Mode Synthesis Research Articles

Free vibration analysis of a rotating graphene nanoplatelet reinforced pre-twist blade-disk assembly with a setting angle

This paper investigates the coupled modeling method and free vibration characteristics of a graphene nanoplatelet (GPL) reinforced blade-disk rotor system in which the blade has a pre-twist angle and setting angle. Based on the Kirchhoff plate theory and the Euler–Bernoulli beam theory, the theoretical model of the rotating blade-disk assembly is carried out accurately. The effective material properties of the blade and disk are assumed to vary continually along their thickness directions and are determined via the Halpin–Tsai micromechanics model together with the rule of mixture. The coupled equations of motion are derived in accordance with the Lagrange's equation. Then, the substructure modal synthesis method and the Galerkin method are employed to calculate the free vibration results of the rotor. Additionally, the effects of the rotating speed, GPL distribution pattern, GPL weight fraction, length-to-thickness ratio and length-to-width ratio of GPLs, inner and outer radius of the disk, and pre-twist angle per unit length, setting angle and length of the blade on free vibration characteristics of the rotating blade-shaft assembly are discussed in detail.

Fast structural dynamic modeling and analysis for horizontally folding wing

To investigate the structural dynamic characteristics of a folding wing effectively, a fast structural dynamic modeling approach is proposed. Firstly, the interface compatible relationship of the traditional fixed interface component modal synthesis method is modified, and the internal force of the interface is completely expressed in the structural dynamic equation, so that the influence of the connection stiffness on the wing structure dynamics can be considered. Then, on the basis of the fixed interface component modal synthesis method, the main mode of fixed-loaded interface is introduced to establish the mixed-loaded interface component modal synthesis method, which makes it feasible to accurately reflect the influence of elasticity and inertia of fuselage and outer wing on inner wing. The structural dynamics modeling method based on two different kinds of component modal synthesis method analyzed and deduced in detail. The application of component modal synthesis method in the fast structural dynamics modeling of folding wing is achieved. The whole program is compiled in MATLAB. At the same time, the dynamic characteristics of the folding wing with different folding angles, different connections and different connection positions is investigated. The results of the method proposed in this paper are compared with the results of the repeated finite model established in MSC.NASTRAN to verify the effectiveness from the aspects of natural frequency and vibration mode.

Newton’s iterative formula for finding an equation solution as an abstract problem of the modal synthesis theory

A new approach to obtain an iterative Newton formula for finding an equation solution, by using modal control theory for linear discrete systems when solving problems of observation or identification is presented. The decomposition method as a modal control method, which allows obtaining analytical solutions, is used. Keywords Newton’s iterative formula; numerical solution of the equation; decomposition method of modal synthesis; linear discrete system

Performances of the double modal synthesis for the prediction of the transient self-sustained vibration and squeal noise

This paper presents an efficient reduction method for predicting the nonlinear transient and steady state squeal events in mechanical systems subjected to friction-induced vibration and noise. This proposed reduction technique is based on the Double Modal Synthesis (DMS) method that involves the use of a classical Craig & Bampton modal reduction on each substructure considering the interface surfaces associated to a condensation at the frictional interface based on complex modes.In this paper, the performances of some reduced bases based on the DMS strategy are investigated in the case of a finite element model of a simplified disc/pads system. The originality of the present work is to propose a comprehensive study on the convergence of the DMS method in order to predict not only the stability or the limit cycles of a simplified brake system but also the transient nonlinear self-excited vibrations, as well as the squeal noise. A special attention is brought to the convergence of the DMS method and more precisely the number of interfaces modes required to provide satisfactory results in regard to various criterion used to characterize the squeal.

Prediction of Squeal Noise Based on Multiresolution Signal Decomposition and Wavelet Representation—Application to FEM Brake Systems Subjected to Friction-Induced Vibration

This paper is devoted to discussion of the efficiency of reduced models based on a Double Modal Synthesis method that combines a classical modal reduction and a condensation at the frictional interfaces by computing a reduced complex mode basis, for the prediction of squeal noise of mechanical systems subjected to friction-induced vibration. More specifically, the use of the multiresolution signal decomposition of acoustic radiation and wavelet representation will be proposed to analyze details of a pattern on different observation scales ranging from the pixel to the size of the complete acoustic pattern. Based on this approach and the definition of specific resulting criteria, it is possible to quantify the differences in the representation of the acoustic fields for different reduced models and thus to perform convergence studies for different scales of representation in order to evaluate the potential of reduced models. The effectiveness of the proposed approach is tested on the finite element model of a simplified brake system that is composed of a disc and two pads. The contact is modeled by introducing contact elements at the two friction interfaces with the classical Coulomb law and a constant friction coefficient. It is demonstrated that the new proposed criteria, based on multiresolution signal decomposition, allow us to provide satisfactory results for the choice of an efficient reduced model for predicting acoustic radiation due to squeal noise.

Topology Optimization for Acoustic–Structure Interaction Systems Using the Model Reduction Method

This paper develops a vibro-acoustic component modal synthesis method to solve the problem of high computation time in solving the eigenvalues and eigenvectors of coupled equations for noise reduction topology optimization. The normalization condition of the left and right eigenvectors of the unsymmetrical vibro-acoustic equation is introduced, and Craig–Bampton method is extended to the structure-acoustic coupled problem, which reduces the time consumed by the response calculation part. The results show that the eigenvalues, the eigenvectors, and the response values calculated by the proposed method match well with the original model. To solve the problem of spurious eigenmodes caused by the traditional penalization model used in the classical bi-directional evolutionary optimization (BESO) method, the linear stiffness penalization model is introduced into the BESO method, and an improved penalization model is proposed. Numerical examples of minimizing the response at a specified target point are studied. The results show that the iterative process is robust. Through these numerical design examples, the effectiveness and reliability of the model reduction method and the improved penalization model are verified.

Vibration characteristics investigation of mistuned blisks with receptance substructure component modal synthesis method

To improve the simulation efficiency of mistuned blisk, a method called receptance substructure component modal synthesis method (RSCMSM) is proposed to reduce its degrees of freedom (DOFs). The advantage of this method is that only the interface DOFs need to be solved, which observably enhances the computational efficiency. The modal frequencies, maximum modal shape and frequency response function are calculated via RSCMSM. It is seen that the smooth frequency band is governed by blade vibration and the steep frequency band is governed by disk or bladed-disk coupling vibration. In addition, a peak is observed for the tuned blisk but many peaks appear for the mistuned blisk and many small wave crests are observed near the peak. To verify validity of this method, the computational time of RSCMSM is compared with high fidelity finite element method (HFFEM) and classical substructure component modal synthesis method (CSCMSM), which manifests that the computational efficiency increases by 32.19 %–80.82 % than that of HFFEM when the mistuned level is 0 %~5 %. Moreover, computational efficiency of RSCMSM is increased by 0.85 %–7.56 % than that of CSCMSM. The validity of RSCMSM is verified for calculating the complex mechanical structure.

Nonlinear aeroelastic analysis of the folding fin with freeplay under thermal environment

The nonlinear aeroelastic behavior of a folding fin in supersonic flow is investigated in this paper. The finite element model of the fin is established and the deployable hinges are represented by three torsion springs with the freeplay nonlinearity. The aerodynamic grid point is assumed to be at the center of each aerodynamic box for simplicity. The aerodynamic governing equation is given by using the infinite plate spline method and the modified linear piston theory. An improved fixed-interface modal synthesis method, which can reduce the rigid connections at the interface, is developed to save the problem size and computation time. The uniform temperature field is applied to create the thermal environment. For the linear flutter analyses, the flutter speed increases first and then decreases with the rise of the hinge stiffness due to the change of the flutter coupling mechanism. For the nonlinear analyses, a larger freeplay angle results in a higher vibration divergent speed. Two different types of limit cycle oscillations and a multiperiodic motion are observed in the wide range of airspeed under the linear flutter boundary. The linear flutter speed shows a slight descend in the thermal environment, but the effect of the temperature on the vibration divergent speed is different under different hinge stiffnesses when there exists freeplay.

