Steady-state Harmonic Response Research Articles

Click-train evoked steady state harmonic response as a novel pharmacodynamic biomarker of cortical oscillatory synchrony

Sensory networks naturally entrain to rhythmic stimuli like a click train delivered at a particular frequency. Such synchronization is integral to information processing, can be measured by electroencephalography (EEG) and is an accessible index of neural network function. Click trains evoke neural entrainment not only at the driving frequency (F), referred to as the auditory steady state response (ASSR), but also at its higher multiples called the steady state harmonic response (SSHR). Since harmonics play an important and non-redundant role in acoustic information processing, we hypothesized that SSHR may differ from ASSR in presentation and pharmacological sensitivity. In female SD rats, a 2 s-long train stimulus was used to evoke ASSR at 20 Hz and its SSHR at 40, 60 and 80 Hz, recorded from a prefrontal epidural electrode. Narrow band evoked responses were evident at all frequencies; signal power was strongest at 20 Hz while phase synchrony was strongest at 80 Hz. SSHR at 40 Hz took the longest time (∼180 ms from stimulus onset) to establish synchrony. The NMDA antagonist MK801 (0.025–0.1 mg/kg) did not consistently affect 20 Hz ASSR phase synchrony but robustly and dose-dependently attenuated synchrony of all SSHR. Evoked power was attenuated by MK801 at 20 Hz ASSR and 40 Hz SSHR only. Thus, presentation as well as pharmacological sensitivity distinguished SSHR from ASSR, making them non-redundant markers of cortical network function. SSHR is a novel and promising translational biomarker of cortical oscillatory dynamics that may have important applications in CNS drug development and personalized medicine.

Effects of Memantine on the Auditory Steady-State and Harmonic Responses to 40 Hz Stimulation Across Species

BackgroundClick trains elicit an auditory steady-state response (ASSR) at the driving frequency (1F) and its integer multiple frequencies (2F, 3F, etc.) called harmonics; we call this harmonic response the steady-state harmonic response (SSHR). We describe the 40 Hz ASSR (1F) and 80 Hz SSHR (2F) in humans and rats and their sensitivity to the uncompetitive NMDA antagonist memantine. MethodsIn humans (healthy control participants, n = 25; patients with schizophrenia, n = 28), electroencephalography was recorded after placebo or 20 mg memantine in a within-participant crossover design. ASSR used 1 ms, 85-dB clicks presented in 250 40/s 500-ms trains. In freely moving rats (n = 9), electroencephalography was acquired after memantine (0, 0.3, 1, 3 mg/kg) in a within-participant crossover design; 65-dB click trains used 5-mV monophasic, 1-ms square waves (40/s). ResultsAcross species, ASSR at 1F generated greater evoked power (EP) than the 2F SSHR. 1F > 2F intertrial coherence (ITC) was also detected in humans, but the opposite relationship (ITC: 2F > 1F) was seen in rats. EP and ITC at 1F were deficient in patients and were enhanced by memantine across species. EP and ITC at 2F were deficient in patients. Measures at 2F were generally insensitive to memantine across species, although in humans the ITC harmonic ratio (1F:2F) was modestly enhanced by memantine, and in rats, both the EP and ITC harmonic ratios were significantly enhanced by memantine. ConclusionsASSR and SSHR are robust, nonredundant electroencephalography signals that are suitable for cross-species analyses that reveal potentially meaningful differences across species, diagnoses, and drugs.

Alternating state-parameter identification of Bouc-Wen hysteretic systems from steady-state harmonic response

• A novel identification method for Bouc-Wen systems based on harmonic balance method is proposed. • An effective initialization method is developed to predetermine the parameters of linear subsystem and Bouc-Wen parameters. • To update the estimation of parameters in linear subsystem and hysteretic force, an iteratively alternating method is further developed. • The simulation and experimental studies validate the effectiveness of the new method. Describing hysteresis behaviors in mechanical engineering has attracted increasing research interest. Among the numerous models that characterize hysteretic nonlinearities, the Bouc-Wen model stands out owing to its versatility. However, there is a critical challenge for Bouc-Wen hysteretic systems identification that the internal hysteretic restoring force is usually not measurable. A novel alternating state-parameter identification method is proposed in this paper to identify Bouc-Wen hysteretic systems from steady-state harmonic response. Based on the interpretation that system response and restoring force can be analytically approximated via Fourier series, the harmonic balance method is leveraged to reconstruct the hysteretic force. To implement the reconstruction process, an effective initialization procedure is developed to predetermine the parameters of linear subsystem and Bouc-Wen parameters. In the initialization procedure, the Duhamel's integral is leveraged to represent the Bouc-Wen hysteretic system identification problem, which avoids the high-order derivative of measured displacement and further reduces the effect of noise. Additionally, to update the estimation of system parameters and hysteretic force, an iteratively alternating procedure is further developed in this paper. The numerical simulation and dynamic experiment validate the effectiveness of the new identification method for Bouc-Wen hysteretic systems.

Balancing of Flexible Rotors Supported on Fluid Film Bearings by Means of Influence Coefficients Calculated by the Numerical Assembly Technique

In this paper, a new method for the balancing of rotor-bearing systems supported on fluid film bearings is proposed. The influence coefficients necessary for balancing are calculated using a novel simulation method called the Numerical Assembly Technique. The advantages of this approach are quasi-analytical solutions for the equations of motion of complex rotor-bearing systems and very low computation times. The Numerical Assembly Technique is extended by speed-dependent stiffness and damping coefficients approximated by the short-bearing theory to model the behavior of rotor systems supported on fluid film bearings. The rotating circular shaft is modeled according to the Rayleigh beam theory. The Numerical Assembly Technique is used to calculate the steady-state harmonic response, influence coefficients, eigenvalues, and the Campbell diagram of the rotor. These values are compared to simulations with the Finite Element Method to show the accuracy of the procedure. Two numerical examples of rotor-bearing systems are successfully balanced by the proposed balancing method.

Steady state harmonic response of nonlinear soil-structure interaction problems through the Preisach formalism

Steady state harmonic response of nonlinear soil-structure interaction problems is addressed in this paper. Due to well-known stiffness degradation phenomenon, the steady-state study is referred to small-medium strain condition whereas no significant pore water pressure change is observed (e.g. clayey soils and sands in drained conditions). A novel cyclic hysteretic model based on the Preisach formalism is proposed to describe the cyclic behavior of soil-foundation interaction. Through a harmonic balance procedure, furthermore, the steady state response of nonlinear soil-structure interaction problems is determined. Equivalent amplitude-dependent stiffness and damping have been derived in closed form to reliably describe the nonlinear soil-structure interaction phenomenon. Those expressions are function of the elastic foundation soil stiffness, the maximum attainable horizontal force and the moment capacity of the foundation. Furthermore, handy expressions of the equivalent fundamental period and damping, also dependent from the level of the excitation, are also derived. The proposed hysteretic model, moreover, has the advantage to be easily calibrated to match the modulus reduction and damping curves determined either from numerical or experimental tests. Numerical applications and comparisons with more advanced models and experimental data available in literature are also presented.

Steady-state linear harmonic vibrations of multiple-stepped Euler-Bernoulli beams under arbitrarily distributed loads carrying any number of concentrated elements

In this paper, general beam vibration problems with several attachments under arbitrarily distributed harmonic loading are solved. A multiple-stepped beam is modelled by the Euler-Bernoulli beam theory and an extension of an efficient numerical method called Numerical Assembly Technique (NAT) is used to calculate the steady-state harmonic response of the beam to an arbitrarily distributed force or moment loading. All classical boundary conditions are considered and several types of concentrated elements (springs, dampers, lumped masses and rotatory inertias) are included. Analytical solutions for point forces and moments and polynomially distributed loads are presented. The Fourier extension method is used to approximate generally distributed loads, which is very efficient for non-periodic loadings, since the method is not suffering from the Gibbs phenomenon compared to a Fourier series expansion. The Numerical Assembly Technique is extended to include distributed external loadings and a modified formulation of the solution functions is used to enhance the stability of the method at higher frequencies. The method can take distributed loads into account without the need for a modal expansion of the load, which increases the computational efficiency. A numerical example shows the efficiency and accuracy of the proposed method in comparison to the Finite Element Method.

Damping mistuning effects on the amplification factor and statistical investigation of vane packet

Abstract Turbine engine compressors consist of many stages. Each stage consists of rotating blades and stationary vanes. Interactions between the rotating blades and the stationary vanes create traveling periodic excitations. The structure is also periodic because its construction has the form of spatially repeated units. Tuned bladed disks are spatially repetitive structures, and their geometry is fully described by that of a sector. The dynamic characteristics of a sector determine the dynamic behavior of the entire tuned structure. Hence, modern structural analysis of bladed disks takes advantage of the assumption of cyclic symmetry. In general, however, vanes consist of several packets; each packet having multiple sectors interconnected with a platform and a casting. The vane system is partially periodic but not cyclic symmetric since the packets are not strongly connected with each other. Due to this fact, the methodologies to understand the dynamic responses of cyclically symmetric structures are not applicable to a vane system. Therefore, a new approach to investigate vane responses under traveling wave excitations and the effect of mistuning between the blades of vanes is presented. A mode-based methodology is used to identify the most responding mode under a traveling wave excitation. Also, a reduced order model (ROM) for calculating the effect of a damping mistuning combined with a stiffness mistuning on a vane packet is presented. A novel methodology for statistical analyses to investigate the effect of damping mistuning in a vane is presented also. The developed ROMs are validated by comparing ROM-predicted steady state harmonic responses to full-order analyses in ANSYS. It is observed that the maximum error of the ROM-predicted amplitude is under 1%. Based on the developed ROMs, a simple algebraic equation is derived for predicting amplification factors by introducing the concept of unit damping mistuning. Amplification factors for all engine order excitations are examined to verify the performance of the proposed methodology. The effects of damping mistuning are studied statistically through Monte Carlo simulations.

Equivalent damping for displacement-based seismic design of pile-supported wharves with soil–pile interaction

Displacement-based seismic design methods have been used in port industry for decades. For determining demand displacements of pile-supported wharves, the substitute structure method, a nonlinear static demand analysis methodology, is used in current practice. In the method, the estimation of equivalent damping does not account for damping from soil–pile interaction. In view of this, a new damping equation for pile-supported wharf system as a whole based on the total energy dissipation of soil and structure is proposed. The energy dissipated by soil is quantified by the p–y curves of soil springs and Masing hysteretic rule, and energy dissipated by structure is the sum of energy dissipated by all plastic hinges appeared in piles. Then, the equivalent damping of wharf as a whole is derived by Jacobsen's approach for steady-state harmonic response. To figure out the contribution of soil–pile interaction on system damping as well as the accuracy and rationality of the equation, a case study is presented. It is found that the soil–pile interaction contributes significantly to the dissipated energy of wharf system, and the displacement demands of wharf based on the proposed equation are closer to those based on time-history analyses than other equations in current practice.

Dynamic Stability Analysis and Critical Speed of Rotor supported by a Worn Fluid film Journal Bearings

In this paper, the effect of wear in the fluid film journal bearings on the dynamic stability of rotor bearing system has been studied depending on the development of new analytical equations for motion, instability threshold speed and steady state harmonic response for rotor with offset disc supported by worn journal bearings. Finite element method had been used for modeling the rotor bearing system. The analytical model is verified by comparing its results with that obtained numerically for a rotor supported on the short bearings. The analytical and numerical results showed good agreement with about 8.5% percentage error in the value of critical speed and about 3.5% percentage error in the value of harmonic response. The results obtained show that the wear in journal bearing decrease the instability threshold speed by 2.5% for wear depth 0.02 mm and 12.5% for wear depth 0.04 mm as well as decrease critical speed by 4.2% and steady state harmonic response amplitude by 4.3% for wear depth 0.02 mm and decrease the critical speed by 7.1% and steady state harmonic response amplitude by 13.9% for wear depth 0.04 mm.
 

Steady-state harmonic response analysis of a stranded wire helical spring-mass system using an iterative harmonic balance method

As a cylindrical spring wound by a wire strand, the stranded wire helical spring has been widely used as key repositioning components for high-end devices. Most of these devices can be simplified as single degree-of-freedom (DOF) stranded wire helical spring-mass systems. The steady-state harmonic response analysis method is an important tool for designing a system with the nonlinear spring. The present work proposes an iterative harmonic balance method to solve for the steady-state harmonic response of the single DOF stranded wire helical spring-mass system. Taking into account the nonlinearity of the stranded wire helical spring, a nonlinear model comprising a modified Bouc-Wen equation was proposed for describing the dynamic nature of the spring-mass system. Based on the harmonic balance method and nonlinear iterative methods, the seeking for the response solution was transformed into a nonlinear optimization issue for the nonlinear spring-mass system. The present method was examined using both simulations and experiments. The results show that the proposed approach is simple yet accurate, and thus being very practical for engineering applications.

Some precautions to consider in using wavelet transformation for damage detection analysis of plates

Over the last two decades Wavelet Transformation (WT) Over the last two decades Wavelet Transformation (WT) method has been widely utilized for the damage identification of structures. The main objective of this paper is to discuss and present some of common shortcomings and limitations of mathematical software, as well as other precautionary measures that need to be considered when using them for wavelet analysis applications. Due to popular usage of MATLAB(R) comparing to other mathematical tools among researchers for data processing of structural responses through WT analysis, this software was chosen for specific study. To the best of the authors\' knowledge, these limitations and observations have not been previously identified or discussed in the literature. In this work, a square plate with a severe damage, in form of a crack, parallel to the left edge of the plate is selected for a pilot study. The steady state harmonic response is used for measuring the deflection shape across the line parallel to one edge and perpendicular to the damage. Several criteria and cases such as the smallest size damage that can be detected, correlation between the crack width and the number of sampling points, and the influence of the damage thickness on the accuracy of the result are investigated.

Reduced Order Models for Blade-To-Blade Damping Variability in Mistuned Blisks

A novel reduced order modeling methodology to capture blade-to-blade variability in damping in blisks is presented. This new approach generalizes the concept of component mode mistuning (CMM), which was developed to capture stiffness and mass mistuning (and did not include variability in damping among the blades). This work focuses on modeling large variability in damping. Such variability is significant in many applications and particularly important for modeling damping coatings. Similar to the CMM based studies, structural damping is used to capture the damping effects due to the mechanical energy dissipation caused by internal friction within the blade material. The steady state harmonic responses of the blades are obtained using the novel reduced order modeling methodology and are validated by comparison with simulation results obtained using a full order model in ANSYS with a maximum amplitude error of 0.3%. It is observed that there is no strong correlation between the engine order of excitation and both the variation in the response from blade to blade and the blade amplification factors. The effects of damping mistuning are studied statistically through Monte Carlo simulations. For this purpose, the blisk model is subjected to multiple traveling wave excitations. The uncertainty in the various mechanisms responsible for dissipation of energy and the uncontrollability of these dissipation mechanisms makes it difficult to assign a reliable value for the loss factor of each blade. Hence, large variations (up to ±80%) in the structural damping coefficients of the blades are simulated.

An Damping Analysis on Local Vibration in Foundation of ‘four in one’ Nitric Acid Plant

‘four in one’ nitric acid plant is a key equipment to manufacture nitric acid. Its foundation is placed between the axial flow compressor and the expander, and has larger amplitude. Choosing the foundation of this machine as study subject, this paper analyses the biggest disturbance value of all vibration source in line with Code for design of dynamic machine foundation and by adding T-shaped kerfs on the box girder between the axial flow compressor and the expander. It also conducts harmonic response analysis on this foundation with the finite element method. By comparatively analyzing the displacement harmonic response figure of nodal points on the X, Y and Z axis in the foundation before and after adding the T-shaped kerfs, it indicates that T-shaped kerfs can dramatically mitigate the steady-state harmonic response amplitude.

Exact three-dimensional piezothermoelasticity solution for dynamics of rectangular cross-ply hybrid plates featuring interlaminar bonding imperfections

An exact three-dimensional (3D) piezothermoelasticity solution is presented for static, free vibration and steady state harmonic response of simply supported cross-ply piezoelectric (hybrid) laminated rectangular plates with interlaminar bonding imperfections. The bonding imperfection is modeled by considering the jump in the displacements, electric potential and temperature across the non-rigid interface proportional, respectively, to the associated tractions, transverse electric displacement and heat flux. The solution includes the case when electric potentials are prescribed at the interfaces for effective actuation. Numerical results are presented for hybrid composite and sandwich plates with varying imperfection compliance. The effect of location of imperfect bonding on the response is investigated for mechanical, electric potential and thermal load cases. The effect of weak bonding at elastic–piezoelectric interface on the actuation authority of the piezoelectric layer is also investigated. These results would serve as benchmark for assessing 2D plate theories incorporating interlaminar bonding imperfections.

Interactive power flow characteristics of an integrated equipment—nonlinear isolator—travelling flexible ship excited by sea waves

A nonlinear interactive system comprising of equipment, nonlinear isolator and travelling flexible ship excited by waves is studied from a vibratory power flow viewpoint to examine its dynamical behaviour and power flow characteristics. The mathematical model describing the dynamics of this nonlinear interactive system is developed. Dynamical interactions between equipment, nonlinear isolator, flexible foundation and water waves are addressed. The nonlinearities of the isolator are characterized by a general pth power model for damping and qth power for stiffness. A harmonic balance method is adopted to derive the steady-state harmonic response of the nonlinear system. A Newton–Raphson iteration process in association with an efficient numerical algorithm is used to obtain the solutions of this nonlinear problem. Through simulations the dynamical behaviour, power flow characteristics and isolation efficiency of this complex nonlinear interaction system are investigated. For different values of power p and q, different wave excitations and flexible or rigid ship, the power transmitted to the equipment and power flow transmission ratios are calculated and analysed. The effect of the vibration source with different wave conditions of the seaway is studied through examining its vibratory power input to the overall system. The effects of the assumptions of flexible or rigid ship, the nonlinearities on the power flows in the system are examined. Nonlinear power flow phenomena and mechanisms are revealed, which provides an insight to the understanding of power flow characteristics in nonlinear systems. Practical guidelines for the design of vibration isolation systems applicable to maritime engineering are suggested.

