There exist many methods to calculate forced response in mechanical systems. Some methods are slow and the errors introduced are unknown. The paper presents a method that uses digital filters and modal superposition. It is shown how aliasing can be avoided as well as phase errors. The parameters describing the mechanical system are residues and poles, taken from FEA models, from lumped MCK systems, from analytic solutions or from experimental modal analysis. Modal damping may be used. The error in the calculation is derived and is shown to be only a function of the sampling frequency used. When the method is applied to linear mechanical systems in MATLAB it is very fast. The method is extended to incorporate nonlinear components. The nonlinear components could be simple, like hardening or stiffening springs, but may also contain memory, like dampers with hysteresis. The simulations are used to generate test data for development and evaluation of methods for identification of non-linear systems.
In this study, the flow stress behavior of ferritic-pearlitic steel (C45E steel) is investigated through isothermal compression testing at different strain rates (1 s-1, 5 s-1, and 60 s-1) and temperatures ranging from 200 to 700 °C. The stress-strain curves obtained from experimental testing were post-processed to obtain true stress-true plastic strain curves. To fit the experimental data to well-known material models, Johnson-Cook (J-C) model was investigated and found to have a poor fit. Analysis of the flow stress as a function of temperature and strain rate showed that among other deformation mechanisms dynamic strain aging mechanism was active between the temperature range 200 and 400 °C for varying strain rates and J-C model is unable to capture this phenomenon. This lead to the need to modify the J-C model for the material under investigation. Therefore, the original J-C model parameters A, B and n are modified using the polynomial equation to capture its dependence on temperature and strain rate. The results show the ability of the modified J-C model to describe the flow behavior satisfactorily while dynamic strain aging was operative. © 2019 by the authors. Licensee MDPI, Basel, Switzerland.
When using electrodynamic vibration exciters to excite structures, the actual force applied to the structure under test is the reaction force between the exciter and the structure. The magnitude and phase of the reaction force is dependent upon the characteristics of the structure and exciter. Therefore the quality of the reaction force i.e. the force applied on the structure depends on the relationship between the exciter and structure under test. Looking at the signal from the force transducer when exciting a structure with a sine wave, the signal will appear harmonically distorted within the regions of the resonance frequencies. This phenomenon is easily observed when performing tests on lightly damped structures. The harmonic distortion is a result of nonlinearities produced by the shaker when undergoing large amplitude vibrations, at resonances. When dealing with non-linear structures, it is of great importance to be able to keep a constant force level as well as a non-distorted sine wave in order to get reliable results within the regions of the resonance frequencies. This paper presents theoretical methods that can be used to create a non-distorted sinusoidal excitation signal with constant force level.
In the field of nonlinear dynamics it is essential to have well tested and reliable tools for estimating the nonlinear parameters from measurement data. This paper presents an identification technique based on using random noise signals, as initially developed by Julius S. Bendat. With this method the nonlinearity is treated as a feedback forcing term acting on an underlying linear system. The parameter estimation is then performed in the frequency domain by using conventional MISO/MIMO techniques. To apply this method successfully it is necessary to have some pre-information about the model structure and thus methods for nonlinear characterization and localization are studied. The paper also demonstrates the various ways the method can be formulated for multiple-degree-of-freedoms. The implementation of the method is illustrated with simulated data as well as a practical application, where the method is used to create a dynamic model of a test-rig with a significant nonlinearity.
With growing demands on product performance and growing complexity of engineering structures, efficient tools for analyzing their dynamic behavior are essential. Linear techniques are well developed and often utilized. However, sometimes the errors due to linearization are too large to be acceptable, making it necessary to take nonlinear effects into account. In many practical applications it is common and reasonable to assume that the nonlinearities are highly local and thus only affect a limited set of spatial coordinates. The purpose of this paper is to present an approach to finding the spatial location of nonlinearities from measurement data, as this may not always be known beforehand. This information can be used to separate the underlying linear system from the nonlinear parts and create mathematical models for efficient parameter estimation and simulation. The presented approach builds on the reverse-path methodology and utilizes the coherence functions to determine the location of nonlinear elements. A systematic search with Multiple Input/Single Output models is conducted in order to find the nonlinear functions that best describe the nonlinear restoring forces. The obtained results indicate that the presented approach works well for identifying the location of local nonlinearities in structures. It is verified by simulation data from a cantilever beam model with two local nonlinearities and experimental data from a T-beam experimental set-up with a single local nonlinearity. A possible drawback is that a relatively large amount of data is needed. Advantages of the approach are that it only needs a single excitation point that response data at varying force amplitudes is not needed and that no prior information about the underlying linear system is needed.
Trial and error and the use of highly time-consuming methods are often necessary for modeling, simulating and characterizing nonlinear dynamical systems. However, for the rather common special case when a nonlinear system has linear relations between many of its degrees of freedom there are particularly interesting opportunities for more efficient approaches. The aim of this thesis is to develop and validate new efficient methods for the theoretical and experimental study of mechanical systems that include significant zero-memory or hysteretic nonlinearities related to only small parts of the whole system. The basic idea is to take advantage of the fact that most of the system is linear and to use much of the linear theories behind forced response simulations. This is made possible by modeling the nonlinearities as external forces acting on the underlying linear system. The result is very fast simulation routines where the model is based on the residues and poles of the underlying linear system. These residues and poles can be obtained analytically, from finite element models or from experimental measurements, making these forced response routines very versatile. Using this approach, a complete nonlinear model contains both linear and nonlinear parts. Thus, it is also important to have robust and accurate methods for estimating both the linear and nonlinear system parameters from experimental data. The results of this work include robust and user-friendly routines based on sinusoidal and random noise excitation signals for characterization and description of nonlinearities from experimental measurements. These routines are used to create models of the studied systems. When combined with efficient simulation routines, complete tools are created which are both versatile and computationally inexpensive. The developed methods have been tested both by simulations and with experimental test rigs with promising results. This indicates that they are useful in practice and can provide a basis for future research and development of methods capable of handling more complex nonlinear systems.
Trial and error and the use of highly time-consuming methods are often necessary for investigation and characterization of nonlinear systems. However, for the rather common case where a nonlinear system has linear relations between many of its degrees of freedom there are opportunities for more efficient approaches. The aim of this thesis is to develop and validate new efficient simulation and experimental methods for characterization of mechanical systems with localized nonlinearities. The purpose is to contribute to the development of analysis tools for such systems that are useful in early phases of the product innovation process for predicting product properties and functionality. Fundamental research is combined with industrial case studies related to metal cutting. Theoretical modeling, computer simulations and experimental testing are utilized in a coordinated approach to iteratively evaluate and improve the methods. The nonlinearities are modeled as external forces acting on the underlying linear system. In this way, much of the linear theories behind forced response simulations can be utilized. The linear parts of the system are described using digital filters and modal superposition, and the response of the system is recursively solved for together with the artificial external forces. The result is an efficient simulation method, which in conjunction with experimental tests, is used to validate the proposed characterization methods. A major part of the thesis addresses a frequency domain characterization method based on broad-band excitation. This method uses the measured responses to create artificial nonlinear inputs to the parameter estimation model. Conventional multiple-input/multiple-output techniques are then used to separate the linear system from the nonlinear parameters. A specific result is a generalization of this frequency domain method, which allows for characterization of continuous systems with an arbitrary number of localized zero-memory nonlinearities in a structured way. The efficiency and robustness of this method is demonstrated by both simulations and experimental tests. A time domain simulation and characterization method intended for use on systems with hysteresis damping is also developed and its efficiency is demonstrated by the case of a dry-friction damper. Furthermore, a method for improved harmonic excitation of nonlinear systems using numerically optimized input signals is developed. Inverse filtering is utilized to remove unwanted dynamic effects in cutting force measurements, which increases the frequency range of the force dynamometer and significantly improves the experimental results compared to traditional methods. The new methods form a basis for efficient analysis and increased understanding of mechanical systems with localized nonlinearities, which in turn provides possibilities for more efficient product development as well as for continued research on analysis methods for nonlinear mechanical structures.
When using electrodynamic vibration exciters to excite structures, the actual force applied to the structure under test is the reaction force between the exciter and the structure. The magnitude and phase of the reaction force is dependent upon the characteristics of the structure and exciter. Therefore the quality of the reaction force i.e. the force applied on the structure depends on the relationship between the exciter and structure under test. Looking at the signal from the force transducer when exciting a structure with a sine wave, the signal will appear harmonically distorted within the regions of the resonance frequencies. This phenomenon is easily observed when performing tests on lightly damped structures. The harmonic distortion is a result of nonlinearities produced by the shaker when undergoing large-amplitude vibrations, at resonances. When dealing with non-linear structures, it's of great importance to be able to keep a constant force level as well as a non-distorted sine wave in order to get reliable results within the regions of the resonance frequencies. This paper presents the method and results from an experimental test creating a nondistorted excitation signal with constant force level.
The industrial demand on good dynamical simulation models is increasing. Since most structures show some form of nonlinear behavior, linear models are not good enough to predict the true dynamical behavior. Therefore nonlinear characterization, localization and parameter estimation becomes important issues when building simulation models. This paper presents identification techniques for nonlinear systems based on both random and harmonic excitation signals. The identification technique based on random excitation builds on the well known reverse-path method developed by Julius S. Bendat. This method treats the nonlinearity as a feedback forcing term acting on an underlying linear system and the parameter estimation is performed in the frequency domain by using conventional MISO/MIMO techniques. Although this method provides a straightforward and systematic way of handling nonlinearities, it has been somewhat limited in use due to the complexity of creating uncorrelated inputs to the model. As is shown in this paper, the parameter estimation will not be improved with conditioned inputs and the nonlinear parameters and the underlying linear system can still be estimated with partially correlated inputs. This paper will also describe a parameter estimation method to be used with harmonic input signals. By using the principle of harmonic balance and multi-harmonic balance it is possible to estimate an analytical frequency response function of the studied nonlinear system. This frequency response function can, in conjunction with measured nonlinear transfer functions, be used to estimate the nonlinearity present in the system. This method is also applicable on nonlinear systems with memory, e.g. systems with hysteresis effects. The above mentioned methods are applied to multi-degree-of-freedom and single-degree-of-freedom systems with different types of nonlinearities. Also, techniques for locating nonlinearities are discussed.
The industrial demand on good dynamical simulation models is increasing. Since most structures show some form of nonlinear behavior, linear models are not good enough to predict the true dynamical behavior. Hysteresis is a highly nonlinear phenomenon which occurs in for example dampers and mechanical joints. This paper presents a method for parameter estimation on nonlinear systems under harmonic excitation. By using the principle of harmonic balance or multi harmonic balance a theoretical frequency response function of the studied system can be estimated. This frequency response function can, in conjunction with measured nonlinear transfer functions, be used to make parameter estimations of the nonlinearity present in the system. A major benefit using this method is the ability to use arbitrary nonlinear functions. This means that the method can be applied to nonlinear systems with memory, for instance systems with hysteresis effects. The method is applied to both simulated systems and an experimental test rig.
When dealing with nonlinear dynamical systems, it is important to have efficient, accurate and reliable tools for estimating both the linear and nonlinear system parameters from measured data. An approach for nonlinear system identification widely studied in recent years is "Reverse Path". This method is based on broad-band excitation and treats the nonlinear terms as feedback forces acting on an underlying linear system. Parameter estimation is performed in the frequency domain using conventional multiple-input-multiple- output or multiple-input-single-output techniques. This paper presents a generalized approach to apply the method of "Reverse Path" on continuous mechanical systems with multiple nonlinearities. The method requires few spectral calculations and is therefore suitable for use in iterative processes to locate and estimate structural nonlinearities. The proposed method is demonstrated in both simulations and experiments on continuous nonlinear mechanical structures. The results show that the method is effective on both simulated as well as experimental data.
Accurate estimates of cutting forces are important in the evaluation of different cutting tool geometries and concepts. However, dynamic influences from the measurement system affect the result, which can make the obtained cutting force data erroneous and misleading. This article presents a method to obtain an inverse filter which compensates for the dynamic influences of the measurement system. Using this approach, unwanted dynamic effects of the measurement system can be counteracted, making it possible to retain information related to the cutting forces contained in the high frequency region. The advantage of the proposed method is illustrated by comparing simulated, inverse- and low-pass filtered forces to unfiltered forces under different cutting conditions. The results show that inverse filtering increases the usable frequency range of the force dynamometer and thereby provide more reliable results compared to both low-pass and unfiltered forces.
Boring bars with tuned mass dampers have a passive damper tuned with respect to the frequency of the first bending mode of the tool. When the tool is clamped into the machine tool there is a stiffness loss that lowers the natural frequency of the bar compared to ideal clamping conditions. For large tools the difference can be more than 35%, depending on clamping structure, tool size and overhang. In this paper we investigate a simple two-degree-of-freedom model for the tool-machine interaction consisting of a bending mode coupled with a rotational stiff mode. The model gives good insight into the system behavior and fits well with measurements. © 2019 The Authors. Published by Elsevier B.V. Peer-review under responsibility of the scientific committee of The 17th CIRP Conference on Modelling of Machining Operations