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Hans Irschik Alexander Belyaev - Dynamics and Control of Advanced Structures and Machines

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Book:
Dynamics and Control of Advanced Structures and Machines
Author:
Hans Irschik Alexander Belyaev / Michael Krommer
Publisher:
Springer International Publishing
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Year:
2017
City:
Cham;Switzerland
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Free and forced vibrations of fuzzy structures -- Resonance of a series of train cars traveling over multi-span continuous -- Stability and supercritical deformation of a circular ring with intrinsic curvature -- Estimation of mechanical properties of micro-lattice panel with irregular cells -- Overview reading and comparing the seismic proof capability of displacement dependent semi-active hydraulic damper and accumulated semi-active hydraulic damper -- On necessary and sufficient conditions for eigenstrain-type control of stresses in the dynamics of force-loaded elastic bodies -- Variational principles for different representations of Lagrangian and Hamiltonian systems -- Hardware-in-the-loop testing of high-speed pantographs using real-time catenary emulation -- Swelling-induced bending of hydrogel bistrips -- Determination of parameters of the external electric circuits providing maximum damping of vibrations of electroelastic bodies -- Forming of woven-reinforced thermoplastic-matrix composites: characterization, modeling, and validation -- Model predictive temperature control of a distribution system for chemicals -- Hidden oscillations in electromechanical systems -- Effect of material layers in a compound circular receiver model design for concentrating solar power -- On multiple support excitation analysis of bridges -- Control of friction by surface microgeometry variation -- Peculiarities of the magnetic behavior of pipe steels with different initial stress-strain states under elastic deformation -- Cable-stayed bridges: a monitoring challenge -- Dynamics and control of motion for systems containing internal moving masses -- Elaborations from the TKB monitoring database -- Reduced order models and localized non-linearity: an approach to the design of meta-structures -- Contact of flexible elastic belt with two pulleys -- Control over internet of oscillations for group of pendulums -- Effect of the load modelling strategy on the dynamic response prediction of bridges subjected to high-speed trains -- Mechanical properties of epoxy resins filled with nano-silica particles.;The papers in this volume present and discuss the frontiers in the mechanics of controlled machines and structures. They are based on papers presented at the International Workshop on Advanced Dynamics and Model Based Control of Structures and Machines held in Vienna in September 2015. The workshop continues a series of international workshops held in Linz (2008) and St. Petersburg (2010).

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Springer International Publishing Switzerland 2017

Hans Irschik , Alexander Belyaev and Michael Krommer (eds.) Dynamics and Control of Advanced Structures and Machines 10.1007/978-3-319-43080-5_1

Free and Forced Vibrations of Fuzzy Structures

Franz Ziegler 1

(1)

Institute for Building Construction and Technology, Vienna University of Technology, Wien, Austria

Abstract

Stochastic free and forced vibrations of layered beams are analyzed that result from a single bounded random stiffness parameter whose probability density function is considered to be zero outside of a given interval, i.e., it is a member of a fuzzy set with bounded uncertainty. The relevant properties of natural vibrations of an ensemble of sandwich beams with three perfectly bonded layers under hard hinged support conditions are worked out in detail when a bounded random shear stiffness of the core material is assigned by employing interval mathematics. The main structure of a compound single-span railway bridge, effectively modeled as a two-layer beam, is subjected to a single moving load as well as to a series of repetitive moving loads traveling with constant speed. It serves as a complex example for the resulting forced random vibrations and resonances under the severe condition of an elastic interface slip of bounded random stiffness. In both cases exact homogenization yields a stochastic sixth-order partial differential equation of motion of the layered beam. Light modal damping is considered. The analysis of the illustrative problems is based on the interval representation with a triangular membership function of the stiffness modulus assigned. A short comment provides information on the limits of such triangular membership functions. Membership functions in the form of envelopes of the random natural frequencies, the dynamic magnification factors, and the phase angles in free vibrations are determined. Both, fuzzy peak deflection and acceleration are derived for the forced single-span compound railway bridge subjected to the moving loads. Approximating superposition of modal maxima is considered by standard routines of reliability analysis.

