Approximate Analytical Determination of Vibrodiagnostic Parameters of Non-Linearity of Elastic Bodies Due to the Presence of a Closing Crack. Part 1. Available and Proposed Methods of Solution

Approximate analytical determination of vibrodiagnostic parameters of a cracked elastic body under subharmonic resonance. Part 1. Weak resonance

“…The ever increasing interest in studying vibrations of elastic bodies with a fatigue crack-like damage is motivated by the necessity of assessing any possible changes in the vibration status of structural elements during their long-term operation as well as by the need to develop efficient methods of vibrodiagnostics for such damage. The related research efforts are pursued along the following three main directions: (i) conventional, i.e., the determination of changes of natural frequencies of structural elements in the presence of cracks [1-7] and tracing a variation of the mode shape [8,9], (ii) determination of parameters of nonlinear effects of vibration processes, which are due to the presence of a closing or "breathing" crack [10][11][12][13][14], and (iii) assessment of the influence of a crack on the damping characteristics of elastic bodies [15][16][17][18].Establishing relations between a closing crack parameters and vibration parameters of an elastic body in crack-induced nonlinear resonances is still one of the less explored lines of inquiry as it is difficult to analytically solve the problem and set up a correct experiment.The previously performed analysis of available methods for calculating the forced vibrations of an elastic body with a closing crack [19] suggests that there are considerable difficulties associated with finding easy-to-use solutions. The resultant sets of complex constitutive equations did eventually require a numerical solution.…”

“…The previously performed analysis of available methods for calculating the forced vibrations of an elastic body with a closing crack [19] suggests that there are considerable difficulties associated with finding easy-to-use solutions. The resultant sets of complex constitutive equations did eventually require a numerical solution.…”

“…Considering that unlike a superharmonic resonance [19] in this case in the lower harmonic period there are four extreme values of the second harmonic, which correspond to the known value of the system's rigidity, we will study the simplest method for the determination of approximate relative amplitude A. In Eqs.…”

“…The unknown parameters A 1 2 / , A 1 , γ 1 2 , and γ 1 are determined by the method [19] which essentially consists in satisfying Eq. (1) at the instants for which the value of the bilinear characteristic of the restoring force is known.…”

“…3) For the principal resonance region (ν ω ≈ ) we use a solution as obtained by an asymptotic method of the nonlinear mechanics [22], where A 0 is the constant component and A n is the higher harmonic amplitudes, The resonance frequency ω 0 is taken equal to the free frequency of an elastic system with an asymmetric bilinear characteristic of the restoring force, which is governed by the α parameter [19], ω α α ω αω 0 2 1…”