Institute of Fundamental Technological Research

The main purpose of the study is an assessment of computational efficiency of selected numerical methods for estimation of vibrational response statistics of a large multi-bearing turbo-generator rotor-shaft system. The effective estimation of the probability distribution of structural responses is essential for robust design optimization and reliability analysis of such systems. The analyzed scatter of responses is caused by random residual unbalances as well as random stiffness and damping parameters of the journal bearings. A proper representation of these uncertain parameters leads to multidimensional stochastic models. Three estimation techniques are compared: Monte Carlo sampling, Latin hypercube sampling and the sparse polynomial chaos expansion method. Based on the estimated values of the first four statistical moments the probability density function of the maximal vibration amplitude is evaluated by the maximal entropy principle method. The method is inherently suited for an accurate representation of the probability density functions with an exponential behavior, which appears to be characteristic for the investigated rotor-shaft responses. Performing multiple numerical tests for a range of sample sizes it was found that the sparse polynomial chaos method provides the best balance between the accuracy and computational effectiveness in estimating the unknown probability distribution of the maximal vibration amplitude.

stochastic moment estimation, sparse polynomial chaos expansion, maximum entropy principle, rotor, uncertainties, hybrid mechanical model, random unbalance distribution

The NUMPRESS System has been developed in IPPT PAN as a result of a project financially supported by European Regional Development Fund (within the Innovative Economy Programme) and is dedicated to small and middle enterprises dealing with sheet metal forming. The program consists of (i) an analytical module for analysis of forming processes with the finite element method, (ii) an optimization module controlling execution of the analytical module and performing optimization with respect to selected process parameters, in both deterministic and robust formulation, (iii) a reliability analysis module controlling execution of the analytical module to assess how random distribution of design parameters affects forming results, and (iv) a graphical user interface enabling communication between modules and easy definition of design parameters and optimization criteria. The analytical module consists of two independent programs up to the user's choice: NUMPRESS-Flow, a faster and less accurate program for implicit quasi-static analysis of rigid-viscoplastic shells (based on the flow approach) and NUMPRESS-Explicit, a program for explicit dynamical analysis of elastic-plastic and elastic-viscoplastic shells. Both programs are interfaced to a well-known commercial graphical pre- and postprocessor GiD. Fundamentals of formulations employed in the system and numerical examples are presented in the paper.

sheet metal forming, finite element method, deterministic and robust design optimization, reliability analysis

The commonly observed nowadays tendency to weight minimization of rotor-shafts of the rotating machinery leads to a decrease of shaft bending rigidity making a risk of dangerous stress concentrations and rubbing effects more probable. Thus, a determination of the optimal balance between reducing the rotor-shaft weight and assuring its admissible bending flexibility is a major goal of this study. The random nature of residual unbalances of the rotor-shaft as well as randomness of journal-bearing stiffness have been taken into account in the framework of robust design optimization. Such a formulation of the optimization problem leads to the optimal design that combines an acceptable structural weight with the robustness with respect to uncertainties of residual unbalances – the main source of bending vibrations causing the rubbing effects. The applied robust optimization technique is based on using Latin hypercubes in scatter analysis of the vibration response. The so-called optimal Latin hypercubes are used as experimental plans for building kriging approximations of the objective and constraint functions. The proposed method has been applied for the optimization of the typical single-span rotor-shaft of the 8-stage centrifugal compressor.

Rotor-shaft system, Robust design optimization, Lateral vibrations, Rubbing effects, Random unbalance distribution

The main objective of the presented study is an evaluation of the effectiveness of various methods for estimating statistics of rotor-shaft vibration responses. The computational effectiveness as well as the accuracy of statistical moment estimation are essential for efficient robust design optimization of the rotor-shaft systems. The compared methods include sampling techniques, the perturbation approach, the dimension reduction method and the polynomial chaos expansion method. For comparison, two problems of the rotor-shaft vibration analysis are considered: a typical single-span rotor-shaft of the 8-stage centrifugal compressor driven by the electric motor and a large multi-bearing rotor-shaft system of the steam turbo-generator. The most important reason for the observed scatter of the rotor-shaft vibration responses is the inherently random nature of residual unbalances as well as stiffness and damping properties of the journal bearings. A proper representation of these uncertain parameters leads to multidimensional stochastic models. It was found that methods that provide a satisfactory balance between the estimation accuracy and computational effectiveness are sampling techniques. On the other hand, methods based on Taylor series expansion in most of the analysed cases fail to approximate the rotor-shaft response statistics.

