1 Introduction
For a plethora of structural systems it is essential to specify their reliability under uncertain environmental loading conditions and most importantly provide design guidelines using knowledge of their response characteristics. This involves accurate estimation of the structural systems probabilistic response. Environmental loads are typically random by nature and are likely to include intermittently occurring components of an extreme magnitude, representing abnormal environmental events or conditions. Although extreme loadings occur with lower probability than typical conditions, their impact is significant and cannot be neglected since these events determine the systems behavior away from the average operating conditions, which are precisely the conditions that are important to quantify for safe assessment and design. Important examples include mechanical and ocean engineering systems. High speed crafts in rough seas
[Riley2011, Riley2012], wave impacts on fixed or floating offshore platforms and ship capsize events [mohamad2016a, belenky07, muller, liu2007, Kreuzer], vibrations of buildings or bridge structures due to earthquakes or strong wind excitations [Lin63, Lin98, lin96, gioffre12] are just a few examples where extreme responses occur infrequently but are critical in determining the overall systems reliability.Numerous research endeavors have been dedicated on the effective suppression and rapid dissipation of the energy associated with extreme impacts on structures. Many of these schemes rely on linear configurations, known as tuned mass damper (TMD) and result in a halving of the resonance frequency. Although the mitigation performance is highly effective when most of the energy is concentrated at the characteristic frequency of the system, their effectiveness drastically drops if there is a mistuning in frequency. Moreover, it is not clear how these configurations perform in the presence of rare impulsive loads. Many of these limitations can be overcome by utilizing small attachments coupled with the primary system through nonlinear springs, also known as nonlinear energy sinks (NES). If carefully chosen these nonlinear attachments can lead to robust, irreversible energy transfer from the primary structure to the attachment and dissipation there [Vakakis01, Vakakis08]. The key mechanism behind the efficient energy dissipation in this case is the targeted energy transfer phenomenon which is an essentially nonlinear mechanism and relies primarily on the energy level of the system, rather then the resonant frequency [Vakakis03, Kerschen05]. Such configurations have been proven to be successful on the mitigation of deterministic impulsive loads on large structures [Shudeifat16, Shudeifat13, Luo14] and their performance has been measured through effective nonlinear measures such as effective damping and stiffness [Sapsis12, Quin12].
Despite their success, nonlinear configurations have been primarily developed for deterministic impulsive loads. To quantify and optimize their performance in the realistic settings mentioned previously it is essential to understand their effects on the statistics of the response and in particular in the heavy tails of the probability distribution function (PDF). However, quantifying the PDF of nonlinear structures under random forcing containing impulsive type extreme events, poses many challenges for traditional methods. Well established approaches for determining the statistics of nonlinear dynamical systems include the FokkerPlanck equation
[Soong93, sobczyk01], the joint responseexcitation method [Sapsis08, venturi_sapsis, Joo16, athan_2016_prsa], Gaussian closure schemes, moment equation or cumulant closure methods [Beran68, wu84], the PolynomialChaos approach [Xiu_Karniadakis02], and stochastic averaging methods [Zhu88]. For systems associated with heavy tails, however, these methods either cannot capture the statistics of rare and extreme type events due to inherent limitations [majda_branicki_DCDS] or are far too computationally expensive in practice, even for lowdimensional systems [Masud_bergman05, di14]. Alternatively, one can study the statistics of the extreme events alone (by ignoring the background ‘nonextreme’ forcing fluctuations) through a Poisson process representation and then analyze the response using the generalized FokkerPlanck or KolmogorovFeller equations [sobczyk01], which governs the evolution of the corresponding PDF, or by applying the path integral formalism [koyluoglu95, Iwankiewicz00], or even through special stochastic averaging techniques [Zeng11]. While attractive, these ideas lead, in general, to analytical results for a very limited number of special cases. Besides, it is still an important aspect to account for the background random fluctuations in the forcing term in order to fully characterize the systems overall probabilistic properties (e.g. this is important in order to fully determine all the moments of the response). Moreover, even