Traditional and Emerging Techniques for Practical Random Vibration Analyses

The paper focuses on available tools for conducting random vibration analyses for practical engineering problems. An inherent aspect of this theme is the simultaneous existence of elements with linear behavior, and of elements with nonlinear behavior within the system. In this regard, techniques both for linear and nonlinear random vibration analyses are discussed. Attention is focused on traditional techniques such as statistical linearization, and Monte Carlo simulation. Further, emerging techniques, such as wavelets, as a tool for signal and response localization; and fractionalcalculus, as a tool for capturingnon-localbehaviorare discussed. Pertinent examples of the application are considered.


INTRODUCTION
Several tools and concepts are necessary for the development of a formalism enabling the capturing and description of uncertainty in the context of random vibration analysis.These must be supplemented by traditional concepts of a vibration analysis such as impulse response function, in the time domain; and transfer function, in the frequency domain.The combination of the probabilistic formalism and of the "solution machinery" of vibrations theory lead to celebrated results such as the relationship between the power spectrum of the input, the power spectrum of the output, and the modulus of the transfer function of a vibratory system.For nonlinear random vibration analysis, these concepts can be extended by using perturbation techniques and generalized tools such as the Wiener-Volterra representation for the response of nonlinear dynamical systems.

RANDOM VIBRATION ANALYSES
Beyond the classical formalism of linear random vibration analysis, four (4) additional tools are discussed within the context of traditional and emerging solution techniques.
First, the concept of statistical linearization is discussed as a tool for conducting random vibration analysis of nonlinear systems.The fundamental concept of the method is the replacement of the nonlinear dynamical system by a surrogate (equivalent) linear system which lends itself to treatment by the standard input/output relationships mentioned above.This replacement is conducted by determining a set of parameters (equivalent linear stiffness and equivalent linear damping) which minimize in a stochastic sense an appropriate measure of the error between the outputs of the linear and nonlinear systems.Further, invoking a reasonable approximation for the system response statistics, the elements of the equivalent linear system are determined by quite convenient formulas.In this particular version, the method of statistical linearization has proved as a quite versatile tool for conducting random analysis for multi-degree-of-freedom systems encountered in engineering practice which are exposed to either stationary or non-stationary excitations, and which exhibit either elastic or inelastic (hysteretic) nonlinearities.
Secondly, and from a certain perspective in conjunction with the concept of the evolutionary power spectrum, the possibility for temporal and spatial localization offered by the wavelets families is discussed.In this context, it is emphasized that many natural phenomena, for instance earthquakes, exhibit frequency content which is changing in time.In this regard, the notion that wavelets can be construed as a "mathematical microscope" is pointed out.They can be viewed as a two-parameter version of the classical Fourier series expansion involving parameters reflecting translation and dilation in the time domain.This eventually affords the representation of the unfolding characteristic of a particular signal both in time and frequency.Various families of wavelets are considered.Further, issues and solution procedures pertaining to the response of linear and nonlinear oscillatory systems exposed to random processes with evolutionary spectra represented via the wavelet transform are discussed.
Thirdly, the concept of fractional (better, noninteger) order derivatives is introduced as a tool for capturing local effects for vibratory systems excitations and responses.In this regard, the fractional derivative of an arbitrary signal is construed, under appropriate conditions, as the inverse transform of the product of the Fourier of the original signal with an arbitrary power of the imaginary axis frequency.From another perspective, it is emphasized that in the discrete domain the fractional order derivative is calculated\ as a linear superposition of previous values of the signal with weights decreasing away from the current point of definition of the derivative.In this context, fractional calculus offers the feature of a representation with "nonlocal" and "fading memory" characteristics.This modeling has proved fruitful in several fields, and appropriate tools have been developed for conducting random vibration analysis of dynamic systems endowed with fractional derivative terms which are exposed to stochastic excitation.It is emphasized that appending a fractional derivative term in the equation of motion of a dynamic system alters both its stiffness and its damping characteristics.
Fourthly, the possibility of encountering randomness in vibratory systems, not only in terms of the excitation, but also in terms of the characteristics of the vibratory system is noted.In this regard, and with emphasis on continuous vibratory systems, uncertainty quantification is discussed within the framework of the Stochastic Finite Element Method (SFEM).In this context, it is noted that a particular representation of the randomness of the continuous afforded by the classical Karhunen-Loéve expansion is particularly expeditious for finite element analyses.

DISCUSSION AND CONCLUSION
Finally, the technique of Monte Carlo simulation is discussed.This is a particularly versatile technique which applies to systems with either deterministic or random parameters, and either deterministic or random excitations.Practically, a statistical population or relevant system responses is generated, in a sense of a "computational laboratory", by sampling from distributions which describe the parameters defining the dynamics systems and its excitation.Obvious relevant issues regarding the synthesis/simulation of random deviates and stochastic processes with specified probability densities are addressed.Further, attention is focused on the problem of generating stationary and non-stationary random processes samples which are compatible with a target power spectrum (stationary or non-stationary).In this context, the significant drawback of the computational burden associated with the repetitive calculations involved in Monte Carlo techniques is pointed out.Nevertheless, it is also noted that it is the most versatile solution tool for any dynamic system which is defined in a stochastic setting.