Vibrations Of Mechanical Systems Research Articles

Lightweight waveguide absorbers based on spiral acoustic black holes for reducing structural vibrations in mechanical systems

We present an ABH-based solution to address challenging industrial problems related to structural vibration and vibration-induced noise. To tackle those problems, we utilize lightweight waveguide absorbers (WGAs) based on spiral acoustic black holes (ABHs). The WGA is designed by combining a spiral acoustic black hole with a rod to absorb flexural waves and dampen the vibrations in dynamic systems. The WGAs are customized to industrial needs, pursuing both high-damping performance and lightness. Specifically, we design non-identical configurations of WGAs by taking multiple or broadband target frequency ranges into account. We then determine optimal attachment positions and directions of the designed WGAs while considering constraints on installation space and total weight of WGAs. Numerical and experimental results verify the feasibility to meet the requirements of industries through the use of WGAs, thereby opening up the possibility of practical applications to real-world problems.

An online reconstruction method of dynamic loading based on adaptive tracking dual nested Kalman filter

Accurately reconstructing the unknown dynamic loading is crucial for controlling the vibration of mechanical systems. Many methods have been proposed for load identification of time-invariant systems, but in some mechanical systems, their structural parameters exhibit time-varying dynamic characteristics during their operation. Therefore, tracking the time-varying structural parameters to update the transfer function is necessary, which is the most important prerequisite for load reconstruction. However, some existing methods are unstable because of the nonlinear conditions. To overcome this issue, an improved dual-nested Kalman filter framework has been proposed. This method allocates unknown variables to two Kalman filters, allowing them to iterate and loop with each other. As a result, it achieves linear reconstruction of the dynamic loading for a time-varying system. The proposed method has been validated in two numerical examples: one is a spring-mass system with abruptly changing mass parameters, and the other is a milling system with slowly changing mass parameters. The results show that the proposed method can reconstruct loading stably and is not susceptible to noise interference.

Improved harmonic balance method for analyzing asymmetric restoring force functions in nonlinear vibration of mechanical systems

The non-linear differential equation governing some mechanical systems, such as the non-linear vibrations of FG beams resting on a nonlinear foundation, behaves similarly to the oscillations of a non-linear oscillator with an asymmetric restoring force function that includes both odd and non-odd non-linear terms. The objective of this research is to obtain higher-order approximate analytical solutions for such problems by introducing an enhanced harmonic balance method. By building upon the original problem, two new symmetric systems with odd non-linear terms are introduced, and their higher-order approximate analytical solutions are derived using a novel approach. In proposed method, the restoring force function is represented by its Fourier series expansion. Unlike previous papers, linearizing the equation or taking the first derivative of the Fourier series in the subsequent iteration is unnecessary. However, to enhance the solution’s accuracy, the remaining error from each iteration is utilized in the next one. Finally, by combining the results from the two introduced systems, the analytical period and corresponding periodic solution of the original problem can be obtained. This method is applied to the governing differential equation of FG beams resting on non-linear foundations, a physical non-natural oscillator, and conservative Toda oscillator. The key advantages of this approach are its simplicity and its ability to provide highly precise solutions for both small and large amplitudes in a single iteration.

Research on decoupled transfer path analysis method and its application in hybrid mechanical systems

As a key technique of vibration control, transfer path analysis (TPA) provides theoretical support for diagnosis, analysis, evaluation, and optimization of vibration in mechanical systems. The coupling phenomena often exists in the vibration transfer paths of complex mechanical systems, which makes most of the existing TPA methods unable to accurately analyze the transfer function of the coupled paths. The classical transfer path analysis (CTPA) is an effective method to solve the problem of path coupling, but it changes the inherent characteristics of the system, the results can not accurately reflect the vibration transfer characteristics of the system. In order to overcome the disadvantages of CTPA method, a decoupled transfer path analysis (DTPA) method is proposed in this paper, and the implementation process of this method is introduced by taking a hybrid mechanical system as an example. The results show that DTPA method can solve the path crosstalk problem and accurately calculate the transfer function of each path.

Penyelesaian khusus persamaan diferensial biasa ordo dua linier tak homogen dengan koefisien konstan untuk fungsi bagian demi bagian

Spring mechanical vibration motion system with a damped degree of freedom and influenced by external forces, is mathematically expressed as an ordinary differential equation of order of two linear constant coefficients that are not homogeneous. If an external force acts on a stationary system expressed as a continuous function f(t) for any time t, then the system will experience mechanical vibrational motion which mathematically the equation of motion can be expressed as a superposition. The equation consists of as a solution to a homogeneous form with mechanical vibrations as a solution to a particular form. In terms of the particular solution this article will show a mathematical way when f(t) is a continuous function section by part which is defined at an interval, such that the mechanical vibration motion equation is at the same time a special solution of the equation the mechanical vibration system.
 Keywords: Vibration, Impulse Functions, ConvolutionMSC2020: 34A37

On developing structural and technical solutions to ensure the vibratory displacement of the working medium

A research study into using vibration technologies for the transport of granular working medium was carried out. A vibration testing machine, whose calculation model includes a mechanical vibration system having two degrees of freedom with a rigid body on elastic supports, was selected as an object of research. The study involved analysing the variation in the system vibrations by using the structural theory for vibration isolation systems, where a dynamic equivalent represented by a structural diagram of an automatic control system is compared to the initial calculation model. The structural diagram of the system is based on the motion equations in operator form obtained using Lagrange differential equations of the second kind. The Laplace transform was used to transform the initial data for the system of differential motion equations. The paper addresses the characteristics of a new structural and technical solution in the field of vibratory displacement of a granular working medium using the working body of a vibration technological machine, which involves introducing a number of additional weights, levers, springs, and hinges. Here, springs comprise generalised structures containing both elastic elements and vibration dampers. To connect the coordinates of the endpoints in the working body of the vibration technological machine, analytical relations were obtained. It was established that varying the parameters of the elements within the elastic-lever blocks allows the dynamic state of the vibration technological machine to be controlled. In addition, it was shown that the obtained structural diagram helps to derive mathematical expressions for transfer functions, comprising the ratio between the motion coordinates of a technical object and an external force disturbance. On the basis of these expressions, the transfer function for the ratio of the motion coordinates of the vibration technological machine was formulated. A mathematical model of a vibration technological machine was obtained in the form of a transfer function, including a large number of additional elastic and massinertial elements, where the parameters of vibration displacement can be adjusted automatically. The research results will allow the existing technical solutions in the field of technological engineering to be modernised.

Equations of motion of the system “structure – reactive vibration dampener”

The problem of damping vibrations of a mechanical system under seismic influences is considered. The construction of a mathematical model of the motion of a mechanical system with kinematic excitation of vibrations and simultaneous operation of a reactive damper is given. Transformations of differential equations of motion based on disturbing forces from a seismogram and damping action are shown. It is assumed that the jet extinguisher works constantly, changing the direction of the pulse depending on the sign of the speed of movement of the unit of its installation. The solution of the system of differential equations of motion is given. These transformations help to model more clearly the task of damping vibrations of mechanical systems during seismic disturbances

Prediction of energy dissipation by analytical solution to combined viscous and Coulomb damping

Damping in mechanical vibration systems primarily falls under (i) linear viscous and (ii) non-linear Coulomb damping. Practically, most systems possess characteristics of both linear viscous and non-linear Coulomb damping. In this work, the method of phase space coordinate averaging is used to obtain an approximate analytical solution for the system’s response when it is subjected to both viscous and Coulomb damping. The accuracy of the proposed solution is examined by comparing it with the numerical simulations and experimental results. The energy dissipated by both damping mechanisms is quantified using semi-analytical integrals. The analysis of energy dissipation quantifies the conditions for natural frequency, damping ratio, and coefficient of friction, at which the energy dissipated by viscous damping is equal to the energy dissipated by Coulomb damping. This has led to the introduction of a non-dimensional quantity called the dissipation number (D). With certainty, it can be stated that if D>1, friction is dominant, and if D<1, viscosity is dominant.

