency. In this research, a fully parameterized model for a ventilated brake rotor is created using the ANSYS Parametric Design Language, to uniquely define the rotor’s geometry. With this parameterized model, two structural optimization cases are studied in this paper. The first one investigated is a modal frequency separation problem: The frequency differences in a tangential mode sandwiched between two nodal diameter modes of the brake rotor model are maximized. An automatic identification scheme for extracting correct mode orders is implemented in the program to track the correct modes during optimization. The second case is a thermal deformation problem: The distortion on the frictional surfaces of the rotor loaded with heat flux generated during the braking process is minimized. The optimization results show that a brake rotor design with a thinner outboard disc and a thicker inboard disc provides a great choice for rotor coning reduction.

Aus rice root architecture variation contributing to grain yield under drought suggests a key role of nodal root diameter class.

The aus rice variety group originated in stress-prone regions and is a promising source for the development of new stress-tolerant rice cultivars. In this study, an aus panel (~220 genotypes) was evaluated in field trials under well-watered and drought conditions and in the greenhouse (basket, herbicideand lysimeter studies) to investigate relationships between grain yield and root architecture, and to identify component root traits behind the composite trait of deep root growth. In the field trials, high and stable grain yield was positively related to high and stable deep root growth (r = 0.16), which may indicate response to within-season soil moisture fluctuations (i.e., plasticity). When dissecting component traits related to deep root growth (including angle, elongationand branching), the number of nodal roots classified as 'large-diameter' was positively related to deep root growth (r = 0.24), and showed the highest number of colocated genome-wide association study (GWAS) peaks with grain yield under drought. The role of large-diameter nodal roots in deep root growth may be related to their branching potential. Two candidate loci that colocated for yield and root traits were identified that showed distinct haplotype distributions between contrasting yield/stability groups and could be good candidates to contribute to rice improvement.

Coupling parametric instability and resonance of a rotating flexible disk with unstressed initial runout under several parametrical excitations

With an improved Hill’s method, which is suitable for harmonic balance processing and coupling of nodal circles and nodal diameters, a dynamics research of a rotating flexible disk is carried out by considering unstressed initial runout and several parametrical excitations. The partial differential equation of disk system is obtained by Galerkin’s method considering three different types of excitations, namely slider loading, speed fluctuation and electromagnetic attraction. The individual and coupling effects of parametrical excitations on system’s instability and resonance are studied via numerical simulation. The instability results are shown by different colors based on RGB color space, and the disk’s steady-state amplitudes under different parameter settings are calculated with different nodal circles and nodal diameters. The calculation results show that these excitations affect the disk system’s stability and resonance in their own way. Meanwhile, this method is universal for the flexible disk’s steady-state vibration problems in practical engineering analysis.

Diagnostic value of LungPoint navigation combined with EBUS-GS & ROSE in peripheral pulmonary nodules.

Recently, there has been a surge to develop new devices and techniques for the diagnosis of peripheral pulmonary lesions such as the combination of LungPoint navigation and endobronchial ultrasound with a guide sheath (EBUS-GS). The present study aimed to explore the diagnostic value of LungPoint navigation in combination with EBUS-GS and rapid on-site evaluation (ROSE) particularly for peripheral pulmonary nodules. Patients (n=108) with pulmonary nodules (10 mm ≤ nodal diameter ≤30 mm) presenting to Henan Provincial People's Hospital were detected using chest computed tomographic (CT) scanning and bronchoscopy. All patients were evaluated using LungPoint navigation, EBUS-GS and ROSE techniques to evaluate the positive rate of combined diagnosis using the three methods. A total of 108 patients participated in this study and successfully underwent all the three procedures. Of these, 82 patients were accurately diagnosed, making the overall diagnostic rate of 75.9 per cent for combined LungPoint navigation, EBUS-GS, and ROSE analyses. Further subgroup analysis of the diagnostic rate of the three combined techniques were conducted based on the size of the nodules which showed a diagnostic rate of 65.3 per cent for 10 mm ≤ nodule diameter ≤20 mm and 85.7 per cent for 20 mm ≤ nodal diameter ≤30 mm. Of the 108 patients, 85 had solid nodules and 23 had ground-glass nodules; the positive rate of diagnosis of solid nodules was the highest. The patients ultimately were diagnosed with lung cancer with a positive rate of 83.5 per cent. The sensitivity, specificity and positive and negative predicted values for ROSE were 90.3, 78.3, 84.8 and 83.6 per cent, respectively. The combined use of the three techniques can effectively shorten the duration of the total diagnosis period and improve the safety of diagnosis without affecting the detection rate.

