Torsional Vibration Frequency Research Articles

Symmetry properties, tunneling splittings of some vibrational energy levels and torsional IR spectra of the trans – and cis – conformers of hydroquinone molecule

Torsional vibrational states of trans − and cis − conformers belonging to point symmetry groups C2H and C2V, respectively, of hydroquinone molecules are classified according to the irreducible representations of the molecular symmetry group D2H(M). A correspondence has been established between the symmetry elements of the D2H(M) group and the symmetry elements on the torsional coordinate plane for two-dimensional surfaces of potential energy, wave functions, kinetic coefficients and dipole moment projections. A correspondence has been established between the symmetry species of the point symmetry groups C2H, C2V and the symmetry species of the molecular symmetry group D2H(M). Conformational states, barriers to internal rotation and the above-mentioned characteristics of the hydroquinone molecule were calculated at the MP2/Aug-cc-pVDZ, MP2/Aug-cc-pVQZ, MP2/Aug-cc-pVTZ, MP2/CBS(aD,aT,aQ), and CCSD(T)/dAug-cc-pVDZ levels of theory. The calculated data sets were approximated using symmetry-adapted sets of basis functions. Using a numerical solution of the vibrational Schrödinger equation of restricted dimensionality, the energies and wave functions of 50 stationary torsional states of the hydroquinone molecule were determined for the first time. The values of tunneling splittings of the ground vibrational and a number of excited torsional states of trans − and cis − conformers were determined. In particular, when calculating at the MP2/CBS(aD,aT,aQ) level of theory, the values of tunneling splittings of the ground vibrational states of trans − and cis − conformers turned were 1.32*10-6 and 1.62*10-6 cm−1, which is consistent with the experimentally established upper limit for this value in the cis − conformer by authors of [W. Caminati, S. Melandri, L. B. Favero, J.Chem.Phys., 100 (1994) 8569 – 8572] (0.2 MHz or 6.67*10-6 cm−1). The calculations of the matrix elements of the dipole moment operator and the partition function made it possible to simulate the torsional IR spectra of the molecule’s conformers at different temperatures. The frequencies of fundamental torsional vibrations in the trans – (267.1 and 269.0 cm−1) and cis – (269.5 and 270.9 cm−1) conformers, calculated at the MP2/CBS(aD,aT,aQ) level of theory, are in good agreement with the experimental value of frequency of this vibration (266 cm−1), established in [W.G. Fateley, G.L. Carlson, F.F. Bentley, J.Phys.Chem., 79 (1975) 199–204.].

On some methods of determining proper frequencies torsional oscillations in metal-working machines

Forced torsional vibrations that occur in machine tool drives are one of the pressing problems of modern machine tool industry. The reasons for these fluctuations may be imbalances of rotating masses, structural imperfections of bearing units, operating or technological process features (for metal-cutting machines – eccentric clamping of the workpiece, processing of non-round workpieces, etc.). In this work, using the example of a real metal-cutting machine, the task of dynamic analysis of the drive of the main movement of the machine with determination of the natural frequencies of torsional vibrations is realized. The real object is replaced by a dynamic model (DM). When constructing the DM, the inertial and rigidity characteristics are determined from the conditions: – the kinetic energy of the object is equivalent to the kinetic energy of the DM (from this condition the axial moments of inertia of the concentrated masses of the row model are obtained); – the potential energy of elastic deformations of the object is equivalent to the potential energy of deformation of the DM (from this condition the rigidity coefficients of the DM are obtained). The natural frequencies of the row DM were determined in three ways: 1) by solving the frequency equation (exact method) in combination with the method of reducing the number of masses of A.P. Cherevkov; 2) partial systems method (fundamental frequency only); 3) method of residuals (Tolle). The following conclusions were made: 1) the main (smaller) frequency can be found by any of the methods considered (since the results of the three methods coincided); 2) when simplifying the DM using the method of partial systems, the criterion for the possibility of further simplifications over the specified frequency range should be taken into account; the transition from a three-mass DM to a two-mass one if the criterion in this calculation was not met led to an error in determining the value of the fundamental frequency of about 10%; 3) dynamic calculations of machines using DM and methods for determining natural frequencies make it possible, based on known frequencies of disturbing forces, to identify possible near-resonant operating modes and avoid them in advance by making changes to the design of the machine at the design stage

