Optimisation of the Parameters of a Vibration Damper Installed on a Historic Bridge

In the case of structures such as historic bridges, structural reinforcement is either not possible or very difficult, as it may alter their historical appearance. Therefore, it is valuable to develop repair techniques that do not interfere with the external appearance of the structure. One of them can be the use of a device such as a dynamic absorber, which has been used for over a century [1, 2], especially in the case of dynamic loads. Numerous ways and means of pre-emptively preventing harmful vibrations are already known, and research on the parameters of optimal absorbers continues for both deterministic [1,2,3,4,5,6,7,8,9,10,11] and stochastic excitations [13,14,15,16,17,18,19,20,21,22,23]. Various types of vibration absorbers are used to reduce vibration levels in different engineering structures, such as bridges [3], tall buildings [4], chimneys [5], pipeline structures [6], traffic signal structures [7], wind turbines [12], thin-walled structures [13] etc. The absorbers play a special role because they can be used during the construction of a structure and also when unsatisfactory dynamic properties appear during its operation. Moreover, absorbers are very common devices for seismic protection [14].

Vibrations of structures can be excited due to a load of a random nature. In such cases, dynamic vibration absorbers (DVAs) or tuned mass dampers (TMDs) are used to reduce the response to random excitation. The parameters of absorbers should be optimised. Many parameters can be adopted as an optimization criterion, for example, the minimum vibration variance [15], the minimum of an excess factor [8], the minimum fatigue of the material [7] etc. The optimisation of absorbers is a very complicated problem, especially when considering the vibrations caused by a random load. Engineers can optimise the parameters of absorbers, their type and number, and also their location on a construction. In the case of stochastic oscillations, the issue of selecting the most appropriate optimisation criterion arises. Identifying the optimal parameters leads to the best design of vibration absorbers, which increases their efficacy in actual applications. Optimisation involves finding the optimal parameters of the absorber with respect to the parameters of the system, the optimization goal being to minimise a given objective function [16].

In general, maximization of the structure’s durability in the assumed time of its use was assumed as the optimisation criterion. Such a criterion can be assumed as 1) minimising the expected rate of exceeding the limit level of stresses or strains, or 2) maximising the lifetime of the structure with respect to its fatigue degradation, or crack propagation. A good criterion, which corresponds to the enlarging reliability of the structure, is the minimisation of the variance of the maximal displacements, strains or stresses. To obtain the best vibration suppression performance, especially for the main continuous structures with multiple dominant vibration modes or antinodes, it is necessary to establish an optimal procedure to evaluate the optimal values of the location, numbers, and parameters of the TMD system efficiently, and/or to verify the validity of directly installing the TMD at the close position of each antinode [9].

In this paper, the problem of the optimal selection of parameters for TMDs that reduce the vibration of a beam subjected to a random train of moving forces with constant velocity is studied. Every force is regarded as a random variable. In addition, the inter-arrival times of the moving forces are also regarded as random variables. The stochastic properties of the load are modelled by means of a filtering Poisson process [24]. In the paper, two different situations and solutions are presented: one for the case where the stream of moving forces is modelled as a filtered Poisson process, and the other for the case when one of the forces is located at such a point on the beam that it causes the maximum response of the beam. In the second model, the random response of the beam is treated as an infinite series of random variables. The problem was solved using the idea of a dynamic influence function [16]. Several optimisation criteria were considered based on the expected values and variance of the beam response.

The structural model in the case of a bridge is often a simply supported beam. Therefore, let us consider the damped vibrations of a simply supported Euler-Bernoulli beam of finite length L, which has an absorber fixed at point x₀, and which is subjected to a random train of moving forces with constant velocity v.

The absorber is described with a single degree of freedom (SDOF) and is attached to the primary structure of a continuous beam that is modelled for dynamic analysis as a system with finite multi-degrees of freedom (MDOF), which is presented in Figure 1.

Figure 1.

The vibrations of this beam are described by the following equations [15]: (1) EI∂4wx,t∂x4+c∂wx,t∂t+m∂2wx,t∂t2++ rtδx−x0=∑k=1NtAkδx−vt−tk, \matrix{ {EI{{{\partial ^4}w\left( {x,t} \right)} \over {\partial {x^4}}} + c{{\partial w\left( {x,t} \right)} \over {\partial t}} + m{{{\partial ^2}w\left( {x,t} \right)} \over {\partial {t^2}}} + } \cr { + \;r\left( t \right)\delta \left( {x - {x_0}} \right) = \mathop \sum \nolimits_{k = 1}^{N\left( t \right)} {A_k}\delta \left[ {x - v\left( {t - {t_k}} \right)} \right],} \cr } (2) rt=kaqa−x0+caq˙a−x˙0=Maq¨a, r\left( t \right) = {k_a}\left( {{q_a} - {x_0}} \right) + {c_a}\left( {{{\dot q}_a} - {{\dot x}_0}} \right) = {M_a}{\ddot q_a}, where EI is the flexural stiffness of the beam, m is the mass per unit length of the beam, c is the damping coefficient, δ(.) is the Dirac delta function, and r(t) is the force response of the absorber acting on the beam. The reaction of the absorber on the beam can be derived from the equilibrium of the absorber and is equal to the sum of forces in the spring and the damping elements, which is equal to the mass of the absorber multiplied by its acceleration. The amplitudes A_k are assumed to be random variables, which are mutually independent and independent of the random instants t_k. It is assumed that the expected values E[A_k]= E[A]= const are known, and that random times t_k constitute a Poisson process N(t) with parameter λ.

In the case of a finite simply supported beam, the boundary conditions have the following forms (3) w0, t=wL, t=0, w\left( {0,\;t} \right) = w\left( {L,\;t} \right) = 0, (4) ∂2wx,t∂x2x=0=∂2wx,t∂x2x=L=0. {\left. {{{{\partial ^2}w\left( {x,t} \right)} \over {\partial {x^2}}}} \right|_{x = 0}} = {\left. {{{{\partial ^2}w\left( {x,t} \right)} \over {\partial {x^2}}}} \right|_{x = L}} = 0.

Similarly, to [15] two dynamic influence functions: H₁(x,t) and H₂(x,t–L/ν) should be considered. Function H₁ (x,t) is the response of the beam at time t (0 ≤ t ≤ L/ν) due to a moving force equal to unity (A_k = 1). Function H₂(x,t–L/ν) is the response of the beam due to the unity force that has already left the beam. These influence functions satisfy the following differential equations: (5) EI∂4H1x,t∂x4+c∂H1x,t∂t++m∂2H1x,t∂t2−rtδx−x0=δx−vt, \matrix{ {EI{{{\partial ^4}{H_1}\left( {x,t} \right)} \over {\partial {x^4}}} + c{{\partial {H_1}\left( {x,t} \right)} \over {\partial t}} + } \cr { + m{{{\partial ^2}{H_1}\left( {x,t} \right)} \over {\partial {t^2}}} - r\left( t \right)\delta \left( {x - {x_0}} \right) = \delta \left( {x - vt} \right),} \cr } (6) EI∂4H2x,t−Lv∂x4+c∂H2x,t−Lv∂t++m∂2H2x,t−Lv∂t2−rt−Lvδx−x0=0, \matrix{ {EI{{{\partial ^4}{H_2}\left( {x,t - {L \over v}} \right)} \over {\partial {x^4}}} + c{{\partial {H_2}\left( {x,t - {L \over v}} \right)} \over {\partial t}} + } \cr { + m{{{\partial ^2}{H_2}\left( {x,t - {L \over v}} \right)} \over {\partial {t ^2}}} - r\left( {t - {L \over v}} \right)\delta \left( {x - {x_0}} \right) = 0,} \cr } as well as the initial conditions and the appropriate boundary conditions, which can be derived from Equations 3–4: (7) H1x,0=∂H1x,t∂tt=0=0, {\left. {{H_1}\left( {x,0} \right) = {{\partial {H_1}\left( {x,t} \right)} \over {\partial t}}} \right|_{t = 0}} = 0, (8) H2x,0=H1x,Lv,∂H2x,t∂tt=0=∂H1x,t∂tt=Lv. \matrix{{{H_2}\left( {x,0} \right) = {H_1}\left( {x,{L \over v}} \right),} \hfill \cr {{{\left. {{{\partial {H_2}\left( {x,t} \right)} \over {\partial t}}} \right|}_{t = 0}} = {{\left. {{{\partial {H_1}\left( {x,t} \right)} \over {\partial t}}} \right|}_{t = {L \over v}}}.} \hfill \cr }

There are two important setups that should be considered at this stage. In the first case, the deflection of the beam w_I(x,t) is calculated at arbitrary time t ≥ L/ν, with none of the force locations being known. In the second case, we consider a situation at the moment t_i when a single force causes an extreme displacement of the beam at point x. The deflection of the beam for the first case [w_I(x,t)] can be written using the Stieltjes integral [15]: (9) wIx, t=∫t−LvtAτH1x, t−τdNτ+ +∫tbt−LvAτH2x, t−τ−LvdNτ, \matrix{ {{w_I}\left( {x,t} \right) = \int_{t-{L \over v}}^t {A\left( \tau \right){H_1}\left( {x,t - \tau } \right)dN\left( \tau \right) + } } \hfill \cr {\;\;\; + \int_{{t_b}}^{t - {L \over v}} {A\left( \tau \right){H_2}\left( {x,t - \tau - {L \over v}} \right)dN\left( \tau \right)} ,} \hfill \cr }

Also, the beam deflection for the second case [w_II(x,t_i)] can be written using the Stieltjes integral: (10) wIIx,ti=AtiH1x, ti+wIx, t. {w_{II}}\left( {x,{t_i}} \right) = A\left( {{t_i}} \right){H_1}\left( {x,{t_i}} \right) + {w_I}\left( {x,t} \right).

In the case of transient vibration, we assumed t_b = 0, and for steady-state, t_b=−∞.

The expected value and variance of the random functions w_I(x,t) amounts to [15]: (11) EwIx, ∞=EAλ∫0LvH1x,τdτ++EAλ∫Lv∞H2x, t−τ−Lvdτ=EAλEw, \matrix{ {E\left[ {{w_I}\left( {x,\infty } \right)} \right] = E\left[ A \right]\lambda \int_0^{{L \over v}} {{H_1}\left( {x,\tau } \right)d\tau } + } \cr { + E\left[ A \right]\lambda \int_{{L \over v}}^\infty {{H_2}\left( {x,t - \tau - {L \over v}} \right)d\tau } = E\left[ A \right]\lambda E\left[ w \right],} \cr } (12) σwI2x, ∞=EA2λ∫0LvH12x,τdτ++EA2λ∫Lv∞H22x, t−τ−Lvdτ=EA2λσ2. \matrix{ {\sigma _{wI}^2\left( {x,\infty } \right) = E\left[ {{A^2}} \right]\lambda \mathop \smallint \nolimits_0^{{L \over v}} H_1^2\left( {x,\tau } \right)d\tau + } \cr { + E\left[ {{A^2}} \right]\lambda \mathop \smallint \nolimits_{{L \over v}}^\infty H_2^2\left( {x,t - \tau - {L \over v}} \right)d\tau = E\left[ {{A^2}} \right]\lambda {\sigma ^2}.} \cr }

The expected value and variance of the random functions w_II(x,t) amounts to (13) EwIIx, ti=EAH1x, ti+EwIx,t, E\left[ {{w_{II}}\left( {x,{t_i}} \right)} \right] = E\left[ A \right]{H_1}\left( {x,{t_i}} \right) + E\left[ {w_I\left( {x,t} \right)} \right], (14) σwII2x, ti=σA2H12x, ti+σwI2x, t==E2AvA2H12x, ti+EA2λσ2. \matrix{ {\sigma _{wII}^2\left( {x,{t_i}} \right) = \sigma _A^2H_1^2\left( {x,{t_i}} \right) + \sigma _{wI}^2\left( {x,t} \right) = } \cr { = {E^2}\left[ A \right]v_A^2H_1^2\left( {x,{t_i}} \right) + E\left[ {{A^2}} \right]\lambda {\sigma ^2}.} \cr }

The symbol E[.] denotes the expected value of the quantity in the brackets, with ν_A being the coefficient of variation and EA2=E2A1+vA2 E\left[ {{A^2}} \right] = {E^2}\left[ A \right]\left( {1 + v_A^2} \right) . These general solutions will be used to optimise the absorber parameters. Time t_i ∈ 〈0, L/v〉 in formulas (10), (13), (14) determines the position of the force on the beam in such a way that it causes the maximum displacement of the beam in the section x ∈ 〈0,L〉. Time t_i is determined by both the velocity of the force and the first few natural frequencies of the coupled beam-absorber system.

In the case of a simply supported beam, one can look for dynamic influence functions in the form of sine series (15) H1x, t=∑n=1∞y1ntsinnπxL, {H_1}\left( {x,t} \right) = \sum\nolimits_{n = 1}^\infty {{y_{1n}}\left( t \right)sin{{n\pi x} \over L}} , (16) H2x, t−τ−Lv=∑n=1∞y2nt−LvsinnπxL. {H_2}\left( {x,t - \tau - {L \over v}} \right) = \mathop \sum \nolimits_{n = 1}^\infty {y_{2n}}\left( {t - {L \over v}} \right)sin{{n\pi x} \over L}.

Substituting expressions (15) and (16) into Eqs. (5) and (6), using the orthogonality method, and taking into consideration the first j eigenforms, one can obtain the following sets of ordinary equations: (17) d2y1ntdt2+2αdy1ntdt++ωn2y1nt+rtsinnπx0L==2mLsinnπvtL, \matrix{ {{{{d^2}{y_{1n}}\left( t \right)} \over {d{t^2}}} + 2\alpha {{d{y_{1n}}\left( t \right)} \over {dt}} + } \cr { + \omega _n^2{y_{1n}}\left( t \right) + r\left( t \right)sin{{n\pi {x_0}} \over L} = } \cr { = {2 \over {mL}}sin{{n\pi vt} \over L},} \cr } (18) d2y2ntdt2+2αdy2ntdt+ωn2y2nt++rtsinnπx0L=0, \matrix{ {{{{d^2}{y_{2n}}\left( t \right)} \over {d{t^2}}} + 2\alpha {{d{y_{2n}}\left( t \right)} \over {dt}} + \omega _n^2{y_{2n}}\left( t \right) + } \cr { + r\left( t \right)sin{{n\pi {x_0}} \over L} = 0,} \cr } where n = 1,2, …, j, 2α=cm 2\alpha = {c \over m} , and ωn2=nπL4EIm \omega _n^2 = {\left( {{{n\pi } \over L}} \right)^4}{{EI} \over m} .

The initial conditions for functions y_1n(t) and y_2n(t) result directly from conditions (7) and (8).

The equations of motion of the SDOF absorber attached to the beam at point x₀ are described by the equation (19) Mad2qatdt2+cadqatdt−dHix0,tdt++kaqat−Hix0,t=0, \matrix{ {{M_a}{{{d^2}{q_a}\left( t \right)} \over {d{t^2}}} + {c_a}\left[ {{{d{q_a}\left( t \right)} \over {dt}} - {{d{H_i}\left( {{x_0},t} \right)} \over {dt}}} \right] + } \cr { + {k_a}\left[ {{q_a}\left( t \right) - {H_i}\left( {{x_0},t} \right)} \right] = 0,} \cr } which after considering (15) or (16) and the first j eigenforms, has the form (20) d2qatdt2+2αadqatdt−∑n=1jdyintdtsinnπx0L++ωa2qat−∑n=1jyintsinnπx0L=0, \matrix{ {{{{d^2}{q_a}\left( t \right)} \over {d{t^2}}} + 2{\alpha _a}\left[ {{{d{q_a}\left( t \right)} \over {dt}} - \sum\nolimits_{n = 1}^j {{{d{y_{in}}\left( t \right)} \over {dt}}sin{{n\pi {x_0}} \over L}} } \right] + } \cr { + \omega _a^2\left[ {{q_a}\left( t \right) - \sum\nolimits_{n = 1}^j {{y_{in}}\left( t \right)sin{{n\pi {x_0}} \over L}} } \right] = 0} \cr } , where 2αa=caMa 2{\alpha _a} = {{{c_a}} \over {{M_a}}} , ωa2=kaMa \omega _a^2 = {{{k_a}} \over {{M_a}}} , and i = 1,2.

After substituting dependence (2) into equations (17) and (18), two separate systems of ordinary differential equations are obtained, together with equation (20), for which solutions are achieved by numerical integration using Wolfram Mathematica [25]. This software uses the Runge–Kutta method, automatically selecting the integration step. However, in calculations, the maximum integration step was limited to one-thousandth of the passing time.

The issue of optimising the parameters of an absorber in the case of stochastic loads has been considered in many studies [12,13,14,15,16,17,18,19,20,21,22,23]. The solution to the optimisation problem depends both on the speed of the moving force and the parameters of the structure. Let us assume the following optimisation criteria: (21) Min EwIx, ∞Min EwIIx, tiMin σwI2x, ∞Min σwII2x, ti. \matrix{ {{\rm{\;Min\;}}E\left[ {{w_I}\left( {x,\infty } \right)} \right]} \cr {{\rm{\;Min\;}}E\left[ {{w_{II}}\left( {x,{t_i}} \right)} \right]} \cr {{\rm{\;Min\;}}\sigma _{wI}^2\left( {x,\infty } \right)} \cr {{\rm{\;Min\;}}\sigma _{wII}^2\left( {x,{t_i}} \right)} \cr } .

For the numerical analysis, a reinforced concrete beam bridge of length L = 30 m with one absorber was considered. Different positions of the absorber, x₀ ∈ (0,L), were considered. Moreover, various parameters of the absorber (mass and stiffness) were also analysed. Parameter µ= M_a/mL, i.e. the ratio of the mass of the absorber to the bridge mass, was used to characterise the absorber mass. Similarly, parameter κ= ω_a/ω_s is the ratio of the absorber’s eigenfrequency to the bridge’s eigenfrequency. The first eigenfrequency of the simply supported beam bridge is equal to ω_s =4 rad/s. After analysing a range of 3 to 20 modes, it was found that 5 eigenforms could be adopted for further calculations (j = 5 in Eq. 17, 18, 20). A constant speed of the moving vehicles was assumed and then compared with the critical speed of the structure, which was determined according to the formula [26]: (22) vcr=πEI/mL2 {v_{cr}} = \pi \sqrt {EI/m{L^2}} where EI denotes the flexural stiffness of the beam, m denotes the mass per unit length of the beam, and L is the span of the bridge beam.

Figures 2 and 3 demonstrate the deflection functions in the middle span of the bridge beam, with an exemplary absorber (with various parameters µ = M_a/mL, κ= ω_a/ω_s being mounted in the central part of the basic system. One can clearly see the different influence of the installed absorbers on the work of the structure, particularly in the vibrations when the vehicle leaves the bridge. The travel time of a force moving at a speed equal to the critical speed (v = v_cr = 137.5 km/h) is 0.78 s. Figure 2 shows the deflection function w_I(L/2,t) as a random response of the beam, which is treated as a filtered Poisson process (in later analyses, it is named “the first case”).

Figure 2.

Figure 3.

The effect of the absorber mass was analysed within a wide range of μ ∈ 〈0,0.3〉, whereas in practice, values of up to 0.1 are generally used. As shown in Figure 3a, in the first case – w_I (L/2,t) – the expected value of the displacement in the middle of the span of the bridge beam does not depend on the mass of the absorber, while in the second case – w_II (L/2,t) – a certain relationship can be seen (Fig. 3b). Similarly, a wide range of the absorber’s frequency κ ∈ 〈 0.5, 2.5〉 was analysed, however, an absorber with a frequency close to that of the structure (κ ≈ 1) is most commonly used. As shown in Figure 4a, in the first case – w_I(L/2,t) – the expected value of the displacement in the midspan does not depend on the frequency of the absorber, while in the second case – w_II (L/2,t) – a slight dependence can be seen (Fig. 4b). However, the impact of the vehicle velocity is very large in each case. The analysed range of velocity for the selected structure is 68–336 km/h. A higher speed corresponds to a smaller deflection, which means that there is a greater influence of the absorber.

Figure 4.

Figures 5 and 6 show the expected values and variances of the deflection in the middle of the beam span with regard to the location of the absorber (x₀), both for the 1st and 2nd case. Results are presented for different velocities of motor vehicles. Similarly to the above, in the first case – w_I(L/2,t) – the expected value of the displacement in the midspan does not depend on the absorber frequency (Fig. 5a), while in the second case – w_II(L/2,t) – slight differences can be seen (Fig. 5b). In turn, the variance of these displacements depends on the location of the absorber. Figure 6 shows that the optimal operation of the absorber is when it is installed within 0.3–0.6 of the beam’s span, both for the first and second cases.

Figure 5.

Figure 6.

As shown in Figures 6–8, the variance of the displacements also depends on velocity. However, this dependency is not the same as in the case of the expected value, as shown in Figures 3–5. When analysing the parameters µ and κ, the largest deflection occurs when velocity v = (0.6–0.8)v_cr. In turn, the mass ratio has a slight influence on the results obtained (see Figure 7). Good results were achieved for µ ∈ 〈5%,15%〉. When µ< 5% and μ> 20% for v = (0.6–0.8)v_cr, the variance increases, both in the first case – w_I(L/2,t) – and second case – w_II(L/2,t). The frequency ratio also has a slight influence on the results (see Figure 8). The smaller the κ, the smaller the deflection within the critical range of the velocity i.e., v = (0.5–1.5)v_cr.

Figure 7.

Figure 8.

Maintenance of historic bridges requires constant inspection of their technical condition and systematic renovation work. The technical wear of a structure is related to, among others: its age, the ageing processes of the constituent materials, rheological phenomena, the quality of the workmanship, and the increasing loading of a bridge due to the intensification of car traffic over time. Monuments are a testimony to our history, traditions, and national pride [27]. Therefore, engineers try to support historical structures. One of them is the idea of reusing whole structural members from disassembled or renovated structures to another [28]. Another can be the use of DVA or TMD dynamic absorbers to reduce vibrations of constructions, especially those subjected to dynamic loading. It is important to properly tune the damper, as a poorly matched device can increase vibrations instead of reducing them.

In the paper, the optimal solution for installing an absorber in a structure to reduce deflections caused by stochastic moving loads was searched. Two different situations were analysed: one – when the stream of moving forces is modelled as the filtered Poisson process; and the other – when one of the forces is located in the point of the beam in which the response of the beam has the maximum value. In most of the analyses carried out, the results were similar for both cases – w_I(L/2,t) and – w_II(L/2,t). The greatest differences were noticed in the case of the expected values within the speed range from 0.5 to 1.5 of the critical speed, which for this analysis corresponds to the speed range of normal vehicle traffic, that is 68–200 km/h. The optimal operation of the absorber is when it is installed within 0.3–0.6 of the beam span, for both the first and the second case. The appropriate parameters of an absorber with regards to its use are: μ= M_a/mL ∈ 〈 5%,15% 〉 and κ = ω_a/ω_s ∈ 〈 0.5,2.5 〉. When considering the fact that the authors in [10] stated that adopting a mass ratio (the absorber ratio of the mass to the mass of the main structure) below 1% is insufficient, it seems reasonable to follow this recommendation.

The modelling of damping plays an important role in obtaining the correct response of a structure (i.e., devoid of harmful effects on the structure). Any deviation in the value of damping (from its actual value) in the analytical model may result in a minor failure or even a construction disaster. Therefore, damping modelling is of great importance with regard to the dynamic analysis of a structure [29].