Experimental Frequency Response Research Articles

Development of an RBFN-based neural-fuzzy adaptive control strategy for an upper limb rehabilitation exoskeleton

The patients of paralysis with motion impairment problems require extensive rehabilitation programs to regain motor functions. The great labor intensity and limited therapeutic effect of traditional human-based manual treatment have recently boosted the development of robot-assisted rehabilitation therapy. In the present work, a neural-fuzzy adaptive controller (NFAC) based on radial basis function network (RBFN) is developed for a rehabilitation exoskeleton to provide human arm movement assistance. A comprehensive overview is presented to describe the mechanical structure and electrical real-time control system of the therapeutic robot, which provides seven actuated degrees of freedom (DOFs) and achieves natural ranges of upper extremity movement. For the purpose of supporting the disable patients to perform repetitive passive rehabilitation training, the RBFN-based NFAC algorithm is proposed to guarantee trajectory tracking accuracy with parametric uncertainties and environmental disturbances. The stability of the proposed control scheme is demonstrated through Lyapunov stability theory. Further experimental investigation, involving the position tracking experiment and the frequency response experiment, are conducted to compare the control performance of the proposed method to those of cascaded proportional-integral-derivative controller (CPID) and fuzzy sliding mode controller (FSMC). The comparison results indicate that the proposed RBFN-based NFAC algorithm is capable of obtaining lower position tracking error and better frequency response characteristic.

A machine-learning based solution for chatter prediction in heavy-duty milling machines

The main productivity constraints of milling operations are self-induced vibrations, especially regenerative chatter vibrations. Two key parameters are linked to these vibrations: the depth of cut achievable without vibrations and the chatter frequency. Both parameters are linked to the dynamics of machine component excitation and the milling operation parameters. Their identification in any cutting direction in milling machine operations requires complex analytical models and mechatronic simulations, usually only applied to identify the worst cutting conditions in operating machines. This work proposes the use of machine learning techniques with no need to calculate the two above-mentioned parameters by means of a 3-step strategy. The strategy combines: 1) experimental frequency responses collected at the tool center point; 2) analytical calculations of both parameters; and, 3) different machine learning techniques. The results of these calculations can then be used to predict chatter under different combinations of milling directions and machine positions. This strategy is validated with real experiments on a bridge milling machine performing concordance roughing operations on AISI 1045 steel with a 125 mm diameter mill fitted with nine cutters at 45°, the results of which have confirmed the high variability of both parameters along the working volume. The following regression techniques are tested: artificial neural networks, regression trees and Random Forest. The results show that Random Forest ensembles provided the highest accuracy with a statistical advantage over the other machine learning models; they achieved a final accuracy of 0.95 mm for the critical depth and 7.3 Hz for the chatter frequency (RMSE) in the whole working volume and in all feed directions, applying a 10 × 10 cross validation scheme. These RMSE values are acceptable from the industrial point of view, taking into account that the critical depth of this range varies between 0.68 mm and 19.20 mm and the chatter frequency between 1.14 Hz and 65.25 Hz. Besides, Random Forest ensembles are more easily optimized than artificial neural networks (1 parameter configuration versus 210 MLPs). Additionally, tools that incorporate regression trees are interesting and highly accurate, providing immediately accessible and useful information in visual formats on critical machine performance for the design engineer.

Non-linear vibrations of a beam with non-ideal boundary conditions and uncertainties – Modeling, numerical simulations and experiments

