Frequency domain analysis and identification of block-oriented nonlinear systems

Block-oriented nonlinear models including Wiener models, Hammerstein models and Wiener-Hammerstein models, etc. have been extensively applied in practice for system identification, signal processing and control. In this study, analytical frequency response functions including generalized frequency response functions (GFRFs) and nonlinear output spectrum of block-oriented nonlinear systems are developed, which can demonstrate clearly the relationship between frequency response functions and model parameters, and also the dependence of frequency response functions on the linear part of the model. The nonlinear part of these models can be a more general multivariate polynomial function. These fundamental results provide a significant insight into the analysis and design of block-oriented nonlinear systems. Effective algorithms are therefore proposed for the estimation of nonlinear output spectrum and for parametric or nonparametric identification of nonlinear systems. Compared with some existing frequency domain identification methods, the new estimation algorithms do not necessarily require model structure information, not need the invertibility of the nonlinearity and not restrict to harmonic inputs. Simulation examples are given to illustrate these new results.     

A new type of semi-active hydraulic engine mount using controllable area of inertia track

Automotive engine mounts function to constrain the engine shake motion resulting at low-frequencies, as well as to isolate noises and vibrations generated by the engine with unbalanced disturbances at the high frequencies. The property of the mount depends on vibration amplitude and excitation frequency. It means that the excitation amplitude is large in low excitation frequency range and small in high frequency range. In this paper, a new hydraulic engine mount with a controllable area of inertia track is proposed and investigated. Theoretical works with the mount model to isolate the engine-related vibrations were conducted by an optimal algorithm to control the area of the inertia track under shocks and multi-signal force excitations. This research clearly gives an analysis of the considerable changes in the mount dynamic properties according to the changes in the inertia track area. Consequently, when the inertia track area is tuned, the transmissibility of the mount is effectively reduced.     

Double-notch single-pumper fluid mounts

Passive fluid mounts are widely used in the automotive and aerospace applications to isolate the cabin from the engine noise and vibration. In the case of aerospace fixed wing applications, when fluid mounts are used, they are placed in between the engine and the fuselage, and the notch frequency of each fluid mount is tuned to either N1 frequency (engine low speed shaft imbalance excitation frequency) or to N2 frequency (engine high speed shaft imbalance excitation frequency) at the cruise condition. Since current passive fluid mount designs have only one notch, isolation is only possible at N1 or at N2, but not both. Here, in this paper, a double-notch passive fluid mount design will be presented, which has two notch frequencies, and therefore can provide vibration and noise isolation at two frequencies. In this paper, the new fluid mount design concept and its mathematical model and simulation results will be presented.     

面向传递路径分析方法研究的试验系统设计*

在传递路径分析理论研究中,经典传递路径分析要求在传递函数测量时拆卸激励源,为了减少工作量,设计一套面向传递路径分析方法研究的实验系统。以简化发动机模型作为激励源,真实汽车发动机悬置作为传递路径,简化车身内的响应为目标响应,建立一套完整的传递路径分析模型。在保留主要激励特征情况下简化发动机机体结构,将NI Compact RIO作为控制核心,激振器作为激励施加装置,以缸压信号和机体振动信号为激励信号对机体加载,实现发动机激励特征的模拟。最终验证该实验系统可模拟出发动机的结构噪声,采集信号中阶次成分明显,主要噪声频段基本吻合,可从阶次信号中提取对应转速信息。整个实验系统结构简单,可作为载体用于传递路径分析方法的研究。     

Experimental Investigation and Semi-Active Control Design of A Magnetorheological Engine Mount

In this paper; the dynamic characteristics of a semi-active magnetorheological fluid (MRF) engine mount are studied. To do so, the performance of the MRF engine mount is experimentally examined in higher frequencies (50~170 Hz) and the various amplitudes (0.01 ~ 0.2 mm). In such an examination, an MRF engine mount along with its magnetically biased is fabricated and successfully measured. In addition, the natural frequencies of the system are obtained by standard hammer modal test. For modelling the behavior of the system, a mass-spring-damper model with tuned PID coefficients based on Pessen integral of absolute error method is used. The parameters of such a model including mass, damping ratio, and stiffness are identified with the help of experimental modal tests and the recursive least square method (RLS). It is shown that using PID controller leads to reducing the vibration transmissibility in the resonance frequency (=93.45 Hz) with respect to the typical passive engine mount by a factor of 58%. The average of the vibration transmissibility decreasing is also 43% within frequency bandwidth (50~170 Hz).     

A novel design of semi-active hydraulic mount with wide-band tunable notch frequency

Hydraulic engine mount is advanced vibration isolator with superior performance to reduce vibration transferred from engine to chassis. As the stiffness at notch frequency is small, some semi-active or active hydraulic mounts tune some parameters to let notch frequency coincide with exciting frequency for better vibration isolation performance. It is discovered the current semi-active mounts can tune the notch frequency in narrow frequency band when only one parameter is tuned. A novel semi-active hydraulic engine mount design which introduces screw thread is proposed and researched in the paper. This hydraulic mount can control both cross section area and the length of inertia track and the theoretical tunable notch frequency band is [0, ∞). Theoretical work is carried out to uncover the capability for the proposed design to tune notch frequency. Simulation work is performed to understand its high vibration isolation performance. For the purpose of energy conservation, the friction self-locking is introduced. This denotes once the mount is tuned at optimal condition, the energy can be cut off and the optimal condition will never change. We also determine the best time to tune the parameters of the proposed mount in order to decrease the acting force. The proposed semi-active mount has capability to obtain wide band tunable notch frequency and has merit of energy conservation.     

A general solution procedure for vibrations of systems with cubic nonlinearities and nonlinear/time-dependent internal boundary conditions

The subject of this paper is the development of a general solution procedure for the vibrations (primary resonance and nonlinear natural frequency) of systems with cubic nonlinearities, subjected to nonlinear and time-dependent internal boundary conditions—this is a commonly occurring situation in the vibration analysis of continuous systems with intermediate elements. The equations of motion form a set of nonlinear partial differential equations with nonlinear, time-dependent, and coupled internal boundary conditions. The method of multiple timescales, an approximate analytical method, is applied directly to each partial differential equation of motion as well as coupled boundary conditions (i.e. on each sub-domain and the corresponding internal boundary conditions for a continuous system with intermediate elements) which ultimately leads to approximate analytical expressions for the frequency-response relation and nonlinear natural frequencies of the system. These closed-form solutions provide direct insight into the relationship between the system parameters and vibration characteristics of the system. Moreover, the suggested solution procedure is applied to a sample problem which is discussed in detail.     

Nonlinear resonant behavior of microbeams over the buckled state

The present study investigates the nonlinear resonant behavior of a microbeam over its buckled (non-trivial) configuration. The system is assumed to be subjected to an axial load along with a distributed transverse harmonic load. The axial load is increased leading the system to lose the stability via a pitchfork bifurcation; the postbuckling configuration is obtained and the nonlinear resonant response of the system over the buckled state is examined. More specifically, the nonlinear equation of motion is obtained employing Hamilton’s principle along with the modified couple stress theory. The continuous system is truncated into a system with finite degrees of freedom; the Galerkin scheme is employed to discretize the nonlinear partial differential equation of motion into a set of ordinary differential equations. This set of equations is solved numerically employing the pseudo-arclength continuation technique; first a nonlinear static analysis is performed upon this set of equations so as to obtain the onset of buckling (supercritical pitchfork bifurcation) and the buckled configuration of the microbeam. The frequency-response and force-response curves of the system are then constructed over the buckled configurations. A comparison is made between the frequency-response curves obtained by means of the modified couple stress and the classical theories. The effect of different system parameters on the frequency-response and force-response curves is also examined.     

NON-LINEAR MODELLING OF HYDRAULIC MOUNTS: THEORY AND EXPERIMENT

This paper focuses on the development of a complete non-linear model of a hydraulic engine mount and the evaluation of the model using a unique experimental apparatus. The model is capable of capturing both the low- and high-frequency behavior of hydraulic mounts. The results presented here provide a significant improvement over existing models by considering all non-linear aspects of a hydraulic engine mount. Enhancements to already published non-linear models include a continuous function that follows a simplistic yet effective approach to capture the switching effect and leakage through the decoupler, and upper chamber bulge damping. It is shown that the model developed here provides the appropriate system response over the full range of loading conditions (frequency and amplitude) encountered in practice. In order to obtain the parameter values for the non-linear model, a unique test apparatus is introduced. Using the experimental set-up, it is possible to verify the model of individual components of the mount, and later on test the behavior of the whole assembly. These data also establish the relative importance of several damping, inertia and stiffness terms. In addition, the measured responses of the mounts to loading at various frequencies and amplitudes are compared to the predictions of the mathematical model. The comparisons generally show a very good agreement (better than 10%), which corroborate the non-linear model of the mount. It is felt that this work will help engineers in reducing mount design time, by providing insight into the effects of various parameters within the mount.     

Flapwise dynamic response of a wind turbine blade in super-harmonic resonance

The flapwise dynamic response of a rotating wind turbine blade in super-harmonic resonance is studied in this paper, while the blade is subjected to unsteady aerodynamic loads. Coupled extensional–bending vibrations of the blade are considered; the governing equations which are coupled through linear and quadratic terms arising from rotating and geometric effects respectively are obtained by applying the Hamiltonian principle. The lth flapwise linear frequency and the rotational frequency are assumed to be in an almost 3:1 ratio, so super-harmonic resonance occurs when this linear frequency is close to the associated nonlinear frequency. By using the first n, no less than l, linear undamped modal functions as a functional basis and applying the Galerkin procedure, a 2n-degree-of-freedom discrete model with quadratic and cubic terms owing to geometric effect is derived. The generalized displacements corresponding to the discrete system are disintegrated into static and dynamic displacements. Perturbation method is adopted to get analytical solutions of the discrete dynamic system for positive aerodynamic dampings. The coning angle and the inflow ratio are chosen as two control parameters to analyze aeroelastic behaviors of the blade. By assuming that the static and dynamic displacements are of the same order in resonance region, and there is no other resonance except the super-harmonic resonance, the multiple-scales method is employed to obtain a set of amplitude modulation equations whose coefficients depend on two control parameters. The frequency-response equation is derived from the amplitude modulation equations. A method to estimate the functional dependence of the detuning parameter on two control parameters is introduced. The amplitude of the harmonic response is derived from the frequency-response equation after knowing the detuning parameter. Then the stability of the steady motion with respect to control parameters can be determined. The evolution of the dynamic response of the resonance mode with decreasing aerodynamic damping is discussed by means of both perturbation and numerical methods.     

A vibration isolation system in low frequency excitation region using negative stiffness structure for vehicle seat

This paper designs and fabricates a vibration isolation model for improving vibration isolation effectiveness of the vehicle seat under low excitation frequencies. The feature of the proposed system is to use two symmetric negative stiffness structures (NSS) in parallel to a positive stiffness structure. Here, theoretical analysis of the proposed system is clearly presented. Then, the design procedure is derived so that the resonance peak of frequency-response curve drifts to the left, the load support capacity of the system is maintained, the total size of the system is reduced for easy practical application and especially, the bending of the frequency-response curve is minimized. Next the dynamic equation of the proposed system is set up. Then, the harmonic balance (HB) method is employed to seek the characteristic of the motion transmissibility of the proposed system at the steady state for each of the excitation frequency. From this characteristic, the curves of the motion transmission are predicted according to the various values of the configurative parameters of the system. Then, the time responses to the sinusoidal, multi frequency and random excitations are also investigated by simulation and experiment. In addition, the isolation performance comparison between the system with NSS and system without NSS is realized. The simulation results reveal that the proposed system has larger frequency region of isolation than that of the system without NSS. The experimental results confirm also that with a random excitation mainly spreading from 0.1 to 10 Hz, the isolation performance of the system with NSS is greatly improved, where the RMS values of the mass displacement may be reduced to 67.2%, whereas the isolation performance of the system without NSS is bad. Besides, the stability of the steady-state response is also studied. Finally, some conclusions are given.     

