Fatigue damage assessment for Sine-on-Random: a comparative analysis of the predictive capability and application limits of spectral methods

In practical engineering scenarios, mechanical and electronic components are rarely subjected to purely stochastic or purely sinusoidal excitations. Most real-life environments are characterized by "mixed-mode" conditions, whereby deterministic and random components coexist. The most notable example of this phenomenon is the Sine-on-Random (SoR) profile, in which one or more sinusoidal tones, representing harmonic components, are superimposed on a broadband Gaussian random vibration background.
The aerospace sector is the primary field of application for SoR profiles. In general aircraft environments, excitation is characterized by a complex combination of sources. The broadband random component is primarily generated by aerodynamic turbulence, boundary layer fluctuations, and jet engine exhaust noise. Concurrently, harmonic components are produced by various rotating masses. In the case of rotary-wing aircraft, these are generated by the main and tail rotors and transmission shafts. In fixed-wing propeller aircraft, they stem from propeller rotation, whereas in jet aircraft, they arise from high-speed turbine stages and auxiliary power units.
The development of SoR test specifications, as outlined in international standards such as MIL-STD-810 [1] and RTCA DO-160 [2], is rooted in the historical necessity to validate components subject to mixed loads. The emphasis of these standards is on replicating the actual effects of environmental conditions on equipment, as opposed to the mere imitation of the signals themselves [3]. While physical testing on electrodynamic shakers remains one of the primary methods for verifying that a component can survive these complex environments, these regulatory profiles are also fundamental during the design phase. Designers employ these standardized loads to perform design verification, ensuring that the structural integrity of a component is sufficient to withstand the prescribed mission profiles, with a specific focus on mechanical fatigue resistance [4].
However, evaluating fatigue damage under SoR conditions poses substantial analytical challenges. Given the inherently nonGaussian nature of SoR loads, the prevailing approach for assessing fatigue damage is the time-domain method, which is based on the rainflow counting algorithm [5]. This approach is considered the "gold standard" as it directly counts stress cycles from the signal's time history without requiring statistical assumptions. However, it is widely acknowledged that timedomain analysis is extremely computationally expensive, which makes its implementation prohibitive for long-duration missions or complex structural models.
For this reason, in environments characterized by purely random vibrations, frequency-domain spectral methods (such as those developed by Dirlik [6] or Rayleigh [7]) are preferred due to their computational efficiency. These classical methods, however, rely on the strong assumption of a stationary Gaussian process [8]- an assumption that is explicitly violated in Sine-on-Random scenarios [9]. The presence of deterministic tones has been shown to fundamentally alter the probability
density function (PDF) of the stress ranges, resulting in inaccurate damage estimates if standard Gaussian formulas are applied [9-10].
To leverage the convenience offered by frequency-domain methods for SoR loads, researchers have proposed two primary strategies: a rigorous, theoretical approach and a more practically oriented engineering approach. The theoretical approach focuses on deriving an analytical PDF of the cycles that accounts for the combined sine-plus-noise statistics. The foundational work of Rice [10] established the theoretical basis for this line of thought, though its practical applications are often constrained to specific, stringent cases, such as when the sine frequency is in proximity to the central frequency of the random noise. However, in numerous structural dynamics applications, the sine waves are positioned well beyond the random band, thereby constituting a perpetual violation of Rice's hypotheses [1,2].
Consequently, practical engineering approaches have become increasingly widespread in industrial applications. These methodologies rely on generating an equivalent random Power Spectral Density (PSD) that represents the SoR phenomenon through a damage-based or energy-based equivalence. Authors such as Lalanne [11], Irvine [12-14], and Jang [15] have developed innovative methods to synthesize a random profile that produces equivalent fatigue damage. In contrast, Cho [16] proposed a method that synthesizes a random profile with the same Root Mean Square (RMS) level as the original SoR signal. These approaches are straightforward to apply; once the equi-damaging or equi-RMS PSD is obtained, the SoR process can be treated as a pure Gaussian random vibration. Thus, the equivalent PSD can be analysed using standard spectral methods [17], effectively transforming the complex mixed-mode problem into a conventional Gaussian scenario. Given the complexity of the Sine on Random phenomenon, the objective of this work is to provide the scientific and engineering communities with a comprehensive knowledge of the problem and the approaches available in the literature. Specifically, the main contributions of the paper consist of the following:

$r$$r$rr$r$his section outlines the established analytical approaches for the fatigue life assessment of components subjected to purely random Gaussian loadings.