EQUIVALENT VISCOUS DAMPING EQUATIONS FOR DIRECT DISPLACEMENT BASED DESIGN

Estimation of equivalent viscous damping is an important step in the methodology of the direct displacement based design. Errors in the estimation of this parameter may lead to consequent errors in the ductility demand of the designed elements. Equivalent viscous damping estimated by Jacobsen's approach for steady-state harmonic response are compared with effective damping factors obtained from an iterative procedure using time — history analyses of single-degree-of-freedom systems. In order to separate the effects of elastic-viscous and hysteretic damping the designs and analyses were carried without elastic viscous damping. Six hysteretic models representing a wide range of hysteretic energy absorption characteristics and six artificial records were used in the analysis. It is found that in general Jacobsen's approach overestimates the values for equivalent viscous damping. Based on the analytical results, modified design equations for equivalent viscous damping are proposed.

WAVE PROPAGATION IN AND VIBRATION OF A TRAVELLING BEAM WITH AND WITHOUT NON-LINEAR EFFECTS, PART II: FORCED VIBRATION

The wave propagation in a simply supported travelling beam, studied in Part I of this paper, has been used to derive the forced responses. Based upon the wave-propagation principles, a simple method for constructing the closed-form transfer function of such a beam has been presented. The use of this transfer function offers an easy alternative to the usual modal analysis for obtaining the steady-state harmonic response. The effects of non-linearities during the steady-state oscillation, maintained by a non-resonant hard harmonic excitation, have also been studied. The present method, when compared to the conventional Galerkin's technique, requires much less computational effort.

A Study on the Hydraulic Excitation Forces Using Transfer Function and Operational Measured Data for the Centrifugal Pump

Operating excitation forces of the linear vibratory system are normally determined by direct measurement techniques using load cells, strain gauges, etc. But, hydraulic forces of the rotating turbomachinery such as centrifugal pumps are exerted on an impeller due to asymmety of the flow by the interaction between pump impeller and volute. So, investigations of wide range of hydraulic designs and geometric deviations are difficult by direct method. This paper presents a hybrid approach for fourier transformed operational excitation forces, which uses pseudo-inverse matrix of the transfer matrix for the system and the measured vibrational data with standard installed pump. The determination of the transfer function matrix is based on a linear rotor/stationary system and steady state harmonic response in finite element analysis. And, vibrational data is collected in both vertical and horizontal directions at inboard and outboard bearing housings. The results of the process may be enhanced by making acceleration measurements at many more locations than there are forces to be determined.

Horizontal impedance of square foundation in layered soil

This paper presents an extensive investigation into the influence of key mechanical and geometrical parameters on horizontal impedance of square foundations resting on or embedded in a two-layer soil deposit. The parameters investigated are the ratio of shear-wave velocities, the thickness of the top layer, the depth of embedment and the degree of contact between the footing-sidewall with backfill-soil. The results are presented in the form of simple and versatile dimensionless graphs, which should prove to be useful in understanding the steady-state harmonic response of square foundations in layered soils due to horizontal excitation. The investigation was conducted using a rigorous boundary element algorithm incorporating isoparametric boundary elements. Higher order quadratic elements were used since they can model the wavy nature of the dynamic problem investigated more accurately.

Frequency Clusters in the Spectrum of Annular Cylinders

As the length of a traction-free annular cylinder is increased, distinct members within any family of radial or longitudinal shear modes have natural frequencies that asymptotically approach a common nonzero value. Such modes, potentially having significantly different numbers of nodes along the cylinder’s generator, can have natural frequencies that are indistinguishable from one another within the resolution of test equipment or numerical simulation. The three-dimensional vibration model discussed here predicts the formation of narrow “frequency clusters” with the cylinder’s increasing length, the converged value of which bounds from below the frequencies of all modes within a particular family. In addition to these spectral characteristics, frequency clusters have implications for the forced response of annular cylinders. For the particular families of modes that are of interest here, the steady-state harmonic response at frequencies near a cluster can be spatially confined with displacements that decay rapidly away from the point of maximum response. At other driving frequencies, the response is distributed more uniformly along the length of the cylinder. The derived analytical model is compared with results from laboratory measurements, and from the predictions of wave propagation theory in the limit of infinite cylinder length.