Closed-form periodic solutions for piecewise-linear vibrating systems

Pilipchuk, Valery N.

doi:10.1007/s11071-009-9469-0

Cited by 7 publications

(6 citation statements)

References 9 publications

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“…As mentioned earlier, the analytical modeling is based on representing cracks by such elastic elements that work normally in compression phases but show no resistance to tension. Note that such an approach is quite common in the reduced order modeling of cracked elastic structures (see, e.g., Chen and Shaw, 1996;Chati et al, 1997;Butcher, 1999;Jiang et al, 2004;Andreaus et al, 2007;Vestroni et al, 2007;Butcher and Lu, 2007;and Pilipchuk, 2009). However, as shown in Sections 2 and 3, the geometrical specifics of T-joints require the additional analytical tools of dynamic modeling.…”

Section: Scope Of the Present Studymentioning

confidence: 99%

“…On the other hand, when the beam deflection is positive everywhere in the area of joint, then the mechanical properties are equivalent to those of the undamaged joint. Note that, on one hand, such kinds of dynamic behavior bring the model into a class of distributed vibrating systems with nonsmooth nonlinearities, which are quite difficult to analyze (see, e.g., Chen and Shaw, 1996;Chati et al, 1997;Butcher, 1999;Jiang et al, 2004;Andreaus et al, 2007;Vestroni et al, 2007;Butcher and Lu, 2007;and Pilipchuk, 2009). On the other hand, it will be shown shortly that the corresponding characteristics of a nonlinear dynamic response can be employed for damage detection in joints.…”

Section: Joint Modeling and Preliminary Remarksmentioning

confidence: 99%

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Analysis of crack formation in T-joint structures under dynamic loading

Pilipchuk

Ibrahim

2010

Journal of Vibration and Control

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T-joints play a vital role in the hulls of ships and bulkheads made of sandwich materials. Their design aspects and crack detection are important in preserving the safety of ships navigating through violent sea waves. In this work, an overview of the state-of-the-art of design aspects of T-joints used mainly in ship structures is first provided. The design aspects are focused on estimating static and quasi-static failure loads, fracture and fatigue characteristics. This study also develops a reduced order dynamic model for identification of cracks in T-joints. The reduced model constitutes three modal equations with piecewise-linear asymmetric characteristics in which the influence of the crack appears in terms of its length parameter. In particular, it is shown that the presence of a crack essentially affects both the amplitude and frequency content of the dynamic response due to nonlinear coupling between normal modes. Under external dynamic loading with a frequency close to the first mode frequency, the development of a crack is identified by the evolution of attractors on the configuration planes created by different combinations of modal coordinates.

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Section: Scope Of the Present Studymentioning

confidence: 99%

Section: Joint Modeling and Preliminary Remarksmentioning

confidence: 99%

Analysis of crack formation in T-joint structures under dynamic loading

Pilipchuk

Ibrahim

2010

Journal of Vibration and Control

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“…Instead the boundary conditions become coupled though. Different examples of solutions in the idempotent basis are given in recent publications [49] and [48].…”

Section: Positive Timementioning

confidence: 99%

Non-Smooth Spatio-Temporal Coordinates in Nonlinear Dynamics

Pilipchuk

2011

Preprint

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This paper presents an overview of physical ideas and mathematical methods for implementing non-smooth and discontinuous substitutions in dynamical systems. General purpose of such substitutions is to bring the differential equations of motion to the form, which is convenient for further use of analytical and numerical methods of analyses. Three different types of nonsmooth transformations are discussed as follows: positional coordinate transformation, state variables transformation, and temporal transformations. Illustrating examples are provided.

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“…Vestroni et al studied a piecewise linear system with two degrees of freedom using the standard Lindstedt–Poincaré method (L–P) method, which works well for small nonlinearity 23 . Pilipchuk used a standard perturbation procedure in conjunction with nonsmooth temporal transformation to obtain analytical asymptotic solutions for piecewise linear single‐ and two degree of freedom dynamic systems with no damping, no excitation, and small stiffness jump 24 …”

Section: Introductionmentioning

confidence: 99%

Reduction of transmissibility and increase in efficacy of vibration isolation using negative stiffness device with enhanced damping

Nagarajaiah

Zou

Herkal

2022

Structural Contr & Hlth

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Effect of passive vibration isolation heavily depends on force-displacement characteristic of the isolation system. In view of this dependence, this paper investigates the influence of negative stiffness device (NSD) on the effect of vibration isolation system. Detailed evaluation of transmissibility is performed.The critical parameters are identified. It is found that with NSD, significant vibration reduction for both absolute and relative displacement transmissibility is obtained. A modified Lindstedt-Poincaré method (modified L-P method) is used to obtain analytical periodic solutions for the approximated piecewise linear dynamic system. The analytical limit cycles by the modified L-P solution agree satisfactorily with the ones by numerical simulation. The most important finding of this study is that larger damping in a system with NSD helps in reducing transmissibility, thus increasing the efficacy of the isolation system; this is in contrast to other conventional isolation systems, wherein increased structural damping decreases the efficacy of vibration control in the frequency range of interest.

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Closed-form periodic solutions for piecewise-linear vibrating systems

Cited by 7 publications

References 9 publications

Analysis of crack formation in T-joint structures under dynamic loading

Analysis of crack formation in T-joint structures under dynamic loading

Non-Smooth Spatio-Temporal Coordinates in Nonlinear Dynamics

Reduction of transmissibility and increase in efficacy of vibration isolation using negative stiffness device with enhanced damping

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