Dynamic critical diameter of an oscillating heat pipe in vertical orientation

Generally, the oscillating heat pipe (OHP) diameter should be kept below a specific critical size. As a result, the working fluid forms a vapor-liquid slug train under the capillary action, producing a gas-spring-mass mechanical vibration system. Determining the critical diameter is essential for the OHP to form this slug train. A conventional critical diameter model only considers the surface tension and gravity. However, the vapor bubble flow in the liquid slug plays a key role in forming this essential slug train. Even if the hydraulic diameter of an OHP is much larger than the conventional critical diameter, the OHP can still function. The dynamic flow significantly affects the formation of the vapor-liquid slug-train. Therefore, a semi-empirical model was developed considering the effects of the dynamic flow and the heat input, filling ratio, and working fluid. The concept of dynamic critical diameter (DCD) was proposed. The model shows that the OHP DCD increases when the heat input increases for a given evaporator temperature. In addition, the model can predict that the OHP DCD decreases when the evaporator temperature increases.

Applications of Fuzzy Differential Equations on Vibrating Spring Mass System

Modelling several real-world issues in the fuzzy world extensively uses ordinary differential equations. In this paper, a mechanical vibration system with the given mass, spring constant, damping and external force is modelled as a second-order ordinary differential equation. Due to measurement errors, the initial displacement of the string is approximate and assumed to be a fuzzy number. A fuzzy version of the Sumudu transform procedure is used to figure out this vibrating spring-mass system with fuzzy initial displacement. The output is displayed as a table at various computational stages. The consequences are visibly presented diagrammatically for different values of r and t. There is a good agreement between the computed results and the analytical solution.

Biologically inspired virtual aperture extension method of small aperture HFSWR multielement array

We propose a method for extending the virtual aperture of the small aperture high-frequency surface wave radar multielement array inspired by a fly named Ormia ochracea. Despite the tremendous incompatibility between its ear and the incoming wavelength, Ormia can accurately local the sound of its host cricket. This ability benefits from the coupled structure of Ormia’s ears which have been modelled as a mechanical vibration system. In this paper, we first design a two-degree of freedom biologically inspired coupled system by mimicking Ormia’s coupled ears. We quantitatively analyze its extension capability to the array aperture and construct the received signal model of the virtual array. We then analyze its response characteristic and available frequency band. To achieve the applications of arbitrary desired frequencies, we propose a frequency conversion algorithm. Moreover, we design two multi-degree of freedom biologically inspired coupled systems for the multielement array We summarize the criteria for extending the degree of freedom and optimize these two systems to address their respective shortcomings. Numerical results give the optimal system parameters for our desired frequency and validate the frequency conversion algorithm. By comparing the radiation pattern of the inspired arrays (arrays with the proposed systems) with that of an ordinary array, we demonstrate the virtual aperture extension capability of our proposed method. We also verify the effectiveness of proposed method by processing the actual received signals of the array.

SIMULATION OF INTERMEDIATE VISCOELASTIC SUPPORT UNDER NON-STATIONARY VIBRATIONS OF TIMOSHENKO BEAMS

When modeling mechanical objects and their systems, mathematical models developed for an elastic domain are most often used, which in some cases can lead to significant inaccuracies in calculations. The use of mathematical models that take into account visco-elastic properties or energy dissipation allows to obtain more realistic models, which will make it possible to obtain more accurate calculation results. In the paper non-stationary loading of a mechanical system, consisting of a beam hinged at the edges, and an additional support installed in the span of the beam is considered. We use a beam deformation model, which is based on the hypotheses of S. P. Timoshenko and takes into account the inertia of rotation and shear. The system of partial differential equations describing the deformation of the beam is solved by expanding the desired functions into the corresponding Fourier series and subsequent using the integral Laplace transform. The additional support is assumed to be realistic rather than absolutely rigid, having linear elastic and viscous components. It is assumed that at the point of attachment of the additional support to the beam their displacements coincide. The reaction between the beam and the additional support is replaced by an external unknown concentrated force applied to the beam, which changes with time. The law of change in time of this unknown reaction is determined from the solution of the Volterra integral equation. A method for obtaining an integral equation for an unknown reaction is described. Analytical relations and calculation results for specific numerical parameters are given. The influence of stiffness and viscosity on the determined reaction of the additional support, as well as on the deflections of the beam at various points, is investigated. The results obtained can also be used for damping forced vibrations of mechanical systems.

