The problem of reducing the level of vibrations in various constructions has been considered for many years. This is especially the case for classic bridge structures when an additional platform on which installations, for example, pipes with hazardous gas, are installed.
Numerous ways and means of preventing unacceptable vibrations are known, with one of these methods being use of various vibration absorbers. Their application has a special role in civil engineering due to the fact that they can be used during a construction's design, as well as later on to modify the structure. Using absorbers is an interesting option for reducing the vibrations of various types of structures, especially in tall buildings  or in bridges, for example, the Millennium Bridge  in London or bridges traversed by high-speed trains . Moreover, absorbers can be used to prevent large vibration amplitudes that occasionally appear, that is, in geotechnical works .
The topic of dynamic vibration absorbers (DVAs) or tuned mass dampers (TMD) is well established in literature. The action of the absorber is accepted as a deterministic [5,6,7,8,9] or random process [10,11,12,13,14,15,16,17,18,19,20]. A large number of studies have been conducted to optimise the design parameters, and hence maximise the performance of vibration absorbers. Although the design concept of a TMD was developed decades ago, its simplicity and effectiveness have made it one of the most popular passive vibration methods to suppress structural vibration .
In the earlier years of the 20th century, Frahm  introduced the use of a linear spring mass attachment to suppress the oscillations of harmonically excited primary structural systems in engineering applications. This early DVA was able to reduce the oscillations of primary structures with a single degree of freedom, but could only reduce vibration transmission in a specific narrow frequency band. A spring-supported mass invented by Frahm is a dynamic absorber without damping, known as a Frahm damper. Ormondroyd and Den Hartog  increased the effectiveness of the absorber to dissipate the energy of the primary structure to harmonic excitations by appending a viscous damper parallel to the linear spring. Later, a semi-empirical design procedure was established by Den Hartog . The DVA proposed by Den Hartog is now known as the Voigt-type DVA dynamic vibration absorber, where a spring element and a viscous element are connected. This is considered as the standard model of the DVA. The main objective in the design of the standard type DVA is to enable the absorber to have optimum parameters. Due to the fact that the mass ratio of the DVA to the primary structure is usually a few percent, the principal parameters of the DVA are its tuning ratio (i.e. the ratio of the DVA's frequency to the natural frequency of the primary structure) and damping ratio .
Ormondroyd and Den Hartog  developed analytical solutions for the optimal design of the classic DVA using the fixed-points technique. Den Hartog  provided an algorithm for selecting the optimal absorber parameters, so that the maximum amplitude of vibration is minimised over frequencies of deterministic sinusoidal excitations, which is essentially an H1 optimisation. Asami and Nishihara used H∞ and H2 optimisation methods based on the perturbation technique and Vieta's theory to derive analytical solutions for the optimal design of the classic DVA [25–26]. Sims  introduced a new analytical solution based on the criterion of minimising either the positive real part or the negative real part of the frequency response function. Shen et al.  proposed a new strategy for obtaining the optimum negative stiffness ratio and to also make the system remain stable.
Most works assume that the load process is deterministic and that it changes harmonically. The problem of the optimisation of absorbers connected with a bridge beam loaded by a moving force was considered in papers [29–30]. The vibrations of some structures are excited by a load of random nature. There are relatively few studies on absorbers in which the load is a stationary stochastic process [31,32,33,34]. In the mentioned works, stochastic vibrations were analysed in the field of correlation theory and spectral density analysis.
The earliest research works on the optimal design of a DVA were presented for a single degree of freedom (SDOF) of a primary structure [5, 22, 23]. In the vast majority of papers, the structure-absorber system is still treated as a system with two degrees of freedom (2DOF). The optimal design of an absorber is not a new topic, and the optimal methodologies and parameters of the optimal DVA or TMD system for SDOF main structures subjected to different loading conditions have been verified by several researchers since the 1960s .
The problem of choosing the optimal parameters for damping absorbers that reduce the random vibrations of a beam subjected to a sequence of moving forces with a constant velocity is studied in this paper. Every force is regarded as a random variable. Moreover, the inter-arrival times of moving forces are regarded as random variables. The stochastic properties of the load are modelled by means of a filtered Poisson process. The problem was solved with the idea of a dynamic influence function . Several optimisation criteria, based on the expected values and variance of the beam response determined in the study, were considered.
Let us consider the damping vibrations of a simply supported Euler–Bernoulli beam of finite length
The vibrations of the beam are described by the equation 
For a finite, simply supported beam, the boundary conditions have the following forms:
In the case of a simply supported beam, one can look for dynamic influence functions in the form of sine series
The equations of motion and the
The issue of optimising absorber parameters has been considered in many studies. The problem cannot always be solved in an analytical way, especially in the case of stochastic loads. Moreover, it is possible to adopt various optimisation criteria. The solution of the optimisation problem depends both on the speed of the moving force and the parameters of the beam.
Let us assume the following optimisation criteria:
Considering that the
The next criterion assumes the probability that the maximum vibrations in the period (0,
The model of the structure refers to a road bridge with a suspended platform equipped with municipal technical infrastructure. It is a reinforced concrete beam bridge with a span of
Figures 2 and 3 show that the deflection of the structure is in three places (
As shown in the figure above, the beam deflection in one-fourth of its length on the left and right sides is almost identical after the travel time has elapsed. Slight changes are visible at the beginning of the observation. A similar situation can be seen for the structure without an absorber (see Figure 3). It is obvious that deflection in the middle is greater than at one-fourth of the beam's length.
Various tuned absorbers were analysed. The influence of
Figure 5 shows three deflections in the middle of the beam: the blue line is for a structure without an absorber, the orange line is for an absorber with parameters
The graphs below show the analysis of the absorber with the parameters
A wide range of analysis of the influence of the absorber mass was adopted for
As shown in Figure 9a and b, the variance of these displacements also depends on velocity, but in a different way than is expected. The biggest variance of deflection is when the velocity of the passage is within the range 0.6–0.8
The subject of the research was also the location of the absorber. Is it really the best solution to install the DVA in the middle of the beam? The authors obtained an answer that the optimal operation of the absorber is when it is installed in the range of 0.3–0.6 of the beam's span. Figure 10 shows the dependence between both the expected value and the variance of the midspan deflection and the absorber's position. The results are presented for different velocities of motor vehicles.
According to Warburton , it is well-known that under the same optimisation procedure, the performance of the TMD system is better with an increase of the total mass of the attached TMD. This study analysed the range of the absorber mass from 0 to as much as 30% of a structure's mass. Years ago, it was accepted that the total mass of the attached absorbers should not exceed 10% of the mass of the main system.
After analysing the influence of the absorber's
In the case of the analysed reinforced concrete beam bridge with a span of 30 m, on which a platform, along with the technical infrastructure, was installed (e.g. hazardous gas), the optimal absorber has parameters in the following ranges: mass
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