Torsional Frequencies Research Articles

Direct Determination of Dynamic Elastic Modulus and Poisson’s Ratio of Timoshenko Rods

In this paper, the exact solution of the Timoshenko circular beam vibration frequency equation under free-free boundary conditions was determined with an accurate shear shape factor. The exact solution was compared with a 3-D finite element calculation using the ABAQUS program, and the difference between the exact solution and the 3-D finite element method (FEM) was within 0.15% for both the transverse and torsional modes. Furthermore, relationships between the resonance frequencies and Poisson’s ratio were proposed that can directly determine the elastic constants. The frequency ratio between the 1st bending mode and the 1st torsional mode, or the frequency ratio between the 1st bending mode and the 2nd bending mode for any rod with a length-to-diameter ratio, L/D ≥ 2 can be directly estimated. The proposed equations were used to verify the elastic constants of a steel rod with less than 0.36% error percentage. The transverse and torsional frequencies of concrete, aluminum, and steel rods were tested. Results show that using the equations proposed in this study, the Young’s modulus and Poisson’s ratio of a rod can be determined from the measured frequency ratio quickly and efficiently.

Determination and Performance Analysis on Miniature and Electric Commercial Vehicle Frames of Carbon Fiber Materials in the Laying Mode

The concept three-dimensional model of the mini commercial vehicle was established, and the vehicle parameters and calculation conditions were defined. The stress direction and value of the beam were calculated by Ansys workbench, and the laying method of the carbon fiber material of the beam was determined to be [0°\\90°\\ 45°\\90°\\0°] of the five-layer layup sequence, using the Abaqus software analyzes the frame made of carbon fiber composite material. It is found that the degree of Hill failure is 23.88% on the beam 3 under the working condition 1, and the damage of the beam 4 is 33.92% under the working condition 4, and the damage degree is less than 100%. The modal analysis results of the whole frame show that torsional mode and bending mode frequency of the first-order are 51.741 Hz and 62.286 Hz, exceeding 48% and 58% of the target value, respectively, which meet the target requirements. Carbon fiber materials provide an important basis for the application of miniature commercial vehicle frames.

Measurements of the Shear Modulus of Materials by the Free-Plate Torsional Mode Shape Method

Abstract In this work, according to the first-order torsional mode shape function computed by ANSYS software and the optimization principle, the relationship between the shear modulus of materials and the first-order torsional frequency of the free plate was established in terms of the energy method. Mode shape coefficients of orthotropic materials (lumbers of Sitka spruce, Scots pine, Beech, et al.) and isotropic materials (steel, aluminum, and glass) were calculated. The correlation between the mode shape coefficients of lumbers in tangential, radial, and cross sections and length-width and width-thickness ratios of the free plate was obtained. Also, the correlation between the mode shape coefficients of isotropic materials and the length-width ratio of the free plate were achieved. The correctness of these correlations was verified by mathematical simulation, dynamic testing, and static testing for shear modulus. The transient excitation method can be used to measure the spectrum of free plates. With the aid of the cross-power spectrum based on the frequency spectrum, the first-order torsional frequency of the free plate can be directly identified. The free-plate torsional mode shape method proposed in this work is applicable not only to testing the shear modulus of isotropic materials such as steel but also to testing shear modulus in the three main sections (GTL, GLR, and GRT) of orthotropic materials such as lumber.

Coupled Bi-Flexural–Torsional Vibration of Fluid-Conveying Pipes Spinning About an Eccentric Axis

This paper presents a dynamical model of a fluid-conveying pipe spinning about an eccentric axis. The coupled bi-flexural–torsional free vibration and stability are analyzed for such a doubly gyroscopic system. The partial differential equations of motions are derived by the extended Hamilton principle, and are then truncated by a 4-term Galerkin technique. The frequency and energy evolutions and representative mode shapes in the two transverse directions and torsional direction are investigated to unveil the essential dynamical attributes of the system. It is indicated that the stability of the present system mainly depends on spinning speed, flow velocity and eccentricity, while the torsional frequencies are almost immune to the flow velocity. The pipe reveals ‘traveling-wave’ modal vibrations in both transverse directions, and a general ‘standing-wave’ modal vibration in the torsional direction.

Sub synchronous torsional interaction of steam turbine under wind power oscillation in wind-thermal power bundled transmission system

The scaled wind energy bundled with thermal power is an essential way for the consumption of renewable energy in wind farms far away from load center. However, practices have shown that sub synchronous torsional interaction may occur in wind–thermal power bundled transmission systems, which threatens the stability of bundled systems. In this paper, the impedance modeling method based on graphical state space is applied to obtain a detailed impedance model of thermal power unit, which contains the characteristics of shaft torsion. Based on time domain simulation, the responses of each torsional mode of thermal power unit to wind power oscillation are analyzed. The results show that when the frequency of wind power oscillation is close to the torsional mode frequency, the shaft torsional oscillation may be excited, but the risks of different torsional modes are different, which can be explained via the characteristics of shaft torsion in the impedance of thermal power units.

Conserved Vibrational Coherence in the Ultrafast Rearrangement of 2-Nitrotoluene Radical Cation.

2-Nitrotoluene (2-NT) is a good model for both photolabile protecting groups for organic synthesis and the military explosive 2,4,6-trinitrotoluene (TNT). In addition to the direct C-NO2 bond-cleavage reaction that initiates detonation in TNT, 2-NT undergoes an H atom attack reaction common to the photolabile 2-nitrobenzyl group, which forms the aci-nitro tautomer. In this work, femtosecond pump-probe measurements with mass spectrometric detection and density functional theory (DFT) calculations demonstrate that the initially prepared vibrational coherence in the 2-NT radical cation (2-NT+) is preserved following H atom attack. Strong-field adiabatic ionization is used to prepare 2-NT+, which can overcome a modest 0.76 eV energy barrier to H atom attack to form the aci-nitro tautomer as soon as ∼20-60 fs after ionization. Once formed, the aci-nitro tautomer spontaneously loses -OH to form C7H6NO+, which exhibits distinctly faster oscillations in its ion yield (290 fs period) as compared to the 2-NT+ ion (380 fs period). The fast oscillations are attributed to the coherent torsional motion of the aci-nitro tautomer, which has a significantly faster computed torsional frequency (86.9 cm-1) than the 2-NT+ ion (47.9 cm-1). Additional DFT calculations identify reaction pathways leading to the formation of the dissociation products C7H6NO+, C7H7+, and C6H6N+. Collectively, these results reveal a rich picture of coherently and incoherently driven dissociation pathways in 2-NT+.

Torsional natural frequencies by Holzer method

In this work is presented the estimation of the torsional natural frequencies, eigenvalues of the equation of motion, as a result of the modal analysis of a mechanical system, by the iterative method Holzer. A numerical model in Workbench ANSYS 14 is used to evaluate the effectiveness of the Holzer method in detecting the torsional natural frequencies. The system consists of a motor coupled to a flywheel by means of a rigid coupling, which has been simplified in a torsional equivalent system of inertial discs and torsional flexible shaft sections. The results obtained by the Holzer method are in good agreement with the results for the numerical case implemented in ANSYS.

Wind Tunnel Test Investigation on Unsteady Aerodynamic Coefficients of Iced 4‐Bundle Conductors

Iced conductor motion is induced by the aerodynamic instability of these conductors. The unsteady aerodynamic characteristics are different from the steady aerodynamic characteristics. The unsteady aerodynamic coefficients of typical iced conductors’ models under torsional motion are measured by the unsteady wind tunnel test. The unsteady aerodynamic coefficients of crescent‐shape and sector‐shape iced 4‐bundle conductors under different torsional motion frequencies, wind velocities, and ice thicknesses are obtained. Wind test results show that there are significant differences between the unsteady and steady aerodynamic coefficients. The unsteady aerodynamic coefficients curve is a loop which is different from the steady aerodynamic coefficients. In addition, the obvious differences exist between unsteady aerodynamic coefficients of crescent‐shape and sector‐shape iced bundle conductors. Critical parameters, including torsional motion frequencies, wind velocity, ice shape, and ice thickness, have significant influences on unsteady aerodynamic coefficients. It shows that the wind tunnel experiment results are able to provide necessary data for the investigation of iced bundle conductor motion and their prevention techniques.

Investigation on glass-epoxy composite drive shaft for light motor vehicle

The composite material is excellent material where a high strength to weight ratio is desirable. It has a wide variety of applications in the aeronautical and automotive field. One of the great features of the composite material is that it can be designed as per the strength requirement of applications. A drive shaft is a rotating shaft that transmits power from the engine to the differential gear of a rear wheel drive vehicles. Generally, an alloy steel drive shaft is used in automotive applications. To improve the fuel economy by reducing inertia loads, steel needs to be replaced by strong and lightweight material. In this research, a glass epoxy composite drive shaft is successfully designed for a light motor vehicle. The replacement of conventional steel by composite material resulted in 73% of weight reduction of the drive shaft. The designed glass epoxy composite drive shaft is investigated experimentally for the torsional strength, natural frequency, and critical speed. Results obtained are compared with the steel drive shaft. The developed composite shaft has proved to be the best alternative to the conventional steel shaft.

Comparative Structural Performance Of Diagrid and Bracing System in Mitigation of Lateral Displacement

The design of high rise building with more than 10 stories is controlled by lateral drift which increases the cost rapidly with the number of stories. Recently, the diagrid structural system has emerged and in fact has been widely used for high rise buildings. How effective is a diagrid system in mitigating the lateral displacement of a high rise building? Thus, the objective of this research is to compare the lateral displacement of high rise building that adopts diagrid system with those that use X-bracing and frame system, due to wind. This research also explored the effect of using diagrid, X-bracing and frame system to the natural frequency of high rise building. Two diagrid systems which are 60°-diagrid and 80°-diagrid that has an angle of inclination of 60 degrees and 80 degrees, respectively, were studied. STAAD Pro software was used to analyse twelve building models to determine the lateral displacement due to wind load and the natural frequency in the along wind, across wind and torsional directions. Building with 80°-diagrid has the least lateral displacement, followed by buildings with the X-bracing, 60°-diagrid and frame for 40 and 60 storey buildings. Further study showed that X-bracing became the most effective system to lower the lateral displacement compared with the diagrid and frame when the number of storey of the building exceeds 71 storeys. Natural frequency was not affected by the different systems used except for torsional natural frequency that increased substantially by the use of 60°-diagrid.

Structural dynamic analysis using wavelets

In this paper, wavelet approach is studied to determine the torsional natural frequencies of a simple discretized structure excited by white noise, emulating environmental conditions. In this work, the implementation of this approach is described and the analytical results of one case are compared with the frequencies determined using Fourier analysis. Results have shown a good agreement between the values reported by Fourier analysis and the obtained by using Wavelet functions.

Study of Torsional Vibrations of Turbomachine Shafts: Part 1. Algorithm Optimization for the Determination of the Parameters of Natural and Forced Torsional Vibrations of Shafts

This article is concerned with the issues of the development and optimization of algorithms for the control and calculation of the torsional vibration parameters of turbomachine shaftings. For the control of shaft torsional vibrations, the discrete phase method (DPM) is used. Depending on the natural shafting vibration forms determined by calculation, the necessary number of informational surfaces (control cross-sections) with labels, which are alternating crest–valley pairs (“gears”), or holes in special perforated disks, are made along its length. The DPM features that must be considered when using it for the control of the torsional vibration parameters are observed. It is shown that the said method can only determine the mean integral angular speed values at angles of rotation of a rotor between labels in informational cross-sections, but it does not provide the direct instantaneous measurement of the shaft torsional vibration parameters. The approximation of the instantaneous torsional vibration parameters by their mean integral values leads, in case when the fast Fourier transform (FFT) is used, to significant errors, especially in the higher harmonics range. In this case, the accuracy of the angular vibration spectrum determination depends significantly on the conversion order, i.e., on the number of labels in an informational cross-section in fact. It is noted that additional errors of the FFT algorithms are caused by the use of data with nonconstant discretization frequency (data on a variable grid). The problems of the DPM determination of the harmonic amplitudes multiple to the rotation frequency (order harmonics) are discussed. It is shown that they can be determined only using angular masks of informational cross-sections. In addition, angular masks that are in compliance with the established rules help to significantly decrease the label pitch difference effect on the calculation results for the torsional vibration parameters. The algorithms for the conversion of the measurement results are presented. The relations for the determination of the angular amplitudes (swirl angles) of a shaft in control cross-sections are obtained with consideration of the chosen number of labels and the torsional vibration frequency. Examples of the results for the numerical modeling and analysis of shaft torsional vibrations using the proposed algorithms are given.

Analysis of Oscillation Frequency Deviation in Elastic Coupling Digital Drive System and Robust Notch Filter Strategy

Mechanical resonance is a common problem in drive systems with elastic coupling. On-line adaptive notch filter is widely used to make systems stable and the key of this method is to identify natural torsional frequency from a speed feedback signal. However, because of common adoption of digital control and expansion of system bandwidth, oscillation frequency of the system is more likely to deviate from natural torsional frequency to a higher one. When oscillation frequency is shifted, the enabled notch filter with erroneous notch frequency causes an oscillation with a lower frequency and even makes resonance more severe. In order to explain this phenomenon, the classical two-mass model based classification of resonances is checked at first. Then, by taking digital control, current loop delay, and saturation nonlinearity into consideration, an improved digital mechanical resonance model is proposed and a criterion for oscillation frequency deviation is finally obtained. Furthermore, a more widely applicable and robust notch filter tuning strategy with no oscillation rebound is presented. In the end, the validity of aforementioned analysis and strategy is verified by experimental results.

Torsional vibration of cracked carbon nanotubes with torsional restraints using Eringen’s nonlocal differential model

Free torsional vibration of cracked carbon nanotubes with elastic torsional boundary conditions is studied. Eringen’s nonlocal elasticity theory is used in the analysis. Two similar rotation functions are represented by two Fourier sine series. A coefficient matrix including torsional springs and crack parameter is derived by using Stokes’ transformation and nonlocal boundary conditions. This useful coefficient matrix can be used to obtain the torsional vibration frequencies of cracked nanotubes with restrained boundary conditions. Free torsional vibration frequencies are calculated by using Fourier sine series and compared with the finite element method and analytical solutions available in the literature. The effects of various parameters such as crack parameter, geometry of nanotubes, and deformable boundary conditions are discussed in detail.

Tool flank wear monitoring using torsional–axial vibrations in drilling

In this paper, monitoring of amplitude variation in the torsional–axial frequency is proposed for evaluating drill flank wear. Vibration signals were captured from the experiments resulting to the drilling process and investigation was focused on the role of torsional–axial coupling on instability predictions arising as a result of drill flank wear in the frequency spectrum. The first and second modes of torsional axial coupling frequencies were found through use of finite element analysis (FEA) and verified using experimental modal analysis (EMA) by the resonance frequency test. The proposed strategy uses dominant peaks of torsional–axial first mode (Tp1) and second mode (Tp2) frequency. The ratio of torsional–axial amplitudes (TP1/TP2) was considered for the monitoring and the evaluation of drill wear and also to nullify process parameter variation. Drill frequencies verified through experimental study showed their capability of predicting drill flank wear. The validation showed the proposed methodology having 80% accuracy and its ability for effective use for monitoring tool wear.

Rotordynamic Analysis of the AM600 Turbine-Generator Shaftline

The AM600 represents the conceptual design and layout of a Nuclear Power Plant Turbine Island intended to address challenges associated with emerging markets interested in nuclear power. When coupled with a medium sized nuclear reactor plant, the AM600 is designed with a unit capacity that aligns with constraints where grid interconnections and load flows are limiting. Through design simplification, the baseline turbine-generator shaftline employs a single low-pressure turbine cylinder, a design which to date has not been offered commercially at this capacity. Though the use of a ‘stiffer’ design, this configuration is intended to withstand, with a margin, the damage potential of torsional excitation from the grid-machine interface, specifically due to transient disturbances and negative sequence currents. To demonstrate the robust nature of the design, torsional rotordynamic analysis is performed for the prototype shaftline using three dimensional finite element modelling with ANSYS® software. The intent is to demonstrate large separation of the shaftline natural frequencies from the dominant frequencies for excitation. The analysis examined both welded drum and monoblock type Low Pressure Turbine rotors for single cylinder and double cylinder configurations. For each, the first seven (7) torsional natural frequencies (ranging from zero–190 Hz) were extracted and evaluated against the frequency exclusion range (i.e., avoidance of 1× and 2× grid frequency). Results indicate that the prototype design of AM600 shaftline has adequate separation from the dominant excitation frequencies. For verification of the ANSYS® modelling of the shaftline, a simplified lumped mass calculation of the natural frequencies was performed with results matching the finite element analysis values.

Dynamics analysis of a rotating plate with a setting angle by using the absolute nodal coordinate formulation

Dynamics analysis of a rotating plate with a setting angle is presented in this study by using the Absolute Nodal Coordinate Formulation (ANCF). The setting angle has a significant influence on distribution of the centrifugal force so that rotating plates can be twisted. However, the rotation-induced torsion deformation has not been discussed in existing literature and the existing approaches are not always appropriate both in transient and modal analysis for this subject. An improvement of the ANCF which is very suitable to solve transient dynamics problems is carried out for modal analysis of rotating plates in this paper. The results obtained from the presented method are validated by comparison with available results from literature and commercial software ANSYS which exhibits a good agreement. The results in different cases indicate that the torsion deformations of rotating plates not only exist in the transient dynamics but also affect significantly the modal characteristics. Incidentally, the effect of setting angle on torsion deformation, frequencies and mode shapes is discussed by parametric analysis.

A CD probe with a tailored cantilever for 3D-AFM measurement

A critical dimensional (CD) probe, such as a boot shaped probe, is widely used for measuring the sidewall profiles of nanostructures in atomic force microscopy (AFM). However, conventional CD probes usually have high spring constants that restrict their ability to operate in critical dimension scanning modes. Furthermore, the flexure spring constant of the tip is much smaller than the torsion spring constant of the cantilever, which probably makes the sidewall measurement unstable. Here, we proposed a method of tailoring the probe where the cantilever was trimmed to reduce the spring constant. The simulation results, using a finite element method (FEM), illustrated that both the flexure and torsion spring constants decreased nearly ten times. Meanwhile, the torsional resonant frequency decreased to below 1 MHz, which could benefit the torsional oscillation scanning. By using a focused ion beam (FIB), a CD probe with a rectangular cantilever was reprocessed. It was mounted to a home-built three-dimensional AFM system (3D-AFM), and had the flexure and torsion sensitivities precisely calibrated. The ratio of the flexure sensitivity to torsion sensitivity showed good accordance with the simulation results. Additionally, a grating standard sample was measured using the tailored probe under a modified step-in scanning mode, and the error due to torsional deformation of the tip was compensated. The measurement results demonstrated the validity of the tailored CD probe in 3D metrology.

Solution for the torsional vibration of the compressor shaft system with flexible coupling based on a sensitivity study

This paper presents the root cause analysis and solving process of the torsional vibration problem in a reciprocating compressor shaft system. The field measured data showed that the fifth-order torsional resonance occurred in the shaft system. The lumped-mass method was adopted to calculate the torsional natural frequencies (TNFs) and mode shapes of the shaft system. A sensitivity study was conducted to investigate the effect of each inertia and stiffness in the calculation model on the first TNF of the shaft system. The uncertainties of the primary data of the calculation model were discussed, and the mass-elastic model was modified by adjusting the stiffness of the flexible disc coupling. After adding an inertia of 4.68 kg·m2, torsional resonance was avoided, and the speed fluctuation of the shaft system was significantly reduced. The compressor package has been operating normally for nearly one year after modification. The analysis methods and conclusions given here can provide guidance for trouble-shooting torsional vibration problems.

Planetary Gear Fault Detection Based on Mechanical Torque and Stator Current Signatures of a Wound Rotor Induction Generator

In this paper, an approach for tooth localized fault detection in the sun gear of a planetary gear, using the measured mechanical torque and the stator current of a wound rotor induction generator (WRIG), is proposed. A theoretical background is developed to demonstrate that the faulty sun gear produces periodic fault signatures in the mechanical torque and consequently fault-related frequencies in the stator current of the WRIG. The employed technique needs knowledge of mechanical system torsional natural frequencies, which aid in identifying the most sensitive fault characteristic harmonics in both mechanical torque and stator current spectra. It is illustrated by simulation and experimental results that the amplitude of these harmonics can be used as a fault index for diagnosis of tooth localized defect in planetary gears. A test rig based on a 5.5 kW WRIG, shaft connected to the sun gear of a planetary gear through a high-speed shaft and couplings, was utilized for validation of the proposed technique.