Reduced Order Modeling of Bolted Joints in Frequency Domain

Most of the structural systems assembled by using bolted joints. Therefore, bolted joint models have a critical importance to estimate the behavior of the overall assembled system. There are several linear bolted joint models which consist of spring and dashpot elements in literature. While they can estimate the resonant frequency of the overall system with a sufficient accuracy, linear bolted joint models are inadequate for approximating the damping which arises from the friction in the contact interface of assembled system. On the other hand, there are examples of nonlinear bolted joint models which utilize 3D contact models to account for the frictional damping behavior in the literature. However, modeling the structures with many bolted joints by using high fidelity 3D contact models is very time consuming. Therefore, reduced order bolted joint models with sufficient accuracy are in need. In this paper, a method for modeling bolted joints in frequency domain is introduced. The joint model consists of microslip friction elements each one of which is constructed by several Coulomb friction elements in parallel and located at both sides of bolt holes

A Novel Computational Method to Calculate Nonlinear Normal Modes of Complex Structures

In this study, a simple and efficient computational approach to obtain nonlinear normal modes (NNMs) of nonlinear structures is presented. Describing function method (DFM) is used to capture the nonlinear internal forces under periodic motion. DFM has the advantage of expressing the nonlinear internal force as a nonlinear stiffness matrix multiplied by a displacement vector, where the off-diagonal terms of the nonlinear stiffness matrix can provide a comprehensive knowledge about the coupling between the modes. Nonlinear differential equations of motion are converted into a set of nonlinear algebraic equations using DFM under harmonic motion assumption. A matrix manipulation based on dynamic stiffness concept was used to localize nonlinearities and reduce the number of nonlinear equations improving the efficiency of the approach, which becomes important in solving large complex structures. The nonlinear algebraic equations are solved numerically by using Newton’s method with Arc-Length continuation. The efficiency of proposed computational approach is demonstrated using a two-degree-of-freedom nonlinear system. The proposed approach has the potential to be applied to large-scale engineering structures with multiple nonlinear elements and strong nonlinearities

Investigation of graphene oxide as highly selective adsorbent in recovery of hydroxytyrosol from olive mill wastewater

Olive mill wastewater (OMW) is produced in large quantities in the production of olive oil with the three-phase method. This waste, which is not biodegradable and characterized by a heavy organic load containing toxic components (such as phenols), is usually given to aqueous receptors or soil, either directly (untreated) or treated inadequately. Biologically active phenolic compounds that inhibit biodegradation of this waste are toxic at high concentrations, especially for microorganisms, but have positive effects with strong antioxidant properties on human health when properly isolated and properly concentrated.Hydroxytyrosol(HT) is one of the most interesting and abundant compounds among the biophenols present in OMW. According to this study, a newly synthesized nanomaterial, graphene oxide (GO), has been utilized for the separation of HT andtotal biophenolic substancefrom OMW. Graphene oxide from graphite oxidation was synthesized using a modified Hummer's method. In order to understand how GO behaves during thermal degradation, thermogravimetric/differential scanning calorimetry analyses were performed. Two-parameter and three-parameter equilibrium isotherm models and kinetic models have been used to determine the mechanism of kinetic sorption and adsorption type. Thermodynamic data indicate that the adsorption process is exothermic, applicable and spontaneous

