Natural Frequency Characteristics Research Articles

Numerical Study of the Strength Radial Turbomachines Based on the Use of Stainless Steel 316L

Numerical Study of the Strength Radial Turbomachines Based on the Use of Stainless Steel 316L

Machine learning analysis of data obtained by finite element method: a new approach in structural design

This study investigates the impact of cutout and added masses on the natural frequencies of a beam structure and employs machine-learning algorithms to predict optimal locations for added masses, achieving desired natural frequency ranges. The evaluation utilizes COMSOL MULTIPHYSICS to analyze a beam structure with cutouts and added mass locations, generating a dataset of original natural frequencies. This dataset is utilized to train machine-learning algorithms, and subsequently tested with desired natural frequencies and cutout locations for forecasting optimal added mass positions. Various machine learning methods are explored, and regression metrics including Mean Squared Error (MSE), Root Mean Squared Error (RMSE), Mean Absolute Error (MAE), and R-squared are employed to assess performance. Results indicate that the Extra Trees Regressor demonstrates superior performance, yielding RMSE, MSE, and R-squared values of 0.000579, 3.35537e-07, and 0.999948, respectively. Additionally, the study explores the influence of employing different natural frequencies (modes) as inputs for machine-learning algorithms. Findings reveal that increasing the number of utilized modes enhances machine-learning performance, albeit at the expense of computational time. Overall, this research establishes a novel approach, leveraging machine learning to optimize the placement of added masses for achieving desired natural frequency characteristics in beam structures.

Vibration model combined with natural frequency characteristics of cylindrical roller bearings with spalling defects

The modeling of early spalling defects in cylindrical roller bearings (CRB) is a challenging problem in fault diagnosis. Due to the absence of prominent "drop-in" excitation, accurately describing the signal characteristics of the early failure mechanism model for CRBs becomes difficult. In this paper, the failure of cylindrical roller bearings due to spalling caused by fatigue is a frequent occurrence. The previous defect models have been centered on sharp-edge impact between the roller and spalling defects. However, they have not provided an adequate depiction of the variation in contact stiffness caused by roller-fault interaction. In this paper, a novel CRB vibration model that takes into account contact stiffness and natural frequency characteristics is proposed. The model considers both the load redistribution and contact stiffness variation caused by spalling defects. This new model provides a spring-damper system based on contact stiffness between the roller and raceway, enabling an estimation of the CRB system's natural frequency characteristics. The outer raceway spalling fault with varying sizes is simulated in this paper, taking into account spalling defect extents and load distribution. The vibration response of the fault signal model is investigated. The simulation results show that the load and contact stiffness will change as the roller passes through the defect zone. Furthermore, the statistical distribution of fault feature signals demonstrates the effects of contact stiffness variation on natural frequency characteristics. The experiments given in this paper have verified the simulation results of the proposed signal model.

Natural frequency analysis of gear system with interval shaft misalignment in multidimensions

Due to the existing tolerances, the shaft misalignment of gear system is inevitably uncertain and shows interval property, which will cause the natural frequency to fluctuate in an interval range, therefore the traditional dynamics analysis that takes it as a deterministic parameter cannot reflect the interval characteristics of natural frequency. To overcome this conundrum, an improved interval eigenvalue algorithm based on matrix non-negative factorization and the Chebyshev inclusion function was proposed to find the natural frequency range of gear system under such uncertainty. First, the average meshing stiffness range of the gears with interval shaft misalignment is obtained by the Chebyshev interval approach. Then combined with the consideration of the remaining excitation parameter deviation coefficients, the global interval stiffness matrix and mass matrix of a single-stage gear system are obtained. Afterwards, the non-negative factorizations of these two interval matrices are carried out and parameter vertex theory is finally employed to obtain the upper and lower bounds of the eigenvalues. The feasibility of the applied algorithm is verified through comparison with the scanning method. Numerical results show that the uncertainty of shaft misalignment will enormously widen the system's resonance bandwidth and intensify oscillation, these will provide a new and valuable perspective for the tolerance allocation in gearbox design.