Under the assumption of Gaussian distribution, any random process $x(t)$$x(t)$x(t)x(t)$x(t)$ can be described in the frequency domain by its statistical properties, namely its PSD $S_{xx}(\omega )$$S_{xx}(\omega )$S_(xx)(omega)S_{x x}(\omega)$S_{xx}(\omega )$, which is defined as the Fourier transform of the autocorrelation function $R_{xx}(\tau )=E[x(t)x(t+\tau )][18]:$$R_{xx}(\tau )=E[x(t)x(t+\tau )][18]:$R_(xx)(tau)=E[x(t)x(t+tau)][18]:R_{x x}(\tau)=E[x(t) x(t+\tau)][18]:$R_{xx}(\tau )=E[x(t)x(t+\tau )][18]:$

Two definitions of PSD exist, the double-sided $S_{xx}(f)$$S_{xx}(f)$S_(xx)(f)S_{x x}(f)$S_{xx}(f)$ and the single-sided $G_{xx}(f)$$G_{xx}(f)$G_(xx)(f)G_{x x}(f)$G_{xx}(f)$, and the relation is provided in Eqn. (2):

The spectral moments $m_n$$m_n$m_(n)m_{n}$m_n$ are important characteristics of a random process; they are defined as:

where the subscript $n$$n$nn$n$ represents the order. Spectral moments represent some important properties of the signal; for example, the moment of order zero, denoted as $m_0$$m_0$m_(0)m_{0}$m_0$, coincides with the variance of the random signal [19].
Combinations of spectral moments of higher order are used to characterize some statistical properties of the random signal that are of interest in fatigue analysis. For a Gaussian process, the number of zero-upcrossings per second $\left({\nu }_0^{+}\right)$$\left({\nu }_0^{+}\right)$(nu_(0)^(+))\left(\nu_{0}^{+}\right)$\left({\nu }_0^{+}\right)$and the peak rate ( ${\nu }_p$${\nu }_p$nu_(p)\nu_{p}${\nu }_p$ ) are defined in Eqns. (4) and (5):

Other crucial spectral moments combinations are the bandwidth parameter $ϵ$$ϵ$epsilon\epsilon$ϵ$, which quantifies the spectral spread of the process, and the irregularity factor $\gamma$$\gamma$gamma\gamma$\gamma$, which serves as an alternative measure of the process bandwidth. They are defined as follows:

The bandwidth parameter ranges from 0 for a strictly narrow-band (NB) process to 1 for a wide-band (WB) process, whereas, by definition, $\gamma$$\gamma$gamma\gamma$\gamma$ approaches unity for NB conditions and tends to zero for WB signals.
The distribution of maxima (or peak values $x_p)p_{x_p}(x,ϵ)$$\left.x_p\right)p_{x_p}(x,ϵ)${:x_(p))p_(x_(p))(x,epsilon)\left.x_{p}\right) p_{x_{p}}(x, \epsilon)$x_p)p_{x_p}(x,ϵ)$ for a zero mean Gaussian process depends on this parameter and follows the formulation provided by Cartwright and Longuet-Higgins [20] and represented, in a normalized form, in Eqn. (8):

where $\zeta =x_p/\sqrt{m_0}$$\zeta =x_p/\sqrt{m_0}$zeta=x_(p)//sqrt(m_(0))\zeta=x_{p} / \sqrt{m_{0}}$\zeta =x_p/\sqrt{m_0}$ represents the normalized maxima and $\sqrt{m_0}$$\sqrt{m_0}$sqrt(m_(0))\sqrt{m_{0}}$\sqrt{m_0}$ is the RMS value of the Gaussian process.
The fatigue characterization of materials is fundamentally described by the S-N curve, often expressed through the powerlaw relationship also known as Basquin's law [21]:

where $N$$N$NN$N$ denotes the number of cycles to failure at a constant stress amplitude $s$$s$ss$s$, while $C$$C$CC$C$ and $m$$m$mm$m$ represent materialspecific fatigue parameters.

For components subjected to stochastic excitations, fatigue damage is typically evaluated under the assumption of linear damage accumulation, as defined by the Palmgren-Miner rule [22]. Under this framework, the expected damage $E[D]$$E[D]$E[D]E[D]$E[D]$ accumulated per unit time, relative to the material's fatigue properties, is expressed by the following generalized integral [23]:

where ${\nu }_p$${\nu }_p$nu_(p)\nu_{p}${\nu }_p$ represents the expected number of cycles per unit time and $p_s(s)$$p_s(s)$p_(s)(s)p_{s}(s)$p_s(s)$ is the PDF of the stress cycle amplitudes, with $s$$s$ss$s$ denoting the stress cycle amplitude.
In cases where the S-N curve intercept constant $C$$C$CC$C$ is unknown, it may be normalized to unity. This modifies Eqn. (10) into a metric defined as pseudo-damage, which remains a valid basis for the comparative assessment of distinct loading histories. While the PDF of stress cycles can be precisely extracted from time-history data using the rainflow counting (RFC) algorithm, such procedures are computationally intensive, especially for long-duration or high-frequency signals. To overcome these limitations, various spectral methods have been developed to estimate the cycle amplitude PDF directly from the PSD of the load. In this context, the damage is treated as an expected value, as the stochastic nature of the process is captured through the spectral moments of the PSD.
The complexity of the spectral characterization depends on the bandwidth of the process. NB processes can be effectively approximated as quasi-sinusoidal signals, where the distribution of peaks is statistically described by the zero-th spectral moment $m_0$$m_0$m_(0)m_{0}$m_0$ and follows a Rayleigh distribution [7]. Conversely, WB and bimodal excitations, typical of complex structural responses, require higher-order spectral moments to accurately characterize the irregularity of the signal and the presence of multiple frequency components [17]. Among the most robust and widely adopted methods for broad-band processes are Dirlik's empirical closed-form solution [6] or the Tovo-Benasciutti method [24].

The Fatigue Damage Spectrum (FDS) is a useful tool in engineering practice since it provides a frequency-domain representation of the expected fatigue damage, thereby enabling a direct comparison between different types of excitations, whether deterministic, impulsive, or stochastic, by reducing them to their cumulative effect on a simplified mechanical model (single degree of freedom system)[25-26].
Since the FDS has been used by some authors as a tool to obtain an equivalent random PSD from a SoR specification, it is fundamental to describe the FDS calculation both in the time domain and in the frequency domain.

Definition and Fundamentals. The FDS is an essential tool for evaluating the damaging potential of a vibration signal. It is defined as the envelope of the fatigue damage experienced by a set of Linear Time-Invariant (LTI) Single-Degree-ofFreedom (SDOF) systems. These systems are characterized by a varying natural frequency $f_0$$f_0$f_(0)f_{0}$f_0$ (or angular frequency ${\omega }_0=2\pi f_0$${\omega }_0=2\pi f_0$omega_(0)=2pif_(0)\omega_{0}=2 \pi f_{0}${\omega }_0=2\pi f_0$ ) and a constant damping ratio $\xi$$\xi$xi\xi$\xi$ (often expressed via the quality factor $Q=\frac{1}{2\xi }$$Q=\frac{1}{2\xi }$Q=(1)/(2xi)Q=\frac{1}{2 \xi}$Q=\frac{1}{2\xi }$ ). The spectrum is calculated for a specific fatigue exponent $m$$m$mm$m$, corresponding to the slope of the material's S-N curve.

Time-Domain Calculation Procedure. Regardless of the signal's nature, the FDS can be computed directly from time-history data. The computational procedure for generating the spectrum follows these fundamental steps:

where $n_i$$n_i$n_(i)n_{i}$n_i$ is the number of cycles at amplitude $Z_{p,i},C$$Z_{p,i},C$Z_(p,i),CZ_{p, i}, C$Z_{p,i},C$ is the fatigue strength coefficient, and $K$$K$KK$K$ is the proportionality factor between displacement and stress ( $s_{\text{inst}}(t)=K\cdot Z(t)$$s_{\text{inst }}(t)=K\cdot Z(t)$s_("inst ")(t)=K*Z(t)s_{\text {inst }}(t)=K \cdot Z(t)$s_{\text{inst }}(t)=K\cdot Z(t)$, with $s_{\text{inst}}$$s_{\text{inst }}$s_("inst ")s_{\text {inst }}$s_{\text{inst }}$ denoting stress instantaneous values).
By assuming a unit proportionality factor ( $K=1$$K=1$K=1K=1$K=1$ ) and normalizing the fatigue strength coefficient ( $C=1$$C=1$C=1C=1$C=1$ ), Eqn. (11) yields a displacement-based pseudo-damage metric. This approach allows for a relative comparison of the fatigue damage potential between different load histories when $C$$C$CC$C$ and $K$$K$KK$K$ are unknown.
By repeating this calculation for a defined range of natural frequencies, $f_0$$f_0$f_(0)f_{0}$f_0$, the function $D\left(f_0\right)$$D\left(f_0\right)$D(f_(0))D\left(f_{0}\right)$D\left(f_0\right)$, known as the Fatigue Damage Spectrum, is obtained.