though the background forcing component does not directly correspond to extreme events, the background term may have important consequences for the initiation of intermittent type extreme responses [mohamad2016b].In this work we consider the problem of nonlinear structural systems under general timecorrelated stochastic forcing that includes extreme, impulsive type random events. We address two important challenges related to this problem. The first is the development of a fast and accurate estimation method for the response statistics, expressed through the PDF, with emphasis on the accurate estimation of the tail form (events far away from the mean). The second is the design and parameter optimization of small attachments that can mitigate or suppress the effects of the extreme forcing events on the system response while they also improve the system behavior during the regular regime. The two problems are connected since extreme event suppression is directly reliant upon a fast and accurate estimation method for the response pdf under different designs or parameters. Indeed, without a fast and reliable method to evaluate response statistics, in particular tail statistics, optimization cannot be performed because of the inherent computational cost associated with typical quantification methods such as MonteCarlo. This aspect highlights the practical utility of the proposed fast PDF estimation scheme. We will illustrate the pdf estimation method and shock mitigation design analysis throughout the manuscript with a practical motivating prototype system related to high speed vehicle motion in rough seas, however we emphasize the proposed method broad applicability.
The probabilistic quantification scheme formulated here is based on the most general probabilistic decompositionsynthesis framework [mohamad2016b, Mohamad15], that has recently been applied in linear systems subjected to stochastic forcing containing extreme events [Joo16a] and can be used to efficiently estimate the PDF for the response displacement, velocity, and acceleration. We begin by formulating the response pdf quantification method (developed for linear multidegreeoffreedom (MDOF) systems in [Joo16a]) for the case of nonlinear MDOF systems. This is achieved by combining the probabilistic decompositionsynthesis framework [mohamad2016b, Mohamad15] with the statistical linearization method [Rob_SPanos03]. The scheme circumvents the rareevent problem and enables rapid design and optimization in the presence of extreme events. We emphasize the statistical accuracy of the derived scheme, which we have validated through extensive comparisons with direct MonteCarlo simulations. Next, we consider two prototype ocean engineering systems and perform a quantitative comparison of the performance of TMD and NES, evaluating their effectiveness at shock suppression under stochastic excitation containing extreme events. Finally, we perform optimization on a very generic, possibly asymmetric family of piecewise linear springs. Previous endeavors in the context of singlesided vibroimpact NES have shown that asymmetries in the NES can improve the shock mitigation properties (see [Shudeifat13]). In agreement with these results, our optimization scheme leads to the derivation of a new asymmetric NES which significantly improves the shock mitigation properties of the system in the realistic setting of stochastic excitation.
The paper is structured as follows. In section 2 we describe the prototype models for high speed craft motion that we utilize throughout the paper as practically relevant example. Next, in section 3 we provide a brief review of the probabilistic decompositionsynthesis (PDS) framework for the response pdf quantification of a linear singledegreeoffreedom system subject to a random forcing term containing extreme impulse type events. Section 4 describes the proposed general semianalytical PDF estimation method for nonlinear MDOF structures and also includes a section on quantifying the conditionally rare response via the effective stiffness and damping framework. In section 5 we present the mitigation of extreme events analysis on the prototype high speed craft designs for both TMD and cubic NES attachments. Next, in section 6 we propose a new piecewise linear and asymmetric NES design that we optimize for extreme event mitigation. Finally in section 7 we offer concluding remarks.
2 Prototype models for high speed vehicle motion in rough seas
Here we describe the prototype models that we apply the quantification method for extreme event analysis and optimization. Specifically, we model the motion of a highspeed craft in random seas through two prototype systems: one being a twodegreeoffreedom system consisting of a suspended seat attached to the hull and the second being a threedegreeoffreedom system where the seat is attached to a suspended deck, which is attached on the hull; both prototypes contain a small linear or nonlinear energy sink (NES) vibration absorber.
