Force-state Mapping Research Articles

Dynamic Modeling and Parameter Identification of Double Casing Joints for Aircraft Fuel Pipelines

Double casing joints are flexible pipe joints used for connecting aircraft fuel pipelines, which can compensate for the displacement and corner of the connected pipes and have complex mechanical characteristics. However, it is difficult to use sensors to directly measure the mechanical connection parameters of flexible joints. In this paper, we construct a coupling dynamics model and parameter identification of a double casing joint. Firstly, we analyze the structure and working principle of double-layer casing joints and establish the dynamics model of a single-layer flexible joint based on the transfer matrix method. Then, we deduce the coupling matrix of the inner and outer pipeline according to the deformation coordination conditions combined with matrix dimension extension. We establish the coupling dynamics model of flow–solid coupling of double casing joints. Furthermore, parameters such as equivalent stiffness and damping of each motion of the double casing joint in the casing unit are identified using the force-state mapping (FSM) method, and an analytical solution in the frequency domain under hammering excitation is given by the dynamics model. Finally, the dynamics test bench of the double casing joint for aircraft fuel is set up, and the free mode test of the double casing joint assembly is carried out. The results show that under free boundary hammering excitation, the theoretical and experimental frequency-domain response results are well matched, both obtaining seven main resonance peaks, and the maximum error is 9.45%, which shows the validity of the pipeline dynamics modeling method with a double casing joint.

Nonlinear Dynamic Modeling of Joints between Solar Panels on Spacecraft

Multiple linear and nonlinear dynamic parameters of the joints at the root of solar panels and between solar panels on spacecraft, both of which have complex nonlinear dynamic properties, were identified by using the force state mapping method for modeling complex nonlinear joints in deployable mechanisms of spacecraft, and the differential equations representing the nonlinear dynamic model of the joints were derived. On the basis of the actual force characteristics of the joints, test systems were developed to investigate the vibration response of the complex nonlinear joints in spacecraft so as to acquire test data necessary for the identification. The relation between the moment and bending angle of the panel root and inter-panel joints on spacecraft was obtained through vibration tests under various frequencies and excitation forces. The validity and effectiveness of the dynamic model have been verified by vibration tests of the joints under pulsed excitation. The parameters identified in this paper reflect the nonlinear stiffness, friction and damping characteristics of the joints. The dynamic model established based on these parameters can describe multiple linear and nonlinear dynamic properties of the joints and can be further applied in modeling and control of the entire spacecraft system.

Demonstration and validation of constructive initialization method for neural networks to approximate nonlinear functions in engineering mechanics applications

This paper is a sequel paper to Pei et al. (Nonlinear Dyn 71(1–2):371–399, 2013) by the authors and their co-authors published in this journal. The main contribution lies in using complex data collected from real-world problems to validate the methodology and techniques given in Pei et al. (Nonlinear Dyn 71(1–2):371–399, 2013) and others to initialize neural networks in a systematic and constructive manner. Two sets of real-world laboratory data are analyzed: One set is from testing a full-sized nonlinear viscous fluid damper with harmonic displacements at different frequencies, and the other set a well-controlled complex nonlinear mechanism involving either rotational or torsional motion under random excitations. The force-state mapping formulation is adopted to analyze these data sets. In addition, we use the fractional derivative Maxwell model for the nonlinear viscous fluid damper application. This study is focused on determining the number of hidden nodes and the initial values of weights and biases for both multilayer feedforward or tapped delay line neural networks, before training them using the well-established backpropagation approach. This critical but subjective design issue is handled in a transparent manner that directly utilizes either the feature of the data, or the governing mathematical expression, and is carried out in a nearly deterministic manner. This contrasts sharply with Nguyen and Widrow (1990), a random initialization approach that prevails in training neural networks for function approximation. The initialization methodology and techniques developed in our prior and current work starting with Pei (2001) are explained and demonstrated, in the context of nonlinear problems encountered in typical applied mechanics applications, by using the two sets of data, and validated by examining the generalization capability of the trained neural networks. Future work and relevant technical hurdles are identified.

