3. Natural frequencies and modes of bridge vibration

Calculation of the natural frequencies and corresponding modes of vibration forms a basis for the determination of the dynamic characteristics of bridge vibration. At present, bridges, especially of larger spans, are complicated space structural systems, comprising many elements which interact with one another, e.g., continuous beams, framed structures, arch construction, suspended structures, and others. Principally, three different simulation models of the bridge can be used: the discrete model, the model with continuously distributed mass and models formulated by the FEM. Advanced numerical methods FEM have been widely developed for practical application in this field, and they are described in the technical literature and available as computer software. Therefore, only two calculation methods are shortly described in this section. The theoretical determination of the natural frequencies and modes of vibration of such structures is fairly difficult in most cases and their verification is advised by experimental measurements.

#### 3.1. Multi-degree of freedom systems

The equations of motion of a multi-degree of freedom system take the form

$$\mathbf{M}\ddot{\mathbf{u}} + \mathbf{C}\dot{\mathbf{u}} + \mathbf{K}\mathbf{u} = \mathbf{f} \tag{34}$$

where f is a column matrix of applied forces and u is a column matrix of displacement components. Both f and u correspond to the same set of points on the structure and the same directions at these points. M, C, and K are the inertia, damping, and stiffness matrices corresponding to the displacement components u. If the applied forces and damping forces are absent, Eq. (34) becomes

$$\left[\mathbf{K} - \omega^2 \mathbf{M}\right] \{\boldsymbol{\Phi}\} = 0\tag{35}$$

if the motion is assumed to be harmonic, that is, u = φ exp{iωt}. This equation is a linear eigen problem similar to n-DOF system. The eigenvalues, ω<sup>i</sup> 2 , represent the squares of the natural frequencies and the eigenvectors, ϕi, represent the shapes of the corresponding modes of free vibration. An eigenvector is arbitrary to the extent that a scalar multiple of it is also a solution of Eq. (35). It is convenient to choose this multiplier in such a way that ϕ has some desirable property. Such eigen vectors are called normalized eigenvectors.

#### 3.2. Numerical procedure application in bridge structure dynamic analysis

3. The vehicle vibration at the moment the vehicle enters the bridge, since the vehicle's energy of vibration is the primary source of the dynamic bridge response. The vertical amplitude of the vehicle vibration is decisive. The initial angular amplitude of the vehicle's

4. The character of the road irregularities (joints, potholes, inserted hinges, frozen snow, etc.). The effects of the damping of the vehicles and bridges, as well as the ratio of the sprung vehicle mass to the bridge mass, are not significant for long-span bridges. However, they are a significant influence on short-span bridges vibration. It was confirmed by the theoretical analysis and the experimental tests that the curve expressing the dependence of the dynamic coefficients on the vehicle speed is not a smooth curve but has many local projections and branching points [26].

Calculation of the natural frequencies and corresponding modes of vibration forms a basis for the determination of the dynamic characteristics of bridge vibration. At present, bridges, especially of larger spans, are complicated space structural systems, comprising many elements which interact with one another, e.g., continuous beams, framed structures, arch construction, suspended structures, and others. Principally, three different simulation models of the bridge can be used: the discrete model, the model with continuously distributed mass and models formulated by the FEM. Advanced numerical methods FEM have been widely developed for practical application in this field, and they are described in the technical literature and available as computer software. Therefore, only two calculation methods are shortly described in this section. The theoretical determination of the natural frequencies and modes of vibration of such structures is fairly difficult in most cases and their verification is advised by experimental

where f is a column matrix of applied forces and u is a column matrix of displacement components. Both f and u correspond to the same set of points on the structure and the same directions at these points. M, C, and K are the inertia, damping, and stiffness matrices corresponding to the displacement components u. If the applied forces and damping forces

if the motion is assumed to be harmonic, that is, u = φ exp{iωt}. This equation is a linear eigen

frequencies and the eigenvectors, ϕi, represent the shapes of the corresponding modes of free

2

ð34Þ

ð35Þ

, represent the squares of the natural

sprung mass vibration can be neglected in the analysis.

3. Natural frequencies and modes of bridge vibration

The equations of motion of a multi-degree of freedom system take the form

measurements.

124 Bridge Engineering

3.1. Multi-degree of freedom systems

are absent, Eq. (34) becomes

problem similar to n-DOF system. The eigenvalues, ω<sup>i</sup>

To avoid creating complicated and sophisticated numerical models involving extensive assumptions in modeling (boundary and initial conditions, mechanisms of bridge flexibility and energy dissipation, inertia, etc.), it is useful to develop an appropriate model with realistic prediction of their dynamic response upon the comparison of the experimental results and theoretical predictions. This enables also the realistic and optimal economical designs. Nowadays, very popular and useful numerical method for engineering analysis is finite element method. FEM is a numerical procedure for obtaining solutions to many of the problems encountered in civil and structure engineering. Numerical solutions of the bridge dynamic analysis problems in many cases need experimental verification in situ, e.g., [20, 27, 29, 30, 31].

To create relevant analytical models with real dynamic bridge structure with input parameters, it is useful to apply experimental modal analysis (EMA) which provides mainly structure natural modes engine frequencies and damping parameters of the tested bridge structure [31]. For such type of bridge dynamic tests performance in most cases, the real bridge service conditions are too restrictive for performance such type bridge tests. In these cases, operational modal analysis (OMA) procedure is applicable, which enables to perform bridge dynamic testing and also bridge health monitoring measurements without interrupting bridge service. A wellpresented review of bridge testing methods explaining their conditions, advantages and limitations was presented by Salawu and Williams [27].

The bridge dynamic analysis programs are commonly available and computational problems are not complicated to solve. A lot of FEM software packages are used in this field mainly for structures modal analysis and dynamic response of bridges (ANSYS Civil FEM Bridge, BRASS, BRIDGES, BridgeSoft, BRIDGADES (ABAQUS), ADINA, DYNSOLV, LUSAS, etc.).