Abstract This paper presents experiments and numerical simulations of a nonlinear clamped-clamped beam subjected to Harmonic excitations and epistemic uncertainties. These uncertainties are propagated in order to calculate the dynamic response of the nonlinear structure via a coupling between the Harmonic Balance Method (HBM) and a non-intrusive Polynomial Chaos Expansion (PCE). The system studied is a clamped-clamped steel beam. First of all, increasing and decreasing swept sine experiments are performed in order to show the hardening effect in the vicinity of the primary resonance, and to extract the experimental multi-Harmonic frequency response of the structure. Secondly, the Harmonic Balance Method (HBM) is used alongside a continuation process to simulate the deterministic response of the nonlinear clamped-clamped beam. Good correlations were observed with the experiments, on the condition of updating the model for each excitation level. Finally, the effects of the epistemic uncertainties on the variability of the nonlinear response are investigated using a non-intrusive Polynomial Chaos Expansion (PCE) alongside the Harmonic Balance Method (HBM). A new methodology based on a phase criterion was developed in order to allow the PCE analysis to be performed despite the presence of bifurcations in the nonlinear response. The efficiency and robustness of the proposed methodology is demonstrated by comparison with Monte Carlo simulations. Then, the stochastic numerical results are shown to envelope the experimental responses for each excitation level without the need for model updating, validating the nonlinear stochastic methodology as a whole.

Electrostatic compensation of structural imperfections in dynamically amplified dual-mass gyroscope

This paper presents a study on dynamics of a dual-mass MEMS vibratory gyroscope in presence of fabrication imperfections and reports a method for precision electrostatic frequency tuning of the operational modes. A number of multi-mass MEMS gyroscopes have emerged in recent years pursuing different goals, such as dynamically balanced structure, increased bandwidth, and dynamic amplification. Along with many perceived advantages of multi-mass devices, several challenges associated with mode-matching in a system with increased number of degrees-of-freedom (DOF) have to be considered. This work shows that it is possible to apply the DC tuning techniques, similar to tuning a conventional single-mass gyroscope, to achieve the precision tuning in a dual-mass sensor, without losing advantages of increased DOF of the system. The presented frequency trimming technique is based on assessing the modes mismatch and cross-coupling between modes by means of fitting the experimental frequency response curves to the analytical solutions of the dual-mass system in presence of imperfections. The tuning algorithm involves two steps. First, the stiffness mismatch along the two axes and the anisoelasticity angles α and β are identified, then the tuning DC voltages for modification of diagonal, off-diagonal, and coupling terms in the stiffness matrix are chosen. The method of electrostatic tuning was validated through the experimental characterization of a dual-mass dynamically amplified gyroscope, where the coupling between the two operational modes was minimized and frequency split was reduced from 26 Hz down to 50 mHz, resulting in 17.5× increase in the gyroscope scale factor and significantly improved noise characteristics. The presented electrostatic compensation method is suitable for both off-line and on-line calibration.

Chatter stability prediction in high-speed micromilling of Ti6Al4V via finite element based microend mill dynamics

High-speed micromilling (spindle speeds 100 000 r/min) can create complex three-dimensional microfeatures in difficult-to-machine materials. The micromachined surface must be of high quality, to meet functional requirements. However, chatter-induced dynamic instability deteriorates the surface quality and can be detrimental to tool life. Chatter-free machining can be accomplished by identifying stable process parameters via stability lobe diagram. To generate accurate stability lobe diagram, it is essential to determine the microend mill dynamics. Frequency response function is required to determine the tool-tip dynamics obtained by experimental impact analysis. Note that application of impact load at the microend mill tip (typically 100 – 500 μm) is not feasible as it would invariably end with tool failure. Consequently, alternative methods need to be developed to identify the microend mill dynamics. In the present work, the frequency response function for the microend mill is obtained by finite element method modal analysis. The frequency response function obtained from modal analysis has been verified from the experimentally obtained frequency response function. The experimental frequency response function was obtained by impacting the microend mill near the taper portion with an impact hammer and measuring the vibration of the tool-tip with a laser displacement sensor. The fundamental frequency obtained from finite element method modal analysis shows a difference of 6.6% from the experimental fundamental frequency. Microend mill dynamics obtained from the finite element method is used for chatter prediction in high-speed micromilling operations. The stability lobe diagram predicts the stability boundary accurately at 60 000 r · min–1 and 80 000 r/min; however, a slight deviation is observed at 100 000 r/min.

Экспериментальное исследование частотных характеристик катушки-конденсатора при различных граничных условиях

The article presents the results from experimental investigations of a coil-capacitor (also known as katkon), which is a new component of electric circuits, under no-load and short-circuit conditions. The article is of scientific (from the viewpoint of the electric circuit theory) and practical interest due to the possibility of using katkons in harmonic filtering and reactive power compensation devices, in short-circuit current limiting devices, in inductive-capacitive converters, and other electrical devices for making them more compact and light. The katkon physical model parameters are given, data on the laboratory equipment are presented, and the experimental frequency responses, volt-ampere and loading characteristics are constructed. The results from comparison of the experimental and theoretical dependences make it possible to estimate the adequacy and correctness of the proposed mathematical model. The course of experimental investigations is described in detail, for carrying out of which well-known methods were used. The experimental data do not contradict the commonly adopted theoretical statements or study results carried out by leading Russian and foreign specialists.

Coupling of structures using frequency response functions

In the field of structural dynamics is common to predict the behaviour of a structure regarding structural modifications. In this context, the frequency based substructuring method is well-known to perform structural modifications based on the coupling of structures. This process gives the possibility to perform the study of a structure at the level of its components and then assess the response of the coupled system. In practice, it is impossible to attain an experimental complete response model, although one can simulate all the responses of a structure using numerical models. Hence, the substructuring process can be enhanced by the combined use of experimental and numerical responses, as it was demonstrated using numerically obtained frequency response functions. This work presents the enhancement of the frequency based substructuring method using a method to expand experimental frequency response functions over the entire set of degrees of freedom in a finite element model. This expansion process, known as modified Kidder’s method, considers that if one can only measure translations due to exciting force, it is possible to obtain the complete response model, including the rotational frequency response functions due to exciting moments. The combined use of the frequency based substructuring and the modified Kidder’s methods has several advantages, as it avoids modal identification or residual compensation. To evaluate the performance of the proposed procedure a numerical example of a beam structure is presented, and its results are discussed.

Nonlinear elastodynamics of piezoelectric macro-fiber composites with interdigitated electrodes for resonant actuation

Macro-fiber composite (MFC) piezoelectric materials are used in a variety of applications employing the converse piezoelectric effect, ranging from morphing and bioinspired actuation to vibration control in flexible structures. Most of the existing literature to date considered linear material behavior for geometrically linear oscillations. However, in many applications, such as bioinspired locomotion using MFCs, material and geometric nonlinearities are pronounced and linear models fail to represent and predict the governing dynamics. The predominant types of nonlinearities manifested in resonant actuation of MFC cantilevers are piezoelectric softening, geometric hardening, and inertial softening. In the present work, we explore nonlinear actuation of MFC cantilevers and develop an experimentally validated mathematical framework for modeling and analysis. In the experimental setting, an in vacuo actuation scenario is considered for a broad range of voltage levels (from low to moderate values) while eliminating nonlinear fluid damping. Experiments are conducted for an MFC bimorph cantilever, and model simulations based on the method of harmonic balance are compared with experimental frequency response curves under resonant actuation. The resulting experimentally validated framework can be used for simulating the dynamics of MFCs under resonant actuation, as well as parameter identification and structural optimization for linear to moderately nonlinear regime.

Validation of vibro-impact force models by numerical simulation, perturbation methods and experiments

The frequency response of a single degree of freedom vibro-impact oscillator is analyzed using Harmonic Linearization, Averaging and Numeric Simulation, considering three different impact force models: one given by a piecewise-linear function (Kelvin-Voigt model), another by a high-order power function, and a third one combining the advantages of the other two. Experimental validation is carried out using control-based continuation to obtain the experimental frequency response, including its unstable branch.

Estimation of beam material random field properties via sensitivity-based model updating using experimental frequency response functions

Abstract Structural parameter estimation is affected not only by measurement noise but also by unknown uncertainties which are present in the system. Deterministic structural model updating methods minimise the difference between experimentally measured data and computational prediction. Sensitivity-based methods are very efficient in solving structural model updating problems. Material and geometrical parameters of the structure such as Poisson’s ratio, Young’s modulus, mass density, modal damping, etc. are usually considered deterministic and homogeneous. In this paper, the distributed and non-homogeneous characteristics of these parameters are considered in the model updating. The parameters are taken as spatially correlated random fields and are expanded in a spectral Karhunen-Loeve (KL) decomposition. Using the KL expansion, the spectral dynamic stiffness matrix of the beam is expanded as a series in terms of discretized parameters, which can be estimated using sensitivity-based model updating techniques. Numerical and experimental tests involving a beam with distributed bending rigidity and mass density are used to verify the proposed method. This extension of standard model updating procedures can enhance the dynamic description of structural dynamic models.

Use of experimental dynamic substructuring to predict the low frequency structural dynamics under different boundary conditions

Flexible structural components can be attached to the rest of the structure using different types of joints. For instance, this is the case of solar panels or array antennas for space applications that are joined to the body of the satellite. To predict the dynamic behaviour of such structures under different boundary conditions, such as additional constraints or appended structures, it is possible to start from the frequency response functions in free-free conditions. In this situation, any structure exhibits rigid body modes at zero frequency. To experimentally simulate free-free boundary conditions, flexible supports such as soft springs are typically used: with such arrangement, rigid body modes occur at low non-zero frequencies. Since a flexible structure exhibits the first flexible modes at very low frequencies, rigid body modes and flexible modes become coupled: therefore, experimental frequency response function measurements provide incorrect information about the low frequency dynamics of the free-free structure. To overcome this problem, substructure decoupling can be used, that allows us to identify the dynamics of a substructure (i.e. the free-free structure) after measuring the frequency response functions on the complete structure (i.e. the structure plus the supports) and from a dynamic model of the residual substructure (i.e. the supporting structure). Subsequently, the effect of additional boundary conditions can be predicted using a frequency response function condensation technique. The procedure is tested on a reduced scale model of a space solar panel.

Estimation accuracy vs. engineering significance of contact parameters for solid dampers

AbstractAll numerical models of friction-damped bladed arrays require knowledge or information of contact-friction parameters. In the literature, these parameters are typically tuned so that the experimental Frequency Response Function (FRF) of a damped blade matches its numerical counterpart. It is well known that there exist multiple combinations of contact parameters capable of satisfying a given experimental-numerical FRF match. A better approach towards a finer tuning could be based on directly measuring contact forces transmitted between blade platforms through the damper: in this case friction coefficients are estimated through tangential over normal force components during those hysteresis segments which are safely identified as being in a slip condition. This has been applied by these authors to rigid bar (solid) dampers. Unfortunately, the four contact stiffness values (left and right damper-platform contact, normal and tangential) are more than the measurements available in the technique presented by these authors. Therefore, the problem is underdetermined. The purpose of this paper is twofold,i.e., to propose an alternative way to estimate contact stiffness values (i.e.thus solving the under-determinacy mentioned above) and to check the effective significance of such estimates from a practical engineering point of view. The contact parameter estimation technique proposed by these authors produces, for each contact parameter, a best-fit value and an uncertainty band. It will be shown that the uncertainty affecting each contact parameter results in an uncertainty on the equivalent damping and stiffness indicators at blade level which is lower than 5%.

Model Identification and Control for a Quarter Car Test Rig of Series Active Variable Geometry Suspension

In this paper, a quarter car test rig is utilized to perform an experimental study of the single-link variant of the Series Active Variable Geometry Suspension (SAVGS). A nonlinear model of the test rig is identified with the use of a theoretical quarter car model and the rig’s experimental frequency response. A linear equivalent modeling method that compensates the geometric nonlinearity is also adopted to synthesize an H-infinity control scheme. The controller actively adjusts the single-link velocity in the SAVGS to improve the suspension performance. Experiments are performed to evaluate the SAVGS practical feasibility, the performance improvement, the accuracy of the nonlinear model and the controller’s robustness.

