1. Introduction

The dynamic characteristics of a structure depend, basically, on its stiffness and mass. With these two elements, the natural frequencies and modes of vibration of the system are determined. However, the initial stiffness of a structure can be affected by the so-called geometric stiffness, a function of the acting normal force. In the case of compression force, the stiffness of the structure decreases, also reducing the natural frequencies of vibration. A class of structures of socio-economic-strategic importance for the national industry are machine bases, which are subject to vibrations induced by the supported equipment. These vibrations can affect the safety of the structure itself and generate detrimental effects on the equipment and the quality of the manufactured product. They can also make the working ambience unsuitable for operators. All industrial sectors are subject to these problems, including oil exploration, production, and refining, mining, wind energy, atomic energy, as well as bridges and viaducts for road and rail use.

Typically, viscoelasticity representation is based on rheological models that are associated with deformations that are deferred over time. These models can be included in a static or dynamic analysis of the structures by relating them to the modulus of elasticity of the material. In the case of dynamic analysis, the stiffness of the structure must be composed of two terms, one of which corresponds to the portion of conventional stiffness and the other to the geometric stiffness parcel [5]. Thus, it is possible to introduce into the first one a modulus of elasticity that is variable over time, according to the rheological model adopted, keeping the stress level constant, and, in the second, to consider the normal stress acting on the system, which includes the self-weight of the structural element. An approximate and satisfactory solution can be

Analytical and Mathematical Analysis of the Vibration of Structural Systems Considering Geometric Stiffness…

http://dx.doi.org/10.5772/intechopen.75615

351

To evaluate these aspects, a numerical simulation has been performed, assuming an idealized section of a beam as an engine base. A rheological model of the three parameters has been used to obtain the variable modulus of elasticity. A model, including geometric stiffness, distributed, and concentrated masses, is derived based on the Rayleigh method and solved for a range of axial compression load values. The results made it allowed us to verify the resonant and non-resonant response of the system. A second analysis has been performed to simulate numerically the variation of the first natural frequency of vibration of an actual structure of reinforced concrete, axially loaded, and considering also the viscoelasticity by means of the

An increase in strain over time under constant stress is a viscoelastic phenomenon. Mathematically, viscoelasticity can be represented by a time-dependent function associated with rheological models capable of describing the phenomenon [6]. The slow deformation for concrete parts is a phenomenon that is related to loads and deformations but is partially reversible [7]. It is a phenomenon that is directly related to the movement of moisture inside concrete. When a sample of concrete is loaded for 90 days and then unloaded, the immediate or elastic recovery is approx-

It is conceptually convenient to consider classic viscoelastic models in which only two types of parameters, relating to elasticity and viscosity, appear [9]. Classic viscoelastic models are obtained by arranging springs and dampers, or dashpots, in different configurations. Springs are characterized by elastic moduli and dashpots by viscosity coefficients. The best known of these mechanical models are the Maxwell model, containing a spring in series with a dashpot, and the Kelvin-Voigt model, containing a spring and dashpot in parallel. One model used to represent the viscoelasticity of solids is the three-parameter model, in which the elastic parameter E0 is connected to the viscoelastic Kelvin-Voigt model with parameters E1 and η1, which is

The three-parameter model sufficiently describes the viscoelastic nature of many solids and is often used to study the phenomenon in various scientific fields. The total deformations of the

imately the same magnitude as the elastic deformation when the first load is applied [8].

a simplification of the Group I Burgers model, as shown in Figure 1.

same rheological model. The loss of stability of the system has been then evaluated.

2. Mathematical solution for representing the viscoelasticity

found by considering viscoelasticity through flexural bending over time.

Although equipment support structures are, as a general rule, over-dimensioned, and therefore not subject to the effects of geometric stiffness, the tendency of modern structural engineering is towards increasingly slender elements, made possible by materials that are more efficient and lightweight, and having more and more powerful structural analysis capabilities. One of these features is prestressed concrete, represented by the presence of a steel bar or cable inside the structure that compresses it, the purpose of which is to reduce the effects of tension on flexion. In the case of beams subjected to periodic excitation, it is assumed that the original design has taken care to distance the natural frequencies of the system from those of the excitation, considering that, by hypothesis, the prestressing force decreases the stiffness of the element and, consequently, its natural frequencies, which may lead to unexpected, potentially dangerous resonance regimes. In the opposite direction, the presence of the prestressing can provide a form of control of this same vibration, where a resource is available to remove the structure of the resonant regime, if perceived in the preliminary stages of design. In one way or another, a satisfactory analysis solution to most engineering problems comes from a consideration that is easily implemented in analytical and numerical-computational formulations: the geometric stiffness. The influence of geometric stiffness has been studied in several contexts, both in laboratory tests and in comparison with the finite element method (FEM) [1–4].