A nonlinear spring mechanism incorporating a bistable composite plate for vibration isolation

The High Static Low Dynamic Stiffness (HSLDS) concept is a design strategy for a nonlinear anti-vibration mount that seeks to increase isolation by lowering the natural frequency of the mount whilst maintaining the same static load bearing capacity. It has previously been proposed that an HSLDS mount could be implemented by connecting linear springs in parallel with the transverse flexure of a composite bistable plate — a plate that has two stable shapes between which it may snap. Using a bistable plate in this way will lead to lightweight and efficient designs of HSLDS mounts. This paper experimentally demonstrates the feasibility of this idea. Firstly, the quasi-static force–displacement curve of a mounted bistable plate is determined experimentally. Then the dynamic response of a nonlinear mass–spring system incorporating this plate is measured. Excellent agreement is obtained when compared to theoretical predictions based on the measured force–displacement curve, and the system shows a greater isolation region and a lower peak response to base excitation than the equivalent linear system.     

System identification from multiple input/output data

For stationary random or transient data representing multicorrelated (multicoherent) input/output data occurring in physical systems, iterative computational algorithms are developed to identify the frequency response functions of optimum constant parameter linear systems connecting this data. Results are obtained from Fourier transform methods and optimum least-squares prediction techniques by changing original arbitrary input records into ordered sets of conditioned input records. These procedures provide the basis for efficient digital computer analysis of general multiple input/output problems. The Appendix contains useful error analysis results for the optimum frequency response estimates determined from measured data.     

Effect of one-way clutch on the nonlinear vibration of belt-drive systems with a continuous belt model

This study focuses on the nonlinear steady-state response of a belt-drive system with a one-way clutch. A dynamic model is established to describe the rotations of the driving pulley, the driven pulley, and the accessory shaft. Moreover, the model considers the transverse vibration of the translating belt spans for the first time in belt-drive systems coupled with a one-way clutch. The excitation of the belt-drive system is derived from periodic fluctuation of the driving pulley. In automotive systems, this kind of fluctuation is induced by the engine firing harmonic pulsations. The derived coupled discrete–continuous nonlinear equations consist of integro-partial-differential equations and piece-wise ordinary differential equations. Using the Galerkin truncation, a set of nonlinear ordinary differential equations is obtained from the integro-partial-differential equations. Applying the Runge–Kutta time discretization, the time histories of the dynamic response are numerically solved for the driven pulley and the accessory shaft and the translating belt spans. The resonance areas of the coupled belt-drive system are determined using the frequency sweep. The effects of the one-way clutch on the belt-drive system are studied by comparing the frequency–response curves of the translating belt with and without one-way clutch device. Furthermore, the results of 2-term and 4-term Galerkin truncation are compared to determine the numerical convergence. Moreover, parametric studies are conducted to understand the effects of the system parameters on the nonlinear steady-state response. It is concluded that one-way clutch not only decreases the resonance amplitude of the driven pulley and shaft's rotational vibration, but also reduces the resonance region of the belt's transverse vibration.     

DESIGN SENSITIVITY ANALYSIS AND OPTIMIZATION OF AN ENGINE MOUNT SYSTEM USING AN FRF-BASED SUBSTRUCTURING METHOD

The FRF-based substructuring method is one of the most powerful methods in analyzing the responses of complex built-up structures with high modal density. In this paper, a general procedure for the design sensitivity analysis of a vibro-acoustic system has been presented using the FRF-based substructuring formulation. For an acoustic response function, the proposed method gives a parametric design sensitivity expression in terms of the partial derivatives of the connection element properties and the transfer functions of the substructures. The derived noise sensitivity formula is combined with a non-linear programming module to obtain the optimal design for the engine mount system of a passenger car. The objective function is defined as the area of the interior noise graph integrated over a concerned r.p.m. range. The interior noise variations with respect to the dynamic characteristics of the engine mounts and bushings have been calculated using the proposed sensitivity formulation and transferred to a non-linear optimization software. To obtain the FRFs, a finite element analysis was used for the engine mount structures and experimental techniques were used for the trimmed body including the cabin cavity. The optimization based on the sensitivity analysis gives the ideal stiffness of the engine mount and bushings. The resultant interior noise in the passenger car shows that the proposed method is efficient and accurate.     

