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This paper presents two experimental investigations of uncertainty in natural frequencies of linear structures estimated from measured frequency response function under dynamic loading. Experiments were conducted on 100 nominally identical realizations of two structures: a fixed--fixed beam and a thin plate with one edge fixed and the remaining three edges free. In the first set of experiments, on a fixed-fixed beam, 12 identical masses were placed at random spatial locations (generated by a computer) along the length of the beam. Each random arrangement of the masses constitutes one realization, and 100 such realizations were individually subjected to impulse loading to obtain frequency response functions. The total random mass is about 2% of the total mass of the beam. In the second set of experiments, 10 spring-mass oscillator units were attached to the cantilevered plate at random spatial locations determined from a computer-generated random matrix. Although the beam experiments represent parametric uncertainty in the mass matrix, the plate experiment pertain to unmodeled dynamics, which in turn results in randomness in both the mass and stiffness matrices. The results obtained from these experiments may be useful for the validation of many random eigenvalue analysis and prediction methods currently available to structural dynamicist. This paper is limited to comparisons with Monte Carlo simulation of deterministic finite elements models of the two structures. It is concluded that the method of estimation of natural frequencies from frequency response functions and the spatial location of the measurements has significant influence upon the first two moments (mean and standard deviation) of the natural frequency ensemble. Furthermore, although the Monte Carlo simulation estimates of the mean and standard deviation are in reasonable agreement with experiments at higher frequencies, the probability density function differ appreciably, within the limits of the sample size investigated in this study. DOI: 10.2514/1.44668

A simple and easy-to-implement algorithm to identify a generalized proportional viscous damping matrix is developed in this work. The chief advantage of the proposed technique is that only a single drive-point frequency response function (FRF) measurement is needed. Such FRFs are routinely measured using the standard techniques of an experimental modal analysis, such as impulse test. The practical utility of the proposed identification scheme is illustrated on three representative structures: (1) a free-free beam in flexural vibration, (2) a quasiperiodic three-cantilever structure made by inserting slots in a plate in out-of-plane flexural vibration, and (3) a point-coupled-beam system. The finite element method is used to obtain the mass and stiffness matrices for each system, and the damping matrix is fitted to a measured variation of the damping (modal damping factors) with the natural frequency of vibration. The fitted viscous damping matrix does accommodate for any smooth variation of damping with frequency, as opposed to the conventional proportional damping matrix. It is concluded that a more generalized viscous damping matrix, allowing for a smooth variation of damping as a function of frequency, can be accommodated within the framework of standard finite element modeling and vibration analysis of linear systems. [DOI: 10.1115/1.2980400]

Rayleigh quotients in the context of linear, nonconservative vibrating systems with viscous and nonviscous dissipative forces are studied in this paper Of particular interest is the stationarity property of Rayleigh-like quotients for dissipative systems. Stationarity properties are examined based on the perturbation theory. It is shown that Rayleigh quotients with stationary properties exist for systems with proportional viscous and non-viscous damping forces. It is also shown that the stationarity property of Rayleigh quotients in the case of nonproportional damping (viscous and nonviscous) is conditional upon the diagonal dominance of the modal damping matrix. [DOI: 10.1115/1.2910898]

The phenomenon of elastic boundary layers under quasistatic loading is investigated using the Floquet-Bloch formalism for two-dimensionaL isotropic, periodic lattices. The elastic boundary layer is a region of localized elastic deformation, confined to the free edge of a lattice. Boundary layer phenomena in three isotropic lattice topologies are investigated: the semiregular Kagome lattice, the regular hexagonal lattice, and the regular fully triangulated lattice. The boundary layer depth is on the order of the strut length for the hexagonal and the fully triangulated lattices. For the Kagome lattice, the depth of boundary layer scales inversely with the relative density. Thus, the boundary layer in a Kagome lattice of low relative density spans many cells. [DOI: 10.1115/1.2775503]