Introduction

In general, the application of the fuzzy finite element method, see, e.g., [].

The analysis of both illustrative examples is based on the interval representation (interval of confidence at a given level of presumption, i.e., -cut) with a triangular fuzzy membership function of the relevant random stiffness prescribed. Fuzzy membership functions of the response are defined using fuzzy set theory [b may even include a worst-case scenario, i.e., min k 2(=0)=0.

Fig. 1

Single bounded random structural parameter, e.g., stiffness. ( a ) Defined interval contains sure design value k 0. Outside of the interval, probability is zero. ( b ) Fuzzy set: interval of confidence at a given level of presumption -cut. Non-dimensional stiffness k 2() is referred to its design value. Isosceles distribution assigned

Effects on Free Vibrations: Example: Fuzzy Sandwich Beam

For linear elastic, symmetrically arranged three-layer beams, Fig. ], an exact homogenization renders the homogeneous partial differential equation (PDE) of sixth order in terms of the common deflection and for the case of free vibrations,

1 see again Fig with the bounded random parameter related to the core - photo 2

(1)

see again Fig., with the bounded random parameter related to the core shear modulus 22 G 2() with its assigned design value at 1 within the level of presumption 01 2 Fig 2 Dimensions of - photo 3

at =1 within the level of presumption 01,

2 Fig 2 Dimensions of the sandwich beam Common cross-sectional - photo 4

(2)

Fig. 2

Dimensions of the sandwich beam. Common cross-sectional rotation after homogenization , as shown, is crucially eliminated. Fuzzy core shear stiffness in layer 2 is considered. Deflection w is referred to the elastic centroid O

For the single-span beam, length l , under hard hinged support conditions, the ortho-normalized mode shapes result with a stochastic normalization factor,

3 The random natural frequencies referred to the sure values at rigid core - photo 6

(3)

The random natural frequencies, referred to the sure values at rigid core shear stiffness, n 2=1 n 4 B , are

4 With light viscous modal damping n 1 understood the random dynamic - photo 7

(4)

With light viscous modal damping n =1 understood, the random dynamic magnification factor n () and the stochastic phase angle n () are expressed by textbook formulas; see, e.g., [],

5 Numerical results for bounds and even for envelope functions are thus - photo 8

(5)

Numerical results for bounds and even for envelope functions are thus straightforwardly derived, putting, e.g., the design values B 0 B =0.1<14 and 2, n =0.25 in Eqs.( ].

Fig. 3

Bounds of the first five natural frequencies when referred to the sure fundamental frequency: worst-case scenario. Note the increased fuzziness of the higher modes []

Fig. 4

Envelopes of the first three random DMFs []. =0-cut; worst-case scenario =0

Fig. 5

Envelopes of the first three random phase angles [

2.1 Effects of Non-symmetric Uncertainty

In the core shear stiffness, Fig. ].

Fig. 6

Model of the main structure of the fuzzy compound bridge []: effective elastic two-layer beam with fuzzy elastic interlayer slip. Layer one refers to the RC-concrete deck. Layer two represents the rigidity of two steel girders. Bounded random shear flow T = k () u . Stiffness center S

Effects on Forced Vibrations: Example: Fuzzy Compound Railway Bridge

The practically very important case of a single-span compound bridge consisting, e.g., of two steel girders connected (elastically) to the concrete deck, when smeared, refers to the model of an asymmetric two-layer elastic beam exhibiting the main defect of elastic interlayer slip with the major uncertainty of its elastic modulus, Fig.), where the random parameters are adapted to the fuzzy elastic modulus of the interlayer slip k () with the sure design value k 0= k (=1). The parameters of the two elastic layers j =1,2, are: elastic modulus E j , effective cross-sectional area A j , and flexural rigidity B j = D j r j 2 with D j = E j A j ,

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