Analytical method for suboptimal design of dynamic absorber for parametrically excited system

Parametric excitation is induced due to systemically unstable equilibrium states and often results in undesired vibration in mechanical systems. Many researchers have shown that the use of a dynamic absorber with an appropriate natural frequency and damping ratio is effective for suppressing parametric excitation. However, unlike the situation in a forced vibration system, it is difficult to find analytical expressions to determine the optimal design of the dynamic absorber for a parametrically excited system, which inevitably makes the design inefficient. This study developed a method for obtaining analytical suboptimal designs, specifically approximate optimal designs, for a dynamic absorber for parametric excitation through two kinds of free vibration systems constructed by approximating the original system. A comparison between the results for the suboptimal designs and an accurate optimal design demonstrated that an effective suboptimal design is achieved using a free vibration system that has an appropriate negative damping term derived from the instability of the original parametrically excited system. The developed approach is expected to provide an efficient option for suppressing parametric excitation.

Homoclinic–Heteroclinic Bifurcations and Chaos in a Coupled SD Oscillator Subjected to Gaussian Colored Noise

The so-called coupled smooth and discontinuous (SD) oscillator whose stiffness term leads to a transcendental function is a simple mass-spring system constrained to a straight line by two parameters, which are the dimensionless distances to the fixed point. This paper studies the homoclinic–heteroclinic chaos in a coupled SD oscillator subjected to Gaussian colored noise. In order to investigate the chaos thresholds analytically, the piecewise linearization approximation is used to fit the transcendental function. Stochastic nonsmooth Melnikov method with homoclinic–heteroclinic orbits is developed to study chaos thresholds of oscillators with tri-stable potential. Based on stochastic Melnikov process, the mean square criterion and the rate of phase-space flux function theory are used to study the chaotic motions of a coupled SD oscillator under weak noise and strong noise, respectively. The obtained results show that it is effective to use the piecewise linear approximation to analyze chaos in the coupled SD oscillator subjected to Gaussian colored noise. It also lays the foundation for chaos research of other nonsmooth mechanical vibration systems under random excitation.

Mechanical fault intelligent diagnosis using attention-based dual-scale feature fusion capsule network

Capsule networks, which have achieved many achievements in machine vision, have also gotten wide attention in machinery fault diagnosis. However, due to the non-stationarity and diversity of mechanical system vibration signals, as well as the dual-scale characteristics of different fault features, it is often difficult for existing single-scale capsule networks to fully mine vital discriminative features in the data, which is not conducive to accurate identification of mechanical faults. This paper studies an attention-based dual-scale feature fusion capsule network for mechanical fault diagnosis. It first extracts dual-scale features from grayscale images obtained from vibration signals using convolutional layers composed of kernels of different sizes; secondly, an attention-based two-branch network is designed to calculate the weights of features at different scales, and accordingly dual-scale feature fusion is performed; finally, the obtained features are entered into the capsule layers, and the classification and identification of mechanical faults are realized by optimizing the model using the classification loss and reconstruction loss. A rolling bearing experimental dataset and a motor fault dataset are adopted to assess the performance of the proposed method, and the comparison results confirm its effectiveness and superiority, indicating that it has the potential to be a useful tool for detecting mechanical faults.

