Vibration Mode Shapes Research Articles

Design, Fabrication and Characterization of L&T Hybrid Mode Transducers for Thermoplastic Welding

ObjectivesThis study aims to present a systematic approach for designing, fabricating and characterizing ultrasonic transducers for thermoplastic sheet welding capable of operating in longitudinal–torsional (L&T) hybrid vibration modes at 20 kHz, driven by an axially poled piezoceramic stack.MethodsMultiple horn models featuring circumferential deep helical grooves were designed to generate longitudinal and torsional vibratory displacements. Modal analyses were conducted to determine the appropriate dimensions for the desired mode shape and resonance frequency. The selected horn geometries were integrated with transducer bodies in a CAD environment to build 3D models of ultrasonic transducers. Modal and harmonic analyses were performed to evaluate their mode shapes, resonance frequencies, and displacement amplitudes under various excitation conditions. Prototypes of the optimized designs were fabricated using predetermined materials and geometries and characterized experimentally for resonance frequencies and tip displacements. The experimental results were compared with numerical analyses to validate the approach.ResultsThe developed transducers demonstrated successful operation in L&T hybrid vibration modes at the targeted frequency, with displacement amplitudes and resonance frequencies aligning well with the numerical predictions. Combining exponential area reduction and helical grooves in the horn appears to be a highly effective method for achieving longitudinal-torsional vibration from a longitudinally vibrating transducer. Finite element analysis proved to be a valuable design tool, facilitating accurate dimensional optimization of the horns to achieve the desired L & T hybrid vibrational mode shapes. The apparent longitudinal and torsional (spring) stiffness of the horn significantly influences the longitudinal displacement amplitude and angular displacement at the horn tip. The resonance frequency of the transducers shifted to higher frequencies with increasing excitation voltage.ConclusionThis research provides a comprehensive and validated framework for designing ultrasonic transducers capable of operating in hybrid vibration modes. The methodology reliably achieved the desired vibratory performance, with prototype results aligning closely with design objectives.

Benchmark exact free vibration solutions of two-dimensional decagonal piezoelectric quasicrystal cylindrical shells

Abstract Quasicrystalline materials with piezoelectric effects show significant potential for advancing actuators, sensors and energy harvesters. In this paper, the free vibration characteristics of two-dimensional decagonal piezoelectric quasicrystal cylindrical shells (PQCSs) are investigated in the framework of symplectic mechanics system. By introducing an original vector and its dual variable vector as the fundamental unknowns, the governing equations are reduced into a set of low-order ordinary differential equations system, thus the free vibration analysis is transformed into an eigenvalue problem within the symplectic space. By using the symplectic mathematics, the exact solutions for free vibration of PQCSs are finally obtained and expanded as a series of symplectic eigensolutions. Finally, accurate natural frequency and analytical vibration mode shapes for arbitrary classical boundary conditions are obtained simultaneously. The accuracy of the obtained solutions is verified by comparing with existing results in open literature. In addition, the effects of geometrical parameters, temperature rise, external voltage and coupling fields on the natural frequency and vibration mode shapes are investigated in numerical examples. Results indicate that the phason field exhibits significant influences on the natural frequencies and cannot be neglected in free vibration analysis of PQCSs. Furthermore, all the results can be served as benchmarks for the development of new analytical and numerical approaches.

Effect of Load-Bearing Wall Material on Building Dynamic Properties.

Nowadays, more and more buildings are being constructed from various types of modern materials. Many works have been written about these materials, which primarily focus on the influence of their properties on the thermal and acoustic insulation of, for example, building walls. However, there are very few publications analyzing the influence of construction materials on the dynamic properties of building structures and their vibration behavior. Yet, vibrations are dangerous for building structures. In the analysis of dynamic issues, the dynamic properties of objects should primarily be taken into account because the dynamic response of a building depends on the values of these parameters. This article focuses on numerically determining and analyzing the impact of load-bearing wall construction material on building dynamic properties-natural vibration frequencies and mode shapes. Seven building construction materials were considered, and then nine variants of building load-bearing walls made from these materials were analyzed. The analyses were carried out on the example of a low-rise administrative building structure. The building was modeled using the finite element method (FEM) with three-dimensional (3D) model analysis. Three variants of 3D FEM models were proposed, validated, and compared. A notable impact of load-bearing wall material properties on the natural frequencies and mode shapes of building structures was found. Two issues could be mentioned as the main new contributions of this paper: numerical analysis and comparing the effect of various building construction materials on dynamic building properties and the proposition and validation of various approaches to 3D FEM building load-bearing wall modeling. The findings of this research are of important significance and should be taken into account when constructing buildings subjected to dynamic loading or analyzing the possible harmful effects of various types of vibrations on buildings.

Standard deviation as a suitable criterion for localizing damages in Nano- composite structures

Wavelet transform is a proper method for detecting damages in signals such as vibration mode shapes. This study introduces the standard deviation as a suitable criterion for selecting and analyzing wavelet function in damage detection of nano-laminated composite structures. In this way, a Fe2O3-carbon epoxy fiber laminated composite structure is studied. Experimental modal analysis is conducted to obtain the natural frequencies and their corresponding mode shapes. Findings show that the standard deviation can effectively find the best analyzing wavelet function for structural damage detection when the damages are led to local and subtle changes in vibration mode shapes.

On the Free Vibration of 3D-Printed Structural Elements

Abstract This paper focuses on using 3D-printing to produce relatively simple, flexible structural components such as cantilevers, rings and plane frames. The relatively high resolution of modern 3D-printers facilitates the production of small-scale but slender structures, and thus provides an opportunity to exploit geometric parameter variations to enhance a practical understanding of stiffness and vibration. This approach has proved successful in both the classroom via demonstration models, as well as in the research lab in which elementary facilities can be utilized to acquire data. An especially useful facet of this approach is the assessment and justification for modeling simplifications, e.g., estimating flexural modulus, or judging the circumstances under which the familiar sway assumption is valid in portal frames. Non-simple structural geometries, for example circular rings, can also be conveniently produced and tested for natural frequencies and vibration mode shapes. We then extend this approach to forced vibration, with examples including the vibrating reed tachometer and sympathetic resonance. The advantage of this approach obviates the need for any external sources of excitation.

Free vibrational mode shapes and frequencies of sandwich folded plates standing on folded foundation and structured by auxetic honeycomb core and FG-GPLRC coating layers

In this article, for the first time, the free vibration and modal shapes of sandwich folded plate with honeycomb core are studied. The current article investigates the free vibrational mode shapes and frequencies of sandwich folded plate made of auxetic honeycomb core and graphene platelet-reinforced functionally graded composite (FG-GPLRC) coating layers which standing on the normal-shear folded supports. The structures are supported by normal-shear folded supports. The negative Poisson’s ratio exhibited by honeycomb materials results in distinct physical and mechanical properties, profoundly impacting the behavior of structures constructed from this material. While structures with basic shapes like rectangular, circular, and spherical configurations find limited applications across various industries, the versatility of folded structures makes them highly suitable for a wide range of industrial applications, given their more complex geometries. Dynamic equations are formulated within the framework of the first-order shear deformation theory (FSDT) for each flat section of the folded plate. The continuous equality conditions for displacements, rotations, and stresses at the junction edges of the two flat sections are rigorously satisfied. To effectively solve these equations, a robust numerical technique known as the generalized differential quadrature method (GDQM) is employed specifically tailored for the 2D analysis of folded plates. Furthermore, the study investigates the influence of various parameters, including material properties and geometry, on the frequencies and mode shapes of the folded structures, shedding light on the intricate interplay between these factors in shaping the dynamic response of the structures.

Stroboscopic sampling moiré microscope (SSMM) for investigating full field in-plane vibration of MEMS mechanical transducers

Precise analysis of the full-field in-plane vibration of microelectromechanical system (MEMS) transducers is crucial for assessing their device functionality and performance. As an example, in the context of frequency/amplitude modulation of Quartz Tuning Fork (QTF)-based atomic force microscopy (AFM) systems, understanding QTF’s in-plane vibration can significantly enhance accurate evaluation of tip-sample forces. Current methods, such as analytical and numerical approaches, have limitations when it comes to providing accurate measurements. To address these limitations, we proposed an experimental approach that combines stroboscopic and sampling moiré (SM) techniques. This method focuses on investigating the in-plane vibration of a QTF and utilizes the obtained results to measure the sensor’s dynamic properties such as vibration mode shape, resonance frequency (f0), and quality factor (Q). Nanometer-scale light pulses, generated using a custom-designed stroboscope, are synchronized with the QTF’s excitation voltage to freeze the vibration effectively, enabling imaging using a standard CCD camera. Subsequently, SM analysis is employed to extract the surface vibration profile, facilitating the measurement of vibration mode shape, f0, and Q. This technique shows promise for analyzing the dynamic behavior of various micro-devices compatible with the sample preparation process.

[formula omitted]-space physics-informed neural network (k-PINN) for compressed spectral mapping and efficient inversion of vibrations in thin composite laminates

The vibrational response of structural components carries valuable information about their underlying mechanical properties, health status and operational conditions. This underscores the need for the development of efficient physics-based inversion algorithms which, given a limited set of sensing data points and in the presence of measurement noise, can reconstruct the response at locations where measurement data is not available and/or identify the unknown mechanical properties. Addressing this challenge, Physics-Informed Neural Networks (PINNs) have emerged as a promising approach. PINNs seamlessly integrate governing equations into their architecture and have gained significant interest in solving inversion problems. In the context of learning and inversion of multimodal, multiscale vibrational responses, this paper introduces a novel spectral extension of PINNs, utilizing Fourier basis functions in the wavenumber domain, commonly known as k-space. The proposed method, referred to as k-space PINN (k-PINN), offers a robust framework for adjusting complexity and wavenumber composition of the response. Notably, the spectral formulation of k-PINN, coupled with the generally sparse representation of vibrations in k-space, facilitate efficient reconstruction and learning of broadband vibrations and alleviate the spectral bias associated with standard PINN. Additionally, the spectral solution space introduced by k-PINN substantially reduces the computational cost associated with computing physics-informed loss terms. We evaluate the effectiveness of the proposed methodology on reconstructing the bending vibrational mode shapes of a thin composite laminate and identifying its effective bending stiffness coefficients. Mode shapes are initially obtained from finite element simulation, and virtual test data with added noise are generated for evaluation purposes. It is shown that the proposed k-PINN methodology outperforms the standard PINN in terms of both learning and computational efficiency. The performance of k-PINN is further demonstrated by show casing its capability in learning different selections of symmetric, anti-symmetric and asymmetric mode shapes.

Experimental and numerical analyses on the modal behaviour of an aluminium alloy sheet

Abstract The study of modal behaviour for vehicle body panels represents a field of interest for automotive engineering, having a major influence on automotive comfort, with respect to vehicle noise, vibration and harshness, but also concerning the durability of vehicle body. The investigation of natural frequencies and vibration mode shapes is crucial in the design phase, because essential information can be obtained concerning the behaviour of the components in dynamic conditions. The current paper focuses on the investigation of vibration modes and frequency response function on an aluminium alloy sheet. The aim of the paper is to perform the study of modal behaviour both experimentally and theoretically, through measurements using the impact hammer method and respectively through finite element analysis. The CAD model of the aluminium alloy sheet has been developed in Catia V5 environment in view of elaborating the finite element model, which has been used for investigation of modal behaviour, taking into account the case of free vibrations. Through the modal finite element analysis performed in Ansys Workbench, the natural frequencies of the aluminium alloy sheet, as well as the corresponding mode shapes have been obtained. Experiments have been performed on an aluminium alloy sheet, mounted on an elastic support with low natural frequency. The impact hammer method has been used for obtaining the natural frequencies, through frequency analysis of accelerations measured with two piezoelectric accelerometers mounted on the aluminium alloy sheet. The frequency response function has been computed from the response acceleration measured on the aluminium alloy sheet and the impact force measured using the force transducer of the impact hammer. Virtual instruments have been developed in LabVIEW environment for data acquisition, time domain and frequency analysis, as well as for determining the frequency response function using the measured signals. The natural frequencies obtained from the finite element analyses have been compared to the results of measurements and very good similarities have been ascertained. The experimental investigation of modal behaviour contributed to the validation of the finite element model, but further research is required to investigate the possibilities of improving the modal behaviour of the aluminium alloy sheet.

3D mode shapes characterization under hammer impact using 3D-DIC and phase-based motion magnification

Modal analysis constitutes a fundamental aspect of structural investigation within diverse engineering domains, encompassing sectors such as automotive, wind energy, and aerospace. The prominence of high-frequency excitation loads, exemplified by the combustion phenomena in liquid rocket engines, necessitates an in-depth examination of the high-frequency vibrational response within structural components. However, the complexity of evaluating high-frequency vibrations arises from the negligible displacement associated with these responses. When using an optical full-field measurement system based on a high-speed camera for vibration measurement, it is usually severely affected by noise. Direct analysis of raw data using an optical measurement system (3D-DIC) is not feasible. In this paper, we combine phase-based motion magnification and digital image correlation methods to obtain the high-frequency vibration modes of the structure. 3D-DIC(3D Digital Image Correlation)analysis is performed on the magnified images to quantify the out-of-plane vibration modes of the structure. Using the cantilever beam as an example, the first five out-of-plane vibration mode shapes were separated from the response video under a single hammer excitation. Especially the 5th order natural frequency is as high as 3503 Hz, and the corresponding structural response was below the noise floor of the camera system. The vibration mode results obtained by this method are highly consistent with the vibration modes obtained by the 3D-SLDV(3D Scanning Laser Doppler Vibrometer) method. Finally, this method was applied to identify the out-of-plane vibration modes of real engine pipe. The combination of motion magnification techniques and DIC can enhance the capability of traditional 3D-DIC, which is beneficial for high-frequency structural identification. Future research could concentrate on optimizing motion amplification factors for different structures and loads, and creating automated algorithms for analyzing and visualizing amplified motion data in real time.

An adaptive continuous scanning laser Doppler vibrometry technique for measuring vibrational mode shapes of structures with holes

Abstract Continuous Scanning Laser Doppler Vibrometry (CSLDV) enables fast and full-field vibration measurement in an intact area of a structure. This paper further proposes a novel adaptive CSLDV method that can obtain the Operational Deflection Shapes (ODS) and mode shape of structures with holes. Firstly, an adaptive path planning method is applied to the surface of the tested structure, which can automatically adapt to the size and position of round and square holes, as well as the resolution requirements for the measurement. Secondly, based on the symmetry and other characteristics of the planned path, the time domain signal is processed by delay optimization and demodulation to obtain the ODS of each vibration mode. Finally, a modal test is conducted on a flat structure with a square and circle holes as an example. A validation experiment is also performed using SLDV. The results show that the mode shapes obtained from both methods are consistent and the Modal Assurance Criterion (MAC) between them are all above 0.96. The method can realize CSLDV of flat plate structures with arbitrary square and round holes, and has the advantages of high efficiency and dense measurement points, which enhances its engineering applicability.

Numerical simulation of a nebulizer with multiple orifices

A vibrating mesh nebulizer is a medical device that can break liquid medication into an aerosol of tiny droplets so that patients can inhale them to treat diseases. The volumetric flow rate of this kind of nebulizer is determined by many factors, including the orifice geometry, the vibration amplitude and the vibration mode shape. Thus, many experiments were designed to assess the effects of these different variables on flow rate and to find the optimal set of device parameters that maximizes the flow rate. However, it can be very expensive to perform such physical experimentation with physical prototypes, which needs different orifice geometries to be fabricated for different test regimes. The orifice sizes are microscopic with diameters typically between 3 µm and 5 µm, thus, they are difficult to be fabricated with the exact shape and diameter that are required. Therefore, virtual prototyping by means of numerical simulation models are required as the orifice geometries and other factors can be easily changed in the simulation models. This study aims to provide a numerical simulation model that is experimentally validated. By comparing the simulated flow rates with the experimentally obtained flow rates, and the simulated droplet diameters with the experimentally obtained droplet diameters, it was found that the errors are all less than 10 % which demonstrates this numerical simulation model is adequately validated by experiment. Thus, instead of carrying out expensive experiments, this simulation model can be used for predicting the volumetric flow rates for different prescribed orifice geometries. Furthermore, this paper also simulates the effects of viscosity and surface tension on the flow rate, where either viscosity or surface tension increases, the flow rate will decrease.

An enhanced stochastic subspace identification incorporating time variables and its application to large-span sea-crossing suspension bridges

Structural health monitoring and assessment are essential to large-span sea-crossing bridges, and modal identification is the most effective method for achieving them, with the stochastic subspace identification (SSI) method dominating. However, when analyzed in open traffic scenarios with SSI for a large-span sea-crossing bridge, the contour of the vibration mode is distorted. Related affecting factors such as the measuring conditions, user-defined parameters, and traffic are explored, and traffic is verified as the essential component. To address this issue, an enhanced SSI algorithm incorporating time variables is created to automatically identify and rectify vibration mode shapes. Feature heterotopic signals are first matched with a common reference signal to perform modal identification. Advanced particle swarm optimization is applied to estimate critical parameters. The SSI algorithm with the density-based spatial clustering of applications with noise (DBSCAN) algorithm is then utilized to extract the modal characteristics. Both simulated misalignment and real records are applied to verify the suggested approach's performance. Comparative analysis with state-of-the-art techniques, such as the error norm, cross correlation, phase transform-β, mean phase deviation, and genetic algorithm, shows that the recommended approach is dependable and successful at identifying the large-span sea-crossing bridge's vibration mode shapes and computationally more accurate and efficient.

Data-driven sensor fault diagnosis for vibration-based structural health monitoring under ambient excitation

In vibration-based structural health monitoring (SHM), the early detection of sensor faults is key to preventing false alarms and misleading conclusions on the condition of monitored structures. Since sensor networks are exposed to hostile environments, they are prone to unexpected errors that might influence the quality of measured data. This paper proposes a novel method for detecting and isolating faulty sensors from vibration response data by establishing an overdetermined system between the measured signals and the actual motion. The method assumes a rigid body motion of the monitored system, describable by a limited number of degrees of freedom (DOFs), to define the overdetermined relation between the sensor outputs and the system’s DOFs. The concept is later extended to systems not governed by rigid body motions by considering their vibration mode shapes. The robustness of the proposed methodology is demonstrated using vibration response data from an experimental monitoring campaign.

Reconstructing flexible body vortex-induced vibrations using machine-vision and predicting the motions using semi-empirical models informed with transfer learned hydrodynamic coefficients

This work assesses the validity of transfer learning the hydrodynamic coefficient database, consisting of the added mass and lift coefficients, applicable to flexible bodies undergoing vortex-induced vibrations. Specifically, the hydrodynamic coefficient database learned on data collected by Braaten and Lie (2005) are used to predict the motions observed during in house bare riser model experiments at the MIT Towing Tank. A fully immersed vertical flexible riser model with a length-to-diameter ratio of 145 is towed at different flow speeds and top tensions. Motion is tracked using underwater cameras and the motions are reconstructed using a machine-vision framework eliminating the need for expensive sensing hardware. The vibration amplitude, frequency, and mode shape are determined and the results are compared with those in the literature. Finally, blind predictions of the in-house observed experiments are made using the software VIVA informed with transfer learned hydrodynamic coefficients learned on the experiments by Braaten and Lie (2005).

Dynamic Response of Steel–Timber Composite Beams with Varying Screw Spacing

Steel–timber composite beams are a relatively new type of composite structure. They have many important advantages, owing to which they may be considered a sustainable solution. Their connectors may be demountable, which makes it possible to separate steel girders from LVL panels at the end of their service life. After disassembly, the structural elements can be recycled. One of their advantages is that they are lighter than steel–concrete composite beams. However, this may result in the poor performance of floors with steel–timber composite elements subjected to dynamic loadings. For this reason, the dynamic characteristics of floors should be investigated to verify the serviceability limit state of human-induced vibrations. In this study, the dynamic response of the three steel–timber composite beams with varying screw spacing was captured and used to validate their numerical models. The frequencies obtained from the numerical analyses correspond to the experimental results. A very high agreement between the vibration mode shapes was obtained because the MAC index values were close to 1. The validated numerical model of a single steel–timber beam may be used in future studies to create a complex numerical model of a steel–timber composite floor.