Using silicon disk resonators to measure mechanical losses caused by an electric field.

Several projects of the next generation gravitational-wave detectors use the high purity monocrystalline silicon test masses. The electric field of the actuator that is applied to correct the position of the silicon test mass causes additional mechanical losses and associated noise. Disk mechanical resonators are widely used to study mechanical losses in multilayer optical coatings that are deposited on the test masses of gravitational-wave detectors. We use silicon disk resonators to study losses caused by an electric field. In particular, the dependence of mechanical losses on the resistivity of silicon is investigated. The resonator is a thin commercial silicon wafer in which a low frequency nodal diameter mode is excited. A DC voltage is applied between the wafer and a nearby electrode. We use two measurement configurations. In the first configuration, the dependence of losses on the resistance in the voltage supply circuit is investigated. The dependence of losses on the resistivity of silicon is investigated in the second configuration. We propose a model that relates the electric field induced mechanical loss in disk resonators to the resistivity of the material. Measurements are carried out for low and high resistivity silicon wafers. The measurement results are compared with calculations. Based on these studies, it is possible to estimate the loss and noise of the test masses of gravitational-wave detectors associated with electrostatic actuators.

Coupling vibration characteristics of the shaft-disk-drum rotor system with bolted joints

In this paper, a finite element (FE) model for rotating-flexible shaft-disk-drum (SDR) system with bolted joints is established based on beam-shell-spring hybrid elements by in-house codes. Rotation-induced effects, that is, centrifugal effect, initial tension, gyroscopic moments, and geometric non-linearity due to large deformation are taken into account. Besides, fixed interface mode synthesis method is applied to improve solving efficiency. The proposed model is verified by the results from literature and experiments. Finally, the effects of the number of bolts, rotating speed, connecting stiffness, and the location of disk-drum on the natural characteristics of SDR system are investigated. The main conclusions are summarized as follows: (1) Modes dominated by different components have different sensitivities to the variations of the number of bolts. (2) The analysis of Campbell diagram indicates that frequency veering phenomenon can be found among the shaft bending (SB) mode, disk transverse (DT) mode, disk swing (DS) mode and drum one-circumferential wave (1CW) mode in low-order modes, and the disk nodal diameter (ND) mode always couples with the drum CW mode in high-order modes. (3) The change of axial connecting stiffness will cause the coupling vibration between the disk and drum, while the change of transverse shear stiffness not only causes the coupling vibration between the disk and the drum, but also the coupling vibration among the shaft, disk and drum. (4) The change of disk-drum location can affect the modes associated with the shaft SB and cause frequency veering phenomena. The disk ND modes and the drum CW modes are not coupled with the shaft SB modes when the number of ND and CW are greater than 2.

Numerical investigations on non-synchronous vibration and frequency lock-in of low-pressure steam turbine last stage

The flow phenomenon of rotating instability (RI) and its induced non-synchronous vibrations (NSV) in the last stage have gradually become a security problem that restricts the long-term flexible operations of modern large-scaled low-pressure steam turbines. Especially, if one structural mode of the last stage moving blade (LSMB) is excited, significant blade vibrations may potentially lead to high-cycle fatigue failure. A loosely coupled computational fluid dynamics reduced model with prescribed blade vibrations has been established to investigate NSV of the LSMB and the potential lock-in phenomenon under low-load conditions. Firstly, calculations with reduced multi-passage domain have been verified by comparing with the results of the full-annulus one, and an appropriate reduced domain is determined. Secondly, a set of calculations by controlling blade vibration parameters indicate that lock-in phenomenon between RI frequency and blade vibration frequency may occur when nodal diameters of cascade vibrations is coincident with the wave number of RI. Furthermore, dynamic modal decomposition technology has been employed to identify the unsteady pressure field around the blade surface and to reveal the interaction relationship between the flow modes of RI and vibration-induced pressure disturbance. Finally, the blade response evaluation based on harmonic analysis shows that in NSV, the global maximum dynamic response level of locked-in case is nearly 20 times than that of unlocked one.

Stress analysis and thermoelastic instability of an annular functionally graded rotating disk

The in-plane behavior, linearized transverse vibration, and stability of a thin heated Functionally Graded (FG) rotating annular disk are investigated. The inner edge of the disk is clamped and heated to a constant temperature. The outer edge of the disk is free. A convective heat transfer is considered acting on the disk due to rotation in an environment with still air. The coefficient of thermal expansion, thermal conductivity, and Young’s modulus vary radially according to the same power law. Assuming linear in-plane strains, the in-plane stresses are obtained and used to solve the linear equation of transverse motion to determine the critical speeds and stability characteristics. The results show that both the material property distribution and the heat convection coefficient have a significant effect on both the critical speed and stability characteristics. Moreover, heating a rotating FG disk can cause the modes with higher number of nodal diameters be excited before modes with lower ones. A convective coolant heat transfer acts as a strong stabilizer for the disk. Results suggests that an FG rotating disk with material properties increasing from the center is favorable to keep the disk stable and stress levels low if a continuous air supply is provided.

Nonlinear dynamic analysis of three-dimensional bladed-disks with frictional contact interfaces based on cyclic reduction strategies

In bladed disks, friction nonlinearities occurring within the contact between the blades and disk or between the blades and underplatform dampers are used to decrease the vibratory energy of the system and extend its lifespan. Modeling these nonlinearities and simulating their effects correctly are challenging: complex nonlinear phenomena such as stick, slip and separation may occur in the contact zones. Efficient numerical methods must be used to compute the dynamics of the system in a reasonable amount of time. This paper proposes a reduction strategy to tackle large cyclically symmetric finite-element models undergoing static preload from centrifugal effects and strong nonlinearities (friction and separation). It combines the nonlinear identification of the possible interacting nodal diameters with a linear component mode synthesis procedure. Its potential and performances are assessed through comparisons with some state-of-the-art methods. Complex and realistic nonlinear finite-element models of bladed disks with underplatform dampers and dovetail or fir-tree blade roots are used. The Dynamic Lagrangian Frequency Time algorithm is employed to capture the nonlinear effects. Using the proposed reduction method, the effect of underplatform dampers on bladed disk contact occurrence and damping efficiency is investigated.

Evaluating the Rationality of the International Guideline's Recommendations Regarding Selective Coverage of Level Ib in Node Clinical Target Volume With Nasopharyngeal Carcinoma: Results From a Real-World Series

<h3>Purpose/Objective(s)</h3> Although the International Guideline for the delineation of clinical target volume (CTV) in nasopharyngeal carcinoma (NPC) patients has provided practical reference for the elective coverage of level 1b, there are marked variations in practice among different institutions, and the rationality of the recommendations remained proven. The purpose of this study was to evaluate its rationality in real-world data and provide clinical evidence for the refinement of nodal CTV delineation in nasopharyngeal carcinoma (NPC). <h3>Materials/Methods</h3> Patients with histologically-proven NPC who received definitive Intensity-modulated radiation therapy (IMRT) with or without chemotherapy at Dr. SJ Lin's attending group in our institution between June 2005 and December 2012 were candidates for this retrospectively study. Other main eligible criteria including the following five situations: 1) level Ib metastasis;2)involvement of the submandibular gland;3) involvement of structures that drain to level Ib as the first echelon site (the oral cavity, anterior half of nasal cavity involvement);4) involvement of level II LNs with extracapsular extension (ECE); 5) level II nodal involvement with maximum nodal axial diameter (MAD) greater than 2 cm. Survival outcomes focused on regional recurrence-free survival (RRFS) and incidence rate of Ib recurrence were analyzed. <h3>Results</h3> A total of 451 patients were finally included according to the above criteria. There were 27, 37, 6, 12, 285 and 414 cases be classified as level Ib metastasis, submandibular gland involvement, oral cavity involvement, anterior half of nasal cavity involvement, level II LNs involvement with ECE and level II nodal involvement with MAD greater than 2 cm, respectively. Among these 451 patients, 60 patients received level Ib-covering IMRT due to the first three principles of the International Guideline according to our attending group's protocol. Of note, patients who had Ib involvement would receive ultrasound guided puncture, only those who had positive pathological results would undergo Ib-covering IMRT in our attending group. For the remaining 391 patients who only fulfilled the last two criteria and/or Ib involvement with negative pathological results, they all received level Ib-sparing IMRT, reported 6-year RRFS was 94.3%. For the 22 patients who had regional recurrence at censorship, only 3 (3/391, 0.78%) were recorded as Ib recurrence. <h3>Conclusion</h3> Level Ib-sparing IMRT should be safe and feasible for patients who only had involvement of level II LNs with ECE, and/or had a MAD of greater than 2 cm in level II LNs, and/or Ib involvement with negative pathological results. Further well-designed multi-center prospective trials should be conducted to confirm our results.

An Evaluation Method for Dry Friction Damping of Ring Damper in Gas Turbine Engines under Axial Vibration

The blisks and labyrinth seals in gas turbine engines are typical rotating periodic structures. Vibration problems will inevitably occur during the operation, which can easily lead to High Cycle Fatigue failure of the structure. Adding ring damper is an effective means of structural vibration reduction. The damper uses the dry friction of the contact surface to dissipate the vibration energy, improve the damping of the system, and then reduce the vibration response of the structure. The structures have a nodal diameter and modal shape, and the forced vibration often presents the characteristics of traveling wave. In this paper, an evaluation method for dry friction damping of ring damper under the axial component of traveling wave vibration is established. For the given vibration stress at the critical location, the equivalent damping ratio provided by the ring damper is calculated based on the friction energy dissipation and the damping characteristic curve that is the equivalent damping ratio varying with the vibration stress is obtained. This method can avoid calculating the nonlinear dry friction forced response and is suitable for the design stage. The damping of split ring dampers with rectangular section for one blisk and labyrinth seal is analyzed in this paper. It is shown that rotating speed, friction coefficient, section area and material density significantly influence the damping characteristics. There are many factors affecting the damping characteristics of the damping, so it is necessary to comprehensively consider various factors and multiple modes for vibration reduction design.

Investigation on the Robustness of Rotor/Stator Contact Interactions With Respect to Small Mistuning

Abstract In this work, the impact of small mistuning on rotor/stator contact interactions is investigated. First, a detailed study of a rotor/stator interaction between the first bending modes and the second engine order is presented in the tuned case. Then, a numerical investigation on the effect of mistuning on the studied rotor/stator contact interaction is carried out. In particular, a stochastic analysis is performed to evaluate the robustness of the interaction with respect to the mistuning level. Simulations are conducted using a reducedo rder model (ROM) of an industrial-bladed disk that combines both physical degrees-of-freedom (along blades tip for contact treatment) and modal coordinates. Mistuning is introduced in the tuned ROM by means of a modified version of the component mode mistuning method that allows to keep physical degrees-of-freedom within the reduced basis. Nonlinear amplification factors, i.e., the amplification factors in the context of contact nonlinearities, are compared with their linear counterparts, and the latter are computed using a linear forcing on each blade using a two nodal diameters traveling wave excitation on the mistuned and the tuned bladed disk. The comparison between the linear and nonlinear amplification factor for each sample highlights that no correlation exists between a mistuning pattern leading to high amplifications in a linear context or when contact nonlinearities are taken into account. Therefore, dedicated analyses on the effect of mistuning should be undertaken with contact nonlinearities considerations at the design stage especially if intentional mistuning is considered.

A Study on Mode Shape and Natural Frequency of Rotating Flexible Cracked Annular Thin Disk

As an important rotating component, the flexible annular thin disk is widely used in mechanical engineering. Cracks may occur in some weak disk parts, which will greatly shorten the equipment service life and even cause equipment failure. Due to the centrosymmetric structure of the flexible annular disk, two typical cracks are studied in this paper; one is radial crack parallel to diameter, including radial closed crack (RC-crack) and radial opening crack (RO-crack); the other one is vertical crack perpendicular to diameter, including circumferential crack (CF-crack) and tangential crack (TG-crack). The effect of crack parameters, such as crack length, direction, and position, on disk vibration characteristics are studied through theoretical simulation and experimental verification. The research shows that the effect of cracks on vibration characteristic gets more obvious with cracks extending in most cases, RO-crack decreases the natural frequency obviously, and vertical cracks would affect mode shapes. In addition, the bigger the nodal diameter is, the more obvious the effect gets. Meanwhile, the most obvious effect appears in the mode of a nodal diameter locating on the crack. The research possesses some guiding significance in industrial production; by comparing with the vibration characteristics of the flawless disk, the integrity of the rotating flexible disk can be judged to prevent possible equipment damage.

116 PROGNOSTIC IMPACT OF THE SHORT-AXIS DIAMETER FOR THE NODAL ASSESSMENT BEFORE PREOPERATIVE CHEMOTHERAPY IN ESOPHAGEAL CANCER

Abstract Clinical N category (cN) is generally assessed by measuring the nodal diameter by CT before the initiation of primary treatment in esophageal squamous cell carcinoma (ESCC). The short-axis diameter is recommended for evaluating treatment response in solid tumors by RECIST. This study aimed to elucidate the prognostic implication of the maximum short-axis diameter of lymph node (cN-size) before preoperative chemotherapy for ESCC. Methods We enrolled a total of 152 patients who underwent preoperative cisplatin/5-fluorouracil therapy (CF) followed by esophagectomy from 2005 to 2011. There were 127 men and 25 women with a median age of 65 years (range: 47–79 years). Clinically metastatic node was defined as follows; the node with cN-size ≥10 mm or that with 5 mm ≤ cN-size &amp;lt;10 mm and contrast enhancement, round shape and/or central necrosis in CT before starting CF. The association between the maximum cN-size and the overall survival (OS) after surgery was statistically investigated. The median follow-up period was 87 months (range: 36–145 months). Results The number of patients with cN0 and cN1–3 was 60 and 92, respectively. Twenty-seven and 65 patients with cN1–3 were classified into cN-size &amp;lt;10 mm and cN-size ≥10 mm group, respectively. The 5-year OS rates in cN0, cN-size &amp;lt;10 mm and cN-size ≥10 mm groups were 70%, 51% and 45%, respectively (P = 0.006). Among Ut-Mt tumors, the OS in the cN-size &amp;lt;10 mm group was significantly worse than that in the cN0 group (5-year OS rate: 45% vs. 74%, P = 0.048). However, there were no significant differences in the OS between these two groups in Lt tumors (67% vs. 64%, P = 0.789). Conclusion The maximum short-axis diameter of lymph node before preoperative chemotherapy is significantly associated with OS in patients with ESCC. Lymph node with 5 mm ≤ cN-size &amp;lt;10 mm in the short axis should be treated as a metastatic node especially in Ut or Mt tumors, considering the poor prognosis.

Clinical Analysis of Video-Assisted Thoracoscopic Surgery for Resection of Solitary Pulmonary Nodules and Influencing Factors in the Diagnosis of Benign and Malignant Nodules.

Purpose This is a retrospective research comparing the clinical outcomes of single-hole versus multi-hole video-assisted thoracoscopic surgical (VATS) resection for solitary pulmonary nodules (SPN) and examining the factors influencing the diagnosis of benign and malignant pulmonary nodules. Method We collected the clinical data, surgical status, outcomes, and corresponding imaging features of 317 patients with SPN who were surgically resected by VATS and diagnosed as benign or malignant by pathology in our hospital from January 2019 to December 2021. Result Among the 317 patients, 124 (39.12%) underwent single-port VATS and 193 (60.88%) underwent multiple-hole VATS. All patients were grouped according to the different surgical methods, and their postoperative indicators were statistically analyzed. The results showed that neither the single-port VATS group nor the multi-port VATS group had any serious adverse events such as death during the perioperative period. The average operation time, intraoperative blood loss, drainage tube indwelling time, and postoperative hospital stay were significantly lower in the two groups. Statistics of postoperative pathological diagnosis showed that 98 cases (30.91%) of all nodules were benign nodules and 219 cases (69.09%) were malignant nodules, and a further single-multivariate analysis showed that age, nodule maximum diameter, lobular sign, burr sign, vascular cluster sign, and pleural depression sign were independent relevant factors for the diagnosis of benign and malignant nodules. Conclusion VATS is less invasive and has fewer complications and is of great clinical value for both diagnosis and treatment of benign and malignant SPN. Age, maximum nodal diameter, lobar sign, burr sign, vascular set sign, and pleural depression sign were independent correlates affecting the diagnosis of benign and malignant SPN, which reminds that great attention should be paid to patients who are older and have risk factors on imaging, and early and timely active treatment or close follow-up should be carried out.

Harmonic Component Analysis of Mode Shapes for Packeted Bladed Disk

Vibration characteristics of packeted bladed disk is complicated. To study the vibration characteristics of the governing stage bladed disk of 660mw steam turbine, cyclic symmetry method is used to calculate the mode shapes. Fourier transformation of vibration distribution in the tangential and axial direction along the circumferential of the bladed disk is applied to obtain the harmonic components. It is found that the Fourier decomposition of each blade displacement contains not only unique component, but also other components. For integrally shrouded blades, the resonance of the m nodal diameter mode can be obtained only when the corresponding natural frequency satisfies the condition ωn = kn = mn, where n is the rotational speed and m is the harmonic number. For packeted bladed disk, there are additional harmonics that resonance occurs. The harmonic components are related to the nodal diameter number when the number of blades groups and the total number of blades are given. The diagrammatic method is applied to predict the wavenumbers present in modulated eigenfunctions.

Vibration analysis of irregular-shaped plates on simple supports

In this work, we propose a general perturbative approach for modal analysis of irregular-shaped plates of uniform thickness with uniform boundary conditions. Given a plate of irregular boundary, first, a uniform circular plate of identical thickness and area, centred at the centroid, is determined. The irregular boundary is then treated as a perturbation with a suitable smallness parameter, and is expressed as a generalized Fourier series. The frequency parameter, shape function and boundary conditions are then perturbed in terms of the smallness parameter. The homogeneous zeroth-order equation corresponds to the circular plate, which is exactly solvable. We show that the inhomogeneous equations in the higher orders can also be solved exactly using a particular solution structure. We can then construct the exact perturbative solution up to any order. The proposed method is demonstrated through the modal analysis of simply supported super-circular plates. The results are validated using the numerical results obtained from ANSYS ® , which are an excellent match. Interestingly, the supposedly degenerate modes with an even number of nodal diameters of super-circular plates are found to split naturally.

A flexible-helical-geared rotor dynamic model based on hybrid beam-shell elements

Considering the flexibility of gear foundations and rotational effects, a geared rotor dynamic model for thin-rimmed helical gears is established. The deformable shafts and gear foundations are modelled by Timoshenko beam elements and Mindlin-Reissner shell elements, respectively. The beam element and the shell element are connected using the ‘interfacial coupling element’. Then the modal superposition method is adopted to improve the computational efficiency. The proposed dynamic model is verified through the comparison between the simulated responses and experimental results in a published literature. Based on the proposed dynamic model, the effects of the web thickness, the rotational effects and the helix angle on the dynamic behaviors are discussed. The results show that the dynamic responses of thin-rimmed gears obtained from the traditional dynamic model and the proposed model are quite different, which emphasizes the significances of the flexible body modeling. Compared to the centrifugal stiffening and the spin softening, the gyroscopic effect has more significant effect on dynamic characteristics. The nodal diameter vibration cannot be excited due to the absence of the axial exciting force when helix angle is zero. The increase of the helix angle will lead to more significant out-of-plane vibration of the gear foundation.