Error Of Natural Frequency Research Articles

Improving mass lumping and stiffness parameters of bar and hinge model for accurate modal dynamics of origami structures.

The bar and hinge framework uses truss elements and rotational springs to efficiently model the structural behaviour of origami. The framework is especially useful to investigate origami metamaterials as they have repeating geometry, which makes conventional finite element simulations very expensive due to a large number of degrees of freedom. This work proposes improvements to the parameters of bar and hinge model within the context of structural dynamics, specifically modal analysis under small deformations, which has not been carried out previously in the literature. A range of low-frequency modes involving origami folding and panel bending deformations that can be accurately captured by the bar and hinge framework are identified. Within this range, bar and hinge parameters like the lumped masses and the rotational spring stiffness values are derived using conservation laws and finite element tests. The best among the proposed schemes is found to predict natural frequencies of the considered origami structures to within 10% maximum error, improving the accuracy by more than three times from existing schemes. In most cases, the errors in natural frequencies are less than 5%. This article is part of the theme issue 'Origami/Kirigami-inspired structures: from fundamentals to applications'.

Dynamic modeling and vibration suppression evaluation of composite honeycomb hemispherical shell with functional gradient protective coating

The vibration reduction performance of composite honeycomb hemispherical shells (CHHSs) coated with functional gradient protection coating (FGPC) are investigated in this work. Using the first-order shear deformation theory and the power-law distribution rule, the virtual spring technique, the regional decomposition method, and the Newmark-Beta approach, etc., a dynamic model of the FGPCCHHSs under base excitation is formulated to solve the inherent characteristics and displacement responses in time and frequency domains. After a set of convergence analyses are completed to ascertain an appropriate segment number and the stiffness values of virtual springs employed in the predictive model, the forecasted vibration parameters are verified using the literature and experimental results that are performed on uncoated and coated shells. The maximal natural frequency errors of the current model compared to the experimental results are 3.8% and 4.8%, and the displacement response errors under different excitation amplitudes are less than 10.3% and 12.7%, respectively, which demonstrate the correctness of such a model. Finally, the impact of key structural and material parameters on the vibration behaviors of the FPGCCHHSs is evaluated. To improve their vibration suppression capability, it is recommended to choose a high gradient index of coating material and a large thickness ratio of the FGPC to the overall shell with a reasonable moduli ratio of Material A to Material B of the FGPC to improve vibration reduction capability. This study offers a practical model tool and several important design recommendations for vibration prediction and dynamic attenuation of honeycomb sandwich hemispherical shell structures in aerospace engineering.

Nonlinear vibration behaviours of foam-filled honeycomb sandwich cylindrical shells: Theoretical and experimental investigations

In this study, the nonlinear vibration behaviours of foam-filled honeycomb sandwich cylindrical shells (FHSCSs) are investigated theoretically and experimentally. Initially, a time-domain minimum residual (TDMR) method is first adopted for the analysis of nonlinear free and forced vibrations of the FHSCSs, which is based on a dynamical model developed, with the effective material parameters of the foam-filled honeycomb core being obtained by employing the Gibson method and the Hamilton equivalence technique. Based on the high-order shear deformation shell principle and von Karman's theory, the nonlinear natural frequencies and resonant responses are solved using the Broyden iterative approach and TDMR method. Furthermore, several literature results are utilized for the rough verification of such a method. Meanwhile, the FHSCS specimens are made manually and comprehensive tests with various base excitation energies are undertaken to provide a thorough validation of the current method. It can be found that the largest calculation errors of natural frequencies and resonant responses are 4.9 and 12.3%, respectively. Finally, the influences of the core form and base excitation amplitudes on the nonlinear vibration behaviours of the FHSCSs are examined. It can be noticed that the foam-filled FHSCS structure exhibits a more excellent anti-vibration capability compared to the non-foam-filled structure. The solving method, fabrication technique, and valuable findings in the current study highlight the path for the design and implementation of such advanced shells.

Validation of the added mass effect in the narrow gaps of nuclear reactor pressure vessels

Nuclear reactor pressure vessels (RPVs) must be robust to external environmental excitations such as seismic loads. Precise numerical models that predict the dynamic responses to external loads are crucial for evaluating the safety of RPVs. Notably, in RPVs submerged in reactor coolant, it is necessary to examine the added mass effect of the fluid via fluid-structure interaction (FSI) analysis, employing either an analytical approach or a numerical analysis. The present study focuses on modeling and analyzing the added mass effect in the narrow gaps associated with the dense structural layout in the reactor. In this context, this paper validates analytical and numerical models that accurately describe the added mass effect for two cases: (i) simple concentric cylinders with a narrow gap filled with fluid and (ii) a complex small-scale RPV with narrow gaps also filled with fluid. Initially, the analytical model, employing Lagrange's equation of motion and velocity potential, and the numerical model, utilizing FE modeling in acoustic FSI through ANSYS, were validated through experimental modal testing for the added mass effect in the narrow gap between simple concentric cylinders. The analytical model for the concentric cylinders indicates that the added mass diverges to infinity as the gap approaches zero, which should be examined carefully. Through experiments for the concentric cylinders, the analytical and FEM-based added mass effects were validated with a natural frequency error of 3 %. Subsequently, based on the results for the concentric cylinders, a complex numerical model of the small-scale reactor with complex boundary conditions was developed and cross-validated through corresponding experimental modal testing of the small-scale RPV. The added mass model for the narrow gaps in the RPV was validated by substantial agreement of eight representative modes, as demonstrated by a 5 % natural frequency error.

Modular approximate discrete modeling for blisk system with triple closed-loop equivalence method

At present, the lumped-parametric model is often used in the preliminary design of bladed disks. However, the lack of systematic modeling methods leads to high errors. In order to develop the systematic modeling method and improve the equivalence accuracy, this paper considers the comprehensive equivalence from the aspects of natural characteristics, transient forced responses, and steady-state forced responses under aerodynamic excitations. This paper includes two novelties. First, based on the modular equation of blisk system, an approximate discrete model is derived for computational efficiency. Second, a triple closed-loop equivalence method with modal weight optimization is developed innovatively for optimal equivalence parameters. The method can significantly improve the accuracy of element parameters in the lumped-parametric model and achieve the comprehensive equivalence to actual blisk. Compared with the existing equivalence method, the average errors of natural frequencies in the whole mode interval and in the high-density mode interval are reduced by 41.75 % and 36.76 %, respectively. The maximum and average displacement errors of the transient response are reduced by 51.76 % and 58.98 %, respectively. Furthermore, vibration characteristics of the mass and stiffness mistuned blisks are analyzed under traveling wake excitation during acceleration and deceleration processes by approximate discrete model and determined parameters.

Mitigating footfall-induced vibrations in cross-laminated timber floor-panels by using beech or birch

Long-span floors are particularly sensitive to footfall-induced vibrations, making their analysis and design critical for ensuring occupant comfort. Cross-laminated timber (CLT) panels have high stiffness relative to their self-weight and are commonly made of spruce. Hardwoods such as birch and beech possess higher stiffness and mass density, which can potentially improve the vibrational performance of CLT panels. In this study, the vibrational performance of CLT panels made of beech and birch with different mechanical properties was investigated using finite element (FE) models calibrated via experimental modal analysis (EMA) tests, yielding calibration with less than 1% error in natural frequencies. The study demonstrated that birch or beech laminations, compared with spruce ones, can provide a significant mitigation of vibrational responses, both for broadband (to approximately 40% of the spruce reference level) and footfall-induced vibrations (to approximately 20%–30% of the spruce reference level). It is noteworthy that low quality beech can result in a marked amplification that may nearly double the level of footfall-induced vibrations.

Modelling and simulation of suspension system based on topological structure

The traditional approach of longitudinal cutting in suspension system simplification introduces redundant degrees of freedom, leading to increased system errors. This study focused on a seven degrees of freedom (7DOF) car model and constructed a suspension system topology model based on an across-cutting approach. To emulate road surface excitation, the filtered white noise method was employed. MATLAB/Simulink was used to create simulation models for the across-cutting, traditional longitudinal cutting and the whole car structure. Comparative analysis of these three topologies was conducted in both the time and frequency domains. Simulation results demonstrated that the performance curve of the across-cutting simplified suspension system closely matched that of the whole car model, validating the accuracy of the proposed across-cutting topology. Furthermore, when compared to traditional longitudinal cutting, the across-cutting simplification method reduced the natural frequency error of body pitch vibration by 25% and decreased the root mean square error of body acceleration by 27%. The suspension topology based on across-cutting more closely resembled the actual car structure, offering a theoretical foundation for enhancing overall ride comfort in automobiles.

A novel forward computational modal analysis method of the motor stator assembly considering core lamination and winding stacking

The forward computational modal analysis (CMA) of the motor stator assembly is gaining attention as a technical challenge in the area of motor noise. There is still no effective method to obtain accurate anisotropic material parameters (AMPs) required for CMA without prototype motors. In this paper, a novel forward calculation method (FCM) for calculating AMPs of the motor stator assembly is proposed, in considering structural discontinuities and composite material properties due to core lamination and winding stacking. The method is based on multi-scale theory. Firstly, the multi-scale equations coupling the macroscopic model and the mesoscopic model of the stator core and winding are established. By decoupling the multi-scale equations, the equivalence and equivalence preconditions of AMPs of the two scale models are described. Accordingly, the finite element method (FEM) for calculating AMPs of the stator core and winding based on the mesoscopic model is proposed. Then, the AMPs of the stator core and winding are calculated by the FCM, and the influencing factors are further studied, respectively. Finally, the effectiveness of this method is verified by modal experiments. Compared with experimental results, the maximum relative error of natural frequencies of the stator assembly is within 1.5%. This method can achieve accurate CMA without prototype motors and modal experiments, which contributes to the accurate prediction and control of electromagnetic noise in the initial motor design stage.

Natural Frequency Analysis on The Base Structure of The CNC Milling Machine Using Simulation and Experimental Methods

The development of national machine tools needs special attention in order to compete with foreign products. Government agency cooperates with industry partners to develop precision CNC milling tools at competitive prices in today's manufacturing industry. One of the problems that often occur in the machine structure is the vibration caused by the resonance between the components that make up the machine, which results in inaccurate machining products. Therefore, it is necessary to do a natural frequency analysis on the milling machine prototype. This research was conducted on the base structure which is the most basic component as a holder for other components. Natural frequency analysis on the base structure uses two methods, namely experimental and numerical simulation using FEA. The results of the two methods are then compared and analyzed for errors that occur. Research parameter so that design does not resonate is that rotational velocity should be 15% above or below the natural frequency. The results show comparison natural frequency error value of numerical simulation method with experimental method and analysis of safety base design against resonance.

Mistuning identification and model updating approach for bladed disks based on blade tip timing

To solve the problem of online mistuning identification of bladed disks, this paper proposes a novel mistuning identification method based on proper orthogonal modes (POM-ID) and the response data. Firstly, the proposed method is analyzed by numerical simulation technology, and the influence of related parameters on the identification results is discussed. Compared with the traditional method, the effectiveness of the proposed method is verified. Secondly, the POM-ID method is applied to identify the mistuning parameters of an actual bladed disk based on the blade tip timing test data. Finally, the identification results of the rotating bladed disk are verified by using static test results, and the dynamic model of the bladed disk is updated. The accuracy of the updated model is confirmed by the aspects of natural frequency and vibration coupling effect. The results indicate that the natural frequency error of the modified model is less than 1%, and the vibration coupling law between blades is consistent with the test results. The mistuning identification method proposed in this paper is especially suitable for test data of the rotating bladed disk and is expected to be applicable for online dynamic updating of the bladed disk model.

Natural frequency prediction of the 3-RPS parallel manipulator using the substructure synthesis technology

AbstractThis paper proposed an elastodynamic modeling method combined with independent displacement coordinates and substructure synthesis technology. Firstly, the connecting rod was discretized, and the elastodynamic control equation for each element was established. The multipoint constraint element theory, linear algebra, and singularity analysis were used to identify the globally independent displacement coordinates of the manipulator. On this basis, the elastodynamic model using the substructure synthesis for the 3-PRS parallel manipulator (PM) was developed, with its natural frequencies distribution in the regular workspace discussed. The comparison with the finite-element results showed that the maximum error of the first three-order natural frequencies was within 1.03%, which verified the correctness of the analytical model. The proposed elastodynamic model included all the kinematic constraints of the manipulator without increasing the Lagrangian multiplier. The method is computationally efficient and assesses the dynamic behaviors of the mechanism at the predesign phase.

Vibrations of joined conical-cylindrical shells with bolt connections: Theory and experiment

This paper investigated the vibration characteristics of joined conical-cylindrical shells (JCCSs) with bolt loosening boundary conditions both theoretically and experimentally. The non-uniform arc constraint model is established to simulate the bolt loosening boundary conditions by the artificial spring technology, which can describe the actual bolt contact pressure distribution. Continuously distributed artificial springs are introduced to realize the connection of the conical and cylindrical shells. Donnell's shell theory and Chebyshev polynomials are adopted to establish the theoretical model, and the dynamic equation is derived from the Lagrange equation. The validity of the present model is verified by the available literature, hammer test and vibration shaker test. The results indicate that the maximum errors of natural frequencies and resonant displacements are less than 4.41% and 18.6%, respectively. It is found that bolt loosening has a significant effect on the fundamental frequency, but it gradually weakens with the increase of the mode order. Increasing bolt loosening degree can lead to an increment of resonant displacement. The developed model is reliable in estimating the dynamic behaviors of bolted JCCSs with different loosening degrees, and it contributes to providing guidance in the design and service process of combined shell structures.

A developed model updating method based on extended frequency response functions and its application study of offshore structures

Accurate finite element (FE) models are extremely vital to the analysis of the dynamic characteristics of engineering structures. However, it is challenging for the frequency response function (FRF)-based model updating method to obtain reliable numerical results as the limited number of FRFs at the selected peak positions usually leads to an unacceptable or failure of model updating. Moreover, the increased number of updating coefficients in complex engineering problems generates nonlinear optimization models with the local solutions by traditional optimization techniques. To tackle these two issues, a new model updating method is proposed to make full use of FRFs extracted from measured field data of structures and the improved particle swarm optimization (PSO) technique for accurately estimating physical parameters of structures. The novelties of this research include: (1) The signature assurance criterion (SAC)-based FRF is evaluated to eliminate the influence of limited frequency points on the accuracy of the updated model using the normalized acceleration component; (2) The enhanced (PSO) is developed to realize the adaptive selection of inertia factors for the better diversity by introducing an average value of the fitness function, and then accurate predictions of updating coefficients within a smaller number of iterations are achieved using the developed constraint factor. The effectiveness of the proposed method is verified by mathematical model of a jacket platform. Results show that the proposed method can accurately obtain the updating coefficients under spatial incomplete condition, and the maximum error of natural frequencies is 0.779% using the accelerations containing 5% noise. To prove the robustness of the proposed method, experimental studies of monopile offshore wind turbine are also conducted and the maximum error of natural frequencies is 1.887% under the consideration of spatial incompleteness represented by the structural stiffness degradation. Finally, the feasibility of the proposed method is evaluated by a test of a complex jacket platform, whose variation is simulated by weakening the connections of some elements. The maximum error of natural frequencies predicted by the updated model is only 4.831% as compared with experimental results. Throughout these examples, the extended FRF model updating method provides engineers and designers with a useful insight into the development of reliable techniques to accurately predict dynamic responses of offshore structures.

Dynamic Modeling and Vibration Characteristics Analysis of a Large-Scale Photoelectric Theodolite Tracking Frame

In this paper, the experimental modes of a large-scale photoelectric theodolite tracking frame are presented. On the basis of the experimental data and the gradient-less optimization approach, the modeling strategy and the parameterized equivalent dynamic finite element model are presented. Shafting, three-point leveling units, and other components are reasonably simplified during the modeling process. Influence factors such as contact stiffness are introduced as dynamic parameters in the model. The optimized parametric model obtained demonstrates that the linearization modeling strategy represents the dynamic response characteristics of this type of structure accurately. The maximum relative error of the first four-order natural frequencies between numerical simulation and experimental data is 4.45% when the consistency of mode shapes is taken into account. The research results in this paper can provide engineering guidance for the dynamic stiffness optimization design of the large-scale photoelectric theodolite tracking frame.

Reconstruction of dynamic wind forces on a transmission steel lattice tower using aeroelastic wind tunnel test data

This article reports a methodology for estimating dynamic wind forces on a steel lattice tower by using limited structural displacements. The methodology involves combining the modal state-space system model and a Kalman filter-based algorithm. For convenient application, the equivalent concentrated forces are taken as the inversing objects, representing the distributed wind forces on a lattice tower. The corresponding equivalent cantilever model of the lattice tower is utilized, in which the stiffness matrix is estimated by a modified flexibility approach. Based on the modal superposition principle, the dynamic wind forces in physical space are reconstructed using the identified input modal wind forces. The scheme of the dynamic wind force reconstruction is designed in detail. A typical transmission steel lattice tower is considered, and the aeroelastic wind tunnel experiments are performed. The performance and effectiveness of the proposed scheme are investigated based on numerical simulations and wind tunnel tests. It is indicated that the reconstructed dynamic wind forces can produce equivalent dynamic responses to the actual. Besides, it performs robustly on the effect of measurement noise. Although the modal force identification is sensitive to the modeling error of structural natural frequency, the effect can be judged according to the change in the power spectra of the identified modal force. The proposed inversing approach can serve as a useful tool to evaluate dynamic wind forces on steel lattice towers with limited response measurements.

Theoretical and experimental investigations of vibration and damping behaviors of carbon fiber-reinforced composite thin shells with partial bolt looseness constraints

In this paper, the vibration and damping behaviors of fiber-reinforced composite thin shells under partial bolt looseness boundary conditions are investigated both theoretically and experimentally. Based on Love's shell theory, the complex modulus approach, and Hooke's law, a theoretical model of fiber-reinforced composite thin shells is constructed. A non-uniform artificial spring technique is proposed to accurately describe the non-uniform looseness in the bolt constraint positions and their interval locations, in which the two kinds of artificial springs, namely the “main spring” and “secondary spring”, are taken into account separately. After the natural characteristics, damping properties, and vibration responses are solved by the Rayleigh-Ritz method, strain energy approach, and Duhamel integral method, detailed measurements are undertaken on a CA 500 carbon/epoxy composite shell specimen to examine vibration and damping characteristics as well as to validate the proposed model. By comparing the theoretical results with the experimental ones, the maximum calculation errors of natural frequencies, resonant responses, and damping ratios are within an acceptable level. In addition, the influences of partial bolt looseness constraints on the vibration characteristics of such shell structures are also investigated here.

Vibration signal diagnosis and conditional health monitoring of motor used in biomedical applications using Internet of Things environment

Vibration, especially basic vibration, may cause loose contacts, open circuits, or other contact problems, which account for a large proportion of system failure causes. Due to the complexity of the chassis structure used in the biomedical motors, the vibration analysis of a single component cannot solve the vibration problems encountered in the work of the system. Therefore, it is necessary to use system simulation and experiments to monitor and enhance the health of the structure. Finite element method is used to carry out the dynamic analysis from the component to the system, and calculate the natural frequency, modal shape and harmonic response. Then, the effectiveness and accuracy of the analysis results are verified by experiments through simulation and experiments. The results show that the measures taken have effectively reduced the vibration of the chassis equipment and improved its safety for better health safety of patients. There is just a 2% variation between the natural frequencies acquired from the experiment and the simulation computation. In terms of natural frequency, the first 6-order natural frequencies all have errors of less than 10%, and the natural frequency error is comparable to that of fireproof board.