A Timoshenko-Ehrenfest beam model for simulating Langevin transducer dynamics

The Langevin transducer is a widely used power ultrasonic device across both medical and industrial applications, from orthopaedic surgery to drilling and welding. It is a sandwich-type device which typically consists of piezoelectric ceramic rings between two metallic end-masses. The transducer is commonly operated at a particular resonant mode to deliver ultrasonic vibrations to a target structure of interest, and is generally modelled using approaches including the transfer matrix method and electromechanical equivalent circuit methods. Here, we propose a variational framework to study the dynamics of the Langevin transducer based on the Timoshenko-Ehrenfest beam theory and Hamilton's principle. The variational equation derived from this model is then discretized by the standard finite element method with spectral elements. To verify the proposed one-dimensional electromechanical model, the computed resonance frequencies, or natural frequencies, of the one-dimensional model are compared to those of a three-dimensional finite element model with respect to varying geometry parameters characterizing the transducer. The results of the reduced one-dimensional and full three-dimensional models are then compared to those measured through an experimental investigation consisting of laser Doppler vibrometry. This is undertaken for the first ten eigenfrequencies, including longitudinal and bending modes, where normalized amplitudes and vibration mode shapes are reported. The close correlation between modelling and experiment demonstrates that the proposed one-dimensional electromechanical model can deliver results consistent with the full three-dimensional model and from experiment, thus verifying a rapid and reliable method for studying the free vibrations and dynamics of the Langevin transducer which accounts for the axial vibrations of the piezoelectric ceramic stack in all cases.

Experimental Study on Dynamic Characteristics of Damaged Post-Tensioning Concrete Sleepers Using Impact Hammer.

Concrete sleepers in operation are commonly damaged by various internal and external factors, such as poor materials, manufacturing defects, poor construction, environmental factors, and repeated loads and driving characteristics of trains; these factors affect the vibration response, mode shape, and natural frequency of damaged concrete sleepers. However, current standards in South Korea require only a subjective visual inspection of concrete sleepers to determine the damage degree and necessity of repair or replacement. In this study, an impact hammer test was performed on concrete sleepers installed on the operating lines of urban railroads to assess the field applicability of the modal test method, with the results indicating that the natural frequency due to concrete sleeper damage was lower than that of the undamaged state. Furthermore, the discrepancy between the simulated and measured natural frequencies of the undamaged concrete sleeper was approximately 1.87%, validating the numerical analysis result. The natural frequency of the damaged concrete sleepers was lower than that of the undamaged concrete sleeper, and cracks in both the concrete sleeper core and the rail seat had the lowest natural frequency among all the damage categories. Therefore, the damage degrees of concrete sleepers can be quantitatively estimated using measured natural-frequency values.

Effect of turbine's torque and speed variation on hydraulic vibration analysis during transient processes

AbstractWith the rapid development of large‐capacity hydraulic turbines and pumped‐storage power stations, hydraulic vibration analysis attracts increasing attention nowadays. To explore the influence of torque and speed variation on hydraulic vibration characteristics during transient processes and ensure the stability of hydropower stations, an improved analytical formula of turbine impedance involving the torque and speed variation was then derived. The free vibration analysis and forced vibration analysis using the transfer matrix method and the hydraulic impedance method were carried out. According to the numerical simulations, the obtained attenuation factors show differences in magnitude which illustrates that the torque variation and speed variation in transients do affect the damping rate of the vibration. In addition, the results indicated the natural angular frequencies of hydropower system were mainly dependent on the length of water conveyance system and pipe characteristics. It can be inferred form vibration mode shapes that for some specific frequencies, the magnitude of oscillatory pressure head and flow rate along the downstream tail tunnel are influenced by the hydraulic turbine. The correlation between the turbine impedance and guide vane openings tells that the hydraulic impedance is getting smaller as the opening increases. What's more, the corresponding free vibration analysis shows that the absolute value of the damping rate is closely related to the value of the flow rate.

Symmetric and asymmetric free vibrations of rotating eccentric annular plate

The symmetric and asymmetric free vibrations of rotating eccentric annular plate in uniform thermal environment are studied in present paper. The priority of research is clarifying the different combined effects of various factors on symmetric and asymmetric vibrations of eccentric annular plate. In order to conveniently obtain the natural frequency and vibration mode of eccentric annular plate with irregular physical domain, the transformed differential quadrature method is employed. Due to the rotational motion, the initial configuration caused by centrifugal force is calculated before vibration analyses. Then the traveling wave vibration arisen from the Coriolis force is investigated. By comparing the vibration characteristics of annular plate with/without eccentricity, the different impacts of interaction between the geometrical size, boundary conditions, temperature rise, rotational motion, and eccentricity on symmetric and asymmetric vibrations are discussed. Results show that eccentricity leads to the splitting phenomena only in asymmetric vibrations. And it does not affect the symmetry of vibration. Temperature rise results in that the fundamental vibration mode shape shifts from symmetric to asymmetric. However, eccentricity inhibits the change. Rotational motion causes the occurrence of traveling wave vibration in asymmetric vibration whereas not in symmetric vibration.