Simulation analysis and collaborative optimization of key components of the precision detection platform for high precision encoder

To realize the error detection of the high-precision photoelectric encoder, the optical small-angle precision measurement principle of the photoelectric autocollimator and single-sided mirror, and the 360° full circle measurement principle of angle inversion of the compound biaxis turntable are used to establish the precision detection platform of the encoder. The error model of the biaxial structure in the photoelectric coding error detection platform is established by introducing the error sources of the biaxial structure, such as the coaxiality error, angular position error, and installation error. At the same time, the biaxis structure and error models are used to optimize the compound biaxis structure system. By analyzing the natural frequency and vibration characteristics of the structure, the rationality and reliability of the compound biaxis system of the high-precision encoder precision detection platform are verified.

Natural Frequency Characteristics of Stiffened FG Multilayer Graphene-Reinforced Composite Plate with Circular Cutout Resting on Elastic Foundation

In this research, natural frequency response of functionally-graded multilayer graphene-reinforced composite plate with circular cutout reinforced by orthogonal stiffeners is investigated for the first time. The structure is surrounded by Winkler-type elastic support. The plate is composed of polymethyl methacrylate (PMMA) as matrix material and reinforced by graphene platelets (GPLs). The material of the orthogonal stiffeners is the same as that for the matrix. Rule of mixtures and Halpin–Tsai approach are applied to estimate the effective material properties of the composite plate. Third-order shear deformation plate theory and finite element procedure is employed to obtain the element matrices of the structure. Natural frequencies and mode shapes of the stiffened plate are reported for different variables such as nanofillers dispersion patterns, width and height of the stiffeners, aspect ratio of plate, plate thickness ratio, weight fraction of nanofillers, number of stiffeners, boundary conditions, elastic foundation stiffness parameter and size of circular cutout. The obtained results denote that with the addition of a set of stiffeners, fundamental frequency enhanced up to 32.3% with just about 10% increase of mass.

Design, analysis, and testing of a new asymmetric vibration-assisted stage for roll-type polishing.

Based on the technological characteristics of roll-type polishing, a new asymmetric vibration-assisted stage is proposed in this paper. This stage is characterized by asymmetric displacement and asymmetric stiffness. With the average particle spacing of roll-type polishing as the constraint, the comprehensive characteristics of structural stiffness, kinematic range, and natural frequency are realized. Thus, to reduce the surface roughness, the removal of simple-directional surface textures generated by roll-type polishing can be achieved. First, the asymmetric structure is designed, modeled, and optimized according to working performance design goals of roll-type polishing. Then, the finite element analysis and actual performance test of the stage are carried out to verify the accuracy of the established model and the effectiveness of the optimization design. The results indicate that the stage can meet the design index. Finally, the asymmetric vibration-assisted polishing experiment is carried out. The results show that the single-directional surface textures of the SiC surface are interrupted and the surface roughness is decreased.

FREE VIBRATIONS OF THIN ELASTIC ORTHOTROPIC CYLINDRICAL PANEL WITH RIGID – CLAMPED EDGE GENERATOR

Using the system of equations corresponding to the classical theory of orthotropic cylindrical shells, the free vibrations of thin elastic orthotropic cylindrical panel with rigid – clamped  edge generator are investigated. In order to calculate the natural frequencies and to identify the respective natural modes, the generalized Kantorovich-Vlasov method of reduction to ordinary differential equations is used. Dispersion equations for finding the natural frequencies of possible types of vibrations are derived. An asymptotic relation between the dispersion equations of the problem at hand and the analogous problem for a cantilever rectangular plate is established.  An algorithm for separating possible boundary vibrations is presented. As an example, the values of dimensionless characteristics of natural frequencies are derived for an orthotropic cylindrical panel.