An improved contact parameter model with elastoplastic behavior for bolted joint interfaces

In this study, considering the elastoplastic behavior of contacting asperities on micro-scale and the contact pressure distribution of bolted joint interfaces on macro-scale, an improved contact stiffness model is presented. Firstly, the bolted T-stub joint is chosen as the object of study, and the distribution of normal contact load over the interface under preload conditions is studied. Then, combining the continuity, smoothness and monotonicity of the constitutive relations of normal contact load-deformation in three different regimes of plastic- elastoplastic-elastic of contacting asperities, the contact stiffness of individual asperity is analytically derived. Meanwhile, the relation between the distribution of contact stiffness and the distribution of normal contact load for bolted joint interfaces is determined based on fractal theory. The presented model overcomes the shortcomings of the contact stiffness without elastoplastic behavior obtained from the MB model and reveals the inhomogeneous distribution features of interfacial contact stiffness in bolted joints. Finally, the finite element (FE) analysis for flange-bolted joints incorporating the proposed model and the relevant experimental verification are performed. The results show that the average error of the first 4-order natural frequencies between the FE results and the experimental ones is only 0.72%, and the predicted results in frequency-domain are consistent with the measured ones, which illustrates that the proposed model can be used in interface modeling of jointed structures.

Free and forced vibrations of composite cylindrical–cylindrical​ shells with partial bolt loosening connections: Theoretical and experimental investigation

In the present study, the free and forced vibrations of composite cylindrical–cylindrical shells with partial bolt loosening connections are investigated both experimentally and theoretically. Initially, a theoretical model is proposed based on the assumption of primary and secondary connection springs with reduced stiffnesses related to different loosening positions of connection bolts. The equation of motion is derived for predicting the fundamental frequencies, mode shapes, and vibration displacement responses on the basis of the Rayleigh–Ritz method, the Jacobian polynomial approach, the modal superposition technique, etc. To ensure the calculation accuracy of the present model, a convergence study is conducted to determine the truncation number. Also, to give a solid validation of the model, some detailed tests, such as the pulse, sweeping-frequency, and resonant excitations, are conducted on a combined shell specimen, which is obtained by assembling two different composite cylindrical shells via twelve M8 bolts, with a torque wrench being adopted to control the torques in the connection bolts to achieve the desired partial bolt loosening effect. By comparing the calculated and tested results, it can be found that the present model is reliable to predict the structural dynamic parameters because the calculation errors of natural frequencies and resonant displacements are less than 7.4 and 11.9 %, respectively. Finally, the impact of critical bolt loosening parameters on vibration characteristics of the combined shells is evaluated, with some important conclusions being highlighted to illustrate the reduction mechanism of dynamic properties as the partial bolt loosening degree and number increase or the loosening pattern becomes more complex.

Hybrid Modeling of Position-Dependent Dynamics of Thin-Walled Parts Using Shell Elements for Milling Simulation

Abstract This article presents a hybrid model to update the position-dependent structural dynamic parameters of thin-walled workpieces as the metal is removed during machining. The initial workpiece is modeled by shell elements, and its full stiffness and mass matrices are used to solve the eigenvalues and mode shapes to predict the frequency response function (FRF) at a fixed location. The model is calibrated using the experimentally measured FRF, which reduces the errors contributed by the uncertainties in the material properties and damping values. The optimized finite element (FE) model is then perturbed at discrete cutting locations to obtain the updated natural frequencies and mode shapes of the part without solving the computationally prohibitive eigenvalue problem. The accuracy of the model is further improved by using either full FE solutions or experimental measurements of FRFs at a few intermediate steps which reduce the accumulated perturbation errors along the tool path. The proposed method is verified in five-axis milling of a thin-walled twisted fan blade. It is shown that using shell elements reduces the computation effort by ∼20 times compared to the conventional three-dimensional (3D) cube elements. The experimental calibration of the numerical model at a few discrete locations reduces the prediction error of natural frequencies by about 50%.