Steady-state Vibration Research Articles

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.

Optimal grounded inerter-based dynamic vibration absorber for controlling inertial force-induced vibrations in rotating machines

ABSTRACT This paper proposes the optimal design of GIDVA for inertial force-induced vibrations reduction in rotating machinery considered as primary system. For this purpose, simple-to-use closed-form suboptimal design solutions were first derived based on the extended fixed point theory (EFPT). Then, the H ∞ optimization problem was formulated, which was reduced to solving a constrained system of multi-variable nonlinear equations. Thus, by considering the analytical solutions based on the EFPT as a starting point in order to accelerate the convergence of the system, the Newton-Raphson method was applied to numerically solve the nonlinear system equations for the range values of the mass ratio 1 % ≤ μ ≤ 10 % . For this range of values, the proposed GIDVA was compared with the traditional dynamic vibration absorber (TDVA) and the non-traditional dynamic vibration absorber (NDVA). The results of comparison shown that the proposed GIDVA is 47.76–70.66% and 44.82–70.54% superior to the optimized TDVA and NDVA, respectively, in the steady state vibrations suppression of primary system under inertial excitation, and can widen the effective frequency band 70.22–80.98% and 64.07–80.95% superior to TDVA and NDVA, respectively. These results were confirmed in the time domain simulation of the primary system response considered as a centrifuge pump. For this range of values, the proposed GIDVA was compared with the TDVA and the NDVA. The results of comparison shown that the proposed GIDVA is 47.76–70.66% and 44.82–70.54% superior to the optimized TDVA and NDVA, respectively, in the steady state vibrations suppression of primary system under inertial excitation and can widen the effective frequency band 70.22–80.98% and 64.07–80.95% superior to TDVA and NDVA, respectively. These results were confirmed in the time domain simulation of the primary system response considered as a centrifuge pump.

Chaos identification and parameter optimization for vibration suppression in composite bistable structures

An innovative composite bistable system (CBs) coupled of prestressed linear and buckled beams with fixed ends has been designed to enhance the damping and suppress the dynamic response. The dynamic equations of the system are derived by applying Hamilton's principle. When the amplitude of the harmonic excitation exceeds the critical value, the buckling beam of the bistable system enters a state of inter- and in-well combined oscillations. Chaotic motion is present. This motion disperses the vibrational energy over a wide frequency band. The vibration in the high-frequency range significantly enhances the dissipation of the viscoelastic layer, leading to an effective suppression of the system's vibration. Approximate analytical solutions and numerical validation of the chaotic and non-chaotic boundaries of the steady state vibration of the bistable system are carried out based on the harmonic balance method (HBM) and the Melnikov method. The Particle Swarm Optimisation (PSO) algorithm is used to optimise system parameters for specific operating conditions. This optimisation process provides guidance for the design and improvement of CBs systems.

Suppression of multi-modal torsional vibration of the long-shafting rotor system with nonlinear piecewise NES

Long-shafting rotor systems tend to cross multi-order natural frequencies at start up and coast down, and the multi-order modes of vibration are excited by various excitations. The bending vibration can be limited to the single rotor, but the torsional vibration is obviously coupled between each single rotor, which leads to more problems. Therefore, the current work proposes to suppress the multi-modal torsional vibration of long-shafting rotor system by a single nonlinear piecewise NES (PNES). The dynamic models of long-shafting rotor system and the long-shafting rotor-PNES system are established in which the PNES is used to suppress the first two-order torsional vibration. The genetic algorithm (GA) is used to optimize the multi-modal transient and steady-state torsional vibration suppression results of PNES to obtain optimal parameters design, respectively. And the influences of the parameters of long-shafting rotor-PNES system and the additional position of PNES on the vibration suppression effect of PNES are also discussed. The experimental verification of the suppression of the first two-order steady-state torsional vibration of the long-shafting rotor system by a PNES is carried out. In numerical simulation, the percentage of accumulated energy dissipation of the PNES can reach 52.2 % for the multi-modal transient torsional vibration, and the inhibitions of the three torsional vibrations by the PNES are about 71.6 % and 77.9 %, 67.5 % and 81.2 %, 68.5 % and 69.5 % in the multi-modal steady-state torsional vibration suppression. In the test, only the suppression of the first torsional vibration by PNES is tested, that the PNES can reduce the first two-order steady-state torsional resonances of the first torsional vibration of the long-shafting rotor system by 62.1 % and 65 % respectively.

Dynamic modeling and nonlinear analysis for lateral–torsional coupling vibration in an unbalanced rotor system

Imbalance is a common problem in rotating machinery. In addition to the unbalanced force, lateral–torsional coupling vibration can also be induced by the mass imbalance. However, the related nonlinear behaviors and coupling mechanism are still not well understood, especially when lateral-angular motion is incorporated in a model. In this paper, a model for coupling lateral and torsional vibrations of an unbalanced rotor is developed by Lagrangian dynamics, in which the coupling of lateral-angular and torsional motions is first considered. Based on the model, the lateral responses, torsional responses, coupling excitation characteristics and coupling mechanisms are investigated in detail. The results show that mass imbalance can lead to nonlinear time-varying characteristics in the dynamic equations. When the imbalance is small, the lateral vibration can be degraded to a traditional imbalance problem. Meantime, very small transient torsional vibration with torsional natural frequency ft in frequency domain can be observed at the beginning. The coupling excitation applying on the torsional DOF caused by the lateral synchronous whirling is demonstrated to behave the feature of static torque, consequently, there actually exists no steady-state torsional vibration in small unbalanced rotor. Under largely unbalanced conditions, both resonant and unstable behavior can be observed in the lateral and torsional responses. The combined resonance is induced by the lateral translational–torsional coupling, while the instability is caused by lateral angular–torsional coupling. By increasing the lateral damping, the instability phenomenon can be suppressed, and the lateral amplitude in the resonance region can be effectively reduced.

A novel test apparatus to study the mechanism of harmonic normal force on fretting wear

Fretting wear significantly affects the hysteresis behavior of the contact surface and leads to joint degradation due to material removal. Although numerous tests have been conducted to investigate fretting wear under constant normal force, the applied mechanical load on the contract surface more or less deviates from that in the operational condition. The normal force is typically time-varying due to the coupling of tangential and normal vibration, rather than remaining constant. The time-varying normal force can be regarded as the superposition of harmonics for steady-state vibration which is widely encountered in engineering. In this paper, we investigate how would the harmonic component of the normal force affect the evolution of interface morphology and contact parameters with fretting wear. To do that, we design a novel fretting wear test apparatus that generates the stable harmonic normal force. The normal force is controlled by a closed-loop control system, ensuring its consistency throughout the entire wear test. In addition, we develop various subsystems to realize different functions like support, constraint and measurement, enabling the acquisition of hysteresis loops under different normal forces. The critical performances of the test apparatus (e.g. force transfer, closed-loop control and repeatability) are also demonstrated. Results indicate that the harmonic normal force can significantly influence the hysteresis loop and the energy dissipation distribution on the contact surface. In comparison to the constant normal force, the worn surface has a higher degree of non-uniformity, characterized by alternating pits and prominences. Furthermore, the friction coefficient exhibits an outstanding upward trend after reaching a steady state, while the tangential contact stiffness continuously weakens as fretting wear progresses. These unique evolution characteristics could be attributed to the non-uniform energy dissipation caused by the harmonic normal force acting across the fretting space. This viewpoint can be partly validated through observing the interesting hysteresis loops, which are captured by the test apparatus.

Integrated transmission vibration reduction technology based on the “isolating-reducing-optimizing” method

Hydro-mechanical transmission devices that can transfer power from the engine to the action mechanism are widely used in heavy-duty vehicles. It has many components, and the operating conditions are complex and changeable. Multi-source excitations and load disturbances easily aggravate vehicle vibrations, which has an adverse impression on the running quality of the whole vehicle. Therefore, it is necessary to carry out research on the vibration problems caused by multiple excitation sources. Thus, in this paper, a new vibration reduction method of “isolating-reducing-optimizing” for hydro-mechanical transmission devices is proposed to optimize such devices in multiple dimensions to improve their dynamic responses. This novel method optimized the coupling stiffness to isolate the vibration caused by the engine fluctuating torque. Moreover, the vibration energy decoupling method and optimization method are used to decouple the rigid body vibration of the mounting system, and gear profile modification method based on a dynamic model is used to attenuate the internal high frequency vibration caused by the gear meshing excitation. Finally, the steady-state conditions and cycle conditions vibration tests of the transmission system before and after optimization are carried out. The time and frequency domain vibration responses of the transmission system before and after optimization are compared to verify the effectiveness of the optimization method.

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.

An Averaging Method for Phase Noise Studies of Quartz Phononic Frequency Combs.

A fast and accurate averaging method was derived and developed for the analysis and design of quartz phononic frequency combs. The phononic frequency combs were obtained from a pair of coupled nonlinear Duffing equations for quartz resonators by solving the equations in the time domain and performing a fast Fourier transformation (FFT) of the steady-state vibrations of the time series. Noise simulations were added to the drive frequency to study noise transfer characteristics between the drive signal and the resonances of the phononic frequency combs produced in 100-MHz quartz shear-mode resonators. Our new method averaged out the carrier frequency, thus allowing for a fast and efficient computation at parts per million accuracies of noise close to the carrier ( ∼ 10 Hz). The goal of our study was to develop methods and resonator requirements for engineering the properties of the phononic frequency combs for low-noise clock 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.

Dynamics of Structures, Frames, and Plates with Viscoelastic Dampers or Layers: A Literature Review

The paper is devoted to a review of recent achievements in the field of dynamic analysis of structures and structural elements, such as beams and plates, with embedded viscoelastic (VE) dampers and/or layers. The general characteristics of VE materials, their rheological models, and methods of parameters identification are discussed. New formulations of dynamic problems for systems with VE elements are also reviewed. The methods of determination of dynamic characteristics, together with the methods of analysis of steady-state and transient vibrations of such systems, are also discussed. Both linear and geometrically non-linear vibrations are considered. The paper ends with a review of the methods of sensitivity and uncertainty analysis, and the methods of optimization, for structures with VE elements.

Vibration response analysis of a two-stage vibration isolation system for large airborne equipment

PurposeThis paper aims to propose a two-stage vibration isolation system for large airborne equipment to isolate aircraft vibration load.Design/methodology/approachFirst, the vibration isolation law of the discrete model of large airborne equipment under different damping ratios, stiffness ratios and mass ratios is analyzed, which guides the establishment of a three-dimensional solid model of large airborne equipment. Subsequently, the vibration isolation transfer efficiency is analyzed based on the three-dimensional model of the airborne equipment, and the angular and linear vibration responses of the two-stage vibration isolation system under different frequencies are studied.FindingsFinally, studies have shown that the steady-state angular vibration at the non-resonant frequency changes little. In contrast, the maximum angular vibration at the resonance peak reaches 0.0033 rad, at least 20 times the response at the non-resonant frequency. The linear vibration at the resonant frequency is at least 2.14 times the response at the non-resonant frequency. Obviously, the amplification factor of linear vibration is less than that of angular vibration, and angular vibration has the most significant effect on the internal vibration of airborne equipment.Originality/valueThe two-stage vibration isolation equipment designed in this paper has a positive guiding significance for the vibration isolation design of large airborne equipment.

Chiral metamaterial -inerter nonlinear energy sink for torsional vibration suppression of the rotor system

This paper explores the use of a chiral metamaterial inerter nonlinear energy sink (CINES) for suppressing torsional vibrations in a rotor system. The compress-torsion coupling effect of the chiral metamaterial is adopted to construct the inerter amplification mechanism, which is then combined with the NES to form CINES. The dynamic model of the rotor-CINES coupled system is established, the transient torsional vibrations and steady-state torsional vibrations are optimized by genetic algorithm and the optimal parameters for numerical simulation are obtained. Then, the vibration suppression capability of CINES is evaluated. Moreover, experiments on the CINES are carried out to verify the theoretical results. The results show that the cumulative energy dissipation rate of CINES in transient torsional vibration suppression can reach 89.2 %, and the percentage of vibration suppression in steady-state torsional vibration suppression can reach 88.5 %. Furthermore, the CINES saves more than 10 times its moment of inertia while achieving the same vibration reduction ratio.

Damping of vibrations of smart-systems incorporating piezoelectric elements and shunt circuits with parameters derived from the models of continuum and discrete mechanics

The paper considers tasks related to the search for the parameters of smart-systems with piezoelectric elements, to the electroded surfaces to which elements of electrical circuits are connected. The required parameters should provide the maximum damping properties of the simulated systems at free and steady-state forced vibrations. The paper presents the results of calculations of smart-system parameters providing maximum damping properties at free and forced vibrations, which were performed in the framework of continuum and discrete mechanics models. The results demonstrate variants for which the parameters found based on continuum mechanics models provide higher vibration damping rates.

Calculation Model for the Steady-State Vibration Amplitude of a New Type of Cascaded Composite Structure-Based Ultrasonic Transducer

The steady-state vibration amplitude is an important performance indicator of high-frequency ultrasonic transducers for ultrasonically assisted manipulating, machining, and manufacturing. This work aimed to develop a calculation model for the steady-state vibration amplitude of a new type of dual-branch cascaded composite structure-based ultrasonic transducer that can be used in the packaging of microelectronic chips. First, the steady-state vibration amplitude of the piezoelectric vibrator of the transducer was derived from the piezoelectric equation. Second, the vibration transfer matrices of the tapered ultrasonic horns were obtained by combining the vibration equation, the continuous condition of the displacement, and the equilibrium condition of the force. Calculation models for the steady-state vibration amplitude of the two working ends of the transducer were then developed. A series of exciting trials were carried out to test the performance of the models. Comparison between the calculated and measured results for steady-state vibration amplitude showed that the maximum deviation was 0.0221 μm, the minimum deviation was 0.0013 μm, the average deviation was 0.0097 μm, and the standard deviation was 0.0046 μm. These values indicated good calculation accuracy, laying a good foundation for the practical application of the proposed transducer.

Damping of Dry Sand in Resonant Column-Torsional Simple Shear Device

The damping ratio values of three different Danube sands were measured in the Resonant Column-Torsional Simple Shear device (RC-TOSS). The distinctive configuration of the RC-TOSS device employed in this investigation enabled the performance of both tests using a single sample. This research estimates and compares the damping ratio values measured with three distinct methods (two of which are in the RC test): The Free Vibration Decay (FVD), the Steady-State Vibration (SSV) methods, and the method of calculating the damping ratio from the hysteretic loops generated in the TOSS test. Both dense and loose samples were tested up to a peak-to-peak amplitude shear strain of 1%. The device provides measurements over a wide range of shear strain amplitudes. The results support the employment of the SSV methods at low strains (below 0.005%), while the FVD method gives a better estimate at higher strains (above 0.03%). The two methods and the TOSS results are in agreement with each other between 0.005% and 0.03%. The effect of the number of cycles on the damping ratio was investigated where a significant decrease was observed in the damping ratio with an increasing number of cycles. A parameter is introduced to describe the rate of this decrease, which should be considered during the structural design to reduce maintenance and life-cycle costs and enhance sustainability.

Dynamic response of nanobeams with randomly distributed multiple vertical cracks

This paper aims to systematically study the steady-state forced vibrations of nanobeams weakened by multiple vertical cracks, which may conform to random or determined distributions. The variable separation and the Laplace transform methods are employed to obtain the Green’s function of beams with one crack for three typical kinds of beams, namely Hinged–Hinged, Clamped–Clamped, and Clamped–Free. Green’s functions for the nanobeams with an arbitrary number of cracks are figured out from the counterparts of beams with one crack by virtue of the massless elastic spring model and the transfer matrix method. The validity of the present solutions is discussed in comparison to those in literature in degenerated situations. The influence of statistical parameters of the random cracks on the dynamic responses of the weakened beams is studied in a systematic fashion. The present work may be beneficial to future structural designs on the nano-scales.

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.

Vibration quality and ride comfort investigation with transient excitation in a ground vehicle simulator environment

Ground vehicle ride comfort is closely associated with vibration quality as perceived by the human occupants of a vehicle. While outstanding vibration quality is a desirable outcome of the vehicle design process, it is challenging to relate customer perceptions of vibration quality to customer-facing vehicle responses measured inside of a vehicle. Previous work with steady-state vibration has shown that human perception of vibration can be compared to how humans perceive sound, where both the amplitude and frequency of the excitation will alter its perception. Thus, a simple measurement of vibration level is insufficient to predict comfort. In this work, an analysis of human perception of transient vibration is undertaken using a ground vehicle simulator. Physical measurements from inside four different production vehicles are presented including accelerometer, microphone, and action camera recordings. The test event performed is travel over a cleat, with varying vehicle speeds and tire inflation pressures. The physical measurements are used as inputs to the simulator. A comparison is made between the perceived vibration quality of the vehicles as reported by an expert human jury and the physically measured vehicle responses. A ranking of 36 cleat test events for vibratory harshness from best to worst is presented.