2.1 2DOF Suspended seat system
In figure 1 we illustrate the first model consisting of a linear primary structure under base excitation that is attached to a small oscillator connected through a nonlinear spring (with cubic nonlineariry). This is a prototype system modeling the suspended seat of a high speed craft [Olausson15, Coe09]. The vibration absorber is attached to the seat with the aim to minimize ocean wave impacts on the operator of the vehicle and naturally we require that the attachment mass is much lower than the seat mass (i.e. ). The equation of motion for this twodegreeoffreedom system is given by:
(1)  
where are the relative displacements of the seat response and attachment response, respectively, with reference to the base motion (that is, and ).
2.2 3DOF Suspended deckseat system
The second prototype system is a suspended deck design for a high speed craft [Townsend12, Ranieri04, KIM97] and is illustrated in figure 2. In this case, the vibration absorber is attached to the suspended deck. The attachment mass is comparable to the seat mass and both are considerably smaller than the deck (i.e. ). The governing equations for this threedegreeoffreedom system are given by:
(2)  
where, again, are the relative displacements of the seat response, the deck response and the attachment response, respectively, with reference to the base motion .
In both prototypes the aim of the vibration mitigating attachment is to minimize extreme impacts on the seat attachment as this represents an operator on the vehicle. We first examine the case of tunedmass damper vibration absorber and the essentially nonlinear energy sink absorber , that has been studied extensively in the context of shock mitigation [Vakakis08]. In the last section we will examine the performance of an asymmetric, piecewise linear, spring.
2.3 The structure of the intermittently extreme stochastic forcing
Motivated by the ocean engineering systems in section 2, we consider base motion of the form,
(3) 
In the expression above, denotes a zeromean smooth motion characterized by a PiersonMoskowitz spectrum,
(4) 
where controls the magnitude of the motion. The second term in equation 3 describes rare and extreme impulses in terms of a random impulse train ( is a unit impulse), occurring due to slamming events. For this component, is a Poisson counting process that represents the number of impulses that arrive in the time interval ,
is the impulse magnitude, which we assume is normally distributed with mean
and variance
, and the constant arrival rate is given by . We take the impulse magnitude as beingtimes larger than the standard deviation of the excitation velocity
:3 Review of the probabilistic decompositionsynthesis (PDS) method
We first provide a brief review of the semianalytical response quantification method for a linear singledegreeoffreedom system [Joo16a] subjected to stochastic excitation containing rare events. The purpose of this section is to provide a selfcontained review of the core ideas, since the scheme for nonlinear structural systems that is described in the following section depends upon these concepts.
Consider the following linear system
(5) 
is the stiffness, is the damping, and is the damping ratio. Despite the simplicity of this system, the structure of the statistical response may be significantly complex and posses heavytails.
The framework to estimate the response PDF of equation 5 is the probabilistic decompositionsynthesis (PDS) method [mohamad2016b]. The basic idea is to decouple the rare events regime from the background fluctuations and then quantify the statistics of the two components separately. The results are then synthesized to obtain the full response PDF by using the total probability law:
(6) 
where is the conditional PDF due to the smooth motion of the base, is the conditional PDF due to the extreme impacts and is the overall probability that the system operates in the extreme events regime.
3.1 Background response PDF
We first obtain the statistical response of the system under the condition that only the background (smooth) forcing component is acting. We have,
(7) 
In this case the analysis is particularly simple since the system is linear and timeinvariant and the response PDF, , is a zeromean Gaussian. The spectral density of the response displacement and the variance are given by:
(8) 
The computations for the response velocity and acceleration can be similarly obtained.
3.2 Numerical histogram for rare events
The next step is to compute the rare event distribution and the rare event probability . Specifically, the rare event distribution can be written as,
(9) 
where is the distribution of the impulse magnitude, and is the conditional PDF of the response for an impact of magnitude .
It is important to note that once an impulse of magnitude hits the system, the momentum of the system right after the impact would be , since the momentum of the system right before the impact is
. As these two variables are both Gaussian distributed and independent, their sum is also Gaussian distributed and is given by,
(10) 
We estimate the conditional PDF by the numerically computed histogram:
(11) 
where is the typical duration of the rare event (see next subsection) and the conditional response is given by,
(12) 
The conditionally extreme event distribution for velocity and acceleration are derived in a similar fashion.
3.3 Numerical estimation of the rare event probability
In order to compute the histogram of a rare impulse event, the duration of a rare response needs to be obtained numerically. We define the typical duration of a rare response by
(13) 
where , or in other words, the histogram is taken over the time it takes for the system response to decay to 10% of its maximum value. The absolute value of the maximum of the response needs to be estimated numerically.
Once this rare event duration has been specified, we can also obtain the probability of a rare event by
(14) 
Note that the extreme event duration for the displacement , velocity , and acceleration are in generally different.
3.4 Semianalytical response probability distributions
With the description above, we obtain the response PDF using the total probability law. The resulting response PDF takes the form,
(15) 
where the argument is either , , or . The validity of this approximation has been thoroughly verified in [Joo16a].
4 PDF quantification method for nonlinear MDOF systems
Here we formulate the probabilisticdecomposition method for multidegreeoffreedom, nonlinear mechanical systems. There are some important differences with respect to the case of linear systems studied in [Joo16a]. Firstly, for the background component the system nonlinearities can be important and to this end we must utilize an appropriate statistical quantification method. Here we employ the statistical linearization approach [Rob_SPanos03]. Secondly, to characterize the statistics in the rare event regime it is even more crucial to take into account the nonlinear properties of the system, since these control the shock mitigation capabilities of the attachment.
To achieve this we use two alternative approaches. The first one is based on the direct simulation of the system for a range of initial conditions corresponding to all possible impact magnitudes. The second is based on the notion of effective stiffness and damping [Sapsis12], which are measures that characterize the system response under various excitation magnitudes taking into account the presence of the nonlinear attachment. We provide comparisons with direct MonteCarlo simulations to demonstrate the accuracy of both approaches. We first present the analysis for the background component
4.1 Quantification of the response pdf for the background component
For the background regime, we must account for nonlinearities and their interaction with the background part of the excitation. We use the statistical linearization method, since we are only interested in resolving the loworder statistics of the background response of the system (the rare events component defines the tails of the PDF).
Consider the response of the suspended seat problem, section 2.1, under the excitation term :
(16)  
(17) 
We first multiply the above two equations by , , at different time instant , and take ensemble averages to write the resulting equations in terms of covariance functions.
(18)  
(19)  
(20)  
(21)  
(22)  
(23) 
Here indicates the partial differentiation with respect to the time difference . We then apply Isserlis’ theorem based on the Gaussian process approximation for response to express the fourthorder moments in terms of secondorder moments [Isserlis18].
(24)  
(25)  
(26) 
This leads to a set of linear equations in terms of the covariance functions. Thus, the WienerKhinchin theorem can be applied to write the equations in terms of the power spectrum, giving
(27)  
(28)  
(29)  
(30)  
(31) 
where,
(32)  
(33)  
(34) 
At this point , , and are still unknown, but can be determined by integrating both sides of equations 29, 28 and 27 and forming the following system of equations:
(35) 
By solving the above we find and . This procedure determines the Gaussian PDF approximation for the background regime response. Further details regarding the special case of a linear attachment and the analysis for the suspended deckseat problem can be found in appendix A.
4.2 Quantification of the response pdf for the extreme event component
We are going to utilize two alternative methods for the quantification of the statistics in the extreme event regime. The first approach is to obtain the conditional statistics based on direct simulations of the system response. The second method is utilizing effective measures [Sapsis12] that also characterize the system nonlinear response in the presence of attachments.
4.2.1 Rare response PDF using direct simulations of the system under impulsive excitation
To compute the conditionally extreme distribution and the probability of rare events we follow the steps described in algorithm 1, which provides a highlevel description for a single mode. The procedure is repeated for each degree of freedom of interest (in this case it is more efficient to simply store all the impulse realizations and then run the procedure for each degree of freedom of interest). We emphasize that the numerical simulation of impulse response for nonlinear systems is efficient, since the integrations are necessarily short due the impulsive nature of the forcing and the condition on the rare event end time in equation 13. Moreover, throughout these simulations we do not take into account the background excitation since this is negligible compared with the effect of the initial conditions induced by the impact.
1:discretize 2:for all values over the discretization do 3: solve ODE system for under impulse , neglecting 4: we set 5: 6:end for 7: 8: 9: 10:output:
Comparison with MonteCarlo Simulations
The full response PDF is composed using the total probability law,
(36) 
where is either the displacement, velocity or acceleration of the seat/attachment response. We utilize a shifted PiersonMoskowitz spectrum for the background forcing term in order to avoid system resonance. Details regarding the MonteCarlo simulations are provided in appendix B.
In figure 3 we show comparisons for the suspended seat problem with parameters and relevant statistical quantities given in table 1. In figure 4 we also show comparisons for the suspended deckseat problem with parameters and relevant statistical quantities in table 2. For both cases the adopted quantification scheme is able to compute the distributions for the quantities of interest extremely fast (less than a minute on a laptop), while the corresponding MonteCarlo simulations take order of hours to complete.
Note that our method is able to capture the complex heavy tail structure many standard deviations away from the mean (dashed vertical line denotes 1 standard deviation). We emphasize that similar accuracy is observed for a variety of system parameters that satisfy the assumptions on the forcing. The close agreement validates that the proposed scheme is applicable and can be accurately used for system optimization and design.
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4.3 Rare response PDF using effective measures
Here we describe an alternative technique to quantify the rare event PDF component using the effective stiffness and damping framework described in [Sapsis12]. These effective measures express any degreeoffreedom of the coupled nonlinear system, for a given initial energy level, as an equivalent linear singledegreeoffreedom system. Specifically, these effective measures correspond to the values of damping and stiffness for a linear system that has (for the same initial conditions) a response that is as close as possible to that of the original system, in the mean square sense.
We focus on the suspended seat problem to illustrate this strategy. It should be pointed out that the accuracy and applicability of this approach has some limitations:

The accurate estimation of the PDF requires the knowledge of the effective measures over a sufficiently large range of initial impulses.

The motion of the system should have an oscillatory character so that it can be captured by effective measures.

The statistics of the attachment motion cannot be obtained directly from the effective measures.
To derive the PDF in the rare event regime we reduce the system to an effective linear system for the degreeoffreedome of interest. Consider the suspended seat system under an impulse,
(37)  
with initial conditions, at an arbitrary time say ,
(38) 
To determine the effective linear system for this system, we follow the strategy in [Sapsis12] and compute the effective stiffness and damping:
(39) 
where
denotes spline interpolation of the local maxima of the time series. We can then compute the weightedaverage effective stiffness and damping:
(40) 
With the weightedaverage effective measures we rewrite the original twodegreeoffreedom system during rare events into an equivalent linear singledegreeoffreedom system with coefficients that depend on the initial impact (or the initial energy level of the system):
(41) 
Using the effective system in equation 41 we can obtain the conditionally rare PDF using the analysis for the linear system in section 3.1. The damping ratio and natural frequency now become functions of the initial impact, :
(42) 
Subsequently, the PDF is obtained by taking a histogram of
(43) 
In figure 5 (top) we present the suppression of the probability for large motions of the primary structure due to the presence of the NES (parameters given in table 1). This suppression is fully expressed in terms of the effective damping measure shown in the lower plot. Note that the suppression of the tail begins when the effective damping attains values larger than one.
We emphasize that in the context of effective measures the motion of the system is assumed oscillatory. Motions with radically different characteristics will not be captured accurately from the last representation and the resulted histograms will not lead to an accurate representation of the tail. This problem is, in general, circumvented if we employ the first approach for the computation of the conditional PDF during extreme impacts. On the other hand, the advantage of the second approach is that we can interpret the form of the tail in the various regimes with respect to the properties of the effective measures (figure 5). This link between dynamics (effective measures) and statistics (heavy tail form) is important for the design process of the NES.
Comparison with MonteCarlo simulations
Here we compare the PDS method combined with the effective measures with direct MonteCarlo simulations. In figure 6 we show the response PDF for the primary structure for parameters given in table 1. Details regarding the MonteCarlo computations are provided in section 4.2. We observe that the PDS method utilizing effective measures performs satisfactorily over a wide range similarly with the first general scheme, based on individual trajectories computation.
4.4 Quantification of the absolute response pdf
The developed PDF quantification schemes provide statistical description for relative quantities (with respect to the base), that is , and . However, for the prototype systems that we consider we are more interested for the suppression of absolute quantities, instead of relative ones. As we illustrate below, the absolute response PDF can be derived from the relative response PDF in a straightforward manner .
Background component
For the background regime, we the absolute motion is expressed as:
(44) 
As the relative motion and base motion are both Gaussian distributed (but not independent), their sum is also Gaussian distributed and it is given by,
(45) 
In the previous section we have derived both and , and what remains is the covariance term whose spectral density function is given in equation 29. This is given by:
(46) 
Extreme event component
For the extreme event component the motion of the motion is assumed very small (compared with the magnitude of the impact), in which case we have:
(47) 
The estimation of the conditional PDF for has already been described in section 4.2.
Comparison with MonteCarlo Simulations
The full absolute response PDF is expressed using eq. (36), where is either relative or absolute displacement, velocity or acceleration of the seat/attachment response. We compare the PDS method with direct MonteCarlo simulations for the case of absolute motions. In figure 7 we show the absolute response PDF for the primary structure for parameters given in table 1. Details regarding the MonteCarlo computations are provided in section 4.2.
5 System optimization for extreme event mitigation
We now consider the problem of optimization in the presence of stochastic excitation containing extreme events. The developed method provides a rapid and accurate semianalytic estimation scheme for the statistical response of the nonlinear structural system. In particular, we can efficiently obtain the response statistics of the primary structure (the seat) for any given shock mitigating attachment and accurately capture the heavytailed structure of the distribution. This allows us to explore rare event mitigation performance characteristics of different attachment parameters and perform optimization. Such analysis is not practically feasible via a direct MonteCarlo approach since a single parameter set takes on the order of hours to compute the resulting response PDF with converged tail statistics.
We consider the prototype systems described in section 2 with the aim to suppress the large energy delivered to the passenger (i.e. the seat). In all cases we optimize the attachment parameters, while the parameters of the primary structure are assumed to be fixed.
5.1 Optimization objective
We adopt the forthorder moment as our measure to reflect the severity of extreme events on the seat:
(48) 
where the argument can be either absolute displacement of the seat or absolute velocity depending on the optimization objective. The goal here is to minimize this measure and analyze the performance characteristics of the attachment when its parameters are varied.
We illustrate the results of the optimization using the following normalized measure:
(49) 
where is either or , and is the corresponding quantity without any attachment. Values of this measure which are less than 1 () denote effective extreme event suppression.
5.2 Optimization of NES and TMD parameters
Results are shown for the suspended seat problem with an attachment mass For a NES attachment () we optimize over and , while for a TMD () we vary and (figure 8). The resulted response PDF that minimize the displacement moments are illustrated in figure 9. The same analysis is performed for the suspended deckseat problem with the same attachment mass for both systems (figure 10). The resulted response PDF are illustrated in figure 11.