Multiformity and Evolution Characteristics of Periodic Motions in Mechanical Vibration Systems with Clearances

Multiformity and Evolution Characteristics of Periodic Motions in Mechanical Vibration Systems with Clearances

Theoretical and Experimental Investigation on Comparing the Efficiency of a Single-Piston Valved Linear Compressor and a Symmetrical Dual-Piston Valved Linear Compressor

The efficiency of the valved linear compressor is very important to the efficiency of the space J-T throttling refrigerator. To compare the efficiency of the single-piston valved linear compressor (SVLC) and the symmetrical dual-piston valved linear compressor (SDVLC), this paper explores the factors that affect efficiency. Firstly, this paper analyzes the mechanical vibration system of the linear compressor, the result shows that the efficiency is highest when the external force (current) is in phase with the speed. Then the numerical solutions of the current and velocity are obtained. By comparing the variance and same direction rate of the current and velocity between the SVLC and SDVLC, the reason for the difference in efficiency is explained. Subsequently, the performance of the SVLC and SDVLC are tested on the experimental system. The result shows that the current and velocity of the SDVLC are more in phase, and the isentropic efficiency, volume efficiency and motor efficiency of the SDVLC are all higher than that of the SVLC.

Linear stability analysis of the condition for vibration during frictional slip

Slip along a frictional contact between elastic bodies can be stable or unstable, leading to stick-slip motion. Frictional slip can also be associated with vibrations. The condition for these vibrations and their characteristics remains poorly understood. To address this issue, which is relevant to engineering and earth science, we carry out a linear stability analysis of a spring-and-slider system obeying rate and state friction. We first identify the solution space for the linearized equation and define the conditions for different slip modes from the real and imaginary parts of the solution. We then derive asymptotic equations for all boundaries between overdamped stable sliding, inertial/non-inertial underdamped oscillation, stick-slip, and harmonic vibration. Finally, we verified the conditions with numerical simulations. Our work provides rigorous criteria regarding the conditions for the various frictional slip modes and the emergence of vibrations. It can help design appropriate approaches for suppressing undesired vibrations in mechanical systems and investigate the mechanisms generating vibrations (tremor) associated with fault slip in nature.

Global dynamics for a class of tristable system with negative stiffness

Tristable system with negative stiffness has attracted extensive attention in the low frequency vibration isolation and vibration energy harvester. As a low frequency vibration isolator, it can achieve high static stiffness and low dynamic stiffness. As a vibration energy harvester, it had a wider bandwidth for resonance than the bistable one. The introduction of negative stiffness may induce subharmonic resonance and chaos in the tristable system. Chaos usually brings disorder to mechanical vibration system. Subharmonic resonance plays the negative effect on low frequency vibration isolation because they will transfer the high frequency energy of the system to the low frequency, but it is beneficial to broaden the working frequency band of vibration energy harvester. In this paper, the subharmonic bifurcation and chaos of a class of tristable system with negative stiffness are studied. The piecewise linearized systems are established to approximate the system with tristable potential. In order to conduct Melnikov analysis, the homoclinic-heteroclinic orbits and periodic orbits for the unperturbed piecewise linearized system are obtained respectively. The subharmonic Melnikov method for nonsmooth systems with four switched manifolds is developed. The thresholds for homoclinic-heteroclinic chaos and subharmonic resonance are derived by using non-smooth Melnikov method. It provides a theoretical support not only for design of the vibration energy harvester to obtain wider working frequency band, but also for design of the vibration isolation system to avoid high frequency energy transfer to low frequency. Moreover, the phenomena for infinite subharmonic bifurcations to chaos from odd order subharmonic orbit and the coexistence for chaotic and subharmonic attractors are revealed. The subharmonic Melnikov method with four switching manifolds developed in this paper lays a foundation for the subharmonic resonance analysis of other nonsmooth tristable systems.

Spherical piezoelectric transducers of functionally graded materials.

In this work, a functionally graded spherical piezoelectric transducer (FG-sPET) is proposed and an accurate theoretical model is constructed, mainly composed of a three-port electromechanical equivalent circuit model (EECM). The EECM of FG-sPET can be connected to that of other vibration systems according to the boundary conditions (force and vibration velocity), making it easier to evaluate the whole mechanical vibration system. The validity of the EECM for FG-sPET is verified by comparison with other literature. The effects of geometric dimensions and non-uniform coefficients on the vibration characteristics (resonance/anti-resonance frequencies and effective electromechanical coupling coefficient) of FG-sPET are also studied, contributing to systematically evaluating the key factors determining the vibration characteristics of FG-sPET. The proposed analytical system is of excellent guidance for the structural optimization design of functionally graded piezoelectric devices.