CAE-Companion-2018-2019

Engineering WISSEN CAE

Operational Strength under Consideration of Random Loads in the Frequency Domain

Dynamic loads characterized by randomness frequently lead to system failure through material fatigue in vibratory mechanical structures. In order to correctly account for this random character, it is necessary to observe the stochastic load over a longer period of time. This requires long mea- surement series for measurement-based determination. In order to ensure that the random process is stationary, it is also necessary to observe the stochastic properties over the course of several measurement series. Description of stochastic processes The aforementioned considerations make only limited allow- ance for the conclusion that determination of the stochastic process can be made using individual realizations (individual measurement series). Consequently, the function p(x) is used to characterize the stochastic process, which indicates the probability that the signal x(t) lies in the interval [ x,x+∆x ]. The temporal correlation of the random process is described using the spectral power density S X (f) , whereby f designates the frequency. Especially in the case that p(x) corresponds to a normal distribution, the distribution parameters (arithmetic mean and variance) can be determined from the spectral power density S X (f) . Linear mechanical systems under stochastic excitation If a linear, time-invariant mechanical system is excited by transfer function h(t) over a normally distributed load signal x(t) , it can be demonstrated that the equivalent stress history ( ) = ∫ ( )ℎ( − ) +∞ −∞ is also distributed normally. The equivalent stress history can be transformed into the frequency domain by means of the Fourier transformation. In particular, Y(f) = H(f) ∙ X(f) results, whereby Y(f), H(f) and X(f) designate the Fourier transforms of the equivalent stress history, the transfer function and the load signal. The power density spectrum of the input signal delivers the power density spectrum of the equivalent stress by means of the relationship S Y (f) = H(f) ∙ S X (f) ∙ H*(f) . Here, H* is the conjugate complex and transposed transfer function. The frequency-dependent transmission behavior H(f) of a linear mechanical structure can be determined effectively by a frequency-based response analysis with a unit load using finite element methods (Fig. 1).

Figure 1: Modal transfer functions (left) and modal stress result for mode 2 (right). Operational strength of stochastic processes For normally distributed equivalent stress histories, it is possible to specify probability densities in particular, which allow counting of the stress cycles which occur (rainflow classification) to be carried out on a probabilistic basis. Following from this, operational strength forecasts can be de- rived. Especially for narrow-band signals, it can be shown by means of analysis that a Rayleigh distribution results for the oscillation amplitudes of the equivalent stress. An estimation of the distribution parameters can be carried out using the spectral equivalent stress power density S Y (f) with the help of moments of the k order = ∫ +∞ −∞ ( ) Dirlik [1] developed an empirical distribution density function for counting stress cycle amplitudes (probability density function) based onMonte-Carlo simulations which is also applicable for broad-band signals. Further approaches for the distribution densities of oscillation amplitudes can be found in the works of Zhao-Baker [2], Tovo – Benasciutti [3], as well as Petrucci – Zuccarello [4], among others. A representation of the overall process is shown in Fig. 2.

Figure 2: Spectral fatigue process

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