# Reliability of controlled linear systems under Gaussian and non-Gaussian loads

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## Abstract

Civil and mechanical engineering systems are often subjected to vibrations which could alter their behaviour or even lead to their damage or failure. Generated either by man-made processes, such as traffic or equipment, or by natural processes, such as seismic or ocean waves, vibrations may be represented by stochastic processes defined over certain ranges of frequencies. Vibration-control systems have been developed to reduce the undesired response of systems subjected to random vibrations. The aim of the current paper is to compare comprehensively the performance of three popular passive vibration-suppression devices installed in linear single-degree-of-freedom (SDOF) systems subjected to Gaussian and non-Gaussian random vibrations with general frequency content, characterised by a power-spectral density function. The vibration-control systems analysed in this study are the viscous dampers (VD), the tuned-mass dampers (TMD) and the tuned-inerter dampers (TID), and their performance is assessed in terms of reliability metrics, such as the probability of failure and the first passage time. The twofold goal of this study is reached through (1) the development of explicit analytical relations of the controlled-systems’ responses to the Gaussian input; and (2) Monte Carlo simulation estimates of the reliability metrics for the systems subjected to the non-Gaussian loads.

## Keywords

Vibration control Non-Gaussian vibrations Random-vibration theory Reliability analysis Mean crossing-rates## 1 Introduction

Mitigation of unwanted vibrations represents an important research topic for many engineering disciplines. This reduction of vibrations is commonly achieved via vibration-isolation systems, i.e. by controlling the supporting structure [1]; or via vibration-suppression systems, i.e. controlling the structure itself by means of supplemental damping [2, 3] or other techniques [4, 5]. This problem can be very challenging given the complexity of engineering systems and the random character of vibrations. The aim of this paper is to provide a comprehensive overview on the performance of vibration-suppression systems, which is achieved through the analysis of the control systems’ performance under (1) Gaussian loads, by providing explicit analytical response-relations; and (2) non-Gaussian loads by using estimates of response statistics using Monte Carlo simulations. Unlike other similar studies, the both types of loads have a general characterisation of the frequency content given by a power-spectral density function.

Three vibration-suppression systems are analysed, that is, the viscous damper (VD), the tuned-mass damper (TMD) and the tuned-inerter damper (TID). Following is a brief presentation of these systems with a non-exhaustive set of selected references. VDs were introduced in the 70’s [6] and are installed to increase the limited inherent structural damping of civil engineering structures [7, 8]. TMDs were proposed by Frahm as early as 1909 [9] and consist of an added mass, mounted in series with a spring and a damper. Several analytical and numerical tuning methods have been derived since, starting with Den Hartog [10]. While VDs and TMDs have been successful in many engineering applications and installed in several buildings around the world, inerter-based systems, such as the TID—where the TMD mass is replaced with an inerter—rely on a newly-introduced device developed by Smith [11]. Since its invention, the inerter has been applied successfully in the automotive industry [12] and in the development of train suspensions [13, 14]. Recent studies have looked into the use of inerters for civil-engineering applications [15, 16, 17, 18].

Previous studies regarding the performance of vibration suppressed systems have been carried out before for VDs [2, 19], TMDs [20, 21], TIDs [4, 22], or other control systems such as liquid column dampers [23], vibration barriers [1], or viscoelastic-mass dampers [24], but they generally refer to very specific loading patterns. For example, some studies quantify the performance of the control systems by considering only harmonic or deterministic ground-motion records [4, 24, 25, 26], while others attempt a probabilistic analysis by using Gaussian white noise [27, 28]. Some studies consider more specific frequency patterns, by using Gaussian vibrations with Kanai-Tajimi frequency spectra [29, 30, 31], or as in the case of [32], where the TMD is designed to respond to the frequency bandwidth of walking crowds. As for reliability studies on control systems, first- and second- order reliability methods (FORM/SORM) have been used in [27] and a more theoretical approach for the reliability of control theory of random vibrations is shown in [33, 34], in which dynamic programming and the stochastic averaging method [35] are used. An analytical solution using crossing theory [36] (Chap. 7.3) for structural control applications is presented in [37], with an application for the tuned mass-damper-inerter in [31].

Similar to some previous studies aforementioned, this paper analyses the response of vibration-controlled linear systems subjected to Gaussian vibrations, but it compares the performances of the VDs, TMDs and TIDs using explicit frequency-response relations for random vibration with general frequency contents. The analytical expressions of the reliability metrics are an essential instrument in designing these systems for random excitation, in order to avoid the expensive Monte Carlo simulations. Explicit analytical relations for reliability metrics have been developed for VD, TMD and TID systems under the assumption of stationary Gaussian input, using elements of random-vibration theory. Analytical solutions are backed by Monte Carlo results, which are further extended for the case of the non-Gaussian vibrations, a relevant distinction between the current research and previous similar studies. The VD-, TMD- and TID-controlled systems’ performances are also analysed in the context of the more general non-Gaussian vibrations, with narrow- and broad-band frequency contents. Performing analyses beyond the Gaussian assumption is essential because realistic excitations do not follow it. The non-Gaussian character of loads can be simply shown by calculating the kurtosis coefficient, which for realistic vibration-inducing loads is different than 3, the value characteristic for Gaussian processes [38]. For example, the kurtosis coefficient is higher than 4 for wind pressure on low-rise structures tested in wind tunnels [39]; is 6.2 for coastal-wave elevations measured in Duck, North Carolina [40]; has an average value of 14.4 for seismic ground-acceleration for earthquakes recorded on rock sites [41]; is only 2.1 for the roughness of a road in South Dakota [38]; and measures 6.2 for the unevenness of a railway track in India [38].

This paper is structured in two main parts: (1) the system and input definitions, that is, the description of the host structure and the vibration-control devices, and the characterisation of the Gaussian and non-Gaussian stochastic processes describing the random vibrations to which the host structure is subjected; and (2) the characterisation of the reliability of the host and vibration-control systems, using random-vibration theory and Monte Carlo simulations, for the Gaussian and non-Gaussian vibrations, respectively. The first part describes the governing equations of motion for the uncontrolled and controlled dynamic systems subjected to random vibrations. For a fair comparison of the controlled systems’ performances, the VDs, TMDs and TIDs are calibrated such that the maximum relative displacements of the controlled structures are similar over the entire frequency spectrum. The second part of the paper defines reliability metrics derived using the crossing theory, metrics used for the comparison of the three types of vibration-control systems analysed. Analytical relations for the reliability metrics are developed for the systems subjected to Gaussian input, supported by numerical simulations, used further on for the analyses under non-Gaussian input. Finally, an overall comparison of the increase in the reliability of the controlled structures using all types of controllers, with respect to the uncontrolled host structures, is shown using exceedance probabilities and first-passage times of the critical response.

## 2 Systems and input definition

The goal of the paper is to provide a comprehensive view of the effects of vibration-suppression devices on the performance of dynamic systems subjected to random vibrations. The performance is quantified in terms of its reliability by using elements of random-vibration theory. The problem solved in this paper consists of three main elements, discussed separately in this section: the host dynamic system, the vibration-suppression systems, and the random-vibration input. The host dynamic system is represented by a viscously-damped linear single-degree-of-freedom (SDOF) system; the vibration-suppression systems analysed are the viscous damper (VD), the tuned mass-damper (TMD) and the tuned inerter-damper (TID); and the random input is represented by stochastic Gaussian and non-Gaussian processes with various frequency contents.

### 2.1 Host dynamic system

*X*(

*t*) denotes the displacement of the linear SDOF system relative to the ground subjected to a stochastic process

*A*(

*t*). Zero initial conditions, \(X(0)=0\) and \({\dot{X}}(0)=0\), are assumed. Coefficients \(\nu _0\) and \(\zeta _0\) are known as the fundamental frequency and damping ratio of the linear SDOF, respectively. Numerical values for the system parameters used for the examples in the following sections are \(\nu _0 = 4\pi \) rad/s and \(\zeta _0=2\%\).

### 2.2 Vibration-suppression systems

Equation 1 is modified to account for the presence of each control system, as detailed in the following subsections. It should be noted that an SDOF host structure embodying a TMD or a TID becomes a two-degrees-of-freedom system. Note that the rest of the equations of motion for the controlled SDOF system are also written in coordinates relative to the ground.

#### 2.2.1 VD

*c*. The response

*X*(

*t*) is the displacement of the VD-controlled SDOF linear system relative to the ground.

#### 2.2.2 TMD

*Y*(

*t*) denotes the displacement response of the TMD system relative to the ground. Zero initial conditions, \(Y(0)=0\) and \({\dot{Y}}(0)=0\), are assumed. Coefficients \(\nu _{TMD}\) and \(\zeta _{TMD}\) are known as the fundamental frequency and damping ratio of the TMD and \(\mu \) is the mass ratio between the TMD and the host structure [10]. Note that the respective responses

*X*(

*t*) and

*Y*(

*t*) are coupled and they represent the displacements of the TMD-controlled SDOF linear and the TMD systems, relative to the ground [42].

#### 2.2.3 TID

*Y*(

*t*) denotes the displacement response of the TID system. Zero initial conditions, \(Y(0)=0\) and \({\dot{Y}}(0)=0\), are assumed. Coefficients \(\nu _{TID}\) and \(\zeta _{TID}\) are known as the fundamental frequency and damping ratio of the TID and \(\mu \) is the inertance-to-mass ratio between the TID and the host structure [4, 22]. Similar to the equations of motion of the TMD,

*X*(

*t*) and

*Y*(

*t*) are coupled and they represent the relative displacements to the ground of the TID-controlled SDOF linear and the TID systems.

#### 2.2.4 Control-systems optimisation

Controlled- and uncontrolled-system parameters

System | \(\nu _0\) [rad/s] | \(\zeta _0\) | \(\mu \) |
---|---|---|---|

Uncontrolled ( | \(4\pi \) | 0.02 | − |

Controlled by | \(4\pi \) | 0.02 | 3.50 |

Controlled by \(TMD_{5\%}\) | \(4\pi \) | 0.02 | 0.05 |

Controlled by \(TID_{5\%}\) | \(4\pi \) | 0.02 | 0.05 |

Controlled by \(TID_{20\%}\) | \(4\pi \) | 0.02 | 0.20 |

### 2.3 Random-Vibration input

*G*(

*t*) is a stationary zero-mean, unit-variance, Gaussian process with standard Gaussian probability density function \(\phi (x)=(2\pi )^{-0.5}\exp \{-0.5x^2\}\) and frequency content defined by the one-sided power-spectral density function \(g(\nu ),~\nu \ge 0\). Note that

*A*(

*t*) covers a large class of Gaussian (for \(f(x)=x\)) and non-Gaussian processes (e.g. log-Normal for \(f(x)=\exp \{x\}\)). For our problem we use a standard Gaussian process and a non-Gaussian process defined by the function \(f(x)=x^3\), to preserve the zero-symmetry of the process. The non-Gaussian process’s marginal distribution

*F*(

*x*) has a functional form such that \(F^{-1}\circ \varPhi (x)=x^3\). The marginal distributions for both types of processes are shown in Fig. 3, and it can be noticed that the non-Gaussian defined by \(f(x)=x^3\) is indeed symmetrical around zero and has considerably larger tails than the Gaussian process. The non-Gaussian distribution chosen has heavier tails than the Gaussian process in this case, as suggested by the data presented in the Introduction section.

*A*(

*t*), i.e. a narrow-band pulse (NBP) frequency with \(\nu _a=3\pi \) rad/s and \(\nu _b=5\pi \) rad/s defined in the vicinity of the natural frequency of the system, \(\nu _0\); and a wide-band frequency with \(\nu _a=0\) and \(\nu _b=10\pi \) rad/s. For the second case, in which \(\nu _a=0\), the process

*A*(

*t*) is also known as band-limited white noise (BLWN). The graphical representation of the power-spectral density functions for the two types of processes is shown in Fig. 4a. Any frequency content, whether it is white or “coloured” noise can be used to replace function \(g(\nu )\). The examples chosen in this paper are two general processes, whose frequency ranges can be found in real processes, such as the frequency of people walking for the NBP [44] or of earthquakes for the BLWN [41].

Samples of the process *A*(*t*) can be obtained via Monte Carlo simulations using the spectral-representation theorem [45] (Chap. 3.9). The procedure consists mainly of representing the process *A*(*t*) as a Fourier series with random coefficients [46].

## 3 Evaluation of system performance

Elements of the random-vibration theory [35, 36] and crossing theory [37, 47] are used to define the system’s performance. Analytical relations can be written for the reliability of systems subjected to Gaussian input [31, 37], while for non-Gaussian problems approximate methods have been suggested [35, 48], even though the only general and reliable method for calculating reliability metrics remains Monte Carlo. This current study develops explicit analytical relations for the frequency response of the structure controlled with VD, TMD and TID, respectively, by using crossing theory, which absolves the need of using Monte Carlo simulations in the case of the Gaussian assumption, and performs a Monte Carlo-based reliability analysis for the systems subjected to non-Gaussian excitation.

*X*(

*t*), which is the solution of Eq. (1). The reliability of the viscously-damped linear SDOF system is defined by the probability that its response stays within a safe set \({\mathcal {D}}\), during its lifetime \(\tau \), i.e.

*X*(

*t*) exits the safe set \({\mathcal {D}}\), is equal to zero, expressed as \(P_R(\tau ) = \mathbb {P}\{(X(0)\in {\mathcal {D}}) \cap (N_{{\mathcal {D}}}(\tau )=0)\}\).

*X*(0) and the number of \({\mathcal {D}}\)-outcrossings \(N_{{\mathcal {D}}}(\tau )\) are independent events; (ii) the probability of an instantaneous failure is zero, i.e. \(\mathbb {P}\{(X(0)\in {\mathcal {D}})\}=1\); and (iii) \(N_{{\mathcal {D}}}(\tau )\) follows a non-homogeneous Poisson distribution with rate \(\eta _{{\mathcal {D}}}(t)=\frac{d}{dt}\mathbb {E}\{N_{{\mathcal {D}}}(\tau )\}\), called the mean crossing-rate at which the process

*X*(

*t*) exits \({\mathcal {D}}\). Finally, under the final assumption that (iv) the response

*X*(

*t*) is stationary, i.e., \(\eta _{{\mathcal {D}}}(t)=\eta _{{\mathcal {D}}}=\mathbb {E}\{N_{{\mathcal {D}}}(\tau )\}/\tau \), the reliability of the system in Eq. (11) reduces to

*X*(

*t*) crosses outside the safe set \({\mathcal {D}}\). It is shown in [36] (Chap. 7.4) that the probability distribution of \(T_{{\mathcal {D}}}\) can only be obtained in limited cases and therefore its

*k*th order moments are of interest

### 3.1 Reliability metrics

*X*(

*t*) of the controlled or uncontrolled linear SDOF system. Since we assumed that the input

*A*(

*t*) and, implicitly,

*X*(

*t*) are symmetric about zero, we can state that the \({\mathcal {D}}\)-outcrossing rate of

*X*(

*t*) is \(\eta _{{\mathcal {D}}}=2\eta _{{\mathcal {D}}}^{+}\), where \(\eta _{{\mathcal {D}}}^{+}\) represents the \(x_{cr}\)-upcrossings rate of

*X*(

*t*), or the rate at which

*X*(

*t*) would cross the upper level \(x_{cr}\) with positive slope. In other words, \(\eta _{{\mathcal {D}}}^{+}\) can be calculated as the probability of

*X*(

*t*) crossing the level \(x_{cr}\) with positive slope in an infinitesimal interval of time \((t,t+\varDelta t)\), i.e.

*X*(

*t*) crosses the lower level \(-x_{cr}\) with negative slope) over the time period \(\tau \). Figure 6a shows the number of \(x_{cr}\)-upcrossings and \(x_{cr}\)-downcrossings for a sample of the response for a critical level \(x_{cr}=0.05\), over a time window of 10

*s*. Figure 6b shows the crossing rates calculated for 10,000 samples of the response

*X*(

*t*) subjected to the non-Gaussian NBP

*A*(

*t*), and the mean crossing-rate of

*X*(

*t*) in black-dashed line.

*A*(

*t*) is Gaussian, it can be inferred that \([X(t),{\dot{X}}(t)]\) is also Gaussian with zero mean and variance \([{\dot{\sigma }}_X^2(t),\sigma _X^2(t)]\), parameters which are calculated further down in this section. The mean crossing-rate

*X*(

*t*) follows to be

*X*(

*t*) and \({\dot{X}}(t)\) used in Eq. (16) are calculated directly:

Transfer function terms for the uncontrolled system (UC), VDs and TMDs

UC | VD | |
---|---|---|

D | 1 | 1 |

E | 0 | 0 |

F | \(1-(\nu /\nu _0)^2\) | \(1-(\nu /\nu _0)^2\) |

G | \(2\zeta _0\nu /\nu _0\) | \(2\zeta _0(1+\mu )\nu /\nu _0\) |

Transfer function terms for TMDs

TMD | |
---|---|

D | \(\mu (\nu /\nu _0)^2 - (1+\mu )\gamma \) |

E | \(2(1+\mu )\beta \zeta _0 \nu /\nu _0\) |

F | \(\mu (\nu /\nu _0)^4-(\gamma +\mu +\gamma \mu +4\beta \zeta _0^2)(\nu /\nu _0)^2+\gamma \) |

G | \(2\zeta _0((\beta +\mu +\beta \mu )(\nu /\nu _0)^3-(\beta +\gamma )(\nu /\nu _0))\) |

*X*(

*t*) may not be stationary, although the input

*A*(

*t*) is, as it can be seen also in Fig. 6a. However, under the stated problem, it has been shown that the second moments converge quickly to the stationary solution even for low damping ratios [36]. Calculation of \(\sigma ^2_{X}(t)\) and \({\dot{\sigma }}^2_{X}(t)\) can be simplified further under the stationarity assumption, which leads to

*X*(

*t*) and

*A*(

*t*), with coefficients in Tables 2, 3 and 4. Coefficients \(\gamma =\mu (\nu _{TMD}/\nu _0)^2\) and \(\beta =\mu \zeta _{TMD}\nu _{TMD}/(\zeta _0\nu _0)\) in Table 3, and \(\gamma =\mu (\nu _{TID}/\nu _0)^2\) and \(\beta =\mu \zeta _{TID}\nu _{TID}/(\zeta _0\nu _0)\) in Table 4 can be calculated simply using the \(\mu \) parameter defined by the calibration of each vibration-control system, shown in Table 1. The mean crossing-rates \(\eta _{{\mathcal {D}}}\) of the uncontrolled response of the system in Eq. (1) have been calculated for both the Gaussian and the non-Gaussian, NBP and the BLWN motions

*A*(

*t*), as defined in Sect. 2.3.

Transfer function terms for TIDs

TID | |
---|---|

D | \(\mu (\nu /\nu _0)^2 -\gamma \) |

E | \(2(1+\mu )\beta \zeta _0 \nu /\nu _0\) |

F | \(\mu (\nu /\nu _0)^4-(\gamma +\mu +\gamma \mu +4\beta \zeta _0^2)(\nu /\nu _0)^2+\gamma \) |

G | \(2\zeta _0((\beta +\mu +\beta \mu )(\nu /\nu _0)^3-(\beta +\gamma )(\nu /\nu _0))\) |

*X*(

*t*), the stationarity assumption can be adopted further on. The results in Fig. 7 support the stationarity assumption given the almost identical mean crossing-rates calculated analytically using Eqs. (19) and (20), and numerically via Monte-Carlo simulations described in detail below.

It must be noted that Eqs. (21)–(23) are valid for any power-spectral density function \(g(\nu )\), and not only for the constant-intensity functions used in this paper. However, the choice of the white-noise functions is relevant for engineering applications with low damping ratios \(\zeta _0\) and relatively smooth power-specral density functions \(g(\nu )\). Under these conditions, \(|h(\nu )|\) is sharply peaked at \(\nu _0\) in the case of the uncontrolled host, and it has been shown that the main contribution of the response can be approximated by setting \(g(\nu )=g(\nu _0)\) to be constant [51].

The same approach can be applied in the case of the controlled structures, by updating \(|h(\nu )|\) accordingly. However, for more complex systems, the number of terms in the expression of the corresponding transfer functions increases significantly. The expressions for coefficients *D*, *E*, *F* and *G* in Eq. 23 can be found in Tables 2–4.

*A*(

*t*). As already mentioned previously, the results shown for the non-Gaussian input, as well as the validation of the analytical solutions are done using Monte Carlo simulations. Thus, if \(a_k(t),~k=1,\ldots ,N\) are

*N*samples of the process

*A*(

*t*), then \(x_k(t)\) are the corresponding response samples of (1) the uncontrolled system in Eq. (1); (2) the VD-controlled system in Eq. (2), (3) the TMD-controlled system in Eq. (3), or (4) the TID-controlled system in Eq. (5), calculated by solving each of these equations respectively. Then, similarly to the approach described earlier, the mean upcrossing-rate in Eq. (15) can be approximated using the Monte Carlo simulations as

*a*(

*t*) is sampled, and function \(\mathbb {1}\) denotes the counting indicator function. The mean upcrossing-rates \(\eta _{{\mathcal {D}}}^{+}\) calculated either by Eq. (15) or by Eq. (24) are used to derive all the reliability metrics described herein. Further on, the expression for the reliability in Eq. (12) is only valid for small \(\eta _{{\mathcal {D}}}^{+}\), but using Monte Carlo simulations, it can be calculated by relaxing this assumption, using the following relation:

### 3.2 Reliability analysis of controlled systems

The mean crossing-rates are a measure of the rate at which the response of a system goes above a threshold \(x_{cr}\). Both figures are calculated for a time window \(\tau =10\) s, and show a decrease in the mean crossing-rates of the controlled systems, which translates into an increase of the system performance. The TMD and TID with \(\mu =5\%\) display almost identical behaviour, a fact foreseen by following the tuning methodology, illustrated graphically in Fig. 2. Figure 2 also suggests that the VD would perform better with respect to the TMD and TID with \(\mu =5\%\), due to its lower response outside a small vicinity around the natural frequency \(\nu _0\). However, this is only reflected in Fig. 8, in the case of the Gaussian input, and less so for the non-Gaussian case. One needs to consider the fact that the control devices have been designed for harmonic oscillations, and not specifically for the random input, which is beyond the purpose of the current study. A considerable improvement in the response of the host structure has been achieved using the analytical tuning methodology described in Sect. 2.2.4, and a design targeted towards random inputs would only reinforce the conclusions regarding the performance of the controlled structures. Given the capability of the inerter to have its apparent mass (i.e. its inertance) increased without a significant increase in its physical mass, the TID with \(\mu =20\%\) represents a more reliable option. This is verified by reduced mean crossing-rates achieved by using this device. This characteristic is valid for both the Gaussian and non-Gaussian excitations, proving the TID to be a more reliable and robust control device in comparison with the VD and the TMD which becomes unrealistic for \(\mu >10\%\).

## 4 Conclusion

This paper offered a comparative overview of the gain in reliability of vibration-controlled systems with commonly-used devices, i.e. VD, TMD and TID, for structures subjected to random vibrations. Linear SDOF systems subjected to both Gaussian and non-Gaussian stationary vibrations with various frequency content were used as host structures. Reliability metrics in terms of probability of failure and moments of the first-passage times of the system to exit the displacement safe range, have been provided using crossing theory. Explicit analytical relations were provided for all types of systems subjected to Gaussian input, and have been backed by numerical results obtained by Monte Carlo simulations, extended also for the non-Gaussian case.

It has been shown that a considerable gain in reliability of the host structure can be achieved for all loading patterns (BLP and BLWN Gaussian and non-Gaussian inputs) using vibration-control systems tuned such that they display similar maximum displacement response over the entire frequency range. The study not only allowed for a parallel comparison between different types of vibration-suppression devices, but also showed that they are efficient for a large range of stochastic processes with different frequency contents and different probability laws. Moreover, the TID is more robust and reliable than the VD and TMD.

## Notes

### Acknowledgements

The work of Alin Radu reported in this paper has been supported by the Marie Skłodowska-Curie Actions of the European Union’s Horizon 2020 Program under the Grant Agreement 704679 - PARTNER. This support is gratefully acknowledged.

### Funding

AR was funded by the Marie Skłodowska-Curie Actions of the European Union’s Horizon 2020 Program under the Grant Agreement No. 704679 - PARTNER.

### Compliance with Ethical Standards

### Conflict of Interest

The authors declare that they have no conflict of interest.

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