FDS of a random signal in the frequency domain. For Gaussian and stationary random processes, the analytical expression for the PDF of the response peaks for an SDOF system is well-defined. This allows for the determination of the FDS directly from the spectral content, significantly reducing the computational effort compared to time-domain simulations. Since such signals are entirely described by their PSD, the FDS can be evaluated through the following analytical procedure:

Although the distribution of rainflow counting cycles is essential for an accurate fatigue damage estimation, the response peaks PDF remains a fundamental tool for NB signals [27]. The response of an SDOF system to a Gaussian excitation is characterized by a Gaussian signal concentrated around the system's natural frequency. As the damping coefficient decreases, the response becomes increasingly NB. Under these conditions, the response $Z(t)$$Z(t)$Z(t)Z(t)$Z(t)$ can be considered a NB signal, allowing the peak distribution to be approximated by a Rayleigh distribution. Under such conditions, the FDS for a Gaussian random signal can be evaluated using the following expression:

where ${\sigma }_z$${\sigma }_z$sigma_(z)\sigma_{z}${\sigma }_z$ is the root-mean-square value of the response, $T$$T$TT$T$ is the exposure time, and $\mathrm{\Gamma }$$\mathrm{\Gamma }$Gamma\Gamma$\mathrm{\Gamma }$ is the gamma function.
Evaluating the fatigue spectrum in the frequency domain provides the expected value of the damage, effectively eliminating the intrinsic statistical variability that arises from calculations based on individual time histories. This method ensures high computational efficiency while maintaining excellent agreement with time-domain results.

The White Noise Approximation. A widely adopted simplification is the White Noise Approximation. By assuming that the input PSD, $G_{xx}(f)$$G_{xx}(f)$G_(xx)(f)G_{x x}(f)$G_{xx}(f)$, is locally constant around the natural frequency $f_0$$f_0$f_(0)f_{0}$f_0$, the RMS stress response can be directly estimated as [19]:

Substituting this into Eqn. (12) yields the analytical closed form for the FDS for the white noise case:

In comparative studies, it is also possible to set $K=1$$K=1$K=1K=1$K=1$ and $C=1$$C=1$C=1C=1$C=1$ for comparative purposes between different loadings. A schematic representation of the FDS evaluation process in the time domain is provided in Fig. 1.

In this section, considerations are made about the statistics of mixed SoR signals and the inapplicability of conventional spectral methods in this context.
A Sine-on-Random signal represents a mixed-mode excitation where one or more deterministic sinusoidal components are superimposed on a stochastic Gaussian process. In its simplest form, a single-tone SoR signal $\succsim (t)$$\succsim (t)$≿(t)\succsim(t)$\succsim (t)$ can be expressed as the sum of a deterministic sine wave $y(t)$$y(t)$y(t)y(t)$y(t)$ and a random Gaussian process $x(t)$$x(t)$x(t)x(t)$x(t)$ :

Statistics of the Sine Component

This section draws upon the seminal work of S.O. Rice [10], who provided the analytical foundations for the study of random processes and signal theory.
A sinusoid is defined by the time-dependent function $y(t)=a\sin ({\omega }_{S}t+\psi )$$y(t)=a\sin \left({\omega }_{S}t+\psi \right)$y(t)=a sin(omega_(S)t+psi)y(t)=a \sin \left(\omega_{S} t+\psi\right)$y(t)=a\sin ({\omega }_{S}t+\psi )$, where $a$$a$aa$a$ is the amplitude, ${\omega }_S$${\omega }_S$omega_(S)\omega_{S}${\omega }_S$ is the angular frequency, and $\psi$$\psi$psi\psi$\psi$ is the phase. By assuming the argument $\theta ={\omega }_{S}t+\psi$$\theta ={\omega }_{S}t+\psi$theta=omega_(S)t+psi\theta=\omega_{S} t+\psi$\theta ={\omega }_{S}t+\psi$ to be a random variable uniformly distributed over $[0,2\pi ]$$[0,2\pi ]$[0,2pi][0,2 \pi]$[0,2\pi ]$, the sine wave $y=a\sin (\theta )$$y=a\sin (\theta )$y=a sin(theta)y=a \sin (\theta)$y=a\sin (\theta )$ represents a transformation of a random variable and, consequently, is a random variable itself.
Since every random variable is uniquely characterized by its PDF and an associated characteristic function ${\varphi }^1$${\varphi }^1$phi^(1)\phi^{1}${\varphi }^1$, the sinusoid can be described analytically through ${\varphi }_y[1]$${\varphi }_y[1]$phi_(y)[1]\phi_{y}[1]${\varphi }_y[1]$ :