Reducing the impact of measurement errors in FRF-based substructure decoupling using a modal model

Abstract As the vibro-acoustic requirements of modern products become more stringent, the need for robust identification methods increases proportionally. Sometimes the identification of a component is greatly complicated by the presence of a supporting structure that cannot be removed during testing. This is where substructure decoupling finds its main applications. However, despite some recent advances in substructure decoupling, the number of successful applications has so far been limited. The main reason for this is the poor conditioning of the problem that tends to amplify noise and other measurement errors. This paper proposes a new approach that uses a modal model to filter the experimental frequency response functions (FRFs). This can reduce the impact of noise and mass loading considerably for decoupling applications and decrease the quality requirements for experimental data. Furthermore, based on the uncertainty of the observed eigenfrequencies, an arbitrary number of consistent (all FRFs exhibit exactly the same poles) FRF matrices can be generated that are all contained within the variation of the original measurement. This way, the variation that is observed within the measurement is taken into account. The result is a distribution of decoupled FRFs of which the average can be used as the decoupled FRF set while the spread on the results highlights the sensitivity or reliability of the obtained results. After briefly reintroducing the theory of FRF-based substructure decoupling, the main problems in decoupling are summarized. Afterwards, the new methodology is presented and tested on both numerical and experimental cases.

Hybrid controller for mid‐power audio application

Driving current of a moving coil loudspeaker, fed from a switched voltage-mode audio amplifier, display non-ideal transfer characteristic relative to the reference audio signal. This is chiefly due to the effective speaker voice coil electrical impedance and phase lag provided by the output filter. This study investigates the deviation in the transfer characteristics from the ideal constant gain and zero-lag properties and proposes a hybrid controller to mitigate the drawback. The hybrid controller comprises a sliding-mode controlled amplifier (SMCA) which has a non-linear control law and ensures maximal command following among voltage-mode algorithms, realisable under switching frequency and filter size constraints. Additionally, a linear controller fine-tunes the speaker current and its design is based on the transfer function of the SMCA and loudspeaker, which are least-square approximated from experimental frequency response data. A 20 W amplifier, driving a 4 Ω speaker load, is developed and experimental results obtained there from establishing the improved electric to acoustic signal conversion using the hybrid controller.

Squeeze-Film Air Damping of a Five-Axis Electrostatic Bearing for Rotary Micromotors.

Air-film damping, which dominates over other losses, plays a significant role in the dynamic response of many micro-fabricated devices with a movable mass suspended by various bearing mechanisms. Modeling the damping characteristics accurately will be greatly helpful to the bearing design, control, and test in various micromotor devices. This paper presents the simulated and experimental squeeze-film air damping results of an electrostatic bearing for use in a rotary high-speed micromotor. It is shown that the boundary condition to solve the three-dimensional Reynolds equation, which governs the squeeze-film damping in the air gap between the rotor and its surrounding stator sealed in a three-layer evacuated cavity, behaves with strong cross-axis coupling characteristics. To accurately characterize the damping effect, a set of multiphysics finite-element simulations are performed by computing both the rotor velocity and the distribution of the viscous damping force acting on the rotor. The damping characteristics varying with several key structure parameters are simulated and discussed to optimize the device structure for desirable rotor dynamics. An electrical measurement method is also proposed and applied to validate the numerical results of the damping coefficients experimentally. Given that the frequency response of the electric bearing is critically dependent on the damping coefficients at atmospheric pressure, a solution to the air-film damping measurement problem is presented by taking approximate curve fitting of multi-axis experimental frequency responses. The measured squeeze-film damping coefficients for the five-axis electric bearing agrees well with the numerical solutions. This indicates that numerical multiphysics simulation is an effective method to accurately examine the air-film damping effect for complex device geometry and arbitrary boundary condition. The accurate damping coefficients obtained by FEM simulation will greatly simplify the design of the five-axis bearing control system and facilitate the initial suspension test of the rotor for various micromotor devices.

Model-updating with experimental frequency response function considering general damping

In order to obtain a more accurate finite element model of a constructed structure, a new frequency response function–based model-updating approach is proposed. A general viscous damping model is assumed in this approach for better simulating the actual structure. The approach is formulated as an optimization problem which intends to minimize the difference between analytical and experimental frequency response functions. Neither dynamic expansion nor model reduction is needed when not all degrees of freedom are measured. State-of-the-art optimization algorithms are utilized for solving the non-convex optimization problem. The effectiveness of the presented frequency response function model-updating approach is validated through a laboratory experiment on a four-story shear-frame structure. To obtain the experimental frequency response functions, a shake table test was conducted. The proposed frequency response function model-updating approach is shown to successfully update the stiffness, mass, and damping parameters in matching the analytical frequency response functions with the experimental frequency response functions. In addition, the updating results are also verified by comparing time-domain experimental responses with the simulated responses from the updated model.

高強度化学繊維によるワイヤ駆動のための基礎的検討

Synthetic fiber ropes, which are lightweight and have high tensile strength and exibility, have attracted attention as an alternative to stainless steel wire rope. This study clarifies various physical characteristics of synthetic fiber ropes to design a tendon-driven mechanism. This paper investigates a viscoelasticity model for Ultra High molecular weight Polyethylene (UHPE) rope and its dynamic property applied to a long distant antagonistic tendon-driven servo system. We confirm that UHPE can be modeled by four element model, and obtained its transfer function of a tendon-driven servo system. The validity of the transfer function was confirmed by a frequency response experiment.

Computational Framework for Online Estimation of Fatigue Damage using Vibration Measurements from a Limited Number of Sensors

This study proposes a computational framework for the online estimation of fatigue damage using operational vibration measurements from a limited number of sensors. To infer the stress response time histories required for fatigue prediction, the measured structural response is driven to a high fidelity finite element (FE) model, which is reconciled using appropriate model updating techniques that minimize the discrepancy between the experimental and analytical frequency response functions (FRFs). Fatigue is accordingly estimated via the Palmgren-Miner rule, while the available FE model allows for stress estimation at unmeasured spots. The method is successfully validated and assessed through an experimental study that pertains to a linear steel substructure supporting the entire body of a pre-beater assembly at a PPC power plant.

Experiments and numerical simulations of nonlinear vibrations of a beam with non-ideal boundary conditions and uncertainties

This paper presents experiments and numerical simulations of a nonlinear clamped-clamped beam subjected to external excitations and uncertainties. A coupling of the Harmonic Balance Method (HBM) with a non-intrusive Polynomial Chaos Expansion (PCE) is proposed in order to determine the dynamic response of the nonlinear structure. The system studied is a steel beam bonded on both sides to a heavy steel block. First of all, increasing and decreasing swept sine experiments are performed in order to show the presence of jumping points in the vicinity of the primary resonance. From the swept-sine experiments, we use a signal processing tool to extract the experimental multi-harmonic frequency response of the structure. Secondly, a numerical method, called the Harmonic Balance Method (HBM), is used to simulate the deterministic vibrational stationary response of the nonlinear clamped-clamped beam. The input of the HBM is either a mono-harmonic signal to update the nonlinearities of the clamped-clamped beam or a multi-harmonic experimental signal to show the effects of the non-ideal input signal on the response. Finally, the quantification of uncertainty effects on the variability of the nonlinear multi-harmonic response is investigated by using a non-intrusive polynomial chaos expansion (PCE) alongside the Harmonic Balance Method (HBM). A new methodology was developed to split the simulated solutions into various intervals (i.e. various branches of continuous nonlinear solutions) in order to allow the PCE analysis to be performed despite the presence of returning points in the nonlinear response. The efficiency and robustness of the proposed methodology is demonstrated by comparison with Monte Carlo simulations.