METHOD OF UNCERTAIN COEFFICIENTS IN PROBLEMS OF OPTIMAL STABILIZATION OF TECHNOLOGICAL PROCESSES

Context. The equivalent transformation method is examined in the given article. Its essence lies in changing of a certain class of non-stationary systems with the stationary ones, for which optimization methods are well processed. Urgency of the method is determined by the fact that in most optimal control methods, developed for continuous systems, tasks are considered in the temporary space using the states space and the matrix theory. All real control objects are known to be non-linear and non-stationary in one way or another. Analysis and synthesis of control systems for such objects is a complex mathematical issue, and its solution is received for some separate occasions for now. As a result of using the suggested method, when the variable coefficients matrix is known, the task of the non-stationary system optimal control is reduced to the task of the equivalent stationary system optimal control for which solution methods are well-known and well processed.Objective. Reducing energy intensity and improving the quality of products of various technological processes is an urgent task of the national economy of Ukraine.Methods. To achieve this goal, we propose a method of modal synthesis of optimal stabilization laws using the method of uncertain coefficients, developed by the authorsResults. Algorithm of synthesis of the optimal controller in the absence and presence of delay in the control loop is developed. The method of selection and correction of the desired spectrum of roots is proposed. To eliminate self-oscillations in the presence of a delay in the control circuit, the R. Bass method is used.Conclusions. The modal synthesis of optimal laws of stabilization of technological processes is proposed on the basis of the original method of uncertain coefficients. The complexity of the choice of the desired eigenvalues is overcome by the proposed procedure of construction and correction of the spectrum of roots in a closed system of optimal control. To eliminate the occurrence of stable self-oscillations (in the presence of a delay) in the stabilization process near a given trajectory, the Bass’s method is proposed to be used. The simulation results confirm the correctness and effectiveness of the results.

Research on muzzle response characteristics of automatic gun on launching based on rigid-flexible coupling dynamics

In order to study the important influence for firing dispersion of muzzle response of automatic gun on launching, the rigid-flexible coupling dynamics model of automatic gun was established based on the rigid-flexible coupling system dynamics theory in software RecurDyn. And the flexible body of gun barrel assembly was set up by the modal synthesis method of dynamic sub-structure. The Hertz contact theory and flexible Bushing forces model were made equivalent to these constrained joints. According to the characteristic of automatic gun structure and continuous launching load, the dynamic vibrating response of simulation and analysis was made for the automatic gun on launching. The muzzle vibration response displacements in vertical and horizontal direction were obtained in the 300 rate of fire. Then the muzzle vibration velocity and acceleration with different structure clearance were also calculated by the finite element analysis program of the rigid-flexible coupling dynamics model of automatic gun. Through the comparison and analysis, the numerical solutions are closely matched with the experimental results. The results show that the vertical angle error of firing dispersion is less than 3.8% and the horizontal angle error is less than 10.2%. Thus, the simulation and experimental results verify the correctness and feasibility of this model. Further the structure design and optimization of automatic gun can be provided a reference foundation.

Dynamic Modeling of Gear System Based on 3D Finite Element Model and Its Application in Spalling Fault Analysis

A reduced-order dynamic model, based on three-dimensional (3D) finite element model (FEM) and component modal synthesis technique (CMS), was presented for simulating the dynamic behavior of the spur gear system. The gear shaft and gear body were established via 3D elements to simulate bending and torsion of the gear system. The CMS technique was used to generate a reduced-order model of a spur gear system. A pair of mating teeth was assimilated to two different foundations (one for the pinion tooth and the other one for the gear tooth) linked in series by some independent springs, which was used to simulate the contact stiffness. The validity of the proposed model was verified by static analysis, dynamic analysis, and experimental analysis. The results show that the proposed model is an effective model. In addition, the proposed model has also been applied to analyze spur gear spalling faults. The results show that the dynamic response of the gear system is periodic vibration shock response due to the alternate meshing of single and double teeth. When the spalling fault occurs, some shock responses with significantly enhanced amplitude will be generated as the result of contact loss.

An iterative state-space component modal synthesis approach for damped systems

Efficiently calculating the high precision modal parameters and/or reduce-order model of large-scale and/or complicated damped structures are important in engineering applications such as optimal design, model modification and updating tasks. In this paper, an iterative procedure based state-space component modal synthesis method is developed to address the problem. The Kron’s substructuring method is modified into a state-space form first to include damping effects and the defectiveness of components and global system. Then an iterative scheme is developed to seek all desired complex eigen-pairs simultaneously in one round of iterations without complex operations. The consistency of the proposed method is proved and the convergency is briefly discussed for integrity. A detailed implementation is given for reference. Compared with other methods for solving modal parameters and/or model order reduction, the proposed method has such merits as high computational efficiency and still in a substructuring scheme. Numerical examples show that the proposed method can provide highly accurate results with low computational cost.

Numerical investigation of the seismic response of a polar crane based on linear complementarity formulation

This paper numerically analyzes the seismic response of a polar crane operating at a nuclear plant, with derailment considered. In this case, the wheels and rails can no longer be treated as being bonded, and thus traditional methods in structural dynamics (e.g. response spectrum method) are no longer suitable for transient contact analysis. On the other hand, contact-impact analysis of flexible multibody systems will often face great difficulties attributed to the complexity of modeling and low computational efficiency. To overcome these problems, the wheel-rail interactions are modeled as multiple frictional unilateral constraints, and the resulting contact-impact problem is formulated as a linear complementarity problem (LCP). Later, a novel smoothing method is applied to smooth the LCP formulation by replacing the normal gap with the time-averaged normal gap over a short time period. A model smoothing method is then combined with modal synthesis to derive the equations of motion without high frequency components. Finally, the validity and feasibility of the proposed method are demonstrated through several numerical examples. The results show the different contact states during the seismic process, including smooth contact, detachment and re-contact.

Sound model of an orchestral kettledrum considering viscoelastic effects

The modeling of the mechanical systems that describe musical instruments has long been a topic of interest for acoustic physicists. The orchestral kettledrum, distinguished by a broad spectrum of sounds, has been the subject of multiple investigations related to the goal of generating a realistic digital sound synthesis. In the present work, we apply the Green function method to estimate modal frequencies considering an air-loaded viscoelastic membrane. We propose a method that includes viscoelasticity to accurately predict the sound spectrum of the modeled kettledrum. Results are compared to real sound recordings of an orchestral kettledrum obtained in controlled conditions. The calculated modal frequencies are found to coincide well with the real values with an absolute mean error of 1.25±0.76 Hz and 1.87±1.83 Hz for the A2 and B2♭ tuned drumheads respectively. The spectral envelope of the synthesized sound spectrum coincides well with the Fourier transform of the real sound. The viscoelastic term was found to generally reduce the amplitude of the sound spectrum and in certain cases, better approximate the modal frequencies and decay times. The modal synthesis method used here is numerically lightweight and can be adapted to be used in real-time applications with low computational resources. To reproduce our experiment, the recorded kettledrum sounds and Python source code of the model are freely available.

Application of extremum response surface method-based improved substructure component modal synthesis in mistuned turbine bladed disk

To improve the computational efficiency of vibration characteristics and reliability analysis for a detailed numerical model of the mistuned turbine bladed disk, a new methodology called extremum response surface method-based improved substructural component modal synthesis (ERSM-ISCMS) is proposed by combining the ISCMS and ERSM. First, the degrees of freedom of the detailed finite element model for the numerical mistuned turbine bladed disk are decreased by ISCMS, which is called as reduced-order model. Compared with high fidelity finite element model, the time saving ratio and the computational accuracy of the first 40 order frequencies are, respectively, 36.37% and 99.99%~99.86% obtained by ISCMS under the same working environment, which can satisfy the engineering requirements. Then, the ERSM is applied to analyze the dynamic probability of the maximum vibration response for the numerical mistuned turbine bladed disk. The investigation indicates that the computational efficiency of ERSM is higher 38.92% than that of traditional RSM in the same computer and the same reduced-order model. Thus, the ERSM-ISCMS is a more effective method to investigate dynamic probabilistic analysis of the mistuned turbine bladed disk, it benefits for the complex structures and develops the theory method for the mechanical reliability design.

The self-excited vibration induced by friction of the shaft-hull coupled system with the water-lubricated rubber bearing and its stick-slip phenomenon

To the underwater vehicle, its self-excited vibration induced by friction in the stern bearing is the non-ignorable component of the propeller vibration and noise. This paper describes how to solve the self-excited vibration, and discuss the effects on the self-excited vibration and its stick-slip phenomenon. After making the analysis about the supporting characteristics between shaft and hull substructures, this paper takes the shaft-hull coupled system as the analysis object, and constructs the FEM (Finite Element Method) model of the system. Based on the FEM model validated by the experiment, the required modal frequencies and the shapes of the shaft and the hull are calculated. Then, the friction force between shaft and water-lubricated rubber bearing is calculated, according to the Stribeck friction curve. Hereafter, through the friction force, the modal shapes of the shaft and the hull are combined to establish the nonlinear differential governing equations of the shaft-hull coupled system, by employing the modal synthesis method of substructures. Subsequently, the self-excited vibration responses induced by the friction can be obtained by solving the nonlinear differential governing equations, by Runge-Kutta method. On this basis, the discussions about the effects of friction coefficient, damping ratio, rotation speed and support stiffness on the self-excited vibration of the shaft-hull coupled system are conducted. The results show that the larger the damping is, or the faster the rotation of shaft is, or the stronger the supporting stiffness is, the more difficultly the shaft-hull coupled system's self-excited vibration will happen. The works on how to solve the self-excited vibration induced by friction in the stern bearing will be of great significance to accurately predict the propeller vibration and noise and optimize the acoustic design of the underwater vehicle.

Synthesis of highly convergent 2D and 3D finite elements for short acoustic wave simulation

New higher-order finite elements of enhanced convergence properties for acoustic wave simulation are presented in the paper. The element matrices are obtained by combining modal synthesis and optimization techniques in order to achieve minimum errors of higher modes of the computational domain. As a result, simulation models of propagating wave pulses require a smaller number of finite element divisions per wavelength compared to the conventional element model thus significantly reducing computational costs. Though finite element matrices are obtained in optimization, the resulting patterns of the matrices are versatile and further can be used in any wave propagation model. The mass matrices of the elements are diagonal, so explicit time integration schemes are applicable. The usage of new elements is especially efficient in situations where wavelengths of the simulated signal are much shorter than the dimensions of the computational domain. This is referred to as short wave propagation analysis. The results of wave propagation simulation for ultrasonic measurements are presented as application examples. The B-scans and computed dispersion curves are provided for visual interpretation of the results.

Modal synthesis of precision control systems

The problem of synthesis of precision modal control systems is considered. It is noted that a common approach to solving this problem is to consistently meet the requirements for the nature of the transient process and for the indicators of its accuracy. This approach to synthesis is faced with the need to make design decisions under incomplete conditions. In practice, this circumstance leads to obtaining synthesis results with undesirable deviations from technical requirements. When designing precision control systems, such deviations are unacceptable. To eliminate the difficulties that arise, a transition to interval methods for formulating and solving modal synthesis problems is proposed. The theoretical possibility of the interval approach is based on the excessive variety of possible placement of eigenvalues in the spectrum of the characteristic matrix of the system. An example of an interval synthesis of a system with a modal controller and additional output feedback is considered. The restrictions on the spectrum of the specified matrix are formed, which determine the fulfillment of the requirements for the monotonicity of the transient process, the regulation time and the accuracy of the response to harmonic influences. It is noted that the variety of solutions obtained creates the preconditions for a multi-alternative approach to modal synthesis of systems.

Generalized modal reduction method for the dynamic analysis ofrotating mechanical systems

The paper proposes modal reduction method of the dynamic systems composed of linear nonconservative sub-systems coupled by nonlinear discrete couplings. Classical approach to the modal reduction is based on thetransformation of the generalized coordinates by the real modal submatrix of the linear conservative part of thewhole system. In case of modal synthesis method, transformation matrices are the real modal submatrices of theconservative part of mutually isolated subsystems. Rotating mechanical systems contain gyroscopic effects andother influences of rotation and damping. The paper introduces a generalized modal reduction method based on thecomplex modal values of the whole system or the isolated subsystems. Their complex eigenvalues and eigenvectorsare used for transformation of the generalized coordinates and reduction of the number of degrees of freedom. Thepresented method is focused on vibrating rotating systems with gyroscopic and dissipative effects and nonlinearinternal couplings.

Modal Analysis and Structure Optimization of Permanent Magnet Synchronous Motor

Permanent magnet synchronous motors are the core components of electric vehicles and widely used in the field of electric vehicles. The existence of vibration will reduce the operating efficiency and service life of the motor, which is the key factor to determine whether the motor is running normally. In this article, a certain type of permanent magnet synchronous motor is taken as the test object. The permanent magnet synchronous motor is divided into three substructures: stator, rotor and shell. The finite element modal results are obtained respectively. The modal parameters of the whole PMSM are calculated by using the substructure modal synthesis method. The weak links of the motor are found out and the structure optimization is carried out. Through the calculation of the modal analysis, it can be seen that each vibration mode of the optimized motor has been reduced, and the amplitude difference between the 2nd order and the 5th order is the largest. Finally, the modal test is carried out on the motor using the exciter method, the test results are analyzed, and the modal parameters of the motor are obtained through the frequency domain analysis method, the FEA result is confirmed by the modal test of the motor.