Vibration of embedded restrained composite tube shafts with nonlocal and strain gradient effects

Torsional vibration response of a circular nanoshaft, which is restrained by the means of elastic springs at both ends, is a matter of great concern in the field of nano-/micromechanics. Hence, the complexities arising from the deformable boundary conditions present a formidable obstacle to the attainment of closed-form solutions. In this study, a general method is presented to calculate the torsional vibration frequencies of functionally graded porous tube nanoshafts under both deformable and rigid boundary conditions. Classical continuum theory, upgraded with nonlocal strain gradient elasticity theory, is employed to reformulate the partial differential equation of the nanoshaft. First, torsional vibration equation based on the nonlocal strain gradient theory is derived for functionally graded porous nanoshaft embedded in an elastic media via Hamilton’s principle. The ordinary differential equation is found by discretizing the partial differential equation with the separation of variables method. Then, Fourier sine series is used as the rotation function. The necessary Stokes' transformation is applied to establish the general eigenvalue problem including the different parameters. For the first time in the literature, a solution that can analyze the torsional vibration frequencies of functionally graded porous tube shafts embedded in an elastic media under general (elastic and rigid) boundary conditions on the basis of nonlocal strain gradient theory is presented in this study. The results obtained show that while the increase in the material length scale parameter, elastic media and spring stiffnesses increase the frequencies of nanoshafts, the increase in the nonlocal parameter and functionally grading index values decreases the frequencies of nanoshafts. The detailed effects of these parameters are discussed in the article.

Application of Maple Mathematical Package to Search for Transfer Gear Oscillation Frequencies

Considers direct spectral problem of determination of frequencies of free torsional oscillations of transfer gearbox, component part of power plants with diesel fuel, gearboxes. The gearbox is modeled by a dynamic mechanical system with three degrees of freedom of bearings and gear. The frequency equation of the direct problem was found, according to which numerical calculations were carried out to search for the frequencies of torsional vibrations of the transfer gearbox. To the conclusion of the equation and the search for the frequencies of the gearbox oscillations, a software implementation of transformations using the functionality and libraries of the Maple mathematical package is given.

An investigation on the torsional vibration of a FG strain gradient nanotube

AbstractIn this work, an attention is paid to the prediction of torsional vibration frequencies of functionally graded porous nanotubes based on the Lam strain gradient elasticity theory. The nanotubes are formed of functionally graded porous nanomaterials that vary in the radial direction. This study also aims to obtain the analytical solution of the strain gradient model presented by Lam for torsional vibration response, in a simple manner, for different rigid or restrained boundary conditions. The torsion angle of a functionally graded nanotube is defined by an infinite Fourier series. Then, the Stokes’ transformation is applied to force the boundary conditions to the desired state. An eigenvalue problem is established with the help of the two systems of equations obtained. This eigenvalue problem, which includes deformable springs at both ends of the nanotube, appears as a general analytical solution that can find torsional vibration frequencies. It is shown that the vibrational responses can be significantly influenced by the through‐radius gradings of material, material length scale parameters and deformable springs of the functionally graded nanotubes and consequently can be predicted by giving proper values to torsional spring parameters.

Elastic medium and torsional spring effects on the nonlocal dynamic of functionally graded porous nanotubes

In this study, Eringen’s nonlocal elasticity theory that applies the small size effects in functionally graded porous nanotubes embedded in an elastic matrix is discussed. The material properties of functionally graded porous nanotubes are taken into account to vary over the radius direction with a rule of mixture. The free torsional vibration relation according to nonlocal elasticity theory, via Hamilton’s principle, is obtained and an eigenvalue solution is constructed for the free torsional vibration response of the presented work. The presented analytical model is validated by comparing the calculated mathematical results for homogeneous nanotubes with rigid and non-rigid boundary conditions. Special attention is given to deformable boundary conditions, porosity coefficient, material grading coefficient and also to the influence of elastic medium on the free torsional vibration frequencies. In this paper, it has been proven that the influence of length, elastic medium, elastic torsional spring rigidities, material grading and porosity coefficients on the vary in the torsional vibration frequency of the functionally graded nanotube is not small.

A novel experimental setup for axial-torsional coupled vibration impact-assisted PDC drill bit drilling.

The geological conditions of hot dry rock (HDR) reservoirs are complex. The geothermal mining of HDR faces major challenges in the drilling and construction of wells, fracturing to create storage, and flowing to extract heat. Vibration impacts help improve the rock-breaking efficiency, where the axial-torsional coupled vibration impact technology can increase the bit penetration depth and reduce the stick-slip effect. To study the feasibility and efficiency of the axial-torsional-coupled vibration impact-assisted Polycrystalline Diamond Compact (PDC) bit to break high-temperature and high-pressure rocks, a new experimental setup was designed. The system includes a drilling fluid circulation system, an axial-torsional coupled impact drilling system, a formation simulation system, and a data acquisition and control system. This setup can produce a rock-breaking torque of 2000 N·m, a drilling speed of 200rpm, a weight on bit of 100kN, an axial vibration frequency of 100Hz, and a torsional vibration frequency of 50Hz. It can simulate the formation pressure of 70 MPa and the rock temperature of 400 °C. A series of rock-breaking drilling experiments were successfully conducted using this setup. The results show that the axial-torsional coupled vibration-impact assisted PDC bit has a good performance in breaking high-temperature and hard rocks, which can accelerate the application of this new technology in deep formation drilling.

Research on high-bandwidth linear active disturbance rejection control method for variable speed turboshaft engine

Abstract The wide flight range and high torsional vibration frequency of high-speed helicopters impose stricter criteria for the high-bandwidth control of turboshaft engines. Consequently, research is underway to implement a high-bandwidth control method for turboshaft engines using the linear active disturbance rejection control (LADRC) theory. Initially, the LADRC is designed based on the mathematical model of the integrated helicopter/engine system. To address the challenge of maintaining control quality with varying speed reference commands for the power turbine, an improved LADRC method with tracking differentiators (TD) is developed. Numerical simulations comparing the control effectiveness of LADRC with TD to cascade PID and conventional LADRC methods are conducted. The results demonstrate that the improved LADRC gains have a wider tuning range than the LADRC controller, and the power turbine speed tracking effect of LADRC with TD is optimal. It is more conducive to accomplish high-bandwidth control of turboshaft engine with variable rotational speed.

Potential and kinetic interaction of two internal tops in molecules belonging to C2v(M) molecular symmetry group

Fourier analysis of the kinetic and potential interactions of non-coaxial internal tops in the molecules C6H4(OH)2, HO(CH2)OH, HOOOH, HOSOH, HSSSH, and HSOSH was carried out. It was found that for the all six molecules, in which both tops are characterized by a period 2 π, the harmonics with a period of π are dominate, with its contribution to the Fourier representation of potential surfaces increasing in the sequence mentioned above. Contributions of internal tops interactions to the kinetic and potential energies were determined for analyzed molecules. The influence of potential and kinetic interactions of internal tops on 1) the features of tunneling splitting of the ground and excited torsional states of molecules, 2) the landscape of 2D potential energy surfaces, 3) frequencies of fundamental torsional vibrations of hydroxyl and thiol groups, 4) the ratio of trans - and cis - conformer energies is also analyzed.

Direct measurement of negative Poisson ratios of auxetic materials via the impulse excitation technique (IET)

The direct measurement of Poisson ratios via the impulse excitation technique (IET) is based on determining the frequency ratio between the resonant frequencies of flexural and torsional vibrations of circular discs. However, the direct measurement of negative Poisson ratios, which occur in auxetic materials, has been impeded until now by the absence of numerical values of Martinček’s solution for frequency ratios smaller than 1.35, because the relevant ASTM standards have not considered the possibility of negative Poisson ratios so far. Here we provide this missing link by reconsidering Martinček’s solution and presenting the necessary values in the forms of graphs and fit relations from which the Poisson ratio, including negative ones (down to −0.273), can be determined from frequency ratios down to 1.1 for a wide range of specimen aspect ratios (down to 4).

Consideration of supports stiffness at torsional and longitudinal vibrations of the rod

The study of the impact of the stiffness of the supports on the value of the first natural frequency of torsional and longitudinal free vibrations of the spring-hinged rod was carried out. In both cases of vibrations the mathematical formulation of the problem of determining the first natural frequency of vibrations reduces to compiling and solving the corresponding differential equation under given boundary conditions. The solution of this complex mathematical problem is performed by the numerical method in a specially created program for many combinations of supports stiffness. To obtain an analytical solution, a quadratic approximation of the obtained values was performed, which made it possible to obtain simple dependencies convenient for engineering applied calculations. The quadratic approximation also allowed solving the inverse problem of determining the required stiffness of supports for the required value of the first natural frequency of vibrations. The obtained solutions were compared with known solutions, which showed that the approximation error is less than 5% if the stiffness of the rod supports of the same order. The obtained results can be used in engineering applied calculations of natural frequencies of torsional and longitudinal vibrations of rods with elastic supports.

Modeling the deadwood bearings wear surface by optical scanning

The development of a methodology for modeling the wear surface of deadwood bearings based on optical scanning is considered in the paper. It is noted that the deadwood bearings of the ship’s shaft line wear out unevenly during operation, which affects the stress-strain state of the shaft and the system parameters responsible for the occurrence of resonance of transverse, torsional and longitudinal vibrations. The methodology and results of an experimental study of bearing wear surfaces are presented. The applied optical scanning scheme makes it possible to obtain high-precision digital models of objects due to synchronous shooting of an object with two cameras. The measurement results are obtained by processing with specialized software. The results obtained are an addition to the methodology for determining the elastic force acting on the shaft from the side of a bearing that is unevenly worn along the length with a non-constant gap. Based on the obtained wear values, a solid-state model is constructed and the natural frequencies of transverse, torsional and longitudinal vibrations are determined. For the constructed model, the forms of natural oscillations for practically significant frequencies are obtained. It is noted that bearing wear affects not only the displacement of the natural frequencies values, but also the oscillations shape. The impact of the wear of the aft bearing on the bearing capacity of the shaft line, taking into account the contact interaction, is evaluated. The simulation has shown that the bearing wear surface forms an area of increased contact pressures that arise from the impact of the propeller shaft. Analysis of the propeller shaft tip has revealed the presence of an area of increased concentration of equivalent stresses formed at the place of bearing wear formation. The proposed method of modeling the deadwood bearings wear allows us to evaluate its effect on the magnitude of the displacement of natural frequencies and the change in the bearing capacity of the shaft line. The results obtained in this work are an addition to the existing calculations in the design and evaluation of the ship’s power plant reliability.

Study of the Influence of the Presence of Local Edge Damage on the Spectrum of Natural Frequencies of Torsional Vibrations of Cantilever Rods

Study of the Influence of the Presence of Local Edge Damage on the Spectrum of Natural Frequencies of Torsional Vibrations of Cantilever Rods

Torsional static and vibration analysis of a non-circular restrained micro/nanowire

This paper aims to study the static and free torsional vibration of non-circular micro/nanowires using the modified couple stress theory with several deformable and non-deformable support conditions. The geometrical properties of the micro/nanowires are considered to have triangular, elliptical and circular profiles. Equation of motion in torsional direction is derived based on torsional equilibrium and systems of linear equations are computed by adopting Fourier sine series with Stokes’ transformation. Moreover, deformable boundary conditions have been taken into account to formulate eigenvalue problem by employing the higher-order force boundary conditions. Material length scale parameter and the shape of cross-section have considerable impacts on torsional vibration frequencies of restrained microwire. The obtained frequency curves of restrained microwires are shown to be dependent on various parameters such as the shape of cross-section, deformable boundaries, material length scale parameter and also warping function.

Identification of pathogenic bacteria using the spectra of native DNA molecules

The purpose of detecting low-frequency (relative to infrared and ultraviolet) resonance absorption of an electromagnetic field in the centimeter wavelength range using DNA molecules at a frequency corresponding to the natural frequency of torsional vibrations of the native DNA helix is to find a relationship between the length of DNA (its spectrum) and the type of a bacterium. Instead of DNA solution, cultures of different bacteria that contained DNA were used and the resonance frequencies of torsional vibrations of the helices in DNA molecules were calculated for them. The cultures were placed in test tubes or in a cuvette and were subjected to microwave irradiation. Theoretical analysis was carried out with respect to the DNA reaction to external centimeter radiation and the resonant DNA frequencies of three types of bacteria such as E. coli M17, M. avium and Mycobacterium tuberculosis were determined. Peak absorption of the ultrahigh-frequency electromagnetic field by bacterial cultures was detected at frequencies close to the calculated natural frequencies of torsional vibrations of DNA helices of the said types of bacteria: 10.271 GHz, 10.317 GHz and 10.356 GHz, respectively.

Dilatant-Fluid Torsional Vibration Damper for a Four-Stroke Diesel Engine Crankshaft

Abstract This paper presents a study of a viscous torsional vibration damper for a crankshaft of a four-stroke diesel engine. The reliable operation of a widely used silicone-type viscous damper depends on the ability of the silicone oil to absorb the energy of torsional vibrations. The non-Newtonian shear flow of the silicone oil interlayer, characterised by a reduction in the shear-rate-dependent viscosity and a moment of the drag forces, negatively affects damping characteristics. A torsional vibration damper, filled with a shear-thickening fluid, was considered and a rheological approach, based on viscosity growth with the shear rate increase, was applied. For such a damper, larger velocity gradients correspond to the higher values of a viscous force, which decreases torsional vibration. The parameter of damper effectiveness (defined by the fluid flow index, values of the damper gaps, torsional vibration amplitude and frequency) was implemented. It has been established that the efficiency of the torsional vibration damper filled with a dilatant fluid does not depend on the damper dimensions and gaps and significantly increases when a shear-thickening fluid is used instead of silicone oil or a Newtonian fluid. At higher values of the flow index, when the non-Newtonian flow becomes distinct, torsional vibrations are damped more effectively. Critical vibration amplitudes at high-velocity gradients, in turn, increase the damping effect as the moment of the drag forces and flow index are power-law related.

Modelling of the torsional IR spectra of the HSSSH, DSSSH, and DSSSD molecules

Equilibrium configurations of trans- and cis- conformers of the HSSSH molecule (trisulfane), as well as the vibrational spectra of both conformers, were calculated at the MP2/Aug-cc-pVTZ, level of theory in the harmonic and anharmonic approximations. A more accurate analysis of torsional vibrations of thiol groups was performed by constructing 2D surfaces of a) potential energy, b) kinetic coefficients, and c) dipole moment components as functions of torsional coordinates of S-H groups. The values of b) and c) were calculated at the MP2/Aug-cc-pVTZ level of theory while the values of a) were calculated at the B3LYP/Aug-cc-pVTZ, MP2/cc-pVTZ, MP2/Aug-cc-pVTZ, MP2/cc-pVQZ, MP2/CBS(T,Q) and CCSD(T)/Aug-cc-pVTZ levels of theory. The energies of the stationary torsional states of the trisulfane molecule conformers were determined by numerically solving the vibrational Schrödinger equation of restricted dimensionality. The frequencies of torsional vibrations of S-H groups, as well as the splittings of torsional energy levels in trans- and cis-conformers due to tunneling between equivalent configurations of the molecule, were determined. The calculated values of tunneling frequencies in the ground torsional state of HSSSH and DSSSD trans-conformers (2.3 * 10−22 and 7.5 * 10−27 cm−1, respectively) are of particular interest due to the fact that trisulfane is considered to be one of the best candidates for establishing the effect of violation of parity conservation law in chiral molecules.

Experimental research on new hole-making method assisted with asynchronous mixed frequency vibration in TiBw/TC4 composites

With the gradual promotion and the application of difficult-to-machine materials such as titanium matrix composites in the aerospace field, high-quality hole-making technology has become a major demand in aviation manufacturing. In order to improve the hole-making quality of TiBw/TC4 composites, asynchronous mixed frequency vibration-assisted hole-making (AMFVAHM) method is proposed. The process consists of two steps: ultrasonic vibration-assisted drilling (UVAD) base hole and low-frequency torsional vibration-assisted helical milling (LFTVAHM) target hole. Based on this process, the cutting trajectory modeling is established, and the hole-making experiment on TiBw/TC4 composites is conducted. The experimental data show that during the hole expansion stage, the maximum XY-plane average milling force decreases by 30.96% and the maximum axial average milling force decreases by 24.49% compared with conventional helical milling (HM) when the torsional vibration frequency and the milling frequency are the same in LFTVAHM. The hole-making experiment shows that AMFVAHM can reduce the chip size, tool wear, and some other defects such as entrance/exit burrs, scratches, and fractures of the hole wall. Comparing with HM and UVAD, the verticality of hole wall increases by 71.43% and 86.21%, the inlet damage decreases by 27.98% and 31.60%, the outlet damage decreases by 2.80% and 14.47%, the hole wall roughness (Ra) decreases by 36.29% and 63.43%, and the maximum white layer thickness decreases by 19.99% and 67.66%. Meanwhile, AMFVAHM process not only reduces the cutting force and cutting temperature but also improves the hole-making quality due to the fretting friction effect of LFTVAHM in secondary hole expansion, which meets the need for high-quality hole-making of difficult-to-machine materials in practical engineering applications.

Impact of shafting angle of F5 cold rolling mill on the dynamics of main drive system

In view of the torsional vibration problem of the main drive system of the F5 rolling mill in a steel plant, two sets of torque monitoring sensors were installed on the cross universal joint shaft on site. Through online monitoring, it is found that the torsional vibration frequency of the cross universal joint shaft is twice the rotational frequency of the drive system, and the vibration amplitude of the upper shaft is greater than that of the lower shaft. Considering the influence of the inclination angle of the cross universal joint shaft, the transmission system can be simplified as a multi-body dynamic model of a five-inertia spring-mass system. Combined with the effect of the rolling torque of the strip and the electromagnetic torque of the main motor on the main drive system, a simulation model considering multi-physics coupled torsional vibration was established MATLAB/Simulink. The simulation shows that the torsional vibration frequency changes linearly with the rolling speed, which is always twice the rotational frequency of the transmission system, and the vibration amplitude also changes with the rolling speed, consistent with the field test results. As the diameter of the work roll increases, the angle of inclination of the universal joint shaft decreases, and the value of the torsional amplitude of the system also decreases. When the inclination angle is not considered at all, the torsional vibration of the system disappears, it shows that the inclination angle of the cross universal joint shaft is the root cause of the double rotational frequency torsional vibration of the main drive system. Since the inclination of the cross universal joint cannot be eliminated, the simulation results show that the torsional vibration of the transmission system can be effectively reduced by using large-diameter work rolls and reducing the stiffness of the upper cross universal joint while increasing the stiffness of the lower universal joint. When the lower shaft stiffness increases to 1.3 times, the upper shaft stiffness decreases to 0.8 times, and the work roll diameter is 425 mm, the torsional vibration amplitude of the upper connecting shaft decreases by 22.3 % at the rolling speed of 1200 m/min.

On-Shaft Wireless Vibration Measurement Unit and Signal Processing Method for Torsional and Lateral Vibration

Microelectromechanical systems accelerometers have opened new possibilities for vibration monitoring of a rotating machinery. They enable mounting accelerometers directly to the rotating component of the machine, e.g., shaft. This enables not only the measurement of a lateral vibration but also a torsional vibration of the machine. This increases the vibration data gathered from the machine by one measurement instrument. This article presents an on-shaft wireless universal measurement unit (UMU) with innovative combination of features, such as a high measurement range and easy mounting. The UMU has a two-sensor configuration where two accelerometers are mounted to the opposite sides of a shaft. This enables to utilize a novel signal processing method to separate torsional and lateral vibration from the data. The signal processing method combines and modifies methods presented in the published literature. The results presented in this article demonstrate that the UMU together with the presented signal processing method can measure frequencies of torsional and lateral vibration accurately. However, the amplitude comparison between the UMU and reference sensors cannot be done adequately because they are measuring either different components of the machine or different physical properties.