An Efficient Uncertainty Quantification Approach for Robust Design of Tuned Mass Dampers in Linear Structural Dynamics

Abstract

The application of tuned mass dampers (TMDs) to high-rise buildings or slender bridges can significantly decrease the dynamical vibrations due to external excitation, such as wind or earthquake loads. However, the individual properties of a TMD such as mass, stiffness and damping have to be designed carefully with respect to the dynamical properties of the investigated structure. In real-world structures, the influence of uncertain system properties might be critical for the performance of a TMD and thus the whole structure. Therefore, the design under uncertainty of such systems is an important issue, which is addressed in the current paper. For our investigations, we consider linear single-degree-of-freedom (SDOF) systems, where analytical formulas for the deterministic design already exist, and linear multi-degree-of-freedom (MDOF) systems, where a time integration and numerical optimization algorithms are usually applied to obtain the optimal TMD parameters. If the numerical optimization should be coupled with a sampling-based uncertainty quantification method, such as Monte Carlo sampling, the design procedure would require the evaluation of a coupled double-loop approach, which is very demanding from the computation point of view. Therefore, we focus the following paper on an efficient analytical uncertainty quantification approach, which estimates the mean and scatter from a Taylor series expansion. Additionally, we introduce an efficient mode decomposition approach for MDOF systems with multiple TMDs, which estimates the maximum displacements using a modal analysis instead of a demanding time integration. Different optimal design problems are formulated as single- or multi-objective optimization tasks, where the statistical properties of the maximum displacements are considered as safety margins in the optimization goal functions. The application of numerical optimization algorithms is straightforward and not limited to specific algorithms. As numerical examples, we investigate an SDOF system with single TMD and a multi-story frame with multiple TMDs. The presented procedure might be interesting for the design process of structures, where the dynamical vibrations reach a critical threshold.

1. Introduction

The application of tuned mass dampers (TMDs) for the reduction in structural vibrations is a well-known procedure that started with the early investigations by Den Hartog [1]. Typically, high-rise buildings under earthquake and wind excitations require the TMD technology to reach new heights or to enable the construction under special loading conditions. An overview of existing famous buildings with TMD applications is given in [2]. Early studies for structures subjected to wind loads can be found in [3,4], as well as for earthquake excitations in [5,6].

Analytical solutions for optimal TMD parameters have been derived by Den Hartog [1] by simplifying the original structure to a linear single-degree-of freedom (SDOF) system and consider the TMD as an additional DOF. The optimality criteria were formulated in this approach in order to minimize the maximum displacements by considering the dynamic amplification function of the 2-DOF system under harmonic excitation. Additional analytical design criteria for harmonic excitations are summarized in [7]. Further investigations on analytical solutions can be found in [8], for harmonic and white noise excitation, and in [7,9,10,11], where different design criteria in the time and frequency domain have been investigated.

The extension for linear and non-linear multi-degree-of-freedom (MDOF) systems requires in the general case numerical procedure such as time integration to calculate the performance criteria, which could be maximum displacements, accelerations, inter-story drift and others. A good summary of the published design criteria and analysis methods is given in [12]. Numerical investigations on this topic can be found, e.g., for MDOF systems with a single TMD [13] and SDOF systems with multiple TMDs [14,15]. The optimal parameters of MDOF systems with multiple TMDs could not be solved analytically in the general case. Therefore, numerical optimization methods such as Particle Swarm optimization [16] and genetic algorithms [17,18] and machine learning methods [19,20] were often used for this task. However, the application of numerical optimization together with time integration methods could be limited due to the significant numerical effort.

Additional to the optimal tuning of the TMDs, the accurate knowledge of the main system properties as well as the TMD parameters are essential for an optimal performance of the TMDs. Therefore, the influence of uncertainties on the TMD performance and reliability is a critical issue. The application of Reliability-based Design and robust design optimization methods summarized in [21] have been published for different types of random excitation [22,23] and random system parameters [24]. Recent applications perform typically numerical time integration methods with random sampling [25,26], which increases the numerical effort even more. Since a coupled robust design optimization would require the evaluation of the statistical performance measure for every design parameter combination, the time integration might be the critical bottle-neck in the numerical analysis. Therefore, approximation methods have been investigated recently for the optimal design of TMDs considering uncertainties [27,28].

In our study, we will focus on efficient methods for the optimal design of TMD parameters for linear SDOF and MDOF systems by considering parameter uncertainty. In the first step, we will introduce a perturbation approach for the uncertainty propagation, where the system responses are linearized by a Taylor series expansion with respect to the random system parameters. This approach is derived and investigated for an SDOF system first, whereby the optimal parameters are obtained by numerical optimization. As excitation, we consider harmonic excitation, which could vary within a defined frequency range, and the maximum value of the corresponding dynamic amplification function is considered as the design criterion. Here, the amplification function of the displacements is chosen as an example similar to early studies by Den Hartog. However, other performance criteria could be considered in the approach in the same manner. We will show that this linearization works well for individual discrete values of the amplification functions of the main system and the TMD relative displacements. Within the optimization objective we consider the safety margin from the estimated mean and standard deviation within a variance-based robustness evaluation approach. The linearization approach may consider continuous scalar random numbers with an arbitrary distribution type, as long as the covariance matrix is known. However, in the numerical examples we will focus on independent and normally distributed random numbers. A Latin Hypercube sampling approach [29] is utilized as a benchmark method to investigate the accuracy of the presented analytical approach. Additional information on the uncertainty source could be obtained by variance-based sensitivity measures, which could be estimated for the linearization approach as a direct post-processing result.

In the second part of the paper, we will focus on linear MDOF systems with multiple TMDs. The damping of the main system is considered as modal damping, which allows for a decoupled modal analysis of the individual vibration modes. No further restrictions are made with respect to the system as long as the mass, stiffness and damping matrix can be defined. The application of the TMDs with arbitrary varying parameters will not fulfill the initial assumption of modal damping, and the more general case of viscous damping has to be considered. In the general case, the displacement solution can be obtained by a numerically demanding time integration procedure only, which makes the presented robust design optimization very impractical. For this reason, we will introduce a decoupled stationary solution of the displacements of the individual DOFs of the main system and the relative displacements of the TMDs. With the help of this very efficient approach, the maximum displacements could be obtained for a given range of harmonic excitations similarly to the SDOF system. The presented approach is of course an approximation, but in the numerical example, we will show that for the critical excitation frequencies, this approach agrees very well with the time integration results. Finally, the presented analytical uncertainty propagation approach based on a linearization of the maximum displacements with respect to the random system parameters is applied and an optimal design under the consideration of a defined safety margin could be obtained in a straightforward manner. As the numerical example, a three-story frame with two TMDs is investigated.

The novelty of the paper is the combination of the efficient uncertainty propagation method and its extension for linear MDOF systems with multiple TMDs. The definition of the optimization goals is straightforward and could consider single- and multi-objective optimization problems with and without constraints. The presented procedure is independent with respect to the choice of the optimization algorithms. Therefore, we focus the paper on the dynamical analysis methods and the uncertainty analysis. The mechanical model and the dynamical analysis including the uncertainty quantification approach were implemented in MATLAB R2024b [30] and are freely available as mentioned in the Data Availability Statement of this paper. As optimization algorithms the simplex Nelder–Mead method [31] was chosen for single-objective optimization and the Non-dominated Sort Genetic Algorithm (NSGA II) according to [32] for multi-objective optimization. For both algorithms the implementation in the Ansys optiSLang 2025R1 optimization software package was used [33].

2. Materials and Methods

2.1. Single-Degree-of-Freedom System

2.1.1. Mechanical Model

In the following section, we assume an SDOF system, which could be a simplified mechanical beam model of a pedestrian bridge as shown in Figure 1.

If only a single natural mode of vibration is considered in the analysis, the main system can be modeled as a single-degree-of-freedom (SDOF) system with the following equation of motion [34]:

$$m_H·\ddot{q}\left(t\right)+c_H·\dot{q}\left(t\right)+k_H·q\left(t\right)=F\left(t\right),$$

(1)

where $m_H$, $c_H$ and $k_H$ are the mass, the viscous damping and the stiffness of the SDOF system, respectively. $q\left(t\right)$ is the corresponding displacement of the simplified degree of freedom. Assuming a harmonic excitation,

$$F\left(t\right)=\widehat{F}·sin(\Omega ·t),$$

(2)

where $\widehat{F}$ is the force amplitude and $\Omega$ is the circular frequency of the excitation. The particular (stationary) displacement solution $q_p\left(t\right)$ for the equation of motion reads [7]

$$q_p\left(t\right)={\widehat{q}}_p·sin(\Omega ·t-{\phi }_1),$$

(3)

with

$$\begin{array}{cccccc}\hfill {\widehat{q}}_p& ={\displaystyle \frac{\widehat{F}}{k_H}}{V}_1,\hfill & \hfill & & \hfill & V_1={\displaystyle \frac{1}{\sqrt{{(1-{\eta }^2)}^2+{\left(2{\zeta }_H\eta \right)}^2}}},\hfill \\ \hfill \eta & ={\displaystyle \frac{\Omega }{{\omega }_H}},\hfill & \hfill & & \hfill & tan{\phi }_1={\displaystyle \frac{2{\zeta }_H\eta }{1-{\eta }^2}},\hfill \end{array}$$

(4)

where $V_1$ is the dynamic amplification function, $\eta$ is the frequency ratio of the excitation and ${\phi }_1$ is the phase shift between the excitation and system response. Furthermore, the natural circular frequency of the undamped system and the damping ratio are given as

$${\omega }_H=\sqrt{{\displaystyle \frac{k_H}{m_H}}},{\zeta }_H={\displaystyle \frac{c_H}{2m_H{\omega }_H}}.$$

(5)

A performance measure within the design process could be the maximum value of the dynamic amplification function $V_1$, as it amplifies the ratio of the force amplitude and system stiffness in the resonance case. Figure 2 shows an example of the amplification function assuming a damping ratio of ${\zeta }_H=0.5\%$.

If a tuned mass damper (TMD) is taken into account as the second degree of freedom, as shown in Figure 1, the following equations of motion are obtained:

$$\begin{array}{cc}\hfill m_H·\ddot{q}\left(t\right)+c_H·\dot{q}\left(t\right)+k_H·q\left(t\right)-c_D·\dot{z}\left(t\right)-k_D·z\left(t\right)& =F\left(t\right),\hfill \\ \hfill m_D(\ddot{q}\left(t\right)+\ddot{z}\left(t\right))+c_D·\dot{z}\left(t\right)+k_D·z\left(t\right)& =0,\hfill \end{array}$$

(6)

where $m_D$, $c_D$ and $k_D$ are the mass, the viscous damping and the stiffness of the additional TMD, and $z\left(t\right)$ describes the relative displacement between the main system and the tuned mass damper. Assuming a harmonic excitation $F\left(t\right)$ on the main system, the stationary displacement solutions $q_p\left(t\right)$ for the main system and $z_p\left(t\right)$ for the secondary system read as follows [7]:

$$\begin{array}{cccccc}\hfill q_p\left(t\right)& ={\widehat{q}}_p·sin(\Omega ·t-{\phi }_H),\hfill & \hfill & & \hfill {\widehat{q}}_p& ={\displaystyle \frac{\widehat{F}}{k_H}}{V}_H,\hfill \\ \hfill z_p\left(t\right)& ={\widehat{z}}_p·sin(\Omega ·t-{\phi }_D),\hfill & \hfill & & \hfill {\widehat{z}}_p& ={\displaystyle \frac{\widehat{F}}{k_H}}{V}_D,\hfill \end{array}$$

(7)

where $V_H$ and $V_D$ are the dynamic amplification functions of the main and secondary system and ${\phi }_H$ and ${\phi }_D$ are the phase shifts with respect to the excitation. In [7] the following equations are given:

$$\begin{array}{cccccc}\hfill V_H& =\sqrt{\frac{b_1^2+b_2^2}{b_3^2+b_4^2}},\hfill & \hfill & & \hfill tan{\phi }_H& ={\displaystyle \frac{b_1{b}_4-b_2{b}_3}{b_1{b}_3+b_2{b}_4}},\hfill \\ \hfill V_D& =\sqrt{\frac{{\eta }^4}{b_3^2+b_4^2}},\hfill & \hfill & & \hfill tan{\phi }_D& ={\displaystyle \frac{b_4}{b_3}},\hfill \end{array}$$

(8)

with the coefficients

$$\begin{array}{cc}\hfill b_1& ={\kappa }^2-{\eta }^2,\hfill \\ \hfill b_2& =2\eta \kappa {\zeta }_D,\hfill \\ \hfill b_3& ={\eta }^4-{\eta }^2\left(1+{\kappa }^2+\mu {\kappa }^2+4\kappa {\zeta }_H{\zeta }_D\right)+{\kappa }^2,\hfill \\ \hfill b_4& =\eta \left[2{\zeta }_H({\kappa }^2-{\eta }^2)+2\kappa {\zeta }_D(1-{\eta }^2-\mu {\eta }^2)\right],\hfill \\ \hfill \mu & ={\displaystyle \frac{m_D}{m_H}},\kappa ={\displaystyle \frac{{\omega }_D}{{\omega }_H}},\eta ={\displaystyle \frac{\Omega }{{\omega }_H}}.\hfill \end{array}$$

(9)

Additionally, ${\omega }_D$ and ${\zeta }_D$ are introduced as follows:

$${\omega }_D=\sqrt{{\displaystyle \frac{k_D}{m_D}}},{\zeta }_D={\displaystyle \frac{c_D}{2m_D{\omega }_D}}.$$

(10)

Decisive parameters in the tuning of the TMD are the mass ratio $\mu$, the frequency ratio $\kappa$ and the damping rate ${\zeta }_D$ of the attached vibration damper.

2.1.2. Optimal Tuning of the TMD Parameters

The optimal frequency ratio ${\kappa }_{opt}$ and damping rate ${\zeta }_{D,opt}$ of a vibration damper with deterministic properties are usually formulated as a function of the mass ratio $\mu$ [7]. The optimal parameters for minimum displacements are given according to Den Hartog as

$${\kappa }_{opt}={\displaystyle \frac{1}{1+\mu }},{\zeta }_{D,opt}=\sqrt{{\displaystyle \frac{3\mu }{8{(1+\mu )}^3}}}.$$

(11)

In Figure 2, the amplification functions from Equation (8) for the displacements of the main system and for the relative displacements are shown for different mass ratios. The figure indicates that the maximum values of both amplification functions decrease with an increasing mass ratio.

Alternatively to the Den Hartog formulas, the optimal values can be obtained by mathematical optimization. If the maximum value of the dynamic amplification function of the main system $V_H\left(\eta \right)$ should be minimized for a given maximum mass ratio ${\mu }_{limit}$, the optimization task can be defined as a single-objective optimization problem as follows:

$$\underset{\mu ,\kappa ,{\zeta }_D}{min}\left(\underset{\eta }{max}\left(V_H(\eta ,\mu ,\kappa ,{\zeta }_D)\right)\right),\mathrm{subjected}\mathrm{to}\mu \le {\mu }_{limit}.$$

(12)

Since an increase in the mass ratio $\mu$ always decreases the amplification function values, the mass ratio can be considered constant as the limit value $\mu ={\mu }_{limit}$ and the frequency ratio $\kappa$ and the damping rate ${\zeta }_D$ can be considered as the remaining optimization variables.

As optimization algorithms, gradient-based approaches such as Quasi-Newton methods [35] as well as gradient-free methods can be applied without special adjustments. In this study an extended Nelder–Mead method [31] from the Ansys optiSLang 2025R1 optimization software package [33] is used, which is very efficient for a small number of optimization variables.