The problem is aggravated when the material itself changes its elastic properties, such as in the case of viscoelasticity, which represents the gradual increase of deformation with time. This is a typical phenomenon of concrete structures because it is a viscoelastic material. It must be considered when verifying the stability of slender pieces compressed under the ultimate limit state (ULS), since these have their stiffness modified in function of the rheology of the material itself. It is important to consider the viscoelastic behavior of concrete structures relative to the characteristics of the structural element under study; this is necessary when verifying the stability of compressed slender pieces, since their stiffness is modified according to the rheology of the material. For this reason, in the specific case of columns in that loading condition, a premature analysis can produce undesired consequences, and the system may even collapse.

Typically, viscoelasticity representation is based on rheological models that are associated with deformations that are deferred over time. These models can be included in a static or dynamic analysis of the structures by relating them to the modulus of elasticity of the material. In the case of dynamic analysis, the stiffness of the structure must be composed of two terms, one of which corresponds to the portion of conventional stiffness and the other to the geometric stiffness parcel [5]. Thus, it is possible to introduce into the first one a modulus of elasticity that is variable over time, according to the rheological model adopted, keeping the stress level constant, and, in the second, to consider the normal stress acting on the system, which includes the self-weight of the structural element. An approximate and satisfactory solution can be found by considering viscoelasticity through flexural bending over time.

1. Introduction

350 Numerical Simulations in Engineering and Science

road and rail use.

The dynamic characteristics of a structure depend, basically, on its stiffness and mass. With these two elements, the natural frequencies and modes of vibration of the system are determined. However, the initial stiffness of a structure can be affected by the so-called geometric stiffness, a function of the acting normal force. In the case of compression force, the stiffness of the structure decreases, also reducing the natural frequencies of vibration. A class of structures of socio-economic-strategic importance for the national industry are machine bases, which are subject to vibrations induced by the supported equipment. These vibrations can affect the safety of the structure itself and generate detrimental effects on the equipment and the quality of the manufactured product. They can also make the working ambience unsuitable for operators. All industrial sectors are subject to these problems, including oil exploration, production, and refining, mining, wind energy, atomic energy, as well as bridges and viaducts for

Although equipment support structures are, as a general rule, over-dimensioned, and therefore not subject to the effects of geometric stiffness, the tendency of modern structural engineering is towards increasingly slender elements, made possible by materials that are more efficient and lightweight, and having more and more powerful structural analysis capabilities. One of these features is prestressed concrete, represented by the presence of a steel bar or cable inside the structure that compresses it, the purpose of which is to reduce the effects of tension on flexion. In the case of beams subjected to periodic excitation, it is assumed that the original design has taken care to distance the natural frequencies of the system from those of the excitation, considering that, by hypothesis, the prestressing force decreases the stiffness of the element and, consequently, its natural frequencies, which may lead to unexpected, potentially dangerous resonance regimes. In the opposite direction, the presence of the prestressing can provide a form of control of this same vibration, where a resource is available to remove the structure of the resonant regime, if perceived in the preliminary stages of design. In one way or another, a satisfactory analysis solution to most engineering problems comes from a consideration that is easily implemented in analytical and numerical-computational formulations: the geometric stiffness. The influence of geometric stiffness has been studied in several contexts, both in laboratory tests and in comparison with the finite element method (FEM) [1–4].

The problem is aggravated when the material itself changes its elastic properties, such as in the case of viscoelasticity, which represents the gradual increase of deformation with time. This is a typical phenomenon of concrete structures because it is a viscoelastic material. It must be considered when verifying the stability of slender pieces compressed under the ultimate limit state (ULS), since these have their stiffness modified in function of the rheology of the material itself. It is important to consider the viscoelastic behavior of concrete structures relative to the characteristics of the structural element under study; this is necessary when verifying the stability of compressed slender pieces, since their stiffness is modified according to the rheology of the material. For this reason, in the specific case of columns in that loading condition, a premature analysis can produce undesired consequences, and the system may even collapse.

To evaluate these aspects, a numerical simulation has been performed, assuming an idealized section of a beam as an engine base. A rheological model of the three parameters has been used to obtain the variable modulus of elasticity. A model, including geometric stiffness, distributed, and concentrated masses, is derived based on the Rayleigh method and solved for a range of axial compression load values. The results made it allowed us to verify the resonant and non-resonant response of the system. A second analysis has been performed to simulate numerically the variation of the first natural frequency of vibration of an actual structure of reinforced concrete, axially loaded, and considering also the viscoelasticity by means of the same rheological model. The loss of stability of the system has been then evaluated.