An improved microslip model for variable normal loads

Detuning of gas turbine blades in order to avoid high cycle fatigue failure due to large resonant stresses is often unfeasible. A possible solution is to add an external source of damping, in the form of dry friction devices such as the under-platform damper. The relative movement between the blades causes possible slip between damper and blade surfaces. Due to the nonlinear nature of dry friction, dynamic analysis of structures constrained through frictional contacts is difficult, commercial finite element codes using time step integration are not suitable given the large computation times. For this reason, ad hoc numerical codes have been developed in the frequency domain. Some authors Yang and Menq (J Eng Gas Turbine Power 120:410–417, 1998) [1], Sanliturk et al. (J Eng Gas Turbine Power 123:919–929, 2001) [2], Csaba (Proceeding of ASME Gas turbine and aeroengine congress and exhibition) [3], Panning et al. (Int J Rotating Mach 9:219–228, 2003) [4] prefer a separate routine in order to compute contact forces as a function of input displacements, others Cigeroglu et al. (J Eng Gas Turbine Power 131:022505, 2009) [5], Firrone et al. (Modelling a friction damper: analysis of the experimental data and comparison with numerical results, 2006) [6], Firrone and Zucca (Numerical analysis—theory and application, 2011) [7] include the damper in the FE model of the bladed array. The available numerical models of dampers require a description of the contact conditions, both in the normal and in the tangential directions. The approach proposed here differs from those available in the literature in that the tangential force-displacement behaviour is described by arrays of springs in parallel, but, unlike pre-existing models, it introduces a variable sharing of normal force according to the approach along the normal. It thus modulates the tangential stick-slip capabilities according to normal force and approach and is capable to reproduce the analytical contact description as originally proposed by Cattaneo (Accademia dei Lincei 6:P I; 342–348, P II; 434–436, P III; 474–478, 1938) [8] and Mindlin and Deresiewicz (J Appl Mech 20:327–344, 1953) [9]. The paper shows how the model can be described and tuned in reference to the analytical Cattaneo and Mindlin’s benchmark for a spherical contact. It is proved that parameters tuned for a certain normal load will correctly simulate the tangential behaviour at any other lower normal load and finally that the transitions between cycles at different normal loads is correctly described. The paper further shows an application to a cylindrical contact where the tangential characteristics are derived from purposely taken experimental measurements

An improved microslip model for variable normal loads

Detuning of gas turbine blades in order to avoid high cycle fatigue failure due to large resonant stresses is often unfeasible. A possible solution is to add an external source of damping, in the form of dry friction devices such as the under-platform damper. The relative movement between the blades causes possible slip between damper and blade surfaces. Due to the nonlinear nature of dry friction, dynamic analysis of structures constrained through frictional contacts is difficult, commercial finite element codes using time step integration are not suitable given the large computation times. For this reason, ad hoc numerical codes have been developed in the frequency domain. Some authors Yang and Menq (J Eng Gas Turbine Power 120:410–417, 1998) [1], Sanliturk et al. (J Eng Gas Turbine Power 123:919–929, 2001) [2], Csaba (Proceeding of ASME Gas turbine and aeroengine congress and exhibition) [3], Panning et al. (Int J Rotating Mach 9:219–228, 2003) [4] prefer a separate routine in order to compute contact forces as a function of input displacements, others Cigeroglu et al. (J Eng Gas Turbine Power 131:022505, 2009) [5], Firrone et al. (Modelling a friction damper: analysis of the experimental data and comparison with numerical results, 2006) [6], Firrone and Zucca (Numerical analysis—theory and application, 2011) [7] include the damper in the FE model of the bladed array. The available numerical models of dampers require a description of the contact conditions, both in the normal and in the tangential directions. The approach proposed here differs from those available in the literature in that the tangential force-displacement behaviour is described by arrays of springs in parallel, but, unlike pre-existing models, it introduces a variable sharing of normal force according to the approach along the normal. It thus modulates the tangential stick-slip capabilities according to normal force and approach and is capable to reproduce the analytical contact description as originally proposed by Cattaneo (Accademia dei Lincei 6:P I; 342–348, P II; 434–436, P III; 474–478, 1938) [8] and Mindlin and Deresiewicz (J Appl Mech 20:327–344, 1953) [9]. The paper shows how the model can be described and tuned in reference to the analytical Cattaneo and Mindlin’s benchmark for a spherical contact. It is proved that parameters tuned for a certain normal load will correctly simulate the tangential behaviour at any other lower normal load and finally that the transitions between cycles at different normal loads is correctly described. The paper further shows an application to a cylindrical contact where the tangential characteristics are derived from purposely taken experimental measurements

Nonlinear time-varying dynamic analysis of a spiral bevel geared system

In this paper, a nonlinear time-varying dynamic model of a drivetrain composed of a spiral bevel gear pair, shafts and bearings is developed. Gear shafts are modeled by utilizing Timoshenko beam finite elements, and the mesh model of a spiral bevel gear pair is used to couple them. The dynamic model includes the flexibilities of shaft bearings as well. Gear backlash and time variation of mesh stiffness are incorporated into the dynamic model. Clearance nonlinearity of bearings is assumed to be negligible, which is valid for preloaded rolling element bearings. Furthermore, stiffness fluctuations of bearings are disregarded. Multi-term harmonic balance method (HBM) is applied on the system of nonlinear differential equations in order to obtain a system of nonlinear algebraic equations. Utilizing receptance method, system of nonlinear algebraic equations is grouped in nonlinear and linear sets of algebraic equations where the nonlinear set can be solved alone decreasing the number of equations to be solved significantly. This reduces the computational effort drastically which makes it possible to use finite element models for gear shafts. In the calculation of Fourier coefficients, continuous-time Fourier transform as opposed to the gear dynamics studies that utilize discrete Fourier Transform is used. Thus, convergence problems that arise when the number of nonlinear DOFs is large are avoided. Moreover, analytical integration is employed for the calculation of Fourier coefficients rather than numerical integration in order to further reduce the computational time required. Nonlinear algebraic equations obtained are solved by utilizing Newton's method with arc-length continuation. Direct numerical integration is employed to verify the solutions obtained by HBM. Several case studies are carried out, and the influence of backlash amount, fluctuation of gear mesh stiffness and variation of bearing stiffness are investigated. In addition to these, the response of the coupled gear system model is compared with that of gear torsional model in order to study the influence of the coupling on dynamics of the system

A high-accuracy model reduction for analysis of nonlinear vibrations in structures with contact interfaces

A highly accurate and computationally efficient method is proposed for reduced modeling of jointed structures in the frequency domain analysis of nonlinear steady-state forced response. The method has significant advantages comparing with the popular variety of mode synthesis methods or forced response matrix methods and can be easily implemented in the nonlinear forced response analysis using standard finite element codes. The superior qualities of the new method are demonstrated on a set of major problems of nonlinear forced response analysis of bladed disks with contact interfaces: (i) at blade roots, (ii) between interlock shrouds, and (iii) at underplatform dampers. The numerical properties of the method are thoroughly studied on a number of special test cases

Vibration reduction by using two tuned mass dampers with dry friction damping

Vibration reduction of a single-degree-of-freedom system connected to two tuned mass dampers (TMDs) equipped with dry friction dampers is considered in this work. The system is subjected to sinusoidal base excitation. Parameters of TMDs are optimized to minimize the peak values of the response of the primary system. Harmonic balance method (HBM) is used to obtain the steady state solution of the three-degrees-of-freedom nonlinear system in frequency domain. Newton’s method with arc length continuation is utilized to solve the resulting nonlinear algebraic equation set. In addition to that, optimum linear system and other nonlinear elements are investigated. Genetic algorithm is used to optimize parameters of TMDs

Free vibrations of moderately thick truncated conical shells filled with quiescent fluid

A novel reduced order formulation is proposed for the vibration analysis of conical shells containing stationary fluid. Hamiltonian approach is followed to obtain the governing equations of motion for the structure. Utilizing the Navier-Stokes equations and simplifying for irrotational, compressible and inviscid assumptions, the final fluid equation is obtained. A general solution based on the Galerkin method is proposed for the conical shell in vacuum. Several boundary conditions are investigated to show the capability of the proposed solution. A novel reduced order formulation based on the finite element method is developed for solution of the fluid equation. Static condensation technique is also utilized to minimize the required number of degrees of freedom and speed up the solution. The main advantage of the current solution method is the use of minimal number of degrees of freedom yet giving highly accurate results. Effects of added mass, semi-vertex angle, boundary conditions and different fluid containments on the natural frequencies of the coupled-field problem are studied and some useful conclusions are drawn