The advantage of an optimization approach compared to the analytical formulas according to Den Hartog and others [7] is that additional constraints such as the maximum relative displacement between the main system and the TMD can be considered in a straightforward manner. In the first example in Section 3.1, the amplification function obtained with the optimum parameters according to Den Hartog is compared to the results of a single-objective optimization. For this purpose, different formulations of the objective function are investigated.

However, if we want to minimize the mass ratio of the vibration damper and the maximum values of the amplification function $V_H$ simultaneously, the optimization task can be solved either by gradually adjusting the limit for the mass ratio within a single-objective problem or as a multi-objective problem using Pareto optimization. The objective functions of the Pareto optimization can be formulated as follows:

$$\underset{\mu ,\kappa ,{\zeta }_D}{min}\left(\mu ,\underset{\eta }{max}\left(V_H(\eta ,\mu ,\kappa ,{\zeta }_D)\right)\right),$$

(13)

where $\mu$, $\kappa$ and ${\zeta }_D$ are the optimization variables. In case of conflicting objectives, no unique optimal solution exists and the optimal designs build a so-called Pareto frontier as shown in Figure 3. In our study, we use the Non-dominated Sort Genetic Algorithm (NSGA II) according to [32] as the Pareto optimization algorithm. This algorithm is available in the Ansys optiSLang 2025R1 optimization software package [33] and is used in this study without modification. The NSGA II algorithm uses a sorting of the Pareto dominance as the performance criterion as shown additionally in Figure 3. In Section 3.1 the results of the Pareto optimization are investigated in detail for an SDOF system.

2.1.3. Uncertainty Propagation and Quantification

When the propagation of uncertainty should be analyzed, it is advisable to assume the basic variables of the system as scattering variables since their variation can usually be determined directly. This means that the mass and stiffness coefficients as well as the damping ratios of the main system and the TMD are considered as random numbers, which could be assembled in a random vector

$$\mathbf{X}=\left[m_H,k_H,{\zeta }_H,m_D,k_D,{\zeta }_D\right].$$

(14)

Each random number $X_i$ can be defined as a scalar random variable by a distribution function and statistical moments, such as the mean value and standard deviation. For normally distributed variables, a linear correlation between the random input parameters can be represented with the Gaussian copula in closed form

$$f_{\mathbf{X}}\left(\mathbf{x}\right)={\displaystyle \frac{1}{\sqrt{{\left(2\pi \right)}^p\left|{\mathbf{C}}_{\mathbf{XX}}\right|}}}exp\left[-{\displaystyle \frac{1}{2}}{(\mathbf{x}-\overline{\mathbf{X}})}^T{\mathbf{C}}_{\mathbf{XX}}^{-1}(\mathbf{x}-\overline{\mathbf{X}})\right],$$

(15)

where $f_{\mathbf{X}}\left(\mathbf{x}\right)$ is the joint probability density function of random vector $\mathbf{X}$ and $\overline{\mathbf{X}}$ is the corresponding mean vector and ${\mathbf{C}}_{\mathbf{XX}}$ the covariance matrix. p is the number of scalar random variables assembled in $\mathbf{X}$. The Nataf model [36] can be used to extend the Gaussian correlation model to non-Gaussian distribution types. Further details on correlation models can be found in [37]. In our study we consider independent normally distributed random numbers.

The resulting values of the amplification functions according to Equation (8) for discrete values of the excitation frequency ratio ${\eta }_i$ are defined in the following as scalar random numbers:

$$\begin{array}{cc}\hfill V_{Hi}\left(\mathbf{X}\right)& =V_H({\eta }_i,\mathbf{X}),\hfill \\ \hfill V_{Di}\left(\mathbf{X}\right)& =V_D({\eta }_i,\mathbf{X}).\hfill \end{array}$$

(16)

The statistical properties of the amplification function values can be investigated by sampling methods such as the Monte Carlo Simulation [38], where a certain number of random samples is generated according to the defined distribution of the input parameters. In our study, an improved Latin Hypercube sampling (LHS) according to [29] is applied, where the marginal distributions are represented with respect to the probability contribution, and spurious correlations between the inputs are minimized accordingly as shown in Figure 4. In our study the LHS implementation of the MATLAB R2024b software package [30] is used to generate discrete samples of the random input vector $\mathbf{X}$ from a given mean vector $\overline{\mathbf{X}}$ and a covariance matrix ${\mathbf{C}}_{\mathbf{XX}}$.

The statistical properties of the amplification function values could be analyzed by histograms and the estimates of statistical moments. Additional to the properties of a single output, the dependence with respect to the random input parameters might be interesting. A quite common approach for such a sensitivity analysis is the global variance-based method. In this method, the contribution of the variation in the input parameters with respect to the variation in a certain model output is analyzed. For further details the interested reader is referred to [39].

Since the scatter of the random response values, e.g., the maximum amplification function value, for a given nominal design of the input parameters should be considered in an outer optimization loop, the sampling-based estimation of the response scatter might be numerically demanding. Therefore, a more efficient analytical approach is used in this study, where the individual amplification function values are linearized with respect to the random input parameters using a Taylor series expansion at the nominal values. The corresponding mean values and the variance of the linearized response values can be estimated in closed form. By assuming the optimal deterministic parameter values as mean values

$$\overline{\mathbf{X}}=\left[{\overline{m}}_H,{\overline{k}}_H,{\overline{\zeta }}_H,{\overline{m}}_D,{\overline{k}}_D,{\overline{\zeta }}_D\right],$$

(17)

the random amplification function values in Equation (16) can be linearized as follows:

$$V_{Hi}\left(\mathbf{X}\right)\approx V_{Hi}^{lin}\left(\mathbf{X}\right)=V_{Hi}\left(\overline{\mathbf{X}}\right)+{\left.{\left({\displaystyle \frac{\partial V_{Hi}\left(\mathbf{X}\right)}{\partial \mathbf{X}}}\right)}^T\right|}_{\overline{\mathbf{X}}}\left(\mathbf{X}-\overline{\mathbf{X}}\right).$$

(18)

Based on this linearization, the mean value and the variance of each amplification function value can be obtained from the standard deviation ${\sigma }_{X_k}$ and the correlation coefficients ${\rho }_{kl}$ of the p random inputs $X_k$ as follows:

$$\begin{array}{cc}\hfill {\overline{V}}_{Hi}^{lin}& =V_{Hi}\left(\overline{\mathbf{X}}\right),\hfill \\ \hfill {\sigma }_{V_{Hi}^{lin}}^2& =\sum _k^p\sum _l^p{\displaystyle \frac{\partial V_{Hi}\left(\mathbf{X}\right)}{\partial X_k}}{\displaystyle \frac{\partial V_{Hi}\left(\mathbf{X}\right)}{\partial X_l}}{\sigma }_{X_k}{\sigma }_{X_l}{\rho }_{kl}.\hfill \end{array}$$

(19)

The required derivatives in Equation (18) are obtained in this study by the central difference method. In case of uncorrelated inputs, Equation (19) simplifies as follows:

$$\begin{array}{c}\hfill {\overline{V}}_{Hi}^{lin}=V_{Hi}\left(\overline{\mathbf{X}}\right),{\sigma }_{V_{Hi}^{lin}}^2=\sum _k^p{\left({\displaystyle \frac{\partial V_{Hi}\left(\mathbf{X}\right)}{\partial X_k}}\right)}^2{\sigma }_{X_k}^2.\end{array}$$

(20)

2.1.4. Optimization Under Uncertainty

By considering uncertain input parameters in the optimization task, we have to distinguish between purely random inputs and design variables, which could be random as well [21]. Let us consider the design variables for the SDOF system similar to the deterministic optimization with the nominal values of $\mu$, $\kappa$ and ${\zeta }_D$

$$\mathbf{d}=\left[{\mu }_d,{\kappa }_d,{\zeta }_{D,d}\right].$$

(21)

The corresponding random numbers for the TMD coefficients are

$${\mathbf{X}}_D=\left[m_D,k_D,{\zeta }_D\right],$$

(22)

where the mean values are adapted by the design variables

$${\overline{m}}_D={\mu }_d·{\overline{m}}_H,{\overline{k}}_D={\mu }_d·{\kappa }_d^2·{\overline{k}}_H,{\overline{\zeta }}_D={\zeta }_{D,d}.$$

(23)

The standard deviation of each parameter could be assumed either as a constant value or could be obtained from the current mean value and a given Coefficient of Variation (CoV)

$${\sigma }_{m_D}={\overline{m}}_D·CoV\left(m_D\right),{\sigma }_{k_D}={\overline{k}}_D·CoV\left(k_D\right),{\sigma }_{{\zeta }_D}={\overline{\zeta }}_D·CoV\left({\zeta }_D\right).$$

(24)

The statistical properties of the pure random parameters of the main system

$${\mathbf{X}}_H=\left[m_H,k_H,{\zeta }_H\right]$$

(25)

remain constant during the optimization.

In Reliability-based Design optimization, the objective function is usually formulated in terms of the deterministic design variables $\mathbf{d}$. Statistical constraints are introduced to consider certain quality requirements [21]

$$\underset{\mathbf{d}}{min}f\left(\mathbf{d}\right),\mathrm{subjected} \mathrm{to}{P}_{F_i}=P[g_i({\mathbf{X}}_D,{\mathbf{X}}_H)\le 0]\le P_{F_i}^{target},$$

(26)

where $g_i({\mathbf{X}}_D,{\mathbf{X}}_H)$ are limit state functions depending on the joint set of ${\mathbf{X}}_D$ and ${\mathbf{X}}_H$. $P_{F_i}$ is the corresponding failure probability, which must not exceed a given target value. In Figure 5 a single random response Y is shown with an indicated limit $Y_{limit}$.

The limit state function can be formulated in this case as follows:

$$g_Y\left(\mathbf{X}\right)=Y_{limit}-Y\left(\mathbf{X}\right).$$

(27)

The evaluation of the failure probability $P_F$ requires an integration of the joint probability density function $f_{\mathbf{X}}$ over the failure domain

$$\begin{array}{cc}\hfill P_F& =P\left[g\left(\mathbf{X}\right)\le 0\right]=\underset{g\left(\mathbf{X}\right)\le 0}{\int \dots \int }{f}_{\mathbf{X}}\left(\mathbf{x}\right)d\mathbf{x},\hfill \end{array}$$

(28)

which might be numerically very demanding for the general case [37].

Within the Design-for-Six-Sigma approach, the optimization constraints are typically formulated in terms of the safety margin of the random output [40]

$$\overline{Y}+\alpha ·{\sigma }_Y\le Y_{limit},$$

(29)

which is usually defined as the standard deviation times a required sigma level $\alpha$. Different procedures for this so-called robust design optimization (RDO) are discussed in [41] and more recently in [21]. Typically, a double-loop approach is necessary, where the outer loop performs the optimization and an inner loop estimates the output mean values and standard deviations with respect to the current nominal values of the design variables.

For the optimization of the TMD parameters, we will apply the linearization approach to estimate the mean value and the standard deviation of the individual amplification function values $V_{Hi}$ and $V_{Di}$. If a certain limit for the dynamic amplification is given, the optimization task can be formulated in terms of a minimization of the nominal TMD mass ratio as follows:

$$\begin{array}{cc}\hfill \underset{\mathbf{d}}{min}{\mu }_d,\mathrm{subjected} \mathrm{to}& \underset{i}{max}({\overline{V}}_{Hi}+\alpha ·{\sigma }_{V_{Hi}})\le V_H^{limit}\hfill \\ \hfill & \underset{i}{max}({\overline{V}}_{Di}+\alpha ·{\sigma }_{V_{Di}})\le V_D^{limit},\hfill \end{array}$$

(30)

where the constraints could be formulated alternatively in terms of the maximum displacements by using Equation (7). If no limit values for the amplification function or displacements are defined, we could either minimize the mass ratio and the maximum amplification values within a multi-objective optimization

$$\begin{array}{c}\hfill \underset{\mathbf{d}}{min}\left({\mu }_d,\underset{i}{max}({\overline{V}}_{Hi}+\alpha ·{\sigma }_{V_{Hi}})\right),\end{array}$$

(31)

or the mass ratio could be kept fixed and the maximum amplification values are minimized. The scaling factor $\alpha$, which defines the safety margin in terms of the standard deviation, is chosen in this study according to the probability levels for the serviceability limit state $\beta \le 2.9$ and the ultimate limit state $\beta \le 4.7$ according to the European design code [42].

2.2. Extension for Multi-Degree-of-Freedom Systems

2.2.1. Mechanical Model

The equation of motion for a linear multi-degree-of-freedom (MDOF) system reads in matrix-vector notation as follows:

$$\begin{array}{c}\hfill {\mathbf{M}}_H·{\ddot{\mathbf{q}}}_H\left(t\right)+{\mathbf{C}}_H·{\dot{\mathbf{q}}}_H\left(t\right)+{\mathbf{K}}_H·{\mathbf{q}}_H\left(t\right)=\mathbf{f}\left(t\right),\end{array}$$

(32)

where the mass matrix ${\mathbf{M}}_H$, the viscous damping matrix ${\mathbf{C}}_H$ and the stiffness matrix ${\mathbf{K}}_H$ correspond to an initial main system without a TMD. $\mathbf{f}\left(t\right)$ is the external force vector acting on the individual DOFs. The displacement vector ${\mathbf{q}}_H\left(t\right)$ contains the displacements of all degrees of freedom of the original main system. For the free vibration mode of the undamped system, the natural circular frequencies ${\omega }_{H_i}$ and the corresponding mode shapes ${\mathit{\varphi }}_i$ can be obtained by solving the following eigenvalue problem:

$$\begin{array}{c}\hfill \left({\mathbf{K}}_H-{\omega }_{H_i}^2·{\mathbf{M}}_H\right)·{\mathit{\varphi }}_i=\mathbf{0}.\end{array}$$

(33)

The mode shapes are orthogonal to each other

$${\mathit{\varphi }}_i^T·{\mathbf{M}}_H·{\mathit{\varphi }}_j=0,{\mathit{\varphi }}_i^T·{\mathbf{K}}_H·{\mathit{\varphi }}_j=0\forall i\ne j,$$

(34)

and could be normalized arbitrarily. In our study, we assume a modal damping according to [43]

$${\mathit{\varphi }}_i^T·{\mathbf{C}}_H·{\mathit{\varphi }}_j=0\forall i\ne j,{\mathit{\varphi }}_i^T·{\mathbf{C}}_H·{\mathit{\varphi }}_i={\tilde{c}}_i,$$

(35)

which leads to decoupled equations of motion in the modal space

$$\begin{array}{ccccccc}\hfill {\mathit{\varphi }}_i^T{\mathbf{M}}_H{\mathit{\varphi }}_i·{\ddot{\tilde{q}}}_{H_i}\left(t\right)& +\hfill & \hfill {\mathit{\varphi }}_i^T{\mathbf{C}}_H{\mathit{\varphi }}_i·{\dot{\tilde{q}}}_{H_i}\left(t\right)& +\hfill & \hfill {\mathit{\varphi }}_i^T{\mathbf{K}}_H{\mathit{\varphi }}_i·{\tilde{q}}_{H_i}\left(t\right)& =\hfill & \hfill {\mathit{\varphi }}_i^T\mathbf{f}\left(t\right),\\ \hfill {\tilde{m}}_i·{\ddot{\tilde{q}}}_{H_i}\left(t\right)& +\hfill & \hfill {\tilde{c}}_i·{\dot{\tilde{q}}}_{H_i}\left(t\right)& +\hfill & \hfill {\tilde{k}}_i·{\tilde{q}}_{H_i}\left(t\right)& =\hfill & \hfill {\tilde{f}}_i\left(t\right),\end{array}$$

