Longitudinal Vibration Response Research Articles

Vibration control of a longitudinally vibrating rod connected through an elastic coupler by installing a nonlinear vibration reduction coupler

To study the potential application of the longitudinal vibration control of a rod coupling system by employing NESs, this study proposes a nonlinear vibration reduction coupler (NVRC), where the NVRC can be regarded as the NES which is installed between the two elastic structures. The introduction of NVRC is aimed at controlling the longitudinal vibration of the rod system. Based on the Lagrange method (LM), the longitudinal vibration analysis model of a rod system connected through a linear elastic coupler and an NVRC is established, in which the longitudinal vibration responses of the rod system with an NVRC are predicted. In the numerical analysis, the correctness and stability of LM in calculating longitudinal vibration responses of the rod system with an NVRC are verified, where the chosen reasons for observation points are explained. It can be concluded that the introduction of NVRC can reduce the vibration of each resonance area of the rod system at the same time. Under certain parameters, the vibration characteristics of the rod system are changed by employing NVRC. Furthermore, nonlinear stiffness and viscous damping of NVRC can be chosen as the adjustable parameters that strengthen the vibration reduction ratio of the rod system. Overall, the longitudinal vibration of the rod system can be effectively controlled by choosing reasonable parameters of NVRC. The introduction of NVRC provides an effective way to control the longitudinal vibration of the rod system.

Longitudinal vibration control of a double-rod system by employing nonlinear energy sinks

This study aims to potential the potential utilization of nonlinear energy sinks (NESs) for controlling longitudinal vibrations in a double-rod system. The research introduces a longitudinal vibration prediction model for a double-rod system equipped with NESs. The generalized Hamilton principle is employed to derive governing equations of the double-rod system. The longitudinal vibration responses of the double-rod system are numerically solved through the application of Galerkin truncation method. The longitudinal vibration responses of the double-rod system are impacted by NESs, as they yield accurate numerical results. The installation of both NES 1 and NES 2 concurrently is recommended for mitigating the vibration of the double-rod system. Under reasonable single-frequency excitations, modifying the parameters of NESs can significantly alter both the vibration state and magnitudes of vibration in the double-rod system. Furthermore, the synchronous optimization of parameters in NES 1 and NES 2 is crucial for effectively controlling vibrations in the double-rod system. Sensitive parameter areas of NESs provide the possibility of controlling the vibration of the double-rod system by utilizing NESs.

A study of analyzing longitudinal dynamic behavior of a double-rod system with longitudinal nonlinear supports

In engineering, shafting systems are typically subjected to longitudinal vibration excitations, which may result in unwanted vibration. To study the control of longitudinal vibration in shafting systems, they can be simplified to rod structures. Currently, engineers have attempted to apply the nonlinear principle to design nonlinear supports to control the vibration of flexible structures. However, the flexible structures referenced in the literature are usually composed of a single component, which limits the application of nonlinear supports to more complex structures. To explore the potential application of nonlinear supports in marine engineering, this work introduces a longitudinal vibration prediction model for a double-rod system equipped with longitudinal nonlinear supports. The generalized Hamilton principle is used to derive the governing equations for the double-rod system with longitudinal nonlinear supports. The longitudinal vibration responses of the double-rod system are numerically solved using the Galerkin truncation method. The numerical results confirm that a 1-term truncation number guarantees the stability of the longitudinal vibration prediction model. Under certain conditions, the longitudinal vibration responses are significantly affected by longitudinal nonlinear supports. It is recommended to install longitudinal nonlinear supports on both Rod 1 and Rod 2 simultaneously to suppress vibration in the first two main resonance orders. With reasonable excitations, the vibration state and magnitudes of the double-rod system can be effectively controlled by adjusting the longitudinal nonlinear supports. Complex longitudinal vibration responses are more readily induced by altering the parameters of the longitudinal nonlinear support installed on Rod 1. Choosing appropriate parameters for the nonlinear supports on Rod 1 and Rod 2 positively contributes to the reduction of vibration in the double-rod system.

Strain Analysis of Vacuum Interrupter Bellows

Abstract The vacuum interrupter (VI) is the main part of the vacuum circuit breaker (VCB), and an essential component of its design is the bellows. In future high-voltage VCBs, faster switching speeds will lead to higher dynamic impact-type loads, and the oscillation of the bellows after the contact opening and closing could shorten the mechanical life of the bellows. This article provides a 1D computer-aided engineering (CAE) analysis of the strain of the bellows oscillation in a VI caused by the axial load from the impact of the contact. A formula for the oscillation is derived using the longitudinal vibration response of a mass-spring 1D model, calculated by the convolution integral of the input and the impulse response function of the vibration modes. The proposed 1D analysis was verified and shown to be effective under four different experimental conditions. This article also provides a 1D CAE analysis of the mechanical life of the bellows. These 1D analyses estimate the effect of stress reducing countermeasures on the bellows' lifetime without increasing its size.

Longitudinal vibration responses of a double-rod system coupled through a nonlinear element

Longitudinal vibration responses of a double-rod system coupled through a nonlinear element

Research on vibratory & oscillatory coexistence nonlinear dynamics based on drum-subgrade coupling model

In order to provide theoretical basis for subgrade continuous compaction monitoring technology and intelligent compaction technology, it is urgent to study the nonlinear coupling dynamics of drum and subgrade in the process of vibratory and oscillatory compaction. In this study, a nonlinear coupling dynamic model of 6-degree-of-freedom vibratory drum-subgrade is proposed, which considers the elastoplastic characteristics of the subgrade during compaction. According to the geometric and physical relationship between the drum and the subgrade in the contact area, the longitudinal dynamic contact force coefficient is derived, which establishes the coupling relationship between the drum-subgrade vertical vibration and longitudinal oscillation. Combined with the evolution law of material properties of subgrade in actual compaction operation, the nonlinear dynamic behavior of coupling system in the whole compaction period is analyzed. The results show that the chaotic compaction state in the vertical direction of the drum may occur during the full compaction cycle, which is related to the excitation parameters of the drum. Furthermore, the excitation parameters are more sensitive to the longitudinal one-cycle vibration response of the system than the vertical vibration response. Through the response characteristics of the drum and the subgrade and the bifurcation diagram of the drum, it is proved that the proposed model can effectively simulate the longitudinal-vertical nonlinear coupling characteristics of the drum and subgrade, and can provide a premise for theoretical exploration for the vibratory & oscillation compaction mode.

Peridynamics with strain gradient for modeling carbon nanotube under static and dynamic loading

This study couples peridynamic (PD) theory with strain gradient elasticity (SGE) theory to investigate the combined effect of PD and SGE length scale parameters on size effect. The SGE theory introduces a length scale parameter in the stress–strain relations to account for size effect. The solution to the classical form of SGE equation of motion requires two additional non-classical boundary conditions arising from the presence of fourth-order spatial derivatives. The wave dispersions level off as the wave number increases as observed in real materials. The PD theory allows for nonlocal interactions of a material point within its horizon which serves as the PD length scale parameter. The equation of motion emerges in the form of an integral equation and the internal forces are expressed through nonlocal interactions (bonds) between the material points within a continuous body. It is a reformulation of classical continuum mechanics which is free of spatial derivatives. The numerical results concern a previously considered carbon nanotube (CNT). Under quasi-static axial loading, the results indicate an increased strengthening effect along the length of the tube. Under dynamic loading, its longitudinal vibration response is captured through explicit time integration. The numerical predictions capture the reference solutions corresponding to the classical SGE equation.

In-situ experiment research on environmental vibration transmission characteristics of air-rail combined airport

The rail transit and civil aviation are the important components of the comprehensive transportation system. The development of the air-rail combined transport is an important effective way for China to become a country with strong transportation network. The transportation intersection form of railway under-passing airport transportation hub gradually becomes popular, thus the environmental vibration due to the under-passing railway cannot be ignored. In this paper, a large-scale integrated transportation hub construction project was taken as an example to analyze the transmission rule of environmental vibration due to the high-speed railway at the speed of 350 km/h under-passing airport in terms of the time domain and frequency domain by means of the on-site in-situ wheel-drop test. The research results show that the vertical vibration response of the ground surface along the normal direction of the railway is greater than those of the other two directions within 40 m from the centerline of the track. A vibration amplification zone appears within 5-40 m. The longitudinal vibration response of the ground surface is greater than those of the other two directions within 40-70 m. The local vibration amplification zones appear within 5-20 m and 30-60 m. The lateral vibration level of the ground surface along the normal direction of the railway increase gradually, but attenuates at the distance of 40 m and 10 m with the maximum attenuation rate of 0.67. The vertical vibration level amplifies at the distance of 60 m. The longitudinal vibration level attenuates at the distance of 20 m with the maximum attenuation rate of 0.3, but amplifies at the distance of 40 m with the maximum amplification rate of 1.8.

Longitudinal Vibration Transmission Control of Marine Propulsion Shafting with Friction Damper Integrated into the Thrust Bearing

Propeller-induced longitudinal vibration resonance in marine propulsion shafting systems causes great harm to the hull structure and is the primary source of shipboard noise. Integrating a friction damper with designed parameters into thrust bearings can prevent these issues. To investigate the performance of the damper-integrated thrust bearing in longitudinal vibration transmission control, an experimental and theoretical study is carried out in a laboratory-assembled test rig, which consists of components similar to the existing marine propulsion system. We developed a prototype of a thrust bearing designed with a friction-damping generation that allows switching from two supporting states, i.e., damper-connected and damper-disconnected states. Furthermore, a nonlinear analysis method for friction dampers is proposed. By this method, the way in which the friction damper changes the dynamic characteristics of the shafting system is analyzed. Based on the test rig, the acceleration frequency response function (AFRF) of the thrust bearing with and without a friction damper is measured. By comparison, the effectiveness of the friction damper is proved. The experimental results show that the friction damper suppresses the shafting longitudinal vibration response in a broadband frequency range and also confirms the stability of the damping effect, which does not change with the shafting rotational speed or static thrust from the propeller.

Longitudinal vibration analysis of FG nanorod restrained with axial springs using doublet mechanics

In the current paper, the free longitudinal vibration response of axially restrained functionally graded nanorods is presented for the first time based on the doublet mechanics theory. Size dependent nanorod is considered to be made of functionally graded material consist of ceramic and metal constituents. It is assumed that the material properties of the functionally graded nanorod are assumed to vary in the radial direction. The aim of this study is that to investigate the influences of various parameters such as functionally graded index, small size parameter, length of the nanorod, mode number and spring stiffness on vibration behaviors of functionally graded nanorod restrained with axial springs at both ends. For this purpose, Fourier sine series are used to define the axial deflection of the functionally graded nanorod. Then, an eigenvalue approach is established for longitudinal vibrational frequencies thanks to Stokes’ transformation to deformable axial springs. Thus, the presented eigenvalue solution method is attributed to both rigid and deformable boundary conditions for the axial vibration of the functionally graded nanorod. With the help of the results obtained with the presented eigenvalue problem, it is observed that the parameters examined cause significant changes in the frequencies of the functionally graded nanorod.

Active Airframe Vibration Control Simulations of Lift-offset Compound Helicopters in High-Speed Flights

This paper studies the simulations of active airframe vibration controls for the Sikorsky X2 helicopter with a lift-offset coaxial rotor. The 4P hub vibratory loads of the X2TD rotor are obtained from the previous work using a rotorcraft comprehensive analysis code, CAMRAD II. The finite element analysis software, MSC.NASTRAN, is used to model the structural dynamics of the X2TD airframe and to analyze the 4P vibration responses of the airframe. A simulation study using Active Vibration Control System(AVCS) with Fx-LMS algorithm to reduce the airframe vibrations is conducted. The present AVCS is modeled using MATLAB Simulink. When AVCS is applied to the X2TD airframe at 250 knots, the 4P longitudinal and vertical vibration responses at the specified airframe positions, such as the pilot seat, co-pilot seat, engine deck, and prop gearbox, are reduced by 30.65 ~ 94.12 %.

Effect of Lift-offset Rotor Hub Vibratory Load Components on Airframe Vibration Responses of High-Speed Compound Unmanned Rotorcrafts

This paper investigates numerically the effect of rotor hub vibratory load components on the airframe vibration responses of high-speed compound unmanned rotorcraft (HCUR) using a lift-offset coaxial rotor, wings, and two propellers. The rotor hub vibratory loads are predicted using a rotorcraft comprehensive analysis code, CAMRAD II, and the airframe vibration responses are calculated by a finite element analysis software, MSC.NASTRAN. It is shown that the vibratory hub pitch moment of a lift-offset coaxial rotor is the most dominant component for both the longitudinal and vertical vibration responses at four specified locations of the airframe. Keywords: HCUR, Lift-offset ë™ì¶•ë°˜ì „ 로터, 로터 진동 하중, 기체 진동 응답 Keywords: High-speed Compound Unmanned Rotorcraft, Lift-offset Coaxial Rotor, Rotor Hub Vibratory Loads, Airframe Vibration Responses

Forced vibrations of size-dependent rods subjected to: impulse, step, and ramp excitations

Forced longitudinal vibration response of nonlocal strain gradient (NLSG) rods is scrutinized and compared with nonlocal (NL), strain gradient (SG), and classical (CL) rods. In this respect, size-dependent kinematics and extended Hamilton’s principle are utilized to derive governing equations of motion. For the first time, forced longitudinal vibratory behavior of non-classical and classical rods is presented including three different types of non-harmonic forcing functions: unit impulse, unit step, and unit ramp. Forcing terms are functions of both temporal and spatial domains. Nonlocal strain gradient and nonlocal rod models take into account the Laplacian of the external forcing term as well. This implies that in models including nonlocal effects (NL, NLSG-h, s, n), forcing term is reduced by the order of its Laplacian with respect to spatial variables. Effect of non-classical parameters over time-domain response of the system is analyzed in details. Results are compared with benchmark for the sake of evaluation. This implies the necessity of adopting proper values of non-classical theory and corresponding parameters through analysis of forced oscillations of small-scaled size-dependent rods. Such external excitations are modeled using spatial Dirac delta function as a concentrated force applying at the critical point of mode shape function. Findings of this research reveal the underlying impact of non-classical theorems along with corresponding non-classical parameters over the prediction of the response of size-dependent rod elements for wide range of applications such as in musical instruments, navigating equipment, gyroscopes, control of robotics, sensors and actuators, resonators, energy harvesters, and vibration control systems.

Nonlinear flow-induced vibration response characteristics of a tubing string in HPHT oil&gas well

This study has established a nonlinear flow-induced vibration (FIV) model of tubing string by using the micro-element method and energy method with the Hamiltonian variational principle, in order to address the vibration failure problem of the tubing string induced by high-speed fluid flow in the tubing of high temperature and high pressure (HPHT) oil&gas well. This model considered factors such as borehole trajectory changes, wellbore temperature and pressure variations, self-weight of the tubing string, and the contact-impact of the tubing-casing. Based on the structural parameters of the M high-yield gas well in the South China Sea, a simulation experiment of the tubing string vibration was conducted according to the similarity principle. A comparison between the experiment data and theoretical data showed that the calculation accuracy exceeded 84%, which verified the correctness and validity of the nonlinear vibration model established in this study. On this basis, the influence of the production rate, well inclination angle, well section length, and packer position on the vibration response characteristics of the tubing string were analyzed systematically. It was found that: ①with the increase of the production rate, the vibration response of the tubing string became clearer. The production rate interval that showed abrupt changes was from 0.9 ~1.2 million m3/day. Thus, during designing, such production rate intervals with abrupt changes should be avoided to ensure the safe usage of the tubing strings; ②with the increase of the well inclination angle, the vibration response of the tubing string decreased to a certain extent. Regarding the influence of the well section length, the longitudinal vibration response of the tubing string was most sensitive to the length of the vertical section, followed by the steady inclined section, and finally the angle building section. Thus, during designing, the length of the vertical section should be reduced, while the length of the angle building section and the well inclination angle of the steady inclined section should be increased; ③As the position of the production packer moved up, the vibration responses of the lower and middle tubing strings both decreased, and the optimal packer position was between 3,549 and 3,666 m. The research results can serve as an effective analysis tool and provide a theoretical basis for the safety design of the tubing strings in the HPHT oil&gas well site.

Vibration reduction simulations of a lift-offset compound helicopter using two active control techniques

This study attempts to alleviate numerically both the rotor and airframe vibrations of a lift-offset compound helicopter using two different active vibration control techniques. The XH-59A helicopter is considered as the lift-offset compound helicopter in the present work. The IBC (Individual Blade pitch Control) and AVCS (Active Vibration Control System) are applied to the lift-offset coaxial rotor and airframe, respectively, of the XH-59A to reduce vibration. Without or with IBC, the hub vibration of the XH-59A rotor is predicted using the rotorcraft comprehensive analysis code, CAMRAD II. For the AVCS simulation with a MIMO (Multi Input and Multi Output) model, various tools such as NDARC, MSC.NASTRAN, and MATLAB Simulink are used. At a flight speed of 200 knots, firstly, the 3/rev hub vibration of XH-59A rotor is reduced by 62% when IBC is used. Secondly, when AVCS is applied to the airframe, which is excited by the rotor vibratory loads reduced using IBC, the 3/rev longitudinal and vertical vibration responses at the specified airframe positions such as the pilot seat, auxiliary propulsions, and rotor-fuselage joint are reduced from 90.32 to 99.55%, when compared to baseline results without active vibration techniques.

Forced vibration analysis of non-local strain gradient rod subjected to harmonic excitations

In this technical paper, forced longitudinal vibration response of non-local strain gradient (NLSG) rods is obtained and compared with that of non-local (NL), strain gradient (SG), and classical (CL) rods. To this end, size-dependent kinematics along with extended Hamilton’s principle are used to derive governing equations of motion. Classical clamped-free and non-classical clamped-forcing free-strained boundary conditions are adopted. For the first time, forced longitudinal vibratory behavior of non-classical and classical rods subjected to harmonic external loading is presented. In investigation process, effects of non-classical parameters over mechanical impedance, resonance frequency, mechanical frequency response function (FRF) (transmissibility), time period, and time-domain response are determined. Effect of non-classical parameters over time-domain response of the system is analyzed in details. Results are compared with benchmark approving good level of accuracy, disclosing the intensifying effects of material length scale parameter over natural frequency as well as mechanical impedance and resonance frequency. Meanwhile, non-local parameter leads to decrement of natural frequency, mechanical impedance and resonance frequency. Furthermore, it is realized that crest value of transmissibility is not predictable based on frequency and/or impedance variations with respect to non-classical parameters; and requires separate analysis for individual cases. In other words, a pattern referring to gradations of natural and resonance frequencies, mechanical impedance and non-classical parameters and rod models is detectable; however, such patterns do not hold for estimation of mechanical frequency response function (transmissibility peak value). Moreover, stiffness-softening and stiffness-hardening effects are the most impressive and influential factors over transmissibility peak values which can be highly important for energy harvesting rod analysis. Findings of this research point to the importance of proper non-classical theorems along with corresponding non-classical parameters to predict the response of small-scaled rods for wide range of applications in control of robotics, musical instruments, navigating equipment, space wheels, fly wheels, gyroscopes, sensors, actuators, resonators, vibration isolators and energy harvesters.

Ground Vibration Test and Dynamic Response of Horseshoe-shaped Pipeline During Tunnel Blasting Excavation in Pebbly Sandy Soil

In order to ensure the safety of horseshoe-shaped pipeline during tunnel blasting excavation, the vibration test and dynamic response of horseshoe-shaped pipeline were investigated. The velocity and frequency of tunnel blasting vibration were analyzed. Sodev's empirical formula was used for regression analysis of the velocity of blasting vibration. 3D numerical model of a horseshoe-shaped pipelines was established with ALE algorithm using ANSYS/LS-DYNA. The propagation law of a blasting seismic wave was analyzed, and the transverse and longitudinal vibration response characteristics of pipelines under tunnel blasting vibration were studied. The velocity of the pipeline increases gradually and the frequency tends to decrease with the decrease of the distance away from the explosion source center under the same charge. The principal frequency of vibration in the Z direction is mainly distributed from 50 to 80 Hz, which is difficult to generate resonance with the pipelines. The maximum relative error between the simulated and measured velocity of X, Y and Z directions was 8.2%. It was reliable to study the dynamic response of pipelines under blasting vibration based on this numerical model. The blasting seismic wave first reached the bottom of the pipeline right above the explosive. Subsequently, seismic waves propagated along the transverse and the longitudinal axes of the pipeline, and the pressure on the pipeline increased gradually. And when it attenuated completely in the soil, the pipeline stopped its response. The peak value of tensile stress of each element of the vault is the largest. However, the velocity of the bottom plate and the arch roof of pipeline are the largest. The peak values of velocity and tensile stress exist in 0 to 4 m away from the explosion source, and gradually decrease as the distance away from the explosion source increases.

Vibration response of railway cable-stayed bridges under high-speed train braking loads

With the increase of the running speed of high-speed trains, the longitudinal vibration of the long span railway cable-stayed bridge under train loads has increased significantly. And the probability of high-speed train braking is greater than earthquake. The excessive vibration response will affect the serviceable of the cable-stayed bridge. Fluid viscous dampers (FVDs) and elastic cables (ECs) which are widely used in seismic design of the bridge are adopted to control the longitudinal vibration response of the cable-stayed bridge induced by train braking loads. The influence of the design parameters of FVDs and ECs on the response of the bridge is studied. And the effectiveness of FVDs and ECs on mitigating the longitudinal response of the bridge is also discussed. It is found that installing FVDs and ECs between the deck and the tower is very efficient in reducing the longitudinal vibration of the railway cable-stayed bridge subjected to train braking loads, especially the longitudinal displacement of the deck and the bending moment of the tower.

Longitudinal vibration responses of axially functionally graded optimized MEMS gyroscope using Rayleigh–Ritz method, determination of discernible patterns and chaotic regimes

Axial vibration analysis of an optimized MEMS gyroscope is investigated in this paper. For this purpose; a model of rotating, axially functionally graded (tapered) nano-rod is represented. Along with classical continuum mechanics, Eringen’s non-local theory is adopted. Using Hamilton’s variational approach along with coupled displacement field concept which is derived from Babaei and Yang, governing equations of motion are derived. Rayleigh–Ritz approximate method is used to solve the equation for both clamped–clamped and clamped-free rod model. Verification of current results is ratified by comparison with results available in technical literature. Incorporation of taper parameter, non-local effect and rotation effects reveals predictable patterns along with unpredictable chaotic behavior of the optimized gyroscope. Such outcomes report necessity of precise and case-by-case perusal as for some specific values of taper and non-local parameters; system results unpredictable responses. In such chaotic regimes, pattern detection based on response of the gyroscope is impossible and a kind of real-time monitoring and analysis is required. Excluding critical values of the mentioned parameters of non-locality and taper; generally rotations around a fixed axis leads to lessening the oscillations, enhancing the non-locality yields weak vibrations, and intensifying the taper effects makes the system oscillate with greater frequencies wherein for quite high values of taper parameter, gyroscope model behavior turns out independent. It is good to mention that only case in which increment of taper parameter suppresses vibrations occurs when cross-section area of clamped-free nano-rod model is increasing. Eventually, influential factors of current model are revealed.

Nonlinear Dynamic Behavior of Winding Hoisting Rope under Head Sheave Axial Wobbles

Axial wobbles of the head sheave due to manufacturing accuracy and installation technology can exert periodic lateral excitations to the winding hoisting rope. A hoisting rope consists of a constant catenary rope from the drum to the head sheave and a variable vertical rope from the head sheave to the conveyance. The longitudinal and lateral vibration responses of the hoisting rope are coupled to each other. In this paper, the coupled lateral‐longitudinal governing equations of the winding hoisting system under the axial wobble excitations of the head sheave are established by the Hamilton principle. The governing equations are nonlinear infinite‐dimensional partial differential equations, which are discretized into the finite‐dimensional ordinary differential equations through the Galerkin method. The dynamic responses of the hoisting rope under the head sheave axial wobbles are given by MATLAB simulation when the winding hoist is lifting or lowering. The results show that lateral vibration displacements of the vertical rope under the head sheave axial wobbles are larger than those of the catenary rope. The axial wobble amplitude range of the head sheave is given to limit the lateral vibration displacements of the hoisting rope.