Spin Softening Effects Research Articles

Effects of rotation on the rolling noise radiated by wheelsets in high-speed railways

Railway rolling noise is produced by the vibration of both the wheelsets and the track; the wheelsets dominate the high frequency noise and become increasingly important at high speed. The effect of rotation on the wheelset vibration and noise radiation is investigated using different wheelset models in a rolling noise prediction model for a wide range of speeds. Each wheelset model takes account of the rotation to a different extent. An axisymmetric finite element model of a flexible rotating wheelset is implemented based on a complex exponential formulation and expressed in either an inertial or a non-inertial frame of reference. The model can include the inertial Coriolis and centrifugal forces and is also extended to include stress stiffening. Modes of the rotating wheel with non-zero number of nodal diameters are split into co- and counter-rotating waves with separated natural frequencies. The extent of the frequency separation depends on the shape of the mode and its dominant component of vibration. At common train speeds the frequency shifts due to stress-stiffening and spin-softening effects are found to be small compared with the gyroscopic effects due to the Coriolis forces. The effect of including the inertial Coriolis and centrifugal forces on the overall A-weighted sound power level is less than 0.3 dB below 400 km/h, while for higher train speeds, differences may exceed 1 dB in some one-third octave bands. Overall, these differences are small compared with other sources of uncertainty in rolling noise modelling, confirming that representing the wheel rotation with a moving load approach provides a suitable approximation for use in rolling noise predictions.

Centrifugal forces enable band gaps that self-adapt to synchronous vibrations in rotating elastic metamaterial

The rotation of mechanical systems can greatly change their modal and wave propagation response. In the classic example of the Foucault pendulum, the Coriolis force from the Earth’s rotation causes the pendulum path to appear deflected to an observer on Earth. More recently, Coriolis forces and gyroscopic effects from rotation have been shown to break time-reversal symmetry, induce non-reciprocity, and change the band structure of phononic crystals. However, rotation also introduces centrifugal forces that can considerably affect the vibration response through stress stiffening and spin softening effects. Although this is well studied in the field of rotor dynamics, the effects of centrifugal forces on the dynamic behavior of acoustic metamaterials (AMs) are still unexplored. In this paper, we study the dependence of torsional band gaps on rotational speed in a rotating shaft with attached local resonances. Different from previous studies, we explicitly consider the effects of stress stiffening in local resonances by introducing a stress stiffening function in a reduced order model of the rotating AM. We then calculate the stress stiffening function for beam-tip-mass resonators and show that these resonators have a resonant frequency that scales linearly with the rotational speed in the asymptotic limit, which in turn causes the band gaps of the AM to depend linearly on the rotational speed. Motivated by the presence of synchronous vibrations in rotating machinery, in which frequency is also linearly dependent on rotational speed, we show that this rotating AM supports band gaps that self-adjust to the synchronous vibration frequency, resulting in attenuation over a wide range of frequencies and thus rotational speeds. Finally, we validate the reduced order model of the rotating AM using 3D finite element analysis. This work contributes a foundational understanding of the effects of rotation in AMs and builds a connection between the AM and rotor dynamics communities, with the potential to introduce novel vibration control in rotor dynamics problems.

Investigation of the effects of frequency veering phenomenon in a pre-twisted and double-tapered rotating Timoshenko beam

The natural frequency curve veering phenomena have been significant and used in several structural dynamic problems. By formulating a pre-twisted and double-tapered rotating Timoshenko beam as a structural problem, variations in mode interactions among coupled axial, chordwise, and flapwise motions were presented. Rotordynamic effects such as gyroscopic couple, centrifugal stiffening, and spin softening effects were taken into account while establishing the equations of motion. Hamilton’s principle was used in deriving the equations of motion and is discretized based on Galerkin finite element method. Using dimensionless parameters, the model was verified with the existing literature for rotation, pre-twist, and double-taper configurations. A comparison of the frequency veering regions among the Euler–Bernoulli beam and Timoshenko beam theories is presented. Two-dimensional mode shape plots were used to show the coupled motions between axial displacement, chordwise bending, and flapwise bending. The effects of pre-twist and double-taper in the non-rotating Timoshenko beam on the modal properties of the rotating beam were examined. Natural frequency maps were drawn to locate the frequency veering curves. Mode interactions around different veering regions were shown with variations in mode shapes. An investigation on 60° pre-twisted and 0.5 ratio double-tapered rotating Timoshenko beam is presented.

Study on the Rotation Effect on the Modal Performance of Wind Turbine Blades

With the large-scale development of wind turbines, large flexible blades bear heavier loads. In the actual rotating work of blades, the coupling of structural deformation and motion produces a dynamic stiffening effect and spin softening effect, which affects the dynamic characteristics of blades. In this study, the finite element method is used to model the NREL 5MW blade, and the dynamic stiffening and spin softening effects are investigated using the modal analysis. The influence of rotating effects on the blade’s natural frequency is revealed. It is concluded that the effect of dynamic stiffening is more significant than that of spin softening, and the comprehensive result of the two effects is not simply the superposition of them but presents obvious nonlinearity.

Finite element modeling and analysis of dynamic characteristics of rotating coated blisks

An effective passive damping treatment with hard coatings has been utilized to suppress the vibration of blisks in previous works. However, the previous investigations have not incorporated the rotational effects (such as Coriolis forces, stress stiffness effect, and spin softening effect). In this paper, a new finite element model is developed to model the rotating coated blisks and is validated by ANSYS. Furthermore, a reduced order model on the basis of an extended subspace iteration method is presented to analyze the rotating blisks with coatings-thickness mistuning and elastic modulus mistuning. The influences of the coatings on the forced responses of rotating tuned blisks and the rotational effects on the forced responses of coated mistuned blisks are analyzed. The results indicate that the coatings can reduce the forced responses significantly of tuned rotating blisks and the rotational effects have significant impact on dynamic characteristics of mistuned blisks.

Vibration Response of Steam Turbine Healthy and Cracked Blade under the Stress Stiffening and Spin Softening Effects

In the present paper, vibration responses of the healthy and cracked blade under the influence of stress-stiffening and spin-softening have been studied. The three-dimensional finite element-based numerical analysis is carried out to study the blade's mode shape and natural frequencies and are validated with the experimental modal analysis test results. In the case of spin softening, for healthy blade at the turbine’s speed over 1000 RPM, natural frequencies increase significantly in both first flap-wise and chordwise bending modes. In both cases of STF and SOF, for the cracked blade, the deviation from the healthy blade in the natural frequency increases with increasing crack size, maximal in a crack size of 90%. It is also observed that irrespective of crack size, deviation in natural frequencies as compared to healthy blade increases as spin speed increases in the cases of chordwise bending mode (Mode II) and higher-order overlapped mode IV, and in the case of fundamental torsion mode (Mode III), higher-order overlapped mode V and VI natural frequencies decrease in both STF and SOF cases.

An Analytical Model of a Rotating Radial Cantilever Beam Considering the Coupling Between Bending, Stretching, and Torsion

Abstract In this paper, the mode coupling between bending, stretching, and torsional deformations is mainly studied by presenting an analytical model of a rotating cantilever beam with pre-twist angle and arbitrary cross section. Equations of motion of the beam are derived using Hamilton's principle. The Coriolis effect due to the coupling of the bending deformation and stretching deformation, the eccentricity caused by inconsistency between elastic center and centroid, spin softening effect, stress stiffening effect, shear deformation, and rotary inertia are included in the model. Equations of motion are solved by the Rayleigh-Ritz method. The natural frequencies obtained by the proposed analytical modal are in good agreement with those obtained by finite element method (FEM) which proved the accuracy of the analytical model. Finally, the coupling between different mode components is studied in detail based on a quantitative method. The transformation/conversion between different mode components is revealed, the influence of rotational speed, setting angle, and pre-twist angle on this conversion mode is studied. Results show that a specific mode shape is usually composed of multiple mode components. The essence of mode coupling is the coupling between different mode components. The influence of rotational speed, setting angle, and pre-twist angle on the mode coupling is that they cause the transformation/conversion between different mode components.

Dynamic analysis of functionally graded rotating thick truncated cone made of saturated porous materials

In the current article, dynamic behavior of functionally graded saturated porous rotating thick truncated cone is investigated for the first time. It is assumed that porosity distributions along the thickness are (1) symmetric nonlinear, (2) nonlinear asymmetric and (3) uniform distributions. The Biot poroelastic law instead of Hooke’s law is used to model the constitutive equations. Also, 2D axisymmetric theory of elasticity instead of simple shell theories is used for the governing motion equations. Graded finite element and Newmark methods have been used to solve the governing motion equations. First, natural frequencies of stationary functionally graded saturated porous cone are obtained, and then by considering the effect of centrifugal force and spin softening effect, dynamic response of porous cone in an undrained condition for different variables such as porosity and Skempton coefficients, rotational velocity, semi vertex angle of the cone and different distribution of porosity has been investigated. Obtained results denote that by increasing Skempton coefficient, natural frequencies are increased, and natural frequencies of PUD distribution are lower than other distributions. Also, rotational velocity and semi-vertex angle have significant effect on the displacement and stress field.

A Reduced-Order Model for the Vibration Analysis of Mistuned Blade–Disc–Shaft Assembly

An effective reduced-order model is presented in this paper for the vibration analysis of a mistuned blade–disc–shaft assembly considering the flexibility of the shaft and the rotordynamic effects. For the sake of accurate modeling and quantitative analysis, three-dimensional (3D) finite element models were employed in obtaining the governing equations of motion with the Coriolis force, centrifugal stiffening, and spin softening effects taken into account. Then, an efficient model order reduction technique based on the coordinate projection by normal modes of tuned assembly and cyclic symmetry analysis was developed for mistuned blade–disc–shaft assembly. The criterion of whether one matrix could be incorporated in cyclic symmetry analysis is presented. During the modeling, the mistuning in blade and disc was taken into account and dealt with independently. In mistuning projection, the blade and disc parts were both projected onto their tuned counterparts of the sector model, where the boundary conditions were set to be fixed and free, respectively. Finally, an example of a blade–disc–shaft assembly was employed to validate the effectiveness of the presented method in free and forced vibration analysis.

An Enhanced Axisymmetric Solid Element for Rotor Dynamic Model Improvement

Abstract The authors present an improved formulation for the axisymmetric solid harmonic finite element (FE) modeling of a flexible, spinning rotor. A thorough comparison of beam-type FE and axisymmetric solid FE rotor models is presented, indicating the errors that result from beam FE usage for various nondimensional rotor topologies. The axisymmetric rotor is meshed in only two dimensions: axial and radial, with both displacement fields being represented with Fourier series expansions. Centrifugal stress-stiffening and spin-softening effects are included in all elements and most importantly in modeling flexible disks. Beam FE and axisymmetric FE natural frequencies, mode shapes, and critical speeds are compared to identify shaft geometries where the beam model yields a significant error. Finally, limitations of beam FE models and guidance for utilizing axisymmetric solid FE models in rotor dynamic simulations are provided.

An experimental verification of the one-dimensional static Willis-form equations

This paper investigates the behavior of a heavy soft spring in steady circular motion. Since the spring is inhomogeneous due to centrifugal force, one can rigorously prove that it follows the one-dimensional static Willis-form equations. The theoretical predictions agree very well with experimental results. It further demonstrates that these equations can give a clear understanding of the stress-stiffening and spin-softening effect. These findings reveal that the Willis-form equations can give very accurate linear approximations of finite deformation problems and are also helpful to clarify the classical concept of the principle of material frame indifference.

The Vibrational Behavior of Coupled Bladed Disks with Variable Rotational Speed

AbstractLow pressure steam turbine blades are subjected to high static and dynamic loads during operation. These loads strongly depend on the turbine's rotational speed, leading to entirely new load conditions. To avoid high dynamic stresses due to the forced vibrations, a coupling of the blades, such as shrouds or snubber coupling, is applied to reinforce the structure. In this work the influence of the rotational speed on the vibration behavior of shrouded blades is investigated. Two fundamental phenomena are considered: the stress stiffening and the spin softening effect. Both effects are caused by centrifugal forces and affect the structural mechanical properties, i.e. the stiffness matrix K, of the rotating system. Since the rotational speed Ω appears quadratically, it is possible to derive the stiffness matrix as a second order matrix polynomial in Ω2 [3].In the case of shrouded blades, contact forces between neighboring blades must be taken into account. The contact status and the pressure distribution in particular is strongly influenced by the rotational speed, respectively, centrifugal forces, caused by the untwisting and radial deformation of the blades. For the calculation, a three dimensional structural mechanical model including a spatial contact model is considered. The solution of the nonlinear equations of motion is based on the well known Multiharmonic Balance Method [2]. Here, the nonlinear forces are computed in the time domain and transferred in the frequency domain by the use of the Fast Fourier Transformation (FFT), also known as the Alternating Frequency Time method (AFT) [1]. (© 2015 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Shaking-Swing Coupled Vibration Analysis of a Laminar Composite Rotating Blade by the Finite Element Method

For studying the vibration characteristics of a laminar composite rotating blade, the spin softening effect calculated by the finite element method (FEM) was incorporated with the prestress effect to analyze shaking, swing, torsion, and coupled vibrations. The mathematical formula was derived. The study shows that the centrifugal force exerts influence on the difference in vibration frequencies. The laminar composite can reduce an influence of the spin softening on the first swing vibration frequency because of its structure and smaller density. With the variation of rotating velocity, the blade spin softening can lead to the coupled first shaking and first swing vibrations. In the coupled vibrations, swing is contained in the shaking mode and shaking is also contained in the swing one, which is the reason of major blade damages. The results confirm the complexity of the dynamic characteristics of the laminar composite rotating blade. This method is also well applicable to the dynamic design of such blades.

The importance of the material properties on the burst speed of turbine disks for aeronautical applications

Abstract The computation of the burst of turbine disks for aeronautical engines is a current problem of great practical interest. Turbine disks are stressed by severe loads, for this reason the knowledge of the burst angular velocity is of primary importance for the whole safety of the aircraft during ordinary missions or extra-ordinary conditions that engines may encounter during service. In this work, a numerical method to predict the burst speed is presented, introducing the physical phenomenon at the base of the burst: the inertial instability. The method is based on Finite Element simulations taking into account the spin-softening effect which leads to instability and therefore to the burst of the disk. Two analytical solutions of rotating structures are used for a preliminary validation of the numerical results that simulate the inertial instability: in both cases the calculated burst speed agrees with the analytical results. The correlation between the nonlinear stress–strain curve and the inertial instability is then addressed by means of simulations that are performed in the end on a real disk subjected to typical loads acting on it. In conclusion, a comparison between the semi-empirical method of Robinson [14] , which is at the base of standard disks certifications, and the FE approach is presented in terms of burst speed prediction to show that FE calculation is more accurate to predict the limit speed.

Dynamics of Turbocharger Rotor with Spin Softening Effect

The solid model of a turbocharger rotor was simulated for the rotordynamic analysis by use of finite element method.The dynamic equations for the solid model were deduced from the rotatory effects and gyroscopic inertia.The program was developed to determine the bending,torsional and their couplied vibration models.Calculations show that both the spin softening and stress stiffening effects have significant influence on the forward and backward whirl modes,and the torsional vibration frequency doesn’t change with the increase of rotor speed.The couplied bending-torsional vibration may be took place as excited.The accuracy of the solid model and the solution techniques have been demonstrated by comparison with other results.

Vibration Analysis of Transmission Shafts Using Solid Models

Fatigue fracture of the intermediate shaft in a transmission was analyzed by finite element method.The solid model has advantages of centrifugal force and spin softening effects over beam models.The results show that the fatigue fracture of the shaft is mainly due to the torsional vibration,the stress stiffening and spin softening have strong influence on the bending vibration medes,and torsional vibration modes do not changed at different speeds.There is accounting for the effects by an adjustment of the stiffness matrix.By using of solid models,the vibration modes of the intermediate shaft were researched,then the model was optimized in order to avoid resonance.

Flexible auto balancer design for the reduction of optical disk drive vibration

This study investigates the design of a flexible auto balancer (FAB) for the compensation of disk eccentricity. The FAB has various advantages over a conventional auto ball balancer, such as low cost, ease of implementation, lack of noise and applicability to standing-type drives. For the FAB design, a finite element model is developed considering the prestress effect and spin-softening effect caused by rotation. The structure is optimized and fabricated based on finite element model verification via modal testing. In addition, the dynamic behavior of the FAB is recorded by high-speed photography, and the vibration transmission to the base deck is measured.

Contact analysis of a flexible bladed-rotor

This paper presents a model of fully flexible bladed rotor developed in the rotating frame. An energetic method is used to obtain the matrix equations of the dynamic behaviour of the system. The gyroscopic effects as well as the spin softening effects and the centrifugal stiffening effects, taken into account through a pre-stressed potential, are included in the model. In the rotating frame, the eigenvalues' imaginary parts of the latter matrix equation give the Campbell diagram of the system and its stability can be analysed through its associated eigenvalues' real parts. The turbo machine casing is also modelled by an elastic ring in the rotating frame through an energetic method. Thus, in some rotational speed ranges the contact problem between the rotor and the stator can be treated as a static problem since both structures are stationary to each other. Prior to the study of the complete problem of contact between the flexible blades of the rotor and the flexible casing, a simple model of an elastic ring having only one mode shape, excited by rotating loads is developed in the rotating frame too, in order to underline divergence instabilities and mode couplings. Then, the complete problem of frictionless sliding contact between the blades and the casing, without rubbing, is studied. The stable balanced static contact configurations of the structure are found as function of the rotational speed of the rotor. Finally, the results are compared to these of the simple model of rotating spring-masses on an elastic ring, showing good adequacy. The present model of rotor appears thus particularly adapted to the study of blades-casing contacts and highlighted an unstable phenomenon near the stator critical speed even in case of frictionless sliding.