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 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

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