96 M. Sheibani et al. enough to produce frequency shifts desired for the method, and low enough to avoid significant changes of mode shapes [8]. Mass-stiffness change method was also proposed to change the first modes of vibration more significantly [9]. The growing accuracy of modal analysis testing equipment and precise methods which have made obtaining the modal parameters possible, as well as the difficulties associated with placing heavy masses on the structure when using the conventional mass change methods, has revealed the need for more convenient and efficient methods. In this article a new method is proposed to exploit the traffic load as the required change in the mass of the structures. The heavy traffic jams or congested traffic situations observed in big cities has appealing characteristics which makes it useful for utilization in the mass change method. Vehicle traffic induces a sufficiently uniform load on the bridges in the case of congested traffic and this condition can be assumed almost time independent for the required time period concerning OMA. Equally distributed mass change strategy is suggested to be the optimum approach in the literature and is provided by the traffic load. On the other hand, there are various urban bridges which are constructed using light-weight materials that promise sufficient frequency shifts desired for the mass change method provided by the traffic stream. Near-congestion traffic has been shown to be best represented by shifted lognormal distribution [10, 11]. To demonstrate, a bridge-like structure is simulated by a finite-element model of a simply-supported Euler-Bernoulli beam. The beam model has been subjected to artificial traffic generated from lognormal distribution and the proposed method has been evaluated. Mass change is applied to the mass matrix of the beam in each time-step to represent the actual case of traffic, and responses of the structure has been determined by the Newmark method. The car-bridge interaction is ignored since vehicles in congested traffic conditions have negligible dynamic interactions with bridges, and therefore a decoupled bridge-vehicle system is considered [12]. The response-only analysis method that is used in this study is the Natural Excitation Technique-Eigensystem Realization Algorithm (NExT-ERA) which has shown promising results [13] and can be exploited for repetitive tests needed in this study. Unscaled mode shapes are derived from response-only modal analysis and the corresponding scale factors are calculated based on the traffic induced mass modification. These scale factors are then compared with the scale factors obtained from finite-element model. 13.2 Theory 13.2.1 Output-Only Modal Identification The method which is used for modal analysis purposes is NExT-ERA. The prevailing assumptions made in this method are that the structure behaves within a linear range, the structure is time-invariant, and the forces applied to the system are uncorrelated to the response of the structure [15]. In this method, response of the structure based on ambient excitation is used for estimating the cross-correlation functions. These functions can represent the impulse response functions of the structure which are used in classical modal analysis. The direct procedure for obtaining the cross-correlation functions between two channels of acceleration i and j is used to this end. In discrete time approach, the correlation function (CF) matrix can be estimated using the formula [14] RRyi;Ryj .k t/ D 1 N k S k XsD1 Ryi.s/Ryj .s Ck/ (13.1) Inwhich S is the total number of data points of the acceleration record and t is the time step. The ERA method is based on the state-space representation of a discrete system. The procedure to obtaining modal parameters from ERA method starts with formation of the Hankel matrix using correlation functions which are calculated by Eq. (13.1). Singular value decomposition (SVD) is then performed and modal parameters are derived with the help of specific equations and guidelines which are described in [15, 16]. 13.2.2 Mass Scaling In order to obtain mass normalized mode shapes from OMA techniques, additional steps are required. The unscaled mode shape vector §i have to be scaled in the interest of modal analysis applications mentioned in the introduction. The scaled mode shapes can be obtained from unscaled modes using the following equation [8]
RkJQdWJsaXNoZXIy MTMzNzEzMQ==