Periodic Forced Vibrations Research Articles

Real-Time Milling Chatter Detection and Control with Axis Encoder Feedback and Spindle Speed Manipulation

This paper introduces a complete real-time algorithm, where the chatter is detected and eliminated by spindle speed manipulation via the chatter energy feedback calculated from the axis encoder measurement. The proposed method does not require profound knowledge of the machining dynamics; instead, the entire algorithm exploits the fact that milling vibrations consist of forced vibrations at spindle speed harmonics and chatter vibrations that are close to one of the natural modes, with sidebands which are spread at the multiples of spindle speed frequency above and below the chatter frequency. The developed algorithm is able to identify the amplitude, phase and frequency of all the harmonics constituting the periodic forced and chatter vibrations. The key challenge is to select dominant chatter frequencies for the calculation of a robust and accurate chatter energy ratio feedback; this is achieved by utilizing the frequency estimation variance of EKF as a novel chatter indicator. Based on the chatter energy ratio feedback, the controller overrides the spindle speed in order to suppress the chatter energy below a particular threshold value. The varying spindle speed challenge is handled by updating the state transition matrices of the Kalman filters and real-time calculation of the band-pass filter coefficients, based on the derived discrete time transfer functions. The developed algorithm is tested on a Deckel FP5cc CNC which is in-house retrofitted and has a PC-based controller for the real-time application of the proposed algorithm. It is shown that the real-time chatter frequency and amplitude estimates are compatible with off-line FFT analysis, and chatter can be successfully eliminated by energy feedback.

Frequency-Domain Sensitivity Analysis of Stability of Nonlinear Vibrations for High-Fidelity Models of Jointed Structures

For the analysis of essentially nonlinear vibrations, it is very important not only to determine whether the considered vibration regime is stable or unstable but also which design parameters need to be changed to make the desired stability regime and how sensitive is the stability of a chosen design of a gas-turbine structure to variation of the design parameters. In the proposed paper, an efficient method is proposed for a first time for sensitivity analysis of stability for nonlinear periodic forced response vibrations using large-scale models structures with friction, gaps, and other types of nonlinear contact interfaces. The method allows using large-scale finite element (FE) models for structural components together with detailed description of nonlinear interactions at contact interfaces. The highly accurate reduced models are applied in the assessment of the sensitivity of stability of periodic regimes. The stability sensitivity analysis is performed in frequency domain with the multiharmonic representation of the nonlinear forced response amplitudes. Efficiency of the developed approach is demonstrated on a set of test cases including simple models and large-scale realistic blade model with different types of nonlinearities, including friction, gaps, and cubic elastic nonlinearity.

Geometrically non-linear periodic forced vibrations of imperfect laminates with curved fibres by the shooting method

Abstract In this paper, the authors study periodic vibrations of variable stiffness composite laminates excited by a harmonic force. The plates have geometrical imperfection in the form of various sinusoidal out-of-plane initial deflections associated with zero stress. The angle of the curvilinear fibre path is introduced as a function of the horizontal Cartesian coordinate. The theory used to extract equations of motion for VSCLs is a third order shear deformation theory that retains rotary inertia. The relations of von Karman for elastic large deflection are used. A p -version finite element is employed and, to find the solution of the equations of motion, the shooting method is applied; frequency response curves are obtained. Static condensation and a modal summation method are applied to reduce the number of degrees of freedom. A damage analysis based on Tsai-Wu criterion is carried out during the studies on vibration. The effects of curvilinear fibres and the influence of modal interactions on the vibration of imperfect VSCLs are investigated. The stability of the periodic solutions is determined by applying Floquet's theory. The effect of geometric imperfections on the vibrational behaviour is studied.

Stability Analysis of Multiharmonic Nonlinear Vibrations for Large Models of Gas Turbine Engine Structures With Friction and Gaps

An efficient method is proposed for the multiharmonic frequency-domain analysis of the stability for nonlinear periodic forced vibrations in gas turbine engine structures and turbomachines with friction, gaps, and other types of nonlinear contact interfaces. The method allows using large-scale finite element models for structural components together with detailed description of nonlinear interactions at contact interfaces between these components. The highly accurate reduced models are applied in the assessment of stability of periodic regimes for large-scale model of gas turbine structures. An approach is proposed for the highly accurate calculation of motion of a structure after it is perturbed from the periodic nonlinear forced response. Efficiency of the developed approach is demonstrated on a set of test cases including simple models and large-scale realistic bladed disk models with different types of nonlinearities: friction, gaps, and cubic nonlinear springs.

On the efficacy of friction damping in the presence of nonlinear modal interactions

This work addresses friction-induced modal interactions in jointed structures, and their effects on the passive mitigation of vibrations by means of friction damping. Under the condition of (nearly) commensurable natural frequencies, the nonlinear character of friction can cause so-called nonlinear modal interactions. If harmonic forcing near the natural frequency of a specific mode is applied, for instance, another mode may be excited due to nonlinear energy transfer and thus contribute considerably to the vibration response. We investigate how this phenomenon affects the performance of friction damping. To this end, we study the steady-state, periodic forced vibrations of a system of two beams connected via a local mechanical friction joint. The system can be tuned to continuously adjust the ratio between the first two natural frequencies in the range around the 1:3 internal resonance, in order to trigger or suppress the emergence of modal interactions. Due to the re-distribution of the vibration energy, the vibration level can in fact be reduced in certain situations. However, in other situations, the multi-harmonic character of the vibration has detrimental effects on the effective damping provided by the friction joint. The resulting response level can be significantly larger than in the absence of modal interactions. Moreover, it is shown that the vibration behavior is highly sensitive in the neighborhood of internal resonances. It is thus concluded that the condition of internal resonance should be avoided in the design of friction-damped systems.

Surface effect on the large amplitude periodic forced vibration of first-order shear deformable rectangular nanoplates with various edge supports

Surface stress and surface inertia effects may play a significant role in the mechanical characteristics of nanostructures with a high surface to volume ratio. The objective of this study is to present a comprehensive study on the surface stress and surface inertia effects on the large amplitude periodic forced vibration of first-order shear deformable rectangular nanoplates. To this end, the Gurtin–Murdoch theory, first-order shear deformation theory (FSDT) and Hamilton׳s principle are employed to develop a non-classical continuum plate model capable of taking the surface stress and surface inertia effects and also the rotary and in-plane inertias into account. To solve numerically the geometrically nonlinear forced vibration of nanoplates with different boundary conditions, the generalized differential quadrature (GDQ) method, numerical Galerkin scheme, periodic time differential operators and pseudo arc-length continuation method are employed. The effects of parameters such as thickness, surface residual stress, surface elasticity, surface mass density, length-to-thickness ratio, width-to-thickness ratio and boundary conditions on the nonlinear forced vibration of rectangular nanoplates are fully investigated. The results demonstrate that surface effects on the nonlinear frequency response of aluminum (Al) nanoplate are more prominent in comparison with the silicon (Si) nanoplate.

Influence of periodic excitation on self-sustained vibrations of one disk rotors in arbitrary length journals bearings

Interaction of forced and self-sustained vibrations of one disk rotor is described by nonlinear finite-degree-of-freedom dynamical system. The shaft of the rotor is supported by two journal bearings. The combination of the shooting technique and the continuation algorithm is used to study the rotor periodic vibrations. The Floquet multipliers are calculated to analyze periodic vibrations stability. The results of periodic motions analysis are shown on the frequency response. The quasi-periodic motions are investigated. These nonlinear vibrations coexist with the periodic forced vibrations.

Shape oscillation and detachment conditions for a droplet on a vibrating flat surface

In this study, we obtain experimental understanding of the mode characteristics of a droplet placed on a flat surface under periodic forced vibrations. The detachment conditions for the droplet on the surface were also studied. In order to estimate the resonance frequency of a droplet placed on a hydrophobic surface, theoretical modelling was combined with experimental approaches. Two high speed cameras were used to observe droplet characteristics, including mode shape, detachment, occurrence of secondary droplet breakup, and horizontal torsional motion. Two cameras were installed to the right above the droplet and at the side of the droplet. There was no more than an 18% discrepancy between the theoretical and experimental resonance frequencies. This discrepancy was likely caused by several factors such as contact line friction, nonlinear wall adhesion, and experimental uncertainty. When applying a relatively low voltage to a speaker, the contact line of a droplet was pinned and shape oscillations of the droplet appeared in a bilaterally symmetric way. In contrast, at higher voltages, the contact line depinned and the shape oscillations became more active. For excitation frequencies identical to the mode frequency, the lobe size of the droplet was relatively larger than that at neighbouring frequencies. The experimental results also indicate that the generation and complete detachment of small-scale droplets occur only at the 2nd mode.

Princípios físicos da elastografia por ressonância magnética

Este trabalho constitui os primeiros esforços em apresentar ao público de língua portuguesa uma nova ferramenta, com potencial uso clínico, para estudar propriedades mecânicas de materiais e tecidos vivos. Busca-se apresentar propostas de exploração de modalidades de aquisição e processamento de imagens por ressonância magnética (IRM). A Elastografia por Ressonância Magnética (ERM) é um novo procedimento que permite a medição in vivo dos parâmetros viscoelásticos dos tecidos, e que se encontra atualmente em fase de pesquisa. Ela baseia-se na Tomografia de Ressonância Magnética (TRM) e não usa radiação inonizante, tal como raios-X que poderiam causar dano ao tecido. A técnica de ERM funciona introduzindo-se ondas mecânicas periódicas na superfície do tecido, utilizando um atuador para excitar o meio. A introdução da onda mecânica senoidal no tecido leva a uma vibração periódica forçada, e as medições feitas pela TRM são sincronizadas com esta onda mecânica. Dessa forma é possível aumentar o contraste da imagem, que é proporcional à amplitude da onda. Assim, a tomografia por ressonância magnética funciona como uma "câmara estroboscópica" que permite a criação de um "instantâneo" da onda mecânica no tecido, sendo que a propagação de uma onda contínua pode ser utilizada para produzir um filme, tendo uma série de instantâneos em intervalos diferentes. As imagens de deslocamento de fase resultantes da propagação das ondas, através do meio, são obtidas utilizando o método de contraste de fase da técnica de imagem por RM. Estas mudanças de fase constituem a fundamentação para a posterior reconstrução dos parâmetros elásticos, que são representados por uma imagem chamada elastograma. Essa nova metodologia quantitativa e não-invasiva de propriedades elásticas de tecidos permite auxiliar os radiologistas nas decisões de diagnósticos corroborando com informações já existentes.

진동 평판 위 액적의 형상 진동 변화 및 모드 특성

Key Words: Droplet(액적), Lobe(둥근돌출부), Resonance Frequency(공진주파수), Secondary Droplet(미소액적), Shape Oscillation(형상진동)초록: 본연구는주기적인강제진동이가해지는액적의모드특성을실험적으로이해하는것을목적으로하고있다.액적의공진주파수예측을수행하여이론및실험적해석을통해두접근방법의타당성을 파악하였으며,초고속카메라를사용하여액적의다양한변형특성-모드형상,분리,미소액적의발생,그리고비틀림의특성을관찰하였다.이론해석및실험결과와의비교에있어공진주파수값의차이가약15%이하라는것이도출되었으며이러한차이의발생원인으로접촉선마찰,비선형벽고착,실험의불확실성등에큰영향을받는것으로판단된다.접촉선이고정되어있을경우와작은진폭조건하에서액적의모양은대칭형상을가졌으며,공진주파수에서의로브의크기는주변부주파수에서의로브크기보다더크게된다는점을확인하였다.Abstract: This study aims to understand the mode characteristics of a droplet under a periodic forcedvibration. To predict the resonance frequency of a droplet, theoretical and experimental approaches wereemployed. A high-speed camera was used to capture the various deformation characteristics of a droplet—mode shape, detachment, separated secondary droplet, and skewed deformation. The comparison between thetheoretical and the experimental approaches shows a ~10% discrepancy in the prediction of the resonancefrequency, which appears to be caused by the effect of contact line friction, nonlinear wall adhesion, andexperimental uncertainty. Owing to contact-line pinning and smaller amplitude, the droplet shape becomessymmetric and the size of each lobe at the resonance frequency exceeds that at the neighbor, which is out ofresonance.

Geometrically nonlinear transverse steady-state periodic forced vibration of multi-degree-of-freedom discrete systems with a distributed nonlinearity

A method based on Hamilton’s principle and spectral analysis has been applied recently to nonlinear free vibrations of two and multi-degree-of-freedom (2-dof) and (N-dof) systems with cubic nonlinearities. This leads to calculation of the nonlinear free modes of vibration and their associated nonlinear frequencies. More recently, this method has also been applied to examine the nonlinear forced transverse steady-state periodic response of 2-dof systems made of two masses and four spiral springs. The objective of the present work was the extension of this method to the nonlinear forced transverse steady-state periodic response of N-dof systems made of N masses and N+2 nonlinear spiral springs. Its dynamic behavior has been described in terms of the mass tensor, the rigidity tensor and the fourth order tensor due to the nonlinearity. In order to solve the nonlinear algebraic system obtained, the corresponding linear free vibration problem was first solved. After determination of the linear eigen vectors and eigen values, a change of basis, from the initial basis – i.e. the displacement basis (DB) – to the modal basis (MB), has been performed using the classical matrix transformation.The new nonlinear algebraic system has then been solved using the so-called first formulation developed in previous works. This has lead to explicit expressions for the nonlinear frequency response function in the neighborhood of each of the N modes. A computer program has been written using Matlab Software. This has allowed the N-dof system nonlinear forced vibrations to be automatically examined, in the neighborhood of each mode, for any distribution of parameters in a systematic and unified way.