In both cases of systems we observe that the TMD and the optimal cubic NES can improve significantly the behavior of the primary structure in terms of reducing the displacement during impacts, with a reduction of 6668% of the fourthorder moment. We also observe that the NES design is more robust to variations in the attachment parameters over the TMD design, which requires more stringent attachment parameter values for best performance with respect to . This is in line with the fact that the NES attachment performs better over a broader excitation spectrum than the TMD configuration, which requires carefully tuning. Note that for the case of the deckseat problem (figure 10) we can achieve much larger mitigation of the absolute velocity at the order of 3234% compared with the simpler system of the seat attached to the hull directly (figure 8), where the suppression is much smaller, 24%.
We performed the grid search for demonstration purposes to illustrate the performance characteristics as the stiffness and damping are varied; clearly, if we are only interested in the optimal attachment the use of an appropriate global optimizer (such a particle swarm optimizer) would be more appropriate. All the results shown where computed using the proposed PDF estimation method. As a further check and validation, we benchmarked the semianalytical PDF estimates and compare them with MonteCarlo results for the extremity measure
over a coarse grid of the attachment parameters.6 Design and optimization of a piecewise linear NES
To further improve the shock mitigation properties of the attachment, we utilize a more generic form of NES consisting of a possibly asymmetric, piecewise linear spring. Similarly with the cubic NES and TMD attachments, we perform parameter optimization on the NES spring restoring characteristics and obtain a new optimal design that outperforms the TMD and cubic NES for the considered problems.
Here, we focus on suppressing large displacements of the seat, although velocity or acceleration would also be appropriate depending on the desired objectives. The general form of the considered spring consists of a linear regime with slope equal to that of the optimal TMD within a range of 4 standard deviations of the expected seat motion (e.g. when the TMD is employed). For motions (displacements) outside this range the spring has also a linear structure but with different slopes, for negative displacements (beyond 4 standard deviations) and for positive displacements (beyond 4 standard deviations). Therefore, the optimal linear stiffness operates for small to moderate displacement values and outside this regime, when the response is very large, we allow the stiffness characteristics to vary. The objective is to determine the optimal values for the curve in the extreme motion regime with respect to optimization criterion.
Therefore, the analytical form of the piecewise linear spring is given by:
(50) 
where, is the standard deviation of the relative displacement between the primary structure (the seat) and the attachment for the case of a TMD attachment. The parameters, and define the slopes in the positive and negative extreme response regimes, which we seek to optimize. Moreover, the values for and are obtained by enforcing continuity:
(51)  
(52) 
The value of the stiffness in the center regime, , is chosen using the optimal TMD attachment.
6.1 Application to the suspended seat and deckseat problem and comparisons
We illustrate the optimization using the fourthorder moment of the seat response, employing the following measure:
(53) 
where is the system response with the optimal TMD attachment (from the previous parametric grid search optimization) and is the response of the system with the piecewise linear NES attachment. Parameters corresponding to values less than 1 () denote additional extreme event suppression, compared with the utilization of optimal TMD.
The result of the optimization for minimum fourthorder moment for the displacement, on the suspended seat problem, is shown in figure 12 while the corresponding PDF for the displacement, velocity and acceleration are shown in figure 13. We note the strongly asymmetric character of the derived piecewise linear spring. This is directly related with the asymmetric character of the impulsive excitation, which is in general positive. The performance of the optimized piecewise linear spring is radically improved compared with the optimal cubic NES and TMD as it is shown in the PDF comparisons. Specifically, for rare events (probability of 1%) we observe a reduction of the motion amplitude by 50%, while for the velocity the reduction is smaller. A representative time series illustrating the performance of the optimal design for the suspended seat problem is shown in figure 14. The PDF for the acceleration for this set of parameters is not changing significantly. Our results are in agreement with previous studies involving singlesided vibroimpact NES that have been shown to improve shock mitigation properties in deterministic setups [Shudeifat13].
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