Scatter analysis, rotor-shaft vibrations

In the paper a stochastic method for fault detection and identification in the shafts of rotating machines is proposed. This approach is based on the Monte Carlo simulations of rotor-shaft lateral–torsional–longitudinal vibrations mutually coupled by transverse cracks of various possible and randomly selected depths and locations on the shaft. For this purpose the structural hybrid model of a real faulty object is applied. This model is characterized by a high practical reliability and great computational efficiency, so important for many hundred thousand single numerical simulations necessary for a creation of the databases applied for inverse problem solution finally leading to crack identification. These databases are created with an arbitrary assumed probability densities of crack parameters which ensures appropriate spread of the dynamic responses of the considered faulty mechanical system. A sufficiently large database determined for the investigated object enable us to estimate almost immediately, i.e. within less than 1 s, the crack depth and axial position with identification errors not exceeding 9% and 5%, respectively. Thus, the proposed method seems to be a very convenient diagnostic tool for engineering applications in the industry.

Rotor-shaft system, Dynamic diagnostics, Crack identification, Monte Carlo simulation, Coupled vibration analysis

In the paper several stochastic methods for detection and identification of cracks in the shafts of rotating machines are proposed. All these methods are based on the Monte Carlo simulations of the rotor-shaft lateral-torsional-longitudinal vibrations mutually coupled by transverse cracks of randomly selected depths and locations on the shaft. For this purpose there is applied a structural hybrid model of a real cracked rotor-shaft. This model is characterized by a high practical reliability and great computational effi-ciency, so important for hundreds of thousands numerical simulations necessary to build databases used in solving the inverse problem, i.e. crack parameter identifications. In order to ensure a good identification accuracy, for creating the Monte Carlo samples of data points there are proposed special probability density functions for locations and depths of the crack. Such an approach helps in enhancing databases corresponding to the most probable faults of the rotor-shaft system of the considered rotor machine. In the presentedstudy six different database sizes are considered to compare identification efficiency and accuracy of considered methods. A sufficiently large database enables us to estimate almost immediately (usually in less than one second) the crack parameters with precision that is in most of the cases acceptable in practice. Then, as a next stage, one of the proposed fast improvement algorithms can be applied to refine identification results in a reasonable time. The proposed methods seem to provide very convenient diagnostic tools for industrial applications.

Crack rotor dynamics, Nonlinear and parametric vibrations, Hybrid modeling, Monte Carlo simulation, Crack identification methods

An assessment of structural reliability requires multiple evaluations of the limit state function for various realizations of random parameters of the structural system. In the majority of industrial applications the limit state functions cannot be expressed explicitly in terms of the random parameters but they are specified using selected outcomes of the FE analysis. In consequence, in order to be useful in practice, a structural reliability analysis program should be closely integrated with a FE module or it should be interfaced with an advanced external FE program. When the FE source code is not available, which is usually the case, the only option is to establish a communication between the reliability analysis program and an external FE software through the batch mechanism of data modification, job submission and results extraction. The main subject of this article is to present the reliability analysis capabilities of STAND software, which is being developed in the Institute of Fundamental Tech no logical Research of Polish Academy of Sciences. A special emphasis is put on the issues related to it s interfacing with external general purpose FE codes. It is shown that when shape type random variables are used, leading to modifications of the FE mesh, or when the limit state function contains numerical noise, standard algorithms for localizing the design point often fail to converge and a special method based on some response surface approximation is needed. A proposition of such a strategy that employs an adaptive response surface approximation of the limit state function is presented in this article. Development of a reliability analysis program is a challenging project and calls for such a code organization, which would facilitate a simultaneous work of many programmers and allow for easy maintenance and modifications. The so-called object-oriented programming seems to provide a convenient framework to realize these objectives. The object-oriented approach is used in STAND development. The advantages of this programming paradigm and a short description of the STAND’s class hierarchy are presented in the text. The study is concluded with two numerical examples of interfacing STAND with state of the art commercial FE programs.

Reliability, optimization software

Possibly the most common application of spot welding is in the automobile manufacturing industry, where it is almost universally used to weld the sheet-metal car components. However, due to manufacturing inaccuracies and fatigue failures an important number of spot welds may be missing in an operational vehicle. It seems that to properly analyse the reliability of such structures, in particular crashworthiness reliability, the spot weld failures must be considered. Representing properties of each spot weld in a stochastic model by corresponding random variables is extremely inefficient. Therefore, in this article an approach is proposed for handling spot-weld defects in the reliability analysis by accounting for their averaged influence on a failure criterion. The approach consists of the appropriate treatment of a random noise component of the limit state function. The noise results from the strategy of deleting a certain number of randomly selected spot-weld elements from the finite element model each time the limit state function value is computed. Dealing with noisy limit state functions in structural reliability analysis is a challenging task. The only method that seems to be insensitive to this phenomenon is Monte Carlo sampling, which for most of the applications of practical interest is prohibitively expensive. Having this in mind, a method based on the algorithm proposed by Zou et al. and published in the journal of Reliability Engineering and System Safety in 2002 is investigated in this article. The method combines the best features of the first-order reliability method, the response surface technique and the importance sampling method to achieve both accuracy and efficiency. A detailed study on the reliability of thin-walled s-rail subjected to crash is performed. Some suggestions concerning the modification of the original algorithm are proposed.

crashworthiness reliability, response surface approximation, adaptive importance sampling, spot weld failures

A crucial component in the statistical simulation of a computationally expensive model is a good design of experiments. In this paper we compare the efficiency of the columnwise–pairwise (CP) and genetic algorithms for the optimization of Latin hypercubes (LH) for the purpose of sampling in statistical investigations. The performed experiments indicate, among other results, that CP methods are most efficient for small and medium size LH, while an adopted genetic algorithm performs better for large LH. Two optimality criteria suggested in the literature are evaluated with respect to statistical properties and efficiency. The obtained results lead us to favor a criterion based on the physical analogy of minimization of forces between charged particles suggested in Audze and Eglais (1977. Problems Dyn. Strength 35, 104–107) over a ‘maximin distance’ criterion from Johnson et al. (1990. J. Statist. Plann. Inference 26, 131–148).

Design of experiment, Latin hypercube, Genetic algorithm

This paper describes the M-Xplore extension of the Radioss software. The module contains facilities for the exploration of a parameterized finite element model design space. It supports facilities for interactive choice of variables and responses, definition of a sampling on a design space, automatic submission of the computations, and post-processing of the results. The computations are run automatically, either locally or in ASP-mode, i.e. as a client of a high-performance computing server. The software is described first in general, then we illustrate its exploration possibilities in terms of a model problem and a more typical application problem of crash simulation.

FE Crash simulation, Latin hypercube sampling, Stochastic analysis software

The commonly observed nowadays tendency to weight minimization of rotor-shafts of the rotating machinery leads to a decrease of shaft bending rigidity making a risk of dangerous stress concentrations and rubbing effects more probable. Thus, a determination of the optimal balance between reducing the rotor-shaft weight and assuring its admissible bending flexibility is a major goal of this study. The random nature of residual unbalances of the rotor-shaft as well as randomness of journal bearing stiffness have been taken into account in the framework of robust design optimization. Such a formulation of the optimization problem leads to the optimal design that combines an acceptable structural weight with the robustness with respect to uncertainties of residual unbalances, the main source of bending vibrations causing the rubbing effects. The applied robust optimization technique is based on using Latin hypercubes in scatter analysis of the vibration response. The so-called optimal Latin hypercubes are used as experimental plans for building kriging approximations of the objective and constraint functions. The proposed method has been applied for the optimization of the typical single-span rotor-shaft of the 8-stage centrifugal compressor.

Rotor-shaft system, robust design optimization, lateral vibrations, rubbing effects, random unbalance distribution

This paper presents NUMPRESS System that has been developed in IPPT PAN as a result of a project financially supported by European Regional Development Fund (within the Innovative Economy Programme) and is dedicated to small and middle enterprises dealing with sheet metal forming. It seems undoubted that efficient design of an industrial sheet forming process requires reliable computer simulations and a tool for numerical optimization of the process parameters. It has to be also admitted that. among small and medium enterprises (SME) in this industrial branch, there are many who do not use any such numerical tools in their practice.
Computer simulation of sheet metal forming processes is a very specific branch of computational mechanics. Finite element systems dedicated strictly to this kind of processes are needed and actually present on the market. Commercial systems (like Autoform, PAM-Stamp, Stampack, etc.) are, due to their prices, usually beyond financial ability of SME.

Design of the drawing process and tools, i.e. choice of proper values of several design parameters, require efficient optimization strategy. In this process, random character of at least some of the parameters has to be taken into account. In view of this fact, the traditional, deterministic approach to optimization is insufficient and elements of robust design optimization techniques and reliability analysis have to be included in the formulation of the optimization problem. It has to be admitted that, even if some of the mentioned commercial simulation systems offer numerical optimization modules, not all of them reach beyond the deterministic concept of the optimization process.

sheet metal forming, finite element method, deterministic and robust design optimization, reliability analysis

This paper presents basic features of the NUMPRESS system and some examples ofuse. The system has been developed at IPPT PAN as a result of a project financially supported by European Regional Development Fund and is dedicated to small and middle enterprises (SME) dealing with sheet metal forming. The program consists of (i) an analytical finite element method module (ii) an optimization module, (iii) a reliability analysis module, and (iv) a graphical user interface enabling communication between modules. The analytical module consists of two independent programs up to the user’s choice: NUMPRESS-Flow, a faster and less accurate program for implicit quasi-static analysis of rigid-viscoplastic shells (based on the flow approach) and NUMPRESS-Explicit, a program for explicit dynamical analysis of elastic-plastic shells. Both programs are interfaced to a well-known commercial graphical pre-and postprocessor GiD.

sheet metal forming, finite element method, deterministic and robust design optimization, reliability analysis

The failure probability estimation of FEM simulated sheet metal forming process is a computationally challenging task. The application of efficient gradient-based reliability techniques is very much limited due to the numerical noise introduced by the explicit dynamic algorithm used to perform the sheet stamping analysis and by the nonlinearity of the failure function. To cope with this difficulty, in the current study a two stage metamodel-based adaptive importance sampling method is employed. In order to assess the reliability of sheet metal forming operations the stochastic character of such parameters as friction, blankholding force, blank thickness, strain hardening parameters of the constitutive law as well as parameters defining the forming limit curve (FLC) are considered. Using the numerical example of a square cup deep drawing, the benchmark problem of the Numisheet’93 conference, it is investigated how the assumptions concerning the probabilistic distribution of the FLC location parameter affect the probability of sheet metal fracture.

Reliability, Metal forming

The main objective of the presented study is an evaluation of the effectiveness of various methods for estimating statistics of rotor-shaft vibration responses. The computational effectiveness as well as the accuracy of statistical moment estimation are essential for efficient robust design optimization of the rotor-shaft systems. The most important sources of the observed response scatter are inherently random rotor-shaft residual unbalances as well as stiffness and damping properties of the journal bearings. A relevant representation of these parameters leads to multidimensional stochastic models. The compared stochastic moment estimation methods include sampling techniques, the dimension reduction method and the polynomial chaos expansion method. Two problems of the rotor-shaft vibration analysis are considered: a typical single-span rotor-shaft of the 8-stage centrifugal compressor driven by the electric motor and a large multi-bearing rotor-shaft system of the steam turbogenerator. It is shown that methods that provide a satisfactory balance between the estimation accuracy and computational effectiveness are sampling techniques. Methods employing polynomial chaos expansion perform well in the case of reduced stochastic models. On the other hand, low accuracy of the methods based on Taylor series expansion very often renders these techniques unsuitable for the robust design optimization of vibrating rotor shafts.

ibrations, robustness, numerical analysis, stochastic phenomena