Nonlinear transverse steady-state periodic forced vibration of 2-dof discrete systems with cubic nonlinearities

A method based on Hamilton’s principle and spectral analysis has been applied recently to nonlinear transverse vibrations of discrete systems with cubic nonlinearities, leading to calculation of the nonlinear free modes of transverse vibration and their associated nonlinear frequencies. The objective of the present work was the extension of this method to the nonlinear forced transverse steady-state periodic response of 2-dof system leading to nonlinear frequency response function in the neighbourhood of the two modes

Nonlinear transverse steady-state periodic forced vibration of 2-dof discrete systems with cubic nonlinearities

A method based on Hamilton’s principle and spectral analysis has been applied recently to nonlinear transverse vibrations of discrete systems with cubic nonlinearities, leading to calculation of the nonlinear free modes of transverse vibration and their associated nonlinear frequencies. The objective of the present work was the extension of this method to the nonlinear forced transverse steady-state periodic response of 2-dof system leading to nonlinear frequency response function in the neighbourhood of the two modes.

Modified shooting approach to the non-linear periodic forced response of isotropic/composite curved beams

Large amplitude periodic forced vibration of curved beams under periodic excitation is investigated using a three-noded beam element. The element is based on the higher-order shear deformation theory satisfying interlayer continuity of displacements and transverse shear stress, and top-bottom conditions on the latter. The periodic responses are obtained using shooting technique coupled with Newmark time marching and arc length continuation algorithm developed. The second order governing differential equations of motion are solved without transforming to the first order differential equations thereby resulting in a computationally more efficient algorithm. The effects of excitation amplitude, support conditions and beam curvature on the frequency versus response amplitude relation are highlighted. The typical frequency response curves for isotropic and cross-ply laminated curved beams are presented. Phenomenon of strong modal interactions is observed.

Steady‐State Oscillation of Hysteretic Differential Model. I: Response Analysis

This study considers the periodic forced vibration of nonlinear hysteretic system. Multiharmonic steady-state responses of a general hysteretic system adopting the Bouc-Wec differential model, subjected to arbitrary periodic excitation, are analyzed by the Galerkin/Levenberg-Marquardt method. The frequency/time domain alternation and the fast Fourier transformation (FFT) techniques are introduced in this frequency domain solution procedure. The frequency-controlled and amplitude-controlled algorithms are simultaneously formulated to obtain complete, probably multivalued, frequency response curves. This approach gives possible harmonic, superharmonic, and subharmonic responses of the Bouc-Wen hysteretic system. The response characteristics of softening, hardening, and quasilinear hysteretic systems are studied through numerical computations. The proposed procedure is also extended to analyze the periodic vibration of degrading hysteretic system with amplitude-dependent stiffness deterioration. In addition, a verification is given by comparing with the results obtained by a numerical integration procedure.

Periodic forced vibration of unsymmetrical piecewise-linear systems by incremental harmonic balance method

A single-degree-of-freedom non-linear system subjected to periodic excitation is considered. The non-linearity is in the restoring force and is represented by unsymmetrical piecewise-linear stiffness. The steady state response are analysed by an extended incremental harmonic balance (IHB) method. All possible subharmonic, harmonic and superharmonic resonances and bifurcations are sought. The results reveal some interesting details of responses of unsymmetrical piecewise-linear systems. Comparisons with available published findings are also made.

Detection of subsurface defects using phase conjugate waves and forced vibration

In four-wave mixing in BSO crystal, a periodic forced vibration is applied to both the object under test and the standard object to provide signal and reference waves, respectively. The system images subsurface defects of the test object by converting the difference of vibration between the test and the standard objects into image intensity. The principle and basic experimental results are given.

Response of rotating rings to harmonic and periodic loading and comparison with the inverted problem

The harmonic and periodic forced vibrations of rotating rings are derived and investigated. The modal expansion technique yields the forced solution, which is characterized by four generalized co-ordinates associated with each n (circumferential wave number). The inextensional assumption is presumed, when flexural vibration is the only important component, to reduce the order of the system. The closed form solutions to the harmonic load cases, once concentrated, once distributed, are demonstrated and interpreted. The approach is then extended to periodic loads, where Fourier sine and cosine series is applied. Examples depict the numerical responses to all the cases being derived. The solutions of a stationary ring subjected to traveling loads are also solved for comparison. Their difference is investigated and interpreted from various viewpoints.

On Relaxation Oscillations Governed by a Second Order Differential Equation for a Large Parameter and with a Piecewise Linear Function(1)

AbstractThis paper deals with the differential equation: ẍ + μF(ẋ) + x = ƒ( X, ẋ, t/Tμ) for μ ≫ 1 where F is a piecewise linear function and f is a periodic function of period μT, where T is to be chosen. It is established that periodic forced vibrations exist in an annular domain R(μ) constructed for the free vibration (ƒ ≡ 0), provided ƒ is not of higher order than Subsequently with ƒ = A cos (2πt/μT*), an asymptotic treatment of the forced vibration problem is carried out, by finding the proper initial conditions and the proper period μT* of f. Finally it is concluded that μT* is close to the period of the free vibration.

Periodic forced vibrations of systems of relaxation oscillators

Periodic forced vibrations of systems of relaxation oscillators