Suggestion of a polynomial model with a convolution integral term for a vibration isolation element with hysteresis and non-linearity for force estimation

Modelling of the force transmitted through vibration isolation elements in terms of the displacement and the velocity is very useful in transfer path or vibration power analysis of dynamic systems. Modelling using the complex stiffness in the frequency domain can be a good solution for linear elements, and force-state mapping can be a good solution for non-linear elements. In this study, it is noted that conventional force-state mapping with a linear part simply consisting of a linear spring and a viscous damper does not work sufficiently well for vibration isolation elements typically made of rubber materials; therefore, modelling of the linear part using the complex stiffness is proposed. The non-linear force-state model with the complex stiffness model for the linear part is applied to a set of simulation data for validation of the proposed approach and to another set of data measured from a cabin mount of an excavator for illustration. A form of the fractional polynomial function of the frequency is utilized for curve fitting of the linear complex stiffness in the frequency domain. The constraints for stability and causality on the coefficients and the order of this form are discussed. Procedures for the construction of the whole model are presented. Based on the modelling of the simulated and measured data, it is claimed that the suggested modelling is promising as a representation of the frequency- and amplitude-dependent complex stiffness of vibration isolation elements.

Mapping some basic functions and operations to multilayer feedforward neural networks for modeling nonlinear dynamical systems and beyond

This study significantly extends the development of an initialization methodology for designing multilayer feedforward neural networks, aimed primarily at modeling nonlinear functions for engineering mechanics applications, as proposed and published in (Pei in Ph.D. dissertation, Columbia University, 2001; Pei and Smyth in J. Eng. Mech. 132(12):1290, 1310, 2006; Pei et al. in Comput. Methods Appl. Mech. Eng. 194(42–44):4481, 2005; Pei et al. in Proc. Int. Joint Conference on Neural Networks (IJCNN’05), pp. 1377–1382, 2005; Pei and Mai in J. Appl. Mech. 2008; Pei et al. in Proc. Int. Joint Conference on Neural Networks (IJCNN’07), 2007). Seeking a transparent and domain knowledge-based approach for neural network initialization and result interpretation, this study examines linear sums of sigmoidal functions as a means to construct approximations to various nonlinear functions including reciprocal, absolute value, the product of absolute value and first-order polynomial, exponential, truncated sinc, Mexican hat, and Gaussian functions as well as the four elementary arithmetic operations (addition, subtraction, multiplication, and division). By extending two initialization techniques (layer condensation and inspiration from high-order derivatives of sigmoidal function), this study advances the previously proposed initialization procedure, thus opening the door to a significantly wider range of nonlinear functions. Specifically, in engineering mechanics, this study directly benefits multilayer feedforward neural networks when modeling nonlinear restoring forces based on the force-state mapping (among others). Application examples are provided to illustrate the importance of studying basic functions and operations, and future work is identified.

Modeling and identification of frictional forces at a contact interface experiencing micro-vibro-impacts

Modeling and identification of restoring forces at contact interfaces is an inevitable part of investigating the dynamic characteristics of mechanical systems. As the vibration amplitude increases, various nonlinear mechanisms such as micro/macro-slip and micro-impacts activate in the interface. This paper considers the nonlinear behavior of a frictional contact in situations where micro-impacts develop in the normal direction to the friction surface. It investigates the effect of variable normal load on the contact frictional forces, and defines a friction model capable of taking into account this effect. In an experimental case study, the contact is excited using a dual sine force to allow a reliable dual-mode identification procedure. The measured data and the normal modes of the corresponding linear system are employed to define the system nonlinear modes used to expand the system response. This provides a reduced order model containing dominant nonlinear effects in the contact interface. Force state mapping method is employed to identify the contact restoring forces. The experimentally obtained forces are employed to determine the parameters of a new friction law defied by modifying the Valanis model. It is shown the model is capable of predicting the main nonlinear characteristics of the contact interface and regenerates the experimental results at different vibration response levels.

Identification of Nonlinear Joint Parameters with Force-state Mapping Method

Identification of the non-linear joint parameters of the complex structure with force-state mapping in frequency domain,the number of the data required in the curve fitting can be drastically reduced compared to the time domain approach.Because the model of the joint is not known in advance,and noise affected the identification accuracy cannot be ignored,when using least square method to solve the over-determined equation,due to the inverse operation of the matrix,the noise may have great effect on the identified parameters,the equation will be ill-condition and the ill-posed problem arise from the noise effect.A novel method to identify the nonlinear joint parameters is proposed from two aspects as follow:the assumed model includes all the exact terms and some surplus term;the assumed model does not include all the exact terms and has some surplus terms.A model combining Tikhonov regularization and iterative method is suggested to update the joint model.It is found that even if the assumed model does not include all the exact terms,as long as model updated continually,meet the convergence criteria,the identification accuracy is still high.Example analysis confirms that this method has high identification accuracy.

Modeling for Dynamic Properties of Honeycomb Paperboard by the Restoring Force-State Mapping Method

The realization of the potential of honeycomb paperboard as an important cushion material in packaging has inspired a close scrutiny of its properties. In present work, the force-state mapping technique is successfully applied to the honeycomb paperboard dynamical properties modeling problem. A experiment system is set up to learn the dynamic properties of the honeycomb paperboard. According to the experiment data and the force-state mapping method, the model governing the dynamics of honeycomb paperboard-mass system is established. This model can be used to predict the vibration transmissibility of the system. The comparison of the measured and predicted vibration transmissibility indicates this model can predict the resonance frequency well, the measured resonance altitude is about 20% greater than the predicted altitude, the possible sources of this error is discussed.

Nonlinear model identification of a frictional contact support

Accurate predictions of a structure dynamics require precise modeling of its boundary conditions including any nonlinear effects. This paper investigates the behavior of frictional supports that are examples of boundary conditions exhibiting nonlinear effects, such as stiction and slip phenomena, depending on the structure vibration amplitudes. The dependency of restoring forces in a frictional contact to the vibration amplitude level is identified in this study using experimental observations. In an experimental case study measured responses of a beam fixed at one end and frictionally supported at the other end were expanded using corresponding nonlinear normal modes of the structure and a reduced order model of the continuous system containing dominant nonlinear effects of the contact was obtained. The obtained discrete model constitutes bases for identification of restoring forces in the contact interface using force state mapping. The identified nonlinear restoring forces are then employed to specify parameters of a predictive contact model for the boundary support. The obtained contact model is capable of predicting damping and softening effects due to micro/macro-slippage and accurately regenerates the experimental results at various response levels.

Force State Maps Using Reproducing Kernel Particle Method and Kriging Based Functional Representations

The problem of identification of nonlinear system parameters from measured time histories of response under known excitations is considered. Solutions to this problem are obtained by using the force state mapping technique with two alternative functional representation schemes. These schemes are based on the application of reproducing kernel particle method (RKPM) and kriging techniques to fit the force state map. The RKPM has the capability to reproduce exactly polynomials of specified order at any point in a given domain. The kriging based methods represent the function under study as a random field and the parameters describing this field are optimally determined based on available observations. The present study investigates the performance of RKPM and kriging based fits to the force state maps for a variety of nonlinear dynamical systems. The study also examines the application of force state maps in (a) determining the fixed points limit cycles of the system and their stability, (b) determining the properties of the linear system which would result if nonlinearity were to be absent, and (c) dealing with nonlinearities that are continuous but not differentiable and nonlinearities that are discontinuous at a set of points within the domain of interest. Illustrative examples on single and multi-degrees of freedom nonlinear systems are presented to demonstrate the scope of the proposed procedures.

