Steady-state Vibration Response Research Articles

Hysteretic Dynamic Modeling and Vibration Characteristics Analysis of Metal-Rubber Isolators

A novel hysteretic constitutive model (HCM) with four parameters was proposed to reproduce the nonlinear elasticity and dry friction characteristics of metal–rubber isolators (MRIs). The HCM accurately characterized the nonlinear hysteretic behavior due to its intrinsic relationship with the geometric properties of the hysteresis loop. A physics-based parameter estimation strategy was developed to provide reasonable initial iteration values. Three MRIs with different design stiffnesses were tested to investigate their hysteretic behavior under various excitation levels. The HCM accurately reproduced the entire experimental hysteresis loops using only a subset of the experimental data. This exhibited higher precision and scalability compared to the classical dynamic model. Friction damping forces were employed to distinguish the stick-slip states of the three MRIs. Additionally, the identified parameters were used to simulate the steady-state vibration response of a complex MRI system. The resulting frequency response functions (FRFs) agreed well with the experimental data. The proposed model can effectively capture the amplitude-dependent effects in MRI systems with various design stiffnesses, enabling more accurate predictions of vibration responses.

Coupling vibration mechanism of rotating shaft–disc–blade system with blade crack—A systematical investigation on the effect of crack, condition, and structure parameters

Rotating shaft–disc–blade (RSDB) system is one of the most important parts of turbomachinery, such as aero-engine, gas turbine, and power plant. Understating and revealing the coupling vibration mechanism of RSDB system is significant for blade health monitoring and blade crack detection. However, the multi-structure, multi-vibration source, and multi-stress coupling makes it a very tough task. This study aims to reveal the coupling vibration mechanism of RSDB system by systematically investigating the steady-state vibration responses under the influences of different blade crack, operation condition, and structure parameters. First and foremost, a coupling vibration model of RSDB system that comprehensively considers the multi-structure, multi-vibration source, and multi-stress coupling effects is presented. And then, a novel condition indicator for blade health monitoring, called blade–blade distance, is proposed to better characterize and analyse the coupling vibration of RSDB system with blade crack. At last, the comprehensive effects of blade crack, operating condition, and structure parameters on the coupling vibration mechanism of the RSDB system with blade crack are systematically studied. The comparative results demonstrate the effectiveness of blade–blade distance and super-harmonic components of shaft torsional vibration and blade bending vibration for indicating the presence of blade crack and tracking the crack severity. Another observation should be noted is that the vibration behaviour including the blade–blade distance, super-harmonic components of shaft torsional vibration and blade bending vibration, the combination frequency components of shaft bending vibration are co-affected by blade crack, operating condition, and structure parameters. It is indicated that the blade health monitoring and crack detection of rotating blade should comprehensively consider different indicators to obtain a more robust and accurate results.

Vibration analysis of a rotor-bearing system using magneto-rheological elastomers

Magneto rheological materials are considered as a promising adaptive means of vibration control. This paper proposes a new configuration of smart bearings comprising magneto-rheological elastomer (MRE) layer inserted between the bearing pedestal and the foundation. This configuration provides a directional vibration controllability in both horizontal and vertical directions of a rotor-bearing system. The effect of using MRE layer on the vibration response and the resonant speeds of a rotor-bearing system is investigated. The finite element method is used to derive a dynamic model for the rotor-bearing system using shaft elements based on the Rayleigh beam theory and accounting for the gyroscopic effect and internal damping of the shaft. Simulation results reveal that the use of MRE materials in passive mode leads to downshifting of the first two resonant speeds and the attenuation of the steady state vibration response of the rotor bearing system in the horizontal and vertical directions at the resonance frequencies. When the MRE layer is subjected to a magnetic field generated by an electric current produced in the magnetic coil, the horizontal and vertical vibration responses are increased at the resonant speeds but decreased at other rotational speed ranges. Furthermore, the first two resonant speeds in the horizontal and vertical directions decrease further with the increase of the electric current.

Finite element based dynamic analysis of a porous exponentially graded shaft system subjected to thermal gradients

Abstract In modern applications, demand for the rotor shafts made of FGM has increased in aerospace, gas turbine and turbo jet engines due to its improved dynamic characteristics at elevated temperatures. The dynamic vibration response of a functionally graded Jeffcott rotor-bearing system with induced porosities has been studied for non-uniform porosity distributions. The functionally graded (FG) shaft’s radial direction exhibits continuous variation in the material qualities. The FG rotor’s cross-sectional material properties have been graded using the exponential law. The temperature gradients have been considered to be varied across the cross-section based on exponential temperature distribution. For the first time, using the FE method, it has been possible to study the impact of induced porosities on the free vibration frequencies and steady-state vibration responses of the functionally graded rotor-bearing system. This has led to the development of a two-node porous functionally graded rotor element with uneven porosity distributions using the Timoshenko beam theory. It has been observed that the free vibration frequencies and the FG rotor-bearing system critical speeds are affected by increased porosity and thermal gradients. The frequency responses demonstrate that as temperature gradient and volume fraction of porosity rise, the FG rotor-bearing system’s critical speeds shift to the left.

Tunable stiffness design of curved-crease origami and extended quasi-zero stiffness vibration isolator

A kind of origami tube based on the curved crease, which has a tunable stiffness, was designed, fabricated, tested, and extended to the concept of a quasi-zero stiffness (QZS) vibration isolator. The regulating function of crease stiffness on the overall origami stiffness without changes in the crease pattern was verified by single-crease models. With various opening ratios along the creases, three tubes composed of mirrored single-crease origami were designed, fabricated by 3D printing, and compressively tested. The test results present the potential of the approach of QZS. Further, the elastic-frictionless origami tubes were redesigned and simulated to obtain the target stiffness. The cubic term fitting of the load curve was adopted by the harmonic balance method to solve the steady-state vibration response, and then the simulation results obtained by the finite element method (FEM) were compared. The study shows that the designed elastic-frictionless isolator has a good low-frequency vibration isolation performance. The concept of the simple stiffness control method of curved-crease origami provides more practice options for high static and low dynamic stiffness systems.

Complex mode superposition method of non-classically damped systems based on hysteretic damping model with frequency-dependent loss factors

For non-classically damped systems, it is easy to construct a damping matrix based on a hysteretic damping model. However, the damped natural frequency is non-physical, and the frequency response function is non-causal. These two drawbacks can be overcome by the hysteretic damping model with frequency-dependent loss factors. In this paper, a complex mode superposition method for non-classically damped systems based on the hysteretic damping model with frequency-dependent loss factors is proposed. Frequency-dependent loss factors of the natural frequencies are considered in the transient vibration response, while frequency-dependent loss factors of the external excitation frequencies are considered in the steady state vibration response. By decoupling complex modes in state space and physical space, non-proportional damping characteristics of hybrid structures can be effectively considered. Therefore, the proposed method can be used to calculate the time-domain dynamic responses of hybrid structures, whose loss factors are sensitive to frequency variation and have strong non-proportional damping characteristics. The time-domain dynamic responses of numerical examples are compared based on different mode superposition methods. Shaking table tests are performed on cantilever plates with different damping characteristics. The numerical and test results validate the proposed method. Moreover, the proposed method has a wide range of applications.

Theoretical dynamic reduction for steady-state nonlinear vibration responses of complex jointed structures having hysteretic contact behavior

A novel nonlinear dynamic reduction method was developed to determine the steady-state vibration responses of complex jointed structures having hysteretic contact behavior. By using the harmonic balance method to reformulate nonlinear dynamic equilibrium equations into a set of nonlinear algebraic ones, a dynamic reduction technique based on nonlinearity transformation was theoretically developed to iterate nonlinear vibration solutions in the local coordinate that was related only to the degree-of-freedoms of nonlinear joints. Only odd-order harmonic components were truncated to approximate the hysteretic nonlinear contact forces of joint interfaces, which was conducive to further dimension reduction of the nonlinear algebraic equations and iteration matrix. Then, a nonlinear dynamic reduction solver was developed to associate the steady-state nonlinear vibration responses of overall structures with the dynamic characteristics of underlying linear substructures, nonlinear joint models and external excitations. Combined with finite element analysis, the steady-state nonlinear vibration responses of a complex assembled structure with four reduced-order nonlinear joint models were numerically simulated to validate the proposed nonlinear dynamic reduction method by comparing with the literature. The comparative results of nonlinear frequency response functions exhibited good agreement, and the proposed method indicated higher computational efficiency. The experimental investigations of a rubber isolator system were also performed to validate the proposed method, which demonstrated good performance.

Influence of Fiber Angle on the Steady State Periodic Response of Annular Sectorial Plates

Structural elements in the form of annular sectorial plates are widely used in aeronautical, biomedical, and marine engineering. When these components are exposed to dynamic load, large-amplitude vibrations occur. The vibration response analysis based on the linear strain-displacement relation typically yields a conservative estimate and can be used as a first approximation to the actual solution for thin structural elements. As a result, geometric nonlinearity must be incorporated for the efficient and fail-proof design of such elements. This paper shows non-linear and linear steady-state forced vibration responses of the annular sectorial plates. The governing equations of motion have been solved in the time domain by using a modified shooting method and an arc-length/pseudo-arc length continuation technique at the bifurcation point to obtain the complete response curve comprised of stable and unstable branches. This work investigates the effect of fibre angle on the non-linear steady-state forced vibration response of annular sectorial plates. The strain/stress fluctuation throughout the thickness of the laminated annular sectorial plate is determined to explain why hardening nonlinearity has increased. The cyclic fluctuation of the non-linear steady-state normal stress during a time period at the centre of the top and bottom surface is also provided in relation to the forcing frequency ratio of peak amplitude in the non-linear response. Due to the change in restoring forces, the frequency spectra reveal much increased harmonic involvement along with the fundamental harmonic for all fibre angles.

A multibody dynamics approach on a tree-shaped acoustic black hole vibration absorber

Acoustic black hole (ABH) has shown great potential and aroused tremendous interest for vibration and sound control in the past two decades. However, efficient computational methods are still lacking for the design of ABH structures with practically required properties. Herein, a multibody system transfer matrix method is proposed to deal with a system comprising a host structure with an attached ABH cluster, exemplified by a novel tree-shaped ABH dynamic vibration absorber (ABH-DVA). Capitalizing on the flexibility offered by the method, the eigen-solutions and the steady-state vibration response of the combined system are derived and validated by finite element results. Observed vibration reduction and the underlying mechanisms are elucidated in terms of the vibrational energy ratio and vibration spatial distribution in the system. Owing to its rich modal dynamics and energy trapping capability, the proposed absorber demonstrates extraordinary and robust low-frequency and broadband vibration reduction performance from 25 Hz to 2500 Hz for different host structures. This can be ensured by designing its threshold frequency as the lower limit of the targeted frequency range. The proposed method and the vibration-reduction-effect are experimentally validated. This study offers a novel yet general approach for addressing complex ABH structures with various structural components and topologies.

Effects of Lamination Schemes on the Periodic Response of the Annular Sectorial Plates

A broad variety of aeronautical, biomedical, and military applications make use of structures in the form of annular sectorial plates. When in use, they are subjected to stress fluctuations and vibrations with considerable amplitudes. Geometric non-linearity is necessary for efficient design of such components, as the vibration responses obtained using the linear strain–displacement relation are often conservative and can only be utilized to approximate the genuine response. This study examined the influence of different lamination schemes on a non-linear/linear steady-state forced vibration response of annular sectorial plates. Both the modified shooting approach and the arc-lengthalgorithm technique were used for solving the governing equations of motion in the time domain. Frequency-response curves were drawn in their entirety, including both the stable and unstable portions. Sectorial plates of varying layer numbers are studied by analyzing their non-linear dynamic behavior by means of frequency response curves, periodic stress variation, phase plane plots, and frequency spectra. The number of layers shown to have an impact on steady-state responsiveness. In a hardening non-linear manner, the annular sector plates show a peak linear displacement amplitude that is significantly larger than the corresponding values in the non-linear analysis.

Steady State and Transient Vibration Analysis of an Exponentially Graded Rotor Bearing System Having a Slant Crack

The dynamic behaviour of a slant-cracked exponentially graded (EG) rotor-bearing system has been investigated using the finite element method for flexural vibrations. A two nodded EG rotor element has been developed based on the Timoshenko beam theory. Local flexibility coefficients (LFCs) of a slant-cracked EG shaft element have been derived using fracture mechanics concepts to develop the stiffness matrix of a cracked EG element. The steady-state and transient vibration responses of cracked and uncracked rotor systems have been simulated using the Houbolt time marching method. When a crack is present in the shaft, the subharmonic frequency peaks are centred on operating speed in the steady-state frequency responses, whereas on critical speed in the transient frequency responses at an interval frequency corresponding to the torsional frequency. It has been found that the crack parameters such as crack depth and location, temperature gradients and torsional frequencies have a significant influence on natural frequencies and dynamic responses, which could be implemented for efficient rotor crack detection methodologies.

Combining CFD and artificial neural network techniques to predict vortex-induced vibration mechanism for wind turbine tower hoisting

In this study, we consider a vortex-induced vibration suppression scheme for wind turbine tower hoisting process based on computational fluid dynamics. We propose a triangular cross-section spiral flow-disturbing device designed to suppress vibration, and we performed a three-dimensional numerical analysis on the steady-state and transient vibration responses of a simulated wind turbine tower. Considering the large amount of calculations and considerable time consumption of CFD simulation, we propose a hybrid meta-model to predict the results of a CFD simulation. The prediction results show that the accuracy of the prediction model met structural engineering design requirements, and the calculation time was effectively reduced. Thus, we establish a set of digital prediction methods for tower hoisting operation and maintenance.

Dynamic Characteristic Analysis and Structural Optimization Design of the Large Mining Headframe

A large headframe is the core structure of a mine hoisting system. In the traditional design, only the static analysis under load is considered, resulting in the resonance phenomenon of the large headframe in later applications. In order to restrain the resonance phenomenon, a novel method for dynamic characteristic analysis and structural optimization design of a large headframe is proposed. First, the eigenfrequencies and vibration modes of the large headframe were obtained through modal analysis. The results showed that the numerical values of the multi-order eigenfrequencies of the system are relatively close. When subjected to alternating loads of similar frequencies, a large headframe is prone to the resonance phenomenon. Second, the steady-state vibration response of the large headframe was obtained through harmonic response analysis. The results showed that when the frequency of the alternating load is close to the first-order eigenfrequency, the vibration amplitude increases. Meanwhile, the fourth-order and the fifth-order eigenfrequencies are very close. When subjected to alternating loads of similar frequencies, the fourth-order and the fifth-order vibration modes of the headframe will be excited simultaneously. At this time, the headframe will have a strong resonance, which may cause structural damage and other problems. Finally, based on the above analysis, nine different structural optimization schemes are proposed in this paper. Through modal analysis and harmonic response analysis, the nine schemes were compared and analyzed, and the optimal scheme was eventually determined as scheme 9. The method proposed in this paper provides a new concept for the structural optimization design of a large mining headframe, and it has great significance for restraining the resonance phenomenon and ensuring the safety of mining operations.

A Study on Dynamic Behaviour of Thermally Distributed Exponentially Graded Rotor System with Induced Porosities

The present work deals with the dynamic analysis of exponential law-based functionally graded (FG) rotor-bearing systems. The effect of thermal gradation and porosity on dynamic characteristics of FG rotor shafts has been studied first time, using exponential law with a novel two-nodded FG rotor element based on Timoshenko beam theory (TBT). Porous material properties are assorted using exponential law and thermal gradation across the cross section of the FG shaft using exponential temperature distribution (ETD). The effects of temperature and porosity on natural frequencies and whirl frequencies are studied. It has been observed that there is a significant reduction in natural frequencies and whirl frequencies with an increase in volume fraction of porosity and temperature. Attempts have been made to obtain suitable reasons for the behaviours based on the material properties. Furthermore, the steady-state and transient vibration responses have been simulated using the Houbolt time marching technique for the ceramic-based FG rotor shaft system. The result shows the maximum amplitude of the steady-state and transient vibration responses is increased, and the critical speed of the FG rotor system shifts towards the left with the increase in volume fraction of porosity and temperature.

Vibration control of a bi-disk rotor using electro-rheological elastomers

Smart materials are widely used for vibration control of rotating machines. This paper considers the control of vibration of a bi-disk rotor bearing system using electro-rheological elastomer (ERE) rings inserted in the bearings. The bi-disk rotor is selected so that vibration response can be studied in a rotating speed range which includes the first two critical speeds. The rotor bearing system is modelled using the finite element method taking into account the gyroscopic effect of the rotor and the internal damping of the shaft. The influence of the active elastomer when it is subjected to different levels of the electric field on the critical speeds is first investigated. Then, the vibration response is determined for both steady-state and transient running up and running down conditions. Simulation results show the potentials of the ERE when used in passive mode for vibration reduction in the steady-state and transient running up and down conditions of the rotor system. In the active mode, the application of an electric field to the EREs shifts slightly the resonant speeds to higher frequencies due to the increase of the stiffness of the bearings while increasing the vibration amplitudes in the steady-state and transient running up and down conditions in the vicinity of the resonant speeds. Nevertheless, it has been shown that the rotor steady-state vibration response can be reduced at other rotating speed range when an electric field is applied.

VIBRATION CHARACTERISTICS TEST AND ANALYSIS OF HANGING CUP TRANSPLANTER

The vibration of the hanging cup transplanter affected the hole size and tray seedling quality. This paper takes the 2ZP-2 hanging cup transplanter as the research object, studies the vibration characteristics of the transplanter when it works, deduces the mathematical model of the transplanter-soil vibration characteristics, and solves the steady-state vibration response. The Danish B&K vibration test system carried out the vibration test. Studies have shown that the vertical vibration of the transplanter is greater than that of the lateral vibration and the forward vibration. The main factors affecting the vertical vibration of the transplanter are the forward speed of the transplanter, the soil compaction, and the planting depth. When the forward momentum of the transplanter is in the range of 0.8~2.4 km/h, the vertical vibration acceleration increases with increasing forward speed of the transplanter. According to the power, spectral density curve, and spectral curve, the spectrum range of the vertical vibration energy peak is 0~10 Hz, the vibration frequency is between 4.5~5.5 Hz, and the corresponding vibration acceleration amplitude is 0.03~0.33 m/s2. The research results can provide a reference for improving the operating speed, comfort, and structural optimization.

Nonlinear assessment of “breathing” fatigue crack using steady-state vibration responses

Abstract Fatigue cracks can occur in structural components under long-term cyclic loads. Such fatigue cracks can open and close like "breathing" under harmonic excitations. Fatigue cracks can develop and jeopardize safety of structures. Therefore, there is a great need to assess fatigue cracks especially in their early stages. Higher harmonics have been widely used as crack indicators, however, slight opening-closing motion can lead to weak higher harmonics with much less amplitudes compared to the first harmonics. In these situations, higher harmonics can be barely identified for crack detection. Addressing this problem, this study develops the damage assessment approach based on singular spectrum analysis to distinguish crack-induced nonlinearities in steady-state vibration responses, whereby nonlinear assessment of the crack can be achieved. The capability of the approach is experimentally validated on a steel beam with a “breathing” fatigue crack, steady-state velocity responses of which are acquired through non- contact laser measurement using a Doppler laser vibrometer.

The vibration response of transformer windings under harmonic excitations and its applications

The power transformer is a key device in the power grid systems. The mechanical degradation of windings represented by the clamping force looseness will cause the decline of the short circuit withstand ability, and cause further damages. This paper proposes a clamping force diagnosis method for operating windings based on the study of the vibration response. In the theory part, the influence of the load current on the natural frequency of windings is discussed, and the influence of the natural frequency change on the steady-state vibration response is studied to obtain the vibration feature related with the clamping force. The subspace method is used to fuse two vibration sequences with the same characteristics to eliminate the measurement error. In the experiment, the free vibration test was performed on a short circuit on-load winding structure. It was proved that the natural frequency change can be extracted from the relationship between the amplitude change of the 100 Hz component and the current change. In the field tests, the vibration sequences of two typical transformers were compared, and the results show that the vibration feature extracted from the relationship between the amplitude variation and the current change contains the structural information of windings.

Dynamic coupling vibration of rotating shaft–disc–blade system — Modeling, mechanism analysis and numerical study

This study aims at revealing the coupling vibration mechanism of RSDB system. First, a comprehensive coupling dynamic model of RSDB system including the shaft bending, shaft torsion, blade bending, and blade radial deformation is formulated based on continuum beam theory and Lagrange equation. The validity of the proposed dynamic model is verified through a comparison with FEM and experimental results. Second, the general coupling mechanism of RSDB system is theoretically interpreted based on the proposed analytical coupling vibration model, which indicates that the blade vibration and shaft vibration, especially the shaft torsion vibration and blade bending vibration, may affects each other significantly. At last, the steady-state coupling vibration responses and the effects of structure parameters on the coupling vibration of RSDB system are comparatively investigated through numerical investigations. The comparative results suggest that the blade setting angle, blade–blade coupling stiffness, and the disc location on shaft will greatly affect the coupling vibrations. It worth more attention that the general coupling vibration mechanism obtained by analytical analysis and the results obtained by numerical study can be cross-verified by each other.

An analytical model for predicting noise radiated by switch reluctance electric motors

Switch reluctance motors (SRM) have become a prominent alternative for electric vehicles in recent years due to their simple, high power density architecture and cost-effective manufacturability. Despite its potential, NVH problems have been one of the biggest challenges for SRM's implementation. Vibration and noise generated by the SRM are mainly caused by phase switching related torque ripple, unbalanced electromagnetic forces from air gap variations and lamination problems. Our proposed model is an analytical noise radiation prediction model which relates geometrical, material and electrical design inputs to radiated sound power. The electromagnetic part of the model is nonlinear with saturation and provides back-emf and flux linkage by receiving design inputs. The computed magnetic energy, radial and tangential rotor forces are utilized as excitation sources to a continuous shell dynamic model to obtain the steady-state vibration response. Finally, surface velocities obtained from the shell model are used to calculate sound power. Utilizing a shell structure provides axial, radial and tangential information on the casing by considering the effect of magneto-restrictive forces of laminations, torque ripples and unbalanced electromagnetic forces. The effect of air gap, lamination error, and stator and rotor geometry on sound radiation are studied through an example case study.