A method has been developed for converting a discrete sequence of extrema into a continuous process. The relevancy of the problem is attributed to the necessity of an approximate estimation of spectral density in in testing materials and structures under random (irregular) loading. A great bulk of available experimental data thus can be used in development and validation of calculation methods for assessing durability in the multi-cycle region. Postulating the continuity of random stress processes and their first derivative we propose to connect piecewise the available starting points (namely, the extrema of the random process) with half-cosine functions under the condition of compatibility at the points of extrema. A distinctive feature of the method is the provision of 100% coincidence of the values and sequences of extrema in the initial discrete and simulated continuous processes. The issue of choosing the magnitude of half-periods for these half-cosine functions is addressed on the basis of information obtained from the analysis of real stress records in the form of a regression equation linking half-periods and half-ranges for some realizations of the random process for transport vehicles. The regression dependences of the half-periods and semi-ranges of bending stresses (part of a railway train) and torsion (torsion shaft of a tracked vehicle) are shown as an example. An analysis of the correlation of two random variables (half-periods and half-ranges) according to empirical data has shown that the correlation exists and is significant for the observed number of points thus providing the basis for using the regression formula for an approximate choice of the frequency composition of the process. Moreover, the lower restrictions are imposed on the number of points (at least 5) in the half-period. Since the extrema of the initial and simulated processes coincide in accordance with the principle of the proposed simulation, the distribution of the amplitudes of complete cycles, as well as the results of schematization by other known methods are identical, therefore, the estimate of the durability by hypotheses based on a linear one is also identical. The validation of the method consists in consideration of the chain: 1) the initial continuous process; 2) the discrete process of extrema; 3) simulated continuous process according to the proposed method. Auxiliary distributions, such as distributions of maximum, minimum and average cycle values also coincide in accordance with the principle of modeling. The method is proposed to be used in analysis of the comparability of two competing approaches in assessing the loading in the problems of assessing durability, namely: those that use cycle-counting methods and methods based on the spectral density of processes. Since the spectral densities of the processes can differ due to an approximate choice of the frequencies on the basis of a regression formula, methods on their base can give estimates of the durability that differ from those obtained by schematization methods. To study this phenomenon, further computational experiments are required. The developed method can be very useful for the experiment design.
Infinite differentiability of Hermitian and positive {$C$}-semigroups and {$C$}-cosine functions