(36)

where ${\tilde{m}}_i$, ${\tilde{c}}_i$ and ${\tilde{k}}_i$ are the modal mass, modal damping and modal stiffness for the mode shape ${\mathit{\varphi }}_i$, and ${\tilde{f}}_i\left(t\right)$ is the corresponding excitation force. The displacement solution in the n original degrees of freedom can be obtained by a superposition of the decoupled modal solutions as follows:

$${\mathbf{q}}_H\left(t\right)=\sum _{i=1}^n{\mathit{\varphi }}_i·{\tilde{q}}_{H_i}\left(t\right).$$

(37)

If we extend the MDOF system by several tuned mass dampers, which are coupled directly to certain degrees of freedom of the initial main system, we get a coupled damping and stiffness matrix in the equation of motion

$$\begin{array}{cc}\left[\begin{array}{cc}{\mathbf{M}}_H& \mathbf{0}\\ \mathbf{0}& {\mathbf{M}}_D\end{array}\right]·\left[\begin{array}{c}{\ddot{\mathbf{q}}}_H\left(t\right)\\ {\ddot{\mathbf{q}}}_D\left(t\right)\end{array}\right]\hfill & +\left[\begin{array}{cc}{\mathbf{C}}_H+{\mathbf{C}}_{DH}& {\mathbf{C}}_{DC}\\ {\mathbf{C}}_{DC}^T& {\mathbf{C}}_{DD}\end{array}\right]·\left[\begin{array}{c}{\dot{\mathbf{q}}}_H\left(t\right)\\ {\dot{\mathbf{q}}}_D\left(t\right)\end{array}\right]\hfill \\ & +\left[\begin{array}{cc}{\mathbf{K}}_H+{\mathbf{K}}_{DH}& {\mathbf{K}}_{DC}\\ {\mathbf{K}}_{DC}^T& {\mathbf{K}}_{DD}\end{array}\right]·\left[\begin{array}{c}{\mathbf{q}}_H\left(t\right)\\ {\mathbf{q}}_D\left(t\right)\end{array}\right]=\left[\begin{array}{c}\mathbf{f}\left(t\right)\\ \mathbf{0}\end{array}\right],\hfill \end{array}$$

(38)

where ${\mathbf{C}}_{DH}$, ${\mathbf{C}}_{DC}$ and ${\mathbf{C}}_{DD}$ as well as ${\mathbf{K}}_{DH}$, ${\mathbf{K}}_{DC}$ and ${\mathbf{K}}_{DD}$ are sub-matrices which represent the damping and stiffness coefficients of the tuned mass dampers and their association to the coupled degrees of freedom of the main system. ${\mathbf{q}}_H\left(t\right)$ is the displacement vector of the original DOFs of the main system and ${\mathbf{q}}_D\left(t\right)$ are the displacements of the additional DOFs of the TMDs. The excitation force vector is assumed to act only on the main system DOFs. This extended equation of motion covers arbitrary structures with symmetric mass, damping and stiffness matrices of the original structure, including the TMDs. The corresponding DOFs are ordered with respect to the main system DOFs and additional DOFs of the TMDs. Since the TMDs are coupled to the original structure only by viscous damper and spring elements, the mass coupling between the main system and the TMDs is zero as indicated by the zero off-diagonal mass sub-matrices.

For arbitrary values of the TMD coefficients, Equation (38) does not necessarily fulfill the assumption of modal damping and a decoupled solution could not be obtained in the general case of viscous damping. Therefore, time integration methods such as the Newmark-Beta approach [43] are necessary to solve Equation (38). However, this procedure is very demanding from the computational point of view. Especially, if the maximum amplification function of a certain degree of freedom should be considered in the objective function, a sweep over the excitation frequency has to be considered.

2.2.2. Approximated Decoupled Stationary Solution

In order to decrease the numerical effort, we assume a decoupled influence of each TMD by designing it only for a single vibration mode of the main system. For example, if we would assume a tuning for the first mode shape, we would approximate the displacement solution as the super position of a 2-DOF system for the first mode and several SDOF systems for the higher modes as illustrated in Figure 6.

The stationary displacements of the original DOFs can be assembled for this assumption as follows:

$${\mathbf{q}}_{p,H}\left(t\right)=\sum _{i=1}^n{\mathit{\varphi }}_i·{\tilde{q}}_{p,H_i}\left(t\right),$$

(39)

where the decoupled solution of an individual vibration mode reads in case of an application of a TMD

$${\tilde{q}}_{p,H_i}\left(t\right)={\displaystyle \frac{{\widehat{\tilde{f}}}_i}{{\tilde{k}}_i}}·V_{H,i}·sin(\Omega ·t-{\phi }_{H,i}).$$

(40)

The modal stiffness ${\tilde{k}}_i$ corresponds to the main system and the force amplitude ${\widehat{\tilde{f}}}_i$ results from a harmonic excitation as follows:

$${\tilde{f}}_i\left(t\right)={\mathit{\varphi }}_i^T\widehat{\mathbf{f}}·sin(\Omega ·t)={\widehat{\tilde{f}}}_i·sin(\Omega ·t).$$

(41)

The amplification function $V_{H,i}$ and the phase shift ${\phi }_{H,i}$ have to consider the modal mass and modal damping as well as the corresponding undamped circular frequency of the main system for the calculation according to Equation (7)

$$\begin{array}{cccccc}\hfill V_{H,i}& =V_H({\eta }_i,{\mu }_i,{\kappa }_i,{\zeta }_{H_i},{\zeta }_{D_i}),\hfill & \hfill & & \hfill {\eta }_i& ={\displaystyle \frac{\Omega }{{\omega }_{H_i}}},\hfill \\ \hfill {\phi }_{H,i}& ={\phi }_H({\eta }_i,{\mu }_i,{\kappa }_i,{\zeta }_{H_i},{\zeta }_{D_i}),\hfill & \hfill & & \hfill {\kappa }_i& ={\displaystyle \frac{{\omega }_{D_i}}{{\omega }_{H_i}}},{\mu }_i={\displaystyle \frac{m_{D_i}}{{\tilde{m}}_i}}.\hfill \end{array}$$

(42)

The corresponding mode shape ${\mathit{\varphi }}_i$ has to be normalized with respect to the DOF, where the TMD is applied. For the other vibration modes, where no TMD is applied, the decoupled stationary displacement solution reads

$${\tilde{q}}_{p,H_j}\left(t\right)={\displaystyle \frac{{\widehat{\tilde{f}}}_j}{{\tilde{k}}_j}}·V_{1,j}·sin(\Omega ·t-{\phi }_{1,j}),\forall j\ne i,$$

(43)

where $V_{1,j}$ and ${\phi }_{1,j}$ are the amplification function and the phase shift of a standard SDOF system without a TMD according to Equation (4)

$$V_{1,j}=V_1({\eta }_j,{\zeta }_{H_j}),{\phi }_{1,j}={\phi }_1({\eta }_j,{\zeta }_{H_j}).$$

(44)

No special normalization is necessary for these mode shapes without TMDs. The maximum stationary displacements of the orginal DOFs can be obtained from the assembled stationary displacements according to Equation (39) for one vibration period while considering the individual phase shifts in Equations (40) and (43). The relative displacements of the TMDs with respect to the coupling DOFs result in the decoupled stationary solution similar to the main system as

$$z_{p,D_i}\left(t\right)={\displaystyle \frac{{\widehat{\tilde{f}}}_i}{{\tilde{k}}_i}}·V_{D,i}·sin(\Omega ·t-{\phi }_{D,i}),$$

(45)

with

$$V_{D,i}=V_D({\eta }_i,{\mu }_i,{\kappa }_i,{\zeta }_{H_i},{\zeta }_{D_i}),{\phi }_{D,i}={\phi }_D({\eta }_i,{\mu }_i,{\kappa }_i,{\zeta }_{H_i},{\zeta }_{D_i}).$$

(46)

In the second example, we will show that the approximated displacement solution agrees very well with the results from a time integration and can be used to estimate the maximum displacements for the full excitation frequency range very efficiently.

2.2.3. Uncertainty Quantification

For the uncertainty quantification, we introduce the maximum stationary displacements of the original DOFs $q_{H_n}$ and maximum relative displacements of the TMDs $z_{D_i}$ for a discrete value of the excitation frequency ${\Omega }_k$ as scalar random outputs

$$Q_{H_{n,k}}\left(\mathbf{X}\right)=\underset{t}{max}\left(\right|q_{H_n}(t,{\Omega }_k,\mathbf{X})\left|\right),Z_{D_{i,k}}\left(\mathbf{X}\right)=\underset{t}{max}\left(\right|z_{D_i}(t,{\Omega }_k,\mathbf{X})\left|\right),$$

(47)

where the linearization approach with respect to the random input vector $\mathbf{X}$ can be applied similarly to the SDOF system

$$Q_{H_{n,k}}\left(\mathbf{X}\right)\approx Q_{H_{n,k}}^{lin}\left(\mathbf{X}\right)=Q_{H_{n,k}}\left(\overline{\mathbf{X}}\right)+{\left.{\left({\displaystyle \frac{\partial Q_{H_{n,k}}\left(\mathbf{X}\right)}{\partial \mathbf{X}}}\right)}^T\right|}_{\overline{\mathbf{X}}}\left(\mathbf{X}-\overline{\mathbf{X}}\right).$$

(48)

The variance of these outputs can be directly estimated using Equation (20). The random input vector contains the random inputs of the main system ${\mathbf{X}}_H$, which define the scatter of the MDOF system itself, and the random properties of the m TMDs

$$\mathbf{X}=\left[{\mathbf{X}}_H,m_{D_1},k_{D_1},{\zeta }_{D_1},\dots ,m_{D_m},k_{D_m},{\zeta }_{D_m}\right].$$

(49)

2.2.4. Optimization Under Uncertainty

As design variables within the robust design optimization, we consider the nominal values of the mass ratio ${\mu }_{D_i}$, the nominal circular frequencies ${\omega }_{D_i}$ and the nominal damping ratio ${\zeta }_{D_i}$ of each TMD

$$\mathbf{d}=\left[{\mu }_{D_1,d},{\omega }_{D_1,d},{\zeta }_{D_1,d},\dots ,{\mu }_{D_m,d},{\omega }_{D_m,d},{\zeta }_{D_m,d}\right].$$

(50)

The choice of design variables is based on the investigations in [44] and is more suitable than the basic mass, stiffness and damping coefficients in the equation of motion.

The objective functions within the robust design optimization are defined similarly to the SDOF system, but the total mass of all TMDs and the safety margin of specific displacements of the main system could be considered for an arbitrary discrete set of excitation frequencies ${\Omega }_k$

$$\underset{\mathbf{d}}{min}\left(\sum _{i=1}^m{m}_{D_i},\underset{k}{max}({\overline{Q}}_{H_{n,k}}+\alpha ·{\sigma }_{Q_{H_{n,k}}})\right).$$

(51)

Additional constraints with respect to the TMD relative displacements could be formulated in a straightforward manner.

3. Results

3.1. Single-Degree-of-Freedom Example

3.1.1. Deterministic Optimization

In the first example we investigated an SDOF system with a TMD as shown in Figure 1. The parameters for the main system have been chosen as examples and are given in

Table 1. SDOF system with TMD: reference and optimization bounds and results of the single-objective optimization of $min\left(max\left(V_H\left(\eta \right)\right)\right)$.

Parameter	Unit	Reference	Den Hartog	Single-Objective		Multi-Objective
				Bounds	Optimum
$m_H$	kg	${10}^4$	${10}^4$	${10}^4$	${10}^4$	${10}^4$
$k_H$	kN/m	${10}^3$	${10}^3$	${10}^3$	${10}^3$	${10}^3$
${\zeta }_H$	-	$0.005$	$0.005$	$0.005$	$0.005$	$0.005$
$\mu$	-	-	$0.020$	$0.020$	$0.020$	0.00–0.20
$\kappa$	-	-	$0.980$	0.70–1.20	$0.979$	0.70–1.20
${\zeta }_D$	-	-	$0.084$	0.00–0.40	$0.087$	0.00–0.40

. Additionally, the optimal parameters according to Den Hartog are given for a mass ratio of $\mu =2.0\%$. In the first step we performed a single-objective optimization with the objective function

$$\underset{\kappa ,{\zeta }_D}{min}\left(\underset{\eta }{max}\left(V_H(\eta ,\mu =0.02,\kappa ,{\zeta }_D)\right)\right).$$

The parameter bounds for the optimization variables $\kappa$ and ${\zeta }_D$ are given in Table 1. In Figure 7 the objective function is shown within the parameter bounds. The figure clearly indicates a unimodal objective function with a single optimum. As the optimization algorithm, the simplex method [31] from the Ansys optiSLang 2025R1 software package [33] was utilized. The optimizer converged within 55 model evaluations. Further details including the full optiSLang projects with the optimization settings and all the results can be found in the accompanying data set. The obtained parameter values of the optimal design agree very well with the optimal values according to Den Hartog as shown in Table 1. The corresponding amplification functions drawn in Figure 8 show very good agreement.

In the second step, we investigated a different objective function by considering the amplification value at the resonance frequency only

$$\underset{\kappa ,{\zeta }_D}{min}\left(V_H(\eta =1.0,\mu =0.02,\kappa ,{\zeta }_D)\right).$$

The obtained amplification function is shown additionally in Figure 7 and Figure 8. As indicated in the second figure, the optimized SDOF+TMD system has two significant resonances which have an amplification close to the original SDOF system. Thus, a different excitation frequency would lead to similar displacements as those without a TMD. Even the relative displacements of the TMD with respect to the main system would be very large compared to the values obtained with the Den Hartog parameters. From this finding, we can summarize that the consideration of a single specific excitation frequency might not lead to optimal amplification functions. Therefore, we consider the maximum amplification function as an objective in the following investigations.

In the next step, a multi-objective optimization was performed with two objective functions,

$$\underset{\mu ,\kappa ,{\zeta }_D}{min}\left(\mu ,\underset{\eta }{max}\left(V_H(\eta ,\mu ,\kappa ,{\zeta }_D)\right)\right),$$

by considering the parameter bounds given in Table 1. The NSGA-II method [32] of the Ansys optiSLang 2025R1 software package was used with 50 generations, each having a population size of 50 in order to assure a good convergence. Further details including the optimization settings and results can be found in the accompanying data set.

Figure 9 compares the resulting Pareto front of both objective functions and the results of several single-objective optimization runs with the optimal values according to Den Hartog. The relationship between the optimal mass ratio $\mu$ and the corresponding values of the frequency ratio $\kappa$ and the damping ratio ${\zeta }_D$ is particularly interesting. The scatter in the parameter values results from the stochastic nature of the NSGA-II. The optimal damping ratio according to Den Hartog is visibly lower than that of the optimizations, whereby the maximum values of the magnification functions hardly differ. As a result of the multi-objective optimization, a suitable choice for the mass ratio could be made: the range of about a 2% to 5% mass ratio represents a good compromise between the conflicting objective functions. The relationship between the maximum values of the relative displacement is analogous to that of the main system. However, a limitation of the installation space and thus of the maximum displacements could be considered directly as a constraint in the single-objective and multi-objective optimization.

3.1.2. Uncertainty Quantification

In the next step, uncertain input parameters were investigated for the given SDOF system. The input parameters were assumed to be independent and normally distributed as given in

Table 2. Statistical properties of the scattering input parameters for the SDOF example with mean values according to the Den Hartog parameters.