where $J_0$$J_0$J_(0)J_{0}$J_0$ is the Bessel function of order zero and $\eta$$\eta$eta\eta$\eta$ is the transform variable associated with the characteristic function of the random variable $y$$y$yy$y$. Starting from ${\varphi }_y$${\varphi }_y$phi_(y)\phi_{y}${\varphi }_y$, the PDF of the instantaneous values is derived via inverse Fourier transform:

As shown in Fig. 2, the sine wave spends most of its time near the extremes $\pm a$$\pm a$+-a\pm a$\pm a$. Even though the PDF of instantaneous values is necessary to describe the process, the statistics of extreme values are very useful in some applications such as fatigue and structural dynamics. The PDF of its maxima (peaks) $p_{y_p}$$p_{y_p}$p_(y_(p))p_{y_{p}}$p_{y_p}$ is represented by a Dirac Delta function centred at the amplitude value $a$$a$aa$a$ [28,29]:

Assuming a stationary Gaussian process $x(t)$$x(t)$x(t)x(t)$x(t)$, the PDF of the background noise, $p_x(x,{\mu }_x,{\sigma }_x)$$p_x\left(x,{\mu }_x,{\sigma }_x\right)$p_(x)(x,mu_(x),sigma_(x))p_{x}\left(x, \mu_{x}, \sigma_{x}\right)$p_x(x,{\mu }_x,{\sigma }_x)$ is represented by a Gaussian function, and its characteristic function is:

where ${\mu }_x$${\mu }_x$mu_(x)\mu_{x}${\mu }_x$ and ${\sigma }_x$${\sigma }_x$sigma_(x)\sigma_{x}${\sigma }_x$ are, respectively, the mean value and the standard deviation of the random signal. It must be noted that, throughout this work, random signals will be assumed having ${\mu }_x=0$${\mu }_x=0$mu_(x)=0\mu_{x}=0${\mu }_x=0$. This means that ${\sigma }_x$${\sigma }_x$sigma_(x)\sigma_{x}${\sigma }_x$ coincides with the signal's RMS.

A SoR signal can be analytically treated as the superposition of two independent random variables. The PDF of the instantaneous values of the combined signal, $z(t)=x(t)+y(t)$$z(t)=x(t)+y(t)$z(t)=x(t)+y(t)z(t)=x(t)+y(t)$z(t)=x(t)+y(t)$, is defined by the convolution of the Gaussian PDF and the sine PDF [10]:

To simplify the evaluation of this convolution integral, the characteristic functions ${\mathit{\varphi }}_x$${\mathit{\varphi }}_x$phi_(x)\boldsymbol{\phi}_{x}${\mathit{\varphi }}_x$ and ${\mathit{\varphi }}_y$${\mathit{\varphi }}_y$phi_(y)\boldsymbol{\phi}_{y}${\mathit{\varphi }}_y$ are involved. According to the properties of Fourier Transform (FT) pairs, the FT of the PDF of a sum of independent variables corresponds to the product of their respective characteristic functions [10]. Given the Gaussian nature of the random noise with zero mean ( ${\mu }_x=0$${\mu }_x=0$mu_(x)=0\mu_{x}=0${\mu }_x=0$ ) and standard deviation ${\sigma }_x$${\sigma }_x$sigma_(x)\sigma_{x}${\sigma }_x$ (representing the RMS value of the stochastic component), the characteristic functions for the two components are expressed in Eqns. (16) and (19). The resulting characteristic function of the SoR signal is therefore ${\varphi }_{\tilde{z}}(\eta )={\varphi }_x(\eta )\cdot {\varphi }_y(\eta )$${\varphi }_{\tilde{z}}(\eta )={\varphi }_x(\eta )\cdot {\varphi }_y(\eta )$phi_( tilde(z))(eta)=phi_(x)(eta)*phi_(y)(eta)\phi_{\tilde{z}}(\eta)=\phi_{x}(\eta) \cdot \phi_{y}(\eta)${\varphi }_{\tilde{z}}(\eta )={\varphi }_x(\eta )\cdot {\varphi }_y(\eta )$. The PDF of the process is then recovered through the inverse Fourier transform of ${\varphi }_z(\eta )$${\varphi }_z(\eta )$phi_(z)(eta)\phi_{z}(\eta)${\varphi }_z(\eta )$, thereby avoiding the direct evaluation of the convolution integral in (20).
The statistical behaviour of the combined process is heavily influenced by the ratio $a/{\sigma }_x$$a/{\sigma }_x$a//sigma_(x)a / \sigma_{x}$a/{\sigma }_x$, which represents the amplitude of the sinusoidal tone relative to the RMS level of the background noise. As this ratio increases, the distribution shows a progressive deviation from the Gaussian bell shape, as illustrated in Fig. 3. The SoR PDF gradually adopts a bimodal profile, which is characteristic of the sinusoidal component, resulting in a significantly higher probability density at the tails of the distribution. This non-Gaussian nature is of critical importance in estimating fatigue life, as it directly affects the distribution of peak values and the subsequent cumulative damage evaluation [2]. Due to the non-Gaussianity of the process, the use of standard spectral methods, such as Dirlik's or narrow-band approximation, will lead to inaccurate life estimations [8].

To overcome the limitations of classical spectral methods imposed by the non-Gaussian nature of SoR signals, the literature proposes two primary frameworks: the theoretical derivation of the cycle PDF and the practical derivation of an equivalent PSD.

Originally pioneered by S.O. Rice [10], this theoretical approach focuses on the rigorous derivation of the PDF of the stress cycle amplitudes. It treats Sine-on-Random excitation as a distinct stochastic process, explicitly modelling the interaction between the deterministic harmonic components and the background Gaussian noise.
The fundamental objective of this approach is to estimate the PDF of the rainflow cycle amplitudes directly in the frequency domain based exclusively on statistical properties. Rice's approach avoids the forced Gaussian assumption, thereby preserving the inherent non-Gaussian characteristics of the SoR signal. These characteristics include the unique peak distribution and the increased probability of high-amplitude cycles.
For a NB random process overlaid with $N_b$$N_b$N_(b)N_{b}$N_b$ harmonics, Rice's PDF for peak values $s_p$$s_p$s_(p)s_{p}$s_p$ is expressed as follows:

where ${\sigma }_{s,r}^2$${\sigma }_{s,r}^2$sigma_(s,r)^(2)\sigma_{s, r}^{2}${\sigma }_{s,r}^2$ is the variance of the zero-mean random component, $a_{S_i}(i=1,2,\dots ,N_b)$$a_{S_i}\left(i=1,2,\dots ,N_b\right)$a_(S_(i))(i=1,2,dots,N_(b))a_{S_{i}}\left(i=1,2, \ldots, N_{b}\right)$a_{S_i}(i=1,2,\dots ,N_b)$ are the sine amplitudes and $J_0$$J_0$J_(0)J_{0}$J_0$ is the Bessel function of order zero.
The only case in which Eqn. (21) offers a closed-form solution is the single-tone scenario: a single sinusoidal signal ( $N_b=1$$N_b=1$N_(b)=1N_{b}=1$N_b=1$ ) of amplitude $a_s$$a_s$a_(s)a_{s}$a_s$ superimposed on a random PSD. The resulting PDF takes the following shape:

where $I_0$$I_0$I_(0)I_{0}$I_0$ represents the modified Bessel function of order zero.
Once the PDF is known, the evaluation of fatigue damage can proceed by means of Miner's integral linear damage evaluation formula (Eqn. (10)). If the random component of the SoR process is NB, the SoR peak values PDF $p_{s_p}\left(s_p\right)$$p_{s_p}\left(s_p\right)$p_(s_(p))(s_(p))p_{s_{p}}\left(s_{p}\right)$p_{s_p}\left(s_p\right)$ represents a good approximation of the distribution of rainflow counted cycles $p_s(s)$$p_s(s)$p_(s)(s)p_{s}(s)$p_s(s)$. It should be noted, however, that the accuracy of this approximation diminishes as the bandwidth of the random component increases.
While a closed-form solution exists for a single sinusoid, the integration of Eqn. (21) must be performed numerically when dealing with multiple pure tones ( $N_b>2$$N_b>2$N_(b) > 2N_{b}>2$N_b>2$ ), possibly leading to a considerable increase in computational times.
A further limitation in Rice's theoretical framework is the requirement for incommensurate harmonic frequencies. For the sine phases under consideration to be treated as independent and uniformly distributed, it is necessary that the frequencies are not rational multiples of one another. However, in practical engineering scenarios, particularly those involving rotating machinery, pure tones are frequently harmonics (integer multiples) of the fundamental rotation speed. This discrepancy between theoretical assumptions and real-world specifications may limit the statistical representativeness of the analytical model in certain operational environments.
For this set of reasons, practical rather than purely theoretical methods will be introduced.
Although these approaches may sacrifice some degree of accuracy, they offer greater ease of implementation and significantly reduced computational time for fatigue life estimation.

The primary objective of these practical approaches is to simplify the complex SoR environment by defining a purely stochastic PSD that yields the same damage potential as the original mixed-mode signal. Despite its inherent approximation, as it imposes a non-Gaussian SoR distribution onto a Gaussian framework, this methodology is employed due to its notable
computational efficiency. The inevitable loss of physical and statistical fidelity is balanced by the significant reduction in processing time compared to exhaustive time-domain analyses, facilitating the rapid synthesis of qualification test profiles. The introduction of the FDS of SoR loadings is necessary to understand the basis of some of the following approaches.

Fatigue damage spectrum of a Sine-on-Random signal. While damage evaluation for purely Gaussian or purely sinusoidal environments is well-established, the mixed nature of SoR processes requires a dedicated methodological approach, as described in the works of Angeli et al. [30] and Lalanne [11].

FDS for signals with a single sinusoid. Consider a SoR acceleration profile defined by a single sinusoidal tone (with amplitude a and frequency $f_S$$f_S$f_(S)f_{S}$f_S$ ) superimposed on a Gaussian stochastic process described by its PSD $G_{xx}(f)$$G_{xx}(f)$G_(xx)(f)G_{x x}(f)$G_{xx}(f)$. Due to the harmonic component, the signal deviates from a Gaussian distribution, leading to the need for a method for estimating the cycle distribution.
The first step involves determining the response of each SDOF system in terms of relative displacement $Z(t)$$Z(t)$Z(t)Z(t)$Z(t)$. The mean square value of the random component of the response, ${\sigma }_{Z,r}^2$${\sigma }_{Z,r}^2$sigma_(Z,r)^(2)\sigma_{Z, r}^{2}${\sigma }_{Z,r}^2$, can be calculated as:

Alternatively, using the white noise approximation:

The amplitude of the sinusoidal response $Z_S$$Z_S$Z_(S)Z_{S}$Z_S$ is determined via the Frequency Response Function (FRF):

The RMS values for the sinusoidal response and its derivatives ( ${\dot{Z}}_{S,RMS},Z_{S,RMS}$${\dot{Z}}_{S,RMS},Z_{S,RMS}$Z^(˙)_(S,RMS),Z_(S,RMS)\dot{Z}_{S, R M S}, Z_{S, R M S}${\dot{Z}}_{S,RMS},Z_{S,RMS}$ ) are defined as follows [19]:

To estimate the expected damage, a simplified assumption is introduced: the SoR signal is treated as a stationary process to estimate the peak rate ${\nu }_{pSoR}$${\nu }_{pSoR}$nu_(pSoR)\nu_{p S o R}${\nu }_{pSoR}$. Assuming statistical independence between the components, the peak rate is expressed as:

The PDF of the peaks $Z_p$$Z_p$Z_(p)Z_{p}$Z_p$ follows the Rice distribution and can be obtained by integrating Eqn. (21). A closed-form solution exists exclusively for a single tone, yielding

where $I_0$$I_0$I_(0)I_{0}$I_0$ is the modified Bessel function of the first kind of order zero. The expected damage $E[D]$$E[D]$E[D]E[D]$E[D]$ is then calculated using Eqn. (10):

For a single sinusoid, the integral yields a closed-form solution: