Estimation of structural stiffness with the use of Particle Swarm Optimization

Abstract

The paper presents the theoretical background and four applications examples of the new method for the estimation of support stiffness coefficients of complex structures modelled discretely (e.g. with the use of the Finite Element Model (FEM) method based on the modified Particle Swarm Optimization (PSO) algorithm. In real-life cases, exact values of the supports' stiffness coefficients may change for various reasons (e.g. order of fastening, state of the contact surfaces, environment changes, etc.). Because of the unknown coefficients, reliable simulations of fixed structure (i.e. mounted, assembled, not in a free-free state) are difficult to perform. The method serves as a tool to obtain good correlation between the FEM of a structure and the experimental data. Simple modal tests are required to estimate the first few modes of the fixed system. The FEM of the structure is considered in a free-free state and the support stiffness coefficients of the FEM are estimated by the proposed method. Further simulations with test-tuned and correlated complete FEM could be performed in time or frequency domain.

Adjusting the Stiffness of Supports during Milling of a Large-Size Workpiece Using the Salp Swarm Algorithm

“…In order to minimize the vibration level, an appropriate set of support settings must be determined. The general procedure for finding these settings is: Identification of stiffness values for different settings for each of the adjustable stiffness supports, for example, by performing static tests at the material testing machine; Preparing the modal model of the workpiece itself and assessing its compliance with the actual object using modal parameters identification methods (for example, ERA—Eigenvalue Realization Algorithm or p-LSCFD—poly-reference-Least Square Complex Frequency Domain methods [ 31 , 56 , 58 ]); Selecting cutting process parameters, i.e., depth of cutting, feed speed, spindle speed; Performing a series of simulations of milling process for given sets of support stiffness settings; Assessing the simulation results by comparing a chosen process quality indicator, for example, average tool–workpiece displacement or Root Mean Square (RMS) of the displacements in the time domain; Choosing the set of support settings that assure the best milling conditions. …”

“…Preparing the modal model of the workpiece itself and assessing its compliance with the actual object using modal parameters identification methods (for example, ERA—Eigenvalue Realization Algorithm or p-LSCFD—poly-reference-Least Square Complex Frequency Domain methods [ 31 , 56 , 58 ]);…”

“…Although one agent is unable to solve the problem alone, owing to the mutual interactions with other agents and with the environment (problem being solved), the whole swarm is able to “intelligently” find the solution. An example of successful application of the swarm intelligence method for Finite Element Model (FEM) updating, which is based on the Particle Swarm Optimization algorithm used for (non-adjustable) support stiffness coefficient estimation, is presented in [ 31 ].…”

“…From the point of view of vibration level during machining, the dynamic properties of the workpiece together with its supporting elements are essential [ 31 , 32 , 33 ] because they determine the workpiece’s susceptibility to vibrations. For example, properties such as resonant frequencies (if excited, chatter phenomena may easily occur) and damping coefficients (high damping prevents vibrations) are important.…”

An Improved Method of Minimizing Tool Vibration during Boring Holes in Large-Size Structures

“…Identification of the parameters of the modal model of the boring tool was performed using the ERA ( Eigenvalue Realisation Algorithm ) [ 4 , 32 , 42 ] and the p-LSCFD ( polyreference—Least Squares Complex Frequency Domain ) [ 32 , 43 , 44 ] methods. Measurements were made by uniaxial accelerometers that were mounted to measure vibration for three orthogonal directions at 5 different points ( Figure 5 ).…”