Parameter	Unit	Mean Value	Coefficient of Variation	Distribution Type
$m_H$	kg	${10}^4$	$1.0\%$	Normal
$k_H$	kN/m	${10}^3$	$2.0\%$	Normal
${\zeta }_H$	-	$0.005$	$5.0\%$	Normal
$m_D$	kg	$200.0$	$1.0\%$	Normal
$k_D$	kN/m	$19.22$	$2.0\%$	Normal
${\zeta }_D$	-	$0.084$	$5.0\%$	Normal

. The mean values were assumed as the optimal Den Hartog parameters for a mass ratio of 2% as given in Table 1. First, an improved Latin Hypercube sampling [29] with 1000 samples was generated as the reference with the MATLAB R2024b software package [30] and the corresponding amplification functions for each sample were calculated. The frequency ratio was discretized with 200 equidistant values within the interval $\eta \in [0.8;1.2]$. The samples and corresponding amplification function values are available in the accompanying data set.

Figure 10 shows the first 100 samples which indicate a significant scattering of the magnification function in the range of the two maximum values corresponding to the natural frequencies.

In Figure 11 the sample estimates of the mean value and standard deviation for each amplification value $V_{Hi}$ and $V_{Di}$ calculated for every frequency ratio ${\eta }_i$ are shown. Additionally, the estimated mean and standard deviation of $V_{Hi}$ and $V_{Di}$ using the analytical linearization approach are plotted. The derivatives required in Equation (18) have been estimated using the central difference method, which required only 12 model calls for the six input parameters. The corresponding derivation interval has been chosen as $1\%$ of the nominal parameter values. The figure indicates very good agreement of both approaches, which indicates a sufficient representation of the scatter of the individual random response values $V_{Hi}$ and $V_{Di}$ by the linearization for the single-degree-of-freedom system.

Additionally, the linearized variance contribution ${\left({\displaystyle \frac{\partial V_H({\eta }_i,\mathbf{X})}{\partial X_k}}\right)}^2{\sigma }_{X_k}^2$ of the input parameters for the individual amplification values was estimated using Equation (20) and is plotted in Figure 12. The figure indicates a significant influence of the stiffness values for the values with the maximum scatter. The influence of the damping of the main system is negligible and the damping of the TMD is mostly significant where the contribution of the mass and stiffness scatter is smallest.

3.1.3. Optimization Under Uncertainty

Finally, the TMD for the SDOF system was optimized considering uncertain input parameters. As the objective function, Equation (31) was considered, whereas the nominal values of $\kappa$ and ${\zeta }_D$ were taken as the optimization variables and the nominal mass ratio was kept as $\mu =2.0\%$. The input scatter was defined by the Coefficients of Variation given in Table 2. The mean values and the standard deviation of the individual amplification function values were estimated by the linearization approach. Figure 13 shows the corresponding maximum values of the mean and standard deviation as a function of the nominal values of $\kappa$ and ${\zeta }_D$. The figure indicates that a decreasing standard deviation correlates with an increasing mean value. Thus, a combination of both measures forms a compromise between the nominal outputs and the scatter of the results. Similar to the deterministic case, the optimization was performed with the simplex method using Ansys optiSLang 2025R1. Further details including the optimization settings and results can be found in the accompanying data set. The $\alpha$-factor for the standard deviation in Equation (31) was assumed with $\alpha =2.9$ and $\alpha =4.7$. Figure 13 shows the optimal values obtained for both cases in comparison to the deterministic solution. It is clearly recognizable that for a higher weighting of the scattering, the optimal parameters tend towards higher damping values.

The amplification function values from the deterministic optimization and the $4.7\sigma$ optimum are compared in Figure 14, where the mean value curves and the mean values + 4.7 times the standard deviation are plotted. The figure clearly indicates an increased maximum mean value of the $4.7\sigma$ optimum compared to the deterministic solution, but the scatter is significantly smaller and thus a reduction in the statistical limit is possible.

As the final investigation for the SDOF example, the single-objective robust design optimization was performed for different nominal values of the mass ratio $\mu$ and different safety margins $\alpha$ by using again the simplex method. In Figure 15 the obtained maximum values for the mean and standard deviation as well as the corresponding nominal values for $\kappa$ and ${\zeta }_d$ are shown, whereby, for $\alpha =0.0$, the nominal values correspond to the deterministic solution presented in Section 3.1.1. The figure indicates that, with an increasing safety margin, the mean values increase, but the standard deviation decreases. Furthermore, the damping ratio increases. This effect is most dominant for small mass ratios, where the influence of the uncertain input parameters is most critical.

3.2. Multi-Degree-of-Freedom Example with Multiple Tuned Mass Dampers

3.2.1. Deterministic Analysis

In the second example, a multi-degree-of-freedom system as shown in Figure 16 was investigated. The deterministic values for the stiffnesses, masses and damping ratios have been taken according to the mean values in

Table 3. Statistical properties of the input parameters for the MDOF example.

Parameter	Unit	Mean Value	Coefficient of Variation	Distribution Type
$m_1$,$m_2$,$m_3$	kg	${10}^4$	$1.0\%$	Normal
$k_1$,$k_2$,$k_3$	kN/m	${10}^3$	$2.0\%$	Normal
${\zeta }_{H_1}$	-	$0.010$	$5.0\%$	Normal
${\zeta }_{H_2}$	-	$0.015$	$5.0\%$	Normal
${\zeta }_{H_3}$	-	$0.020$	$5.0\%$	Normal
$m_{D_1}$	kg	$500.0$	$1.0\%$	Normal
$m_{D_2}$	kg	$500.0$	$1.0\%$	Normal
$k_{D_1}$	kN/m	$9.386$	$2.0\%$	Normal
$k_{D_2}$	kN/m	$73.69$	$2.0\%$	Normal
${\zeta }_{D_1}$	-	$0.097$	$5.0\%$	Normal
${\zeta }_{D_2}$	-	$0.097$	$5.0\%$	Normal

. The damping was assumed as modal damping as explained in Section 2.2.1. The three natural circular frequencies of the main system are ${\omega }_{H_1}=4.45{\displaystyle \frac{1}{s}}$, ${\omega }_{H_2}=12.47{\displaystyle \frac{1}{s}}$ and ${\omega }_{H_3}=18.02{\displaystyle \frac{1}{s}}$ and the corresponding mode shape are shown in Figure 17. The amplification functions for a harmonic excitation $F\left(t\right)$ are plotted in Figure 18. The peak values decrease for the larger natural frequencies due to the higher damping ratios. The TMDs have been designed for the first and second vibration mode and are coupled to the DOF 3 and 1, respectively, which have the maximum value of the corresponding mode shape. Both mode shapes ${\mathit{\varphi }}_1$ and ${\mathit{\varphi }}_2$ were normalized with respect to the coupling DOFs $q_{H_3}$ and $q_{H_1}$ as shown in Figure 17, which results in the modal masses of ${\tilde{m}}_1={\tilde{m}}_2=18411.7$ kg. The optimal parameters for the nominal mass of the TMDs are taken according to Den Hartog as ${\mu }_1={\mu }_2=0.027$, ${\kappa }_1={\kappa }_2=0.974$ and ${\zeta }_{D_1}={\zeta }_{D_2}=0.097$, which results in the nominal stiffnesses given in Table 3. The corresponding amplification functions for the first and second modes including the TMDs are shown additionally in Figure 18.

In a first step, the accuracy of the approximated displacement solution using the decoupled stationary approach according to Section 2.2.2 was investigated. A harmonic excitation at DOF $q_{H_3}$ was assumed as $F\left(t\right)=1kN·sin(\Omega ·t)$. As the benchmark method, the Newmark time integration was used for the full 5-DOF system including the TMDs within a time range of 100 excitation periods and a time step of ${\displaystyle \frac{1}{200}}$ of the smallest free vibration period of the initial main system. Figure 18 shows the displacements of DOF $q_{H_3}$ for the first 25 excitation periods with $\Omega =4.35{\displaystyle \frac{1}{s}}$. The figure indicates very good agreement of the approximated stationary solution with the Newmark results in the steady state.

This analysis was repeated for different excitation frequencies, and the maximum amplitudes of the Newmark results have been extracted from the last 25 out of 100 excitation periods. In Figure 19 the vibration amplitudes in the steady state are compared for the displacements of all DOFs of the main system as well as for the relative displacements $z_{D_1}$ and $z_{D_2}$ of the TMDs. Additionally, the maximum values of the drift displacement at the second floor $q_{H_3}-q_{H_2}$ are plotted. The figure indicates very good agreement for the main DOFs. Small deviations could be observed for the drift displacements. The relative displacements of the TMDs are represented very well for the corresponding first and second vibration resonance, but larger deviations could be observed at the additional peaks for the other resonances. This is the case since no interaction of the TMDs with the other vibration modes was considered in the stationary approach. However, the maximum relative displacements could be approximated sufficiently.

3.2.2. Uncertainty Quantification

In the next step the uncertainty propagation was investigated for the MDOF example by using 1000 Latin Hypercube samples and the linearization approach according to Equation (48). The samples were generated again with the MATLAB R2024b software package [30]. The random input parameters are assumed as normally distributed independent random variables with mean values and the CoV according to Table 3. As model responses, the maximum displacements of the three original DOFs and the maximum drift $q_{H_3}-q_{H_2}$ between the second and third DOF were evaluated. The estimated mean values and standard deviations are compared for both approaches depending on the excitation frequency in Figure 20 for the original DOFs and in Figure 21 for the maximum drift. The figure indicates very good agreement in the range of the first two natural frequencies, where the TMDs are active. Larger deviations could be observed in the range of the third natural frequency. The variance contribution of the inputs indicate a significant influence of the stiffness and mass coefficients of the main system as well as the modal damping ${\zeta }_3$ on this displacement variation. However, the scatter of the TMD coefficients is significant only for the variation in the drift displacement for an excitation frequency around the first and second natural frequency.

3.2.3. Optimization Under Uncertainty

In the final analysis, a robust design optimization was performed with different displacement measures. First, the total mass of both TMDs was minimized and the mean value and standard deviation of the maximum displacements at the third main DOF ${\displaystyle Q_{H_{3,k}}\left(\mathbf{X}\right)=\underset{t}{max}\left(\right|q_{H_3}(t,{\Omega }_k,\mathbf{X})\left|\right)}$ were considered in the second objective:

$$\underset{\mathbf{d}}{min}\left(m_{D_1}+m_{D_2},\underset{0<{\Omega }_k\le 15{\displaystyle \frac{1}{s}}}{max}({\overline{Q}}_{H_{3,k}}+\alpha ·{\sigma }_{Q_{H_{3,k}}})\right).$$

The evaluated discrete excitation frequency values were limited to the range of $0<{\Omega }_k\le 15{\displaystyle \frac{1}{s}}$ to consider only the first two resonance frequencies in the tuning of the TMD parameters. The maximum displacements for the third resonance frequency remain almost constant if the TMD parameters are modified within the optimization. In Figure 22 the maximum mean values and standard deviations of the displacements are shown, dependent on the total mass of the TMDs for the deterministic optimization with $\alpha =0$ and for the optimization under uncertainty assuming $\alpha =4.7$. As in the SDOF example, a significant reduction in the scatter can be observed for the RDO case while the mean values are slightly increased. The TMD tuned for the second vibration mode gets more mass in the RDO case since the scatter of the displacements around the second resonance is larger than for the first resonance frequency as shown in Figure 23.

Second, we considered the maximum drift in the upper floor

$$D_{H_{2-3,k}}\left(\mathbf{X}\right)=\underset{t}{max}\left(\right|q_{H_2}(t,{\Omega }_k,\mathbf{X}))-q_{H_3}(t,{\Omega }_k,\mathbf{X})\left)\right|)$$

in the second objective

$$\underset{\mathbf{d}}{min}\left(m_{D_1}+m_{D_2},\underset{0<{\Omega }_k\le 15{\displaystyle \frac{1}{s}}}{max}({\overline{D}}_{H_{2-3,k}}+\alpha ·{\sigma }_{D_{H_{2-3,k}}})\right).$$

In Figure 24 the obtained mass distribution of the TMDs is shown with the mean values and standard deviations depending on the excitation frequency. The figure indicates that the TMD for the second vibration mode obtained much more mass as in the first investigated case. The mean values and the scatter of the maximum displacements are significantly reduced for the considered frequency range while the values around the third resonance remain almost constant. For the robust optimum obtained with $\alpha =4.7$, the scatter of the maximum drift displacements are significantly reduced compared to the solutions of the deterministic optimization.

4. Discussion

4.1. Single-Degree-of-Freedom Systems

The deterministic optimization of the TMD parameters is straightforward if the maximum value of the dynamic amplification function for the main system displacements is considered as the objective function. Further constraints could be the maximum relative displacements of the TMD, which would just limit the design space. The single- and multi-objective optimization lead to similar results; therefore, a subsequent single-objective optimization with defined mass ratios seems to be the most promising approach.

The presented analytical uncertainty quantification approach could estimate the mean value and the standard deviation of individual amplification function values with sufficient accuracy for the design process. For the final design, a sampling-based proof of the estimates should be considered. The TMD parameters were assumed within the investigated examples to be independent and normally distributed. Theoretically, the variance estimation is independent of the distribution type as long as the covariance of the random number is available. Here, further investigations for other distribution types and correlated inputs might be necessary.

The definition of the safety margin as the optimization goal or constraint is straightforward and could be directly considered by standard optimization algorithms, such as the simplex Nelder–Mead. However, the calculation of the failure probability itself would require more information, like the definition of a limit state function. Nevertheless, the obtained robust design shows significantly less scatter in the critical values of the amplification function than the pure deterministically optimized design. The optimal nominal parameters changed slightly; especially, the TMD damping is larger than that obtained by the deterministic optimization or the Den Hartog formulas.

4.2. Multi-Degree-of-Freedom Systems

In the MDOF example, we could show that the decoupled stationary solution in the modal space results in similar estimates for the maximum displacements as the time integration approach. For the robust design procedure, this approach seems sufficiently accurate. Nevertheless, the final design should be analyzed with a full time integration for verification purposes, especially since the interaction of the TMDs with the other modes is not represented in the decoupled approach. This may be critical if several natural frequencies of the original system are close to each other.

The implementation of the uncertainty quantification approach is straightforward, whereby the linearization is realized directly for the maximum displacements of the original DOFs and the investigated maximum relative displacements for each discretized excitation frequency. However, the estimated standard deviation shows significant deviations for the higher-resonance frequencies without TMD tuning. Within the design optimization it is necessary to limit the considered frequency range in order to focus on the investigated resonances damped by the TMDs. Within this region, the estimated standard deviation was sufficiently accurate for all the investigated displacement values.

Similarly, as for the SDOF system, the application of the optimization algorithms is straightforward and does not require special attention in the investigated simple example. However, this statement can not be generalized to more complex structures without further investigations. Additionally, the definition of a limit state function and a final reliability analysis might be necessary to proof the robustness of the obtained optimal design. Nevertheless, the results showed a clear difference in the obtained optimal parameters and mass distribution between the robust and the deterministic design. Additionally, the standard deviation of the investigated displacement values could be significantly reduced while the mean values have been increased slightly.

5. Conclusions

The presented uncertainty quantification method seems to be an efficient approach to avoid the typical double loop in sampling-based robust design optimization methods. Since the accuracy of linearization is very sensitive to the investigated structural response values, it should be applied to the individual amplification function or displacement values as presented in the paper. However, an extension for non-normally distributed and dependent input parameters might be the focus of future research. For a large number of random inputs, the central difference approximation of the derivatives may limit its efficiency, and sampling-based methods might be more suitable.

The underlying structural analysis is limited to linear SDOF and MDOF systems with modal damping, which enables the decomposition of the displacement solution in the modal space. For non-linear models and other damping models, the presented decomposition approach might be not suitable anymore and further research would be necessary.

However, the presented approximation could be considered during the robust design optimization to avoid numerically demanding simulation runs. For the final design, a full time integration of the structural system including TMDs should be performed in any case to validate the approximated results. So far, we could show that the procedure itself works for rather simple systems. In further studies, the presented approach should be applied for more sophisticated structures, such as high-rise buildings and pedestrian bridges.