Finite Element Analysis of Curved Beam Elements Employing Trigonometric Displacement Distribution Patterns

A finite element analysis (FEA) model was developed for Euler and Timoshenko curved beam elements by incorporating trigonometric displacement distribution patterns. Local polar coordinate stiffness matrices were derived based on force-displacement relations and static equilibrium conditions. By employing the kinetic energy theorem and triangular displacement functions, an expression for the consistent mass matrix of a curved beam element was obtained. A coordinate transformation matrix for the curved beam element was established by relating the local polar coordinate system to the global polar coordinate system. Calculation programs were implemented in the Fortran language to evaluate the static–dynamic performance and natural frequency characteristics of curved beam bridges. The obtained results were then compared with those obtained using ANSYS solid models and “replace curve with straight” beam element models. The comparison demonstrated a strong agreement between the results of the Euler and Timoshenko curved beam element models and those of the ANSYS solid models. However, discrepancies were observed when comparing with the results of the “replace curve with straight” beam element model, particularly in terms of lateral static displacement. This discrepancy suggests that the characteristic matrix derived in this study accurately represents the stiffness and mass distribution of the curved beam, making it suitable for mechanical performance analysis of curved beam bridges. It should be noted that the “replace curve with straight” method overlooks the initial curvature and the bending–torsion coupling effects of a curved beam, resulting in calculation deviations. On the other hand, the use of curved beam elements in numerical analysis provides a simple and practical approach, which facilitates further research in areas such as vehicle–bridge coupling vibrations and seismic analysis of curved beam bridges.

Dynamical system of a time-delayed ϕ6-Van der Pol oscillator: a non-perturbative approach

A remarkable example of how to quantitatively explain the nonlinear performance of many phenomena in physics and engineering is the Van der Pol oscillator. Therefore, the current paper examines the stability analysis of the dynamics of ϕ6-Van der Pol oscillator (PHI6) exposed to exterior excitation in light of its motivated applications in science and engineering. The emphasis in many examinations has shifted to time-delayed technology, yet the topic of this study is still quite significant. A non-perturbative technique is employed to obtain some improvement and preparation for the system under examination. This new methodology yields an equivalent linear differential equation to the exciting nonlinear one. Applying a numerical approach, the analytical solution is validated by this approach. This novel approach seems to be impressive and promising and can be employed in various classes of nonlinear dynamical systems. In various graphs, the time histories of the obtained results, their varied zones of stability, and their polar representations are shown for a range of natural frequencies and other influencing factor values. Concerning the approximate solution, in the case of the presence/absence of time delay, the numerical approach shows excellent accuracy. It is found that as damping and natural frequency parameters increase, the solution approaches stability more quickly. Additionally, the phase plane is more positively impacted by the initial amplitude, external force, damping, and natural frequency characteristics than the other parameters. To demonstrate how the initial amplitude, natural frequency, and cubic nonlinear factors directly affect the periodicity of the resulting solution, many polar forms of the corresponding equation have been displayed. Furthermore, the stable configuration of the analogous equation is shown in the absence of the stimulated force.

Higher-Order Finite Element Vibration Analysis of Circular Raft on Winkler Foundation

On a Winkler foundation, solid circular plate vibration is examined using a higher-order finite element in polar co-ordinate system. The present formulation has developed a Mat-lab code to handle any boundary conditions. Validation of the code is carried out after the convergence studies. The results are compared to other researchers and show excellent conformity. Furthermore, a parametric analysis gave the first 10 natural frequency characteristics in tabular and graphical form. The authors conclude that the present formulation is straightforward, behaves exceptionally well for thin solid circular plates on elastic foundations with reasonable convergence rate and accuracy, and requires less computational effort, resources, and time.

Research on pendulum-type tunable vibration energy harvesting

Sensors, as the most common electronic devices in electromechanical systems, are often scattered in huge numbers in the environment to be monitored. When the power of the sensor comes from batteries, the long-term continuous power supply and the replacement of the battery under severe working conditions are the main problems. Compared with other energy sources, vibration energy is more universal and stable, so it is often harvested and utilized by specific devices. Most of the traditional vibration energy harvesting devices have the disadvantage that the natural frequency cannot be tuned, which will lead to significant decline of energy harvesting efficiency when the environmental frequency changes. Although some energy harvesting devices are with tunable nature frequency, the linearity is relatively poor when tuning. In order to solve the problems above, a structural model of an electromagnetic energy harvesting device using motion of a pendulum rod is constructed. Natural frequency characteristics and electromagnetic field finite element simulation analysis of the model are carried out. An experimental platform of the device and a corresponding electric energy storage circuit are designed and built. Simulation and experimental results show that the natural frequency of the device can be tuned linearly in the range of 7 Hz–13.5 Hz. The maximum induced electromotive force that the device can output is 1.37 V and the maximum power is 521 mW. When the device is connected to the electric energy storage circuit, the device can output 3.4 V DC voltage. Compared with the traditional devices that can only harvest energy efficiently from a single vibration frequency, the advantage of this model is that its natural frequency can be adjusted to adapt to different ambient vibration frequencies and the linearity of tuning has been verified, which makes its energy harvesting efficiency relatively higher. The device model proposed in this paper would have potential application value in the field of vibration energy harvesting and wireless micro sensors power supply.

Nonlinear dynamics of a vertical pipe subjected to a two-phase, solid-liquid internal flow

In this paper, a numerical study is carried out on a simply supported pipe model which conveys a two-phase, solid-liquid flow. First, a system of nonlinear, partial differential equations of dimensionless, two-phase, solid-liquid flow pipeline is established. Then, the above mentioned partial differential equation system is discretized by the Galerkin method of finite order to obtain a system of ordinary differential equations. Finally, the natural frequency characteristics of the corresponding linear system are solved to approximate the stability range of the nonlinear system, the structural response characteristics of which are solved by the Newton-Raphson method. The dimensionless aspect ratio, the dimensionless liquid-phase volume, and the dimensionless gravity coefficient are chosen to analyze the impact on the critical velocity of buckling and structural nonlinear response characteristic in this research. The research results indicate that a change to the above parameters will cause the critical speed to change in a linear or non-linear form. Moreover, at subcritical speeds, the vibrations are damped; at critical speeds, steady-state vibrations with extremely small amplitudes with a single frequency are seen; and at supercritical speeds, the pipeline model will produce buckling in different directions according to initial values.

Experimental Study on the Vibration Characteristics of Dangerous Rock Mass under Constant Micromotion

In mountainous areas, dangerous rock mass collapse is a serious threat to human life and property safety. However, how to effectively prevent and control the instability of dangerous rock mass is still an urgent unsolved problem. In this study, the dynamic characteristics of dangerous rock mass under constant micromotion were analysed from the perspective of elastic wave propagation. When the slip plane of the dangerous rock mass is damaged, many micropores and cracks will appear in the medium composed of the slip surface. With constant micromotion as the vibration source, the elastic wave propagates to the dangerous rock mass through the damaged slip plane. The high-frequency components of elastic waves scatter in pores and fissures, where energy dissipates. According to this characteristic, a laboratory simulation experiment was designed. In the experiment, the damage process of the slip plane was simulated by the freeze‒thaw process of frozen hydrosol. The experimental analysis showed that the centre frequency of the high-frequency part of the dangerous rock mass model and bedrock mass model decreased as the frozen surface continued to melt. As the dangerous rock mass model and the bedrock mass model continued to fit, the centre frequency of the high-frequency part of the two rock mass models rebounded. This phenomenon showed that the damage degree of the slip plane between the dangerous rock mass and the bed rock mass can be effectively reflected by the centre frequency of the high-frequency part of the two rock mass models. During the experiment, the dangerous rock mass did not slide in the whole process, indicating that the deformation index has difficulty reflecting the stability of the hidden dangerous rock mass. In addition, the application conditions of using the natural frequency characteristics of dangerous rock mass and the scattering characteristics of elastic wave in the damage identification of structural plane were analysed: (1) when the structural plane has macrofracture, the change of natural frequency of dangerous rock mass should be used to analyse the damage degree of structural plane; (2) when there is no macrofracture of the structural plane, the characteristics of elastic wave scattering should be used to analyse the damage degree of the structural plane. This study provided a new idea for the prevention and control of dangerous rock masses and is expected to provide a useful reference for the automation of dangerous rock mass prevention and control.

Free Vibration Characteristics of CFRP Laminate with One-Dimensional Periodic Structures.

This paper proposes an approach of stacking prepreg periodically for carbon fiber-reinforced polymer composites (CFRP) laminate. This paper will discuss the natural frequency, modal damping, and vibration characteristics of CFRP laminate with one-dimensional periodic structures. The damping ratio of CFRP laminate is calculated using the semi-analytical method which combines modal strain energy with the finite element method. The finite element method is used to calculate the natural frequency and bending stiffness which are verified with experiments. The numerical results of the damping ratio, natural frequency, and bending stiffness are in good agreement with the experiment results. Finally, the bending vibration characteristics of CFRP laminate with one-dimensional periodic structures and traditional CFRP laminate are investigated with experiments. The finding confirmed that the CFRP laminate with one-dimensional periodic structures exists band gaps. This study provides theoretical support for the promotion and application of CFRP laminate in the field of vibration and noise.

Identification Procedure for Design Optimization of Gravitational Electromagnetic Energy Harvesters

Energy harvesting is a promising technique for supplying low-power devices as an alternative to conventional batteries. Energy harvesters can be integrated into Autonomous Internet of Things (AIoT) systems to create a wireless network of sensor nodes for real-time monitoring of assets. This paper shows a design and optimization methodology for gravitational vibration-based electromagnetic energy harvesters (GVEHs) of different sizes considering the design constraints of its real application. The configuration, analytical model, and electro-mechanical coupling of these devices are described in detail. A numerical model is developed in the Ansys Maxwell FEM environment to derive the non-linear stiffness and damping of the asymmetric magnetic suspension. Experimental laboratory tests on three harvester prototypes are compared to numerical results of dynamic simulations in MATLAB/Simulink for the validation of the proposed model through error estimation. The fully-parametric validated model is used to perform sensitivity analyses on the device’s mechanical characteristics of natural frequency and magnet equilibrium position by varying the fixed and moving magnets dimensions. The set of magnets composing the magnetic spring is chosen complying with the application design constraints of size and resonance frequency tuning. Coil parameters of length and number of turns are optimized for maximum output power generation. The optimized device simulated performances are compared to other devices in the literature in terms of NPD, a significant index that evaluates power density under different excitation amplitudes. The optimized harvester presents the highest NPD value of 2.61, achieving an improvement of 52% with respect to the best harvester amongst the three tested prototypes.

Influence of Electrostatic Force Nonlinearity on the Sensitivity Performance of a Tapered Beam Micro-Gyroscope Based on Frequency Modulation.

Electrostatic force nonlinearity is widely present in MEMS systems, which could impact the system sensitivity performance. The Frequency modulation (FM) method is proposed as an ideal solution to solve the problem of environmental fluctuation stability. The effect of electrostatic force nonlinearity on the sensitivity performance of a class of FM micro-gyroscope is investigated. The micro-gyroscope consists of a tapered cantilever beam with a tip mass attached to the end. Considering the case of unequal width and thickness, the motion equations of the system are derived by applying Hamilton's principle. The differential quadrature method (DQM) was used to analyze the micro-gyroscope's static deflection, pull-in voltage, and natural frequency characteristics. We observed that from the onset of rotation, the natural frequencies of the drive and sense modes gradually split into a pair of natural frequencies that were far from each other. The FM method directly measures the angular velocity by tracking the frequency of the drive and sense modes. Then, based on the linear system, the reduced-order model was used to analyze the influence of the shape factor and DC voltage on the sensitivity performance. Most importantly, the nonlinear frequency of system was obtained using the invariant manifold method (IMM). The influence of electrostatic force nonlinearity on the performance of the FM micro-gyroscope was investigated. The results show that the different shape factors of width and thickness, as well as the different DC voltages along the drive and sense directions, break the symmetry of the micro-gyroscope and reduce the sensitivity of the system. The sensitivity has a non-linear trend with the rotation speed. The DC voltage is proportional to the electrostatic force nonlinearity coefficient. As the DC voltage gradually increases, the nonlinearity is enhanced, resulting in a significant decrease in the sensitivity of the micro-gyroscope. It is found that the negative shape factor (width and thickness gradually increase along the beam) can effectively restrain the influence of electrostatic force nonlinearity, and a larger dynamic detection range can be obtained.

