Steady-State Response to Periodic Excitation in Fractional Vibration System

2016 ◽  
Vol 32 (1) ◽  
pp. 25-33 ◽  
Author(s):  
C. Huang ◽  
J.-S. Duan
Keyword(s):  
Steady State ◽  
Fractional Derivative ◽  
Fractional Power ◽  
Steady State Response ◽  
Periodic Excitation ◽  
Vibration System ◽  
Principle Of Superposition ◽  
State Response ◽  
Harmonic Excitations ◽  
Frequency Relation

AbstractThe steady-state response to periodic excitation in the linear fractional vibration system was considered by using the fractional derivative operator . First we investigated the response to the harmonic excitation in the form of complex exponential function. The amplitude-frequency relation and phase-frequency relation were derived. The effect of the fractional derivative term on the stiffness and damping was discussed. For the case of periodic excitation, we decompose the periodic excitation into a superposition of harmonic excitations by using the Fourier series, and then utilize the results for harmonic excitations and the principle of superposition, where our adopted tactics avoid appearing a fractional power of negative numbers to overcome the difficulty in fractional case. Finally we demonstrate the proposed method by three numerical examples.

scholarly journals Vibration Equation of Fractional Order Describing Viscoelasticity and Viscous Inertia

Open Physics ◽  
2019 ◽  
Vol 17 (1) ◽  
pp. 850-856 ◽  
Author(s):  
Jun-Sheng Duan ◽  
Yun-Yun Xu
Keyword(s):  
Fractional Derivative ◽  
Fractional Order ◽  
Fractional Derivatives ◽  
Harmonic Excitation ◽  
Steady State Response ◽  
Vibration System ◽  
Derivative Operator ◽  
Vibration Equation ◽  
State Response ◽  
Frequency Relation

Abstract The steady state response of a fractional order vibration system subject to harmonic excitation was studied by using the fractional derivative operator ${}_{-\infty} D_t^\beta,$where the order β is a real number satisfying 0 ≤ β ≤ 2. We derived that the fractional derivative contributes to the viscoelasticity if 0 < β < 1, while it contributes to the viscous inertia if 1 < β < 2. Thus the fractional derivative can represent the “spring-pot” element and also the “inerterpot” element proposed in the present article. The viscosity contribution coefficient, elasticity