Analysis of nonlinear oscillators using Volterra series in the frequency domain

The Volterra series representation is a direct generalisation of the linear convolution integral and has been widely applied in the analysis and design of nonlinear systems, both in the time and the frequency domain. The Volterra series is associated with the so-called weakly nonlinear systems, but even within the framework of weak nonlinearity there is a convergence limit for the existence of a valid Volterra series representation for a given nonlinear differential equation. Barrett (1965) [1] proposed a time domain criterion to prove that the Volterra series converges within a given region for a class of nonlinear systems with cubic stiffness nonlinearity. In this paper this time-domain criterion is extended to the frequency domain to accommodate the analysis of nonlinear oscillators subject to harmonic excitation. A common and severe nonlinear phenomenon called jump, a behavior associated with the Duffing oscillator and the multi-valued properties of the response solution, is investigated using the new frequency domain criterion of establishing the upper limits of the nonlinear oscillators, to predict the onset point of the jump, and the Volterra time and frequency domain analysis of this phenomenon are carried out based on graphical and numerical techniques.     

W-band gunn second subharmonically locked oscillator

In this paper, the averaging method is used to analyse the performance of second subharmonically injection locked Gunn oscillator. Some useful expressions such as the locking range, output response, output impedance of nonlinear device in fundamental and subharmonic frequency are obtained. a W — band subharmonically locked Gunn oscillator is developed and experimental result demonstrates the validity of this analysis.     

Indirect measurement of dynamic force transmitted by a nonlinear hydraulic mount under sinusoidal excitation with focus on super-harmonics

Direct measurement of forces is not practical in many real-life applications since the interfacial conditions may change. Thus indirect force estimation methods must be developed though they pose special difficulty for nonlinear mounts or isolators. The hydraulic engine mount is examined as an illustrative example in this article since it exhibits spectrally varying and amplitude-sensitive parameters. First, we propose linear time-invariant, nonlinear and quasi-linear fluid and mechanical system models. Second, models are utilized to predict the transmitted force time history under sinusoidal excitation conditions given measured (or calculated) motion and/or internal pressure time histories. Experimental data from the non-resonant dynamic stiffness test is investigated in both time and frequency domains. In particular, the super-harmonic contents in fluid chamber pressure and force time histories are investigated using both measurements and mathematical models. This paper examines several alternate indirect schemes for estimating dynamic forces and highlights their strengths. The quasi-linear model with effective system parameters, say in terms of force to pressure or force to motion transfer functions, is found to correlate well with measured dynamic forces though linear and nonlinear models could be employed as well.     

新型高频消音器对Whoosh噪声的优化

为了诊断匹配涡轮增压器的汽油车型急加速过程中产生的Whoosh噪声,并分析噪声产生的原理,确定噪声的频率特性以及噪声产生的工况,本文通过对Whoosh噪声在进气系统中贡献量的分析,按照"源-路径-响应"原则,设计出频率相应的高频穿孔消音器并将其插入到进气系统中,从噪声传递路径上进行优化与控制。通过整车道路客观数据分析和主观驾评,确定该方案切实可行,可推广至多款增压车型上应用。     

Experimental study of hydraulic engine mounts using multiple inertia tracks and orifices: Narrow and broad band tuning concepts

Hydraulic engine mount tuning concepts with one inertia track and one decoupler are well understood. However, the dynamic response with multiple tracks or orifices is not. To overcome this void in the literature, dynamic tuning concepts of hydraulic engine mounts, with emphasis on multiple (n-)inertia tracks/orifices, are experimentally examined. A new prototype mount concept is designed, built, and experimentally evaluated in a controlled manner. Refined linear time-invariant models of fixed decoupler-type designs are developed to critically assess the dynamic stiffness measurements and to explore a family of alternate designs. Three narrowband devices are investigated for accurately predicting the frequencies corresponding to peak loss angles for the first time, in addition to examining and validating an n = 3 track mount. Two broadband devices are also successfully evaluated by tuning damping introduced by orifice-type tracks. A special broad-tuned design utilizing a controlled ‘leakage’ path flow area is then suggested, and the role of fluid resistance in achieving the desired performance is clarified. Finally, a production mount with unknown configuration is diagnosed using the proposed models with n tracks.