Research on Inherent Frequency and Vibration Characteristics of Sandwich Piezoelectric Ceramic Transducer

Great progress has been made in the field of ultrasonic processing in recent years, and piezoelectric ceramic transducers have been widely used as drive sources. In this paper, a sandwich piezoelectric ceramic transducer is designed, and the vibration of each part of the transducer is analyzed by elastic mechanics and piezoelectric theory. According to its mechanical and electrical boundary conditions, the vibration model of the piezoelectric transducer was established. Based on the equivalent elastic modulus method for simplifying the pre-stressed bolts into a one-dimensional transducer vibration model, the relationship between the one-dimensional axial response frequency of the transducer and the length of each component was obtained. Based on the half wavelength theory, a transducer with the vibration node in the crystal stack and an inherent frequency of 15 kHz was designed and fabricated. In order to verify the natural frequency and vibration characteristics of the piezoelectric transducer, a laser vibration measurement system was built in this study. The vibration characteristics of the transducer under different parameters such as voltage and frequency were analyzed, and the accuracy of the vibration model was verified. The vibration states of the end surface of the transducer and the radial surface were evaluated at the first-order inherent frequency and second-order inherent frequency. The results show that the equivalent simplified model established in this study can effectively design the inherent frequency of the transducer, and the operation at the first-order inherent frequency meets the one-dimensional assumptions of this study. The transducer operating conditions measured in this study also provide a more detailed reference for ultrasonic processing applications.

Application of a Modified Differential Quadrature Finite Element Method to Flexural Vibrations of Composite Laminates with Arbitrary Elastic Boundaries

This paper formulates a modified differential quadrature finite element method (DQFEM) by a combination of the standard DQFEM and the virtual boundary spring technique, which makes it easy to implement arbitrary elastic restraints by assigning reasonable values to the boundary spring stiffnesses. This new formulated method can offer a unified solution for flexural vibrations of composite laminates subjected to general elastic boundary combinations including all the classical cases. The influences of the number of Gauss–Lobatto nodes and the boundary spring stiffnesses on the convergence characteristics of natural frequencies are investigated, and some conclusions are drawn in terms of the minimum number of unilateral nodes required to generate convergent solutions and the optimal values of the boundary spring stiffnesses to simulate classical boundaries. Numerical examples are performed for composite laminates under various classical boundary conditions. Excellent accuracy, numerical stability, and reliability of the present method are demonstrated by comparisons with available exact and numerical solutions in open literatures. Additionally, for elastically constrained composite laminates, which are beyond the scope of most existing approaches, numerous new results obtained by the present method may serve as reference values for other research.

Numerical investigation on natural vibration characteristics of ship propulsion shafting under uncertainty based on nonparametric approach

In order to investigate the natural vibration characteristics of ship propulsion shafting under uncertainties, the random matrixes of mass, damping, and stiffness are constructed based on a nonparametric approach. The effect of uncertainties on the natural vibration characteristics of ship propulsion shafting is discussed by the distribution characteristics of natural frequency. Meanwhile, the natural frequency identification experiment of ship propulsion shafting was used to illustrate the effectiveness of the nonparametric modelling method. The results show that different levels of uncertainties in stiffness, mass and damping matrixes could be obtained by controlling the divergence parameter. The natural frequency could fluctuate significantly under uncertainties. Furthermore, the uncertainties of stiffness and mass have significant impact on the natural frequency. However, the uncertainty in damping has little influence on it. Therefore, it is essential to consider the uncertainties in the analysis of natural vibration characteristics of the ship propulsion shafting.