Deterministic Excitation Forces for Simulation and Identification of Nonlinear Hysteretic SDOF Systems

This paper investigates how to design deterministic excitation forces in studying nonlinear single-degree-of-freedom systems, especially those with rate and path dependency and strength and stiffness degradation. One frequency-modulated periodic excitation and its amplitude-modulated counterpart are proposed as a solution, and a series of numerical exercises are carried out to show that these forces can be designed for sufficient forcing functions to study the complex nonlinear hysteresis. To rapidly reveal the underlying characteristics of the system and also to further lead to an effective system identification, four evaluation tools are proposed to be utilized together with the proposed excitation forces. These tools include the response curves, force-state map, intercycle drift, and intercycle pattern change, based on which some distinctive “patterns” are obtained to reveal the existence of nonlinearities, types of nonlinearities, existence of memory, and degradation. By using both Bouc-Wen and Bouc-Wen-Baber-Noori models for the system in all the simulations, the writers compare the commonly used forces with the proposed excitation forces to further demonstrate the advantages of the proposed excitation forces and evaluation tools. The writers also explore challenges in terms of implementing the proposed excitation forces. The results of this study are expected to benefit both physical testing and numerical simulation of complex nonlinear hysteretic systems in a time- and cost-effective manner, as well as leading to efficient schemes for system identification.

Model and Parameters Identification of Non-Linear Joint by Force-State Mapping in Frequency Domain

AbstractGenerally, the Force-State Mapping (FSM) is an effective method to identify the parameters of nonlinear joints provided that the joint model is exactly known in advance. However, the variation of the non-linear joints is so large that the mathematical models of non-linear joints generally are not known in advance. Therefore, the model and the parameters of a non-linear joint should be identified simultaneously in practice. In this work, a new identification procedure which was based on the FSM method in frequency domain was proposed to identify the mathematical model and parameters of a non-linear joint simultaneously. Generally, there are many feasible combinations of models and parameters which can satisfy the measurement data within an allowable range of error. In this work, an iteration procedure was used to update the feasible models to result in an optimal model with its parameters. The simulation results show that a proper mathematical model and accurate parameters can be identified simultaneously by the new procedure even that the measurement data are contaminated by noise.

Identification of nonlinear bolted lap-joint parameters by force-state mapping

Joints and fasteners often have a significant effect on the dynamical behaviour of assembled mechanical structures and the analytical prediction of structural responses therefore depends upon the accuracy of joint modelling. Detailed constitutive models that fully describe the behaviour of frictional interfaces are often unduly complicated; in which case simpler phenomenological models having parameters identified from vibration tests may be preferable. Unfortunately the direct measurement of forces transmitted between two contacting surfaces and their relative displacements are not possible in practice and it is therefore necessary to rely on measurements remote from joints. In this paper, the parameters of an assumed nonlinear joint model are identified by force-state mapping from time-domain acceleration records in response to single-frequency excitation close to the first natural frequency. The problem of lack of accessibility for measurement at the joint is overcome by casting the governing equation of the system in modal coordinates so that modal parameters are identified to represent the nonlinear behaviour of the joint. A particular result from the experimental programme is the identification of viscous damping coefficients dependent upon displacement amplitude. The significance of this result is that the complex phenomenon of energy dissipation in lap joints can be represented by a simple analytical model capable of producing accurate results.

Investigation of the nonlinear behaviour of damping controlled fluidelastic instability in a normal triangular tube array

The fluidelastic behaviour of two normal triangular tube arrays subject to cross flow have been tested. A novel feature of the test facility is an electromagnetic device which has been used to control the system damping and to provide additional excitation. A linearized model of the fluid force is identified for both arrays and it is shown that nonlinear effects are more significant in the denser array. An empirical nonlinear model is proposed for this array and the associated parameters are quantified directly from free response data using the force state mapping technique. Although, the predicted critical velocity is accurately predicted, the limit cycle amplitude is systematically over estimated. Nonetheless, the resulting model shows good qualitative agreement with experimentally determined system behaviour.

VIBRATION CONTROL OF SANDWICH BEAMS USING ELECTRO-RHEOLOGICAL FLUIDS

Electro-rheological (ER) fluids are a class of smart materials exhibiting significant reversible changes in their rheological and hence mechanical properties under the influence of an applied electric field. Efforts are in progress to embed ER fluids in various structural elements to mitigate vibration problems. The present work is an experimental investigation of the behaviour of a sandwich beam with ER fluid acting as the core material. A starch–silicone-oil-based ER fluid is used in the present study. Significant improvements in the damping properties are achieved in experiments and the damping contributions by viscous and non-viscous forces are estimated by force-state mapping (FSM) technique. With the increase in electric field across the ER fluid from 0 to 2 kV, an increase of 25–50% in equivalent viscous damping is observed. It is observed that as concentration of starch is increased, the ER effect grows stronger but eventually is overcome by applied stresses.

APPLICATION OF FORCE-STATE MAPPING TO A NON-LINEAR FLUID–ELASTIC SYSTEM

The damping mechanisms associated with flow/structure interactions are, in general, considered to be non-linear. In this paper, the fluid–elastic instability of a tube in a heat exchanger is controlled using a tunable damper and the stiffness and damping associated with the flow-induced non-linearities are determined using the force-state mapping technique. The analysis of the results clearly indicates a linear destabilising fluid mechanism but also identifies non-linear damping and stiffness forces which are consistent with both the experimental behaviour and theoretical models for the instability.

FORCE-STATE MAPPING METHOD OF A CHAOTIC DYNAMICAL SYSTEM

This paper is concerned with the system identification of an experimental chaotic system. A conventional double-well potential vibrator is modified to accommodate an external air dash-pot damper. Damping can be controlled and set to have either high or low damping. The system exhibits very rich dynamics from periodic to chaotic states. The force-state mapping identification method is applied to both linear and chaotic states with the emphasis on quantifying the damping property for both high and low damping. This points to the potential use in condition monitoring of chaotic systems.

PSEUDO FORCE–STATE MAPPING METHOD: INCORPORATION OF THE EMBEDDING AND FORCE–STATE MAPPING METHODS

It is shown that the embedding method can be incorporated with the force state mapping technique, and the resulting surface form of the pseudo force-state map preserves the structure of the original non linearity of the system. Thus, the pseudo force state mapping method may be used in a practical situation when one cannot obtain all the variables

Experimental identification of nonlinear model parameter by frequency domain method

In this work, a frequency domain method is tested numerically and experimentally to improve nonlinear model parameters using the frequency response function at the nonlinear element connected point of structure. This method extends the force-state mapping technique, which fits the nonlinear element forces with time domain response data, into frequency domain manipulations. The force-state mapping method in the time domain has limitations when applying to complex real structures because it needd a time domain lumped parameter model. On the other hand, the frequency domain method is relatively easily applicable to a complex real structure having nonlinear elements since it uses the frequency response function of each substurcture. Since this mehtod is performed in frequency domain, the number of equations required to identify the unknown parameters can be easily increased as many as it needed, just by not only varying excitation amplitude bot also selecting excitation frequency domain method has some advantages over the classical force-state mapping technique in the number of data points needed in curve fit and the sensitivity to response noise.

On an Innovative Method of Modeling General Nonlinear Mechanical Systems, Part 2: Experiments

In this the second part, the attention will he focused on the practical implementation of the techniques proposed in Part 1. Firstly, the problem of generating a periodic excitation signal with a specific spectrum in the experimental conditions will be solved by the use of a programmable signal analyzer. Secondly, using this spectral control technique and applying the generalized linear path identification method proposed in Part 1, two simple nonlinear structures are tested and analysed. Finally, the application of this new package of nonlinear system identification techniques is experimentally demonstrated on a multiple DOF structure. It includes the practical implementation of the spectral controller, the generalized linear path identification method, the practical location scheme of nonlinear elements and the in-situ nonlinear force state mapping technique.