Research Article: Study on optimization algorithm of tuned mass damper parameters to reduce vehicle-bridge coupled vibration

Date Published: April 23, 2019

Publisher: Public Library of Science

Author(s): Jianwei Liu, Dejian Li, Peng Yu, Fang-Bao Tian.


A vehicle-bridge tuned mass damper (TMD) coupled dynamic analysis and vibration-control model was established to optimize TMD damping effects on a steel-box girder bridge bearing vehicle loads. It was also used to investigate optimization efficiency of different algorithms in TMD design parameters. This model simulated bridges and vehicles with the use of a 7 degrees of freedom curved-beam element model and a 7 degrees of freedom vehicle model, respectively. The TMD system was simulated with the use of multiple rigid-body systems linked with springs and dampers. Road surface condition, as a vibration source, was simulated with the use of a frequency equivalent method based on a power spectrum. A variably-accelerated pattern search algorithm was proposed in line with the initial TMD parameters calculated by Den Hartog formula. Visual software was compiled by Fortran and used for an optimization study of vibration reduction. A three-span, curved, continuous steel-box girder bridge was used as the numerical example. Optimized effects and computational efficiency of vibration reduction under different methods were compared. The comparison included a single variable optimization based on Den Hartog formula, an ergodic search method, an integer programming method, a traditional genetic algorithm, a traditional pattern search algorithm, and a variably-accelerated pattern search algorithm. The results indicate that variably-accelerated pattern search algorithm is more efficient at improving TMD optimal parameter design. Final TMD parameter optimization values obtained by different methods are quite close to each other and tends verify the reliability of the optimization results.

Partial Text

Vehicles moving on irregular bridge road surfaces produce coupling vibrations in the vehicle-bridge system. Large amplitude causes noise and driving discomfort and generates fatigue damage to the bridge structure. Presently steel-box girder bridges are widely used as urban overpasses as they can be rapidly constructed and do not need full construction support. Vibration and noise remain as drawbacks. Vibration control research on steel-box girder bridge dynamic responses under vehicle loads is of great theoretical significance and practical engineering value. Prior studies tended to analyze the coupled dynamic response of vehicle-bridge system [1–5]. Recent studies [6–8] suggested that studies of coupled-systems vibration control should be included and that system damping could be achieved through control techniques such as tuned mass damper (TMD). Kwon et al., in their study of TMD damping and optimization, analyzed TMD vibration control effects on a three-span continuous girder bridge simulated with six degrees of freedom (DOF) beam element [9]. Guo and Lu [10] concluded that a Den Hartog (DH) optimization formula [11] could be applied to vibration control using TMD on a simple-supported beam bridge for high-speed trains. Miguel et al. proposed a novel hybrid stochastic-deterministic algorithm for optimal multiple tuned mass dampers (MTMD) design under seismic excitation [12]. Tubino and Piccardo proposed a numerical optimization criterion based on an efficiency factor maximization defined as “the ratio between the uncontrolled acceleration standard deviation and the controlled one to optimize TMD parameters and mitigate human-induced vibrations of pedestrian bridges” [13]. Fan et al. solved MTMD optimal parameters under arbitrary distribution of damping ratio and frequency ratio using a genetic algorithm (GA) [14]. Li et al. established a power balance equation for a structure-TMD system and obtained a power consumption formula for the main structure when the structural substrate was subjected to harmonic load [15]. They obtained TMD optimal frequency ratios and damping ratios by establishing an optimal goal of minimizing main structure energy consumption and comparing it to the four analytical optimization methods proposed by Den Hartog [11], Warburton [16], Tsai and Lin [17], and Sadek et al. [18].

Steel structure low-frequency vibration takes displacement-based fatigue strength failure into account [25]. A TMD system design parameter optimization goal (mass ratio μ, frequency ratio α, and damping ratio ζ) is to find a suitable combination of parameters {μopt,αopt,ζopt} and to minimize the bridge displacement response Dz under the vehicle load:

This paper proposes a variable, accelerated pattern search method (VAPS) which uses improved initial values. A Den Hartog (DH) formula is used to determine better initial search values and accelerate the convergence speed of traditional pattern search algorithm (PS). A traditional PS algorithm step length, and acceleration and deceleration, principles have been modified to improve search efficiency. Overall optimization is strengthened by improving initial values and traditional pattern search algorithm search processes. Partial search capabilities are improved and realize an optimal parameter solution for a TMD system.

This section discusses different TMD parameter optimization methods, considering both displacement response damping effect and optimization calculations time costs. It compares and evaluates various methods to obtain the most suitable TMD damping optimization method for a three-span curve continuous steel-box girder bridge.

Comparing the operating conditions before, and after installing, the TMD system, the optimal 3-TMD displacement damping effects are calculated and shown in Fig 22. By analyzing time-displacement curves, it can be seen intuitively that displacement dynamic response after the TMD installation is weakened. Comparing the different stages of the displacement time history curves before and after TMD system installation, it can be seen that there are significant differences: 1) When the motorcade initially enters the bridge, the difference in displacement response between the presence and absence of the TMD system is very small, due to the TMD damping activation delay. In the first few vibration cycles, there is almost no vibration damping effect; 2) As the motorcade is on the bridge, i.e. the 8~16s part of the time history curve, there is a stable vibration phase, and bridge displacement response maximizes. TMD system has significant vibration damping effects; 3) After the motorcade exits the bridge, the bridge freely vibrates. After a TMD system installation, amplitude decreases and decay speeds up.

For the optimization problem of vehicle-bridge coupling damping for steel box-girder bridges, a dynamic analysis model of vehicle-bridge-TMD coupled system was established. A variably accelerated pattern search algorithm based on a Den Hartog formula was proposed. The VBTS-1 software with a visual interface, programmed in Fortran language, was used for an optimization study of vibration damping. A three-span curved continuous steel box girder bridge situated on Hongqi Road, Changsha was taken the example. A discussion on the parameter optimization of TMD system was performed. Their optimized effects under different methods were compared. The comparison included a single variable optimization method based on a Den Hartog formula, an ergodic search method, an integer programming method, a traditional genetic algorithm, a traditional pattern search algorithm, and a variably accelerated pattern search algorithm. Finally, the vibration reduction effects before and after the optimized TMD system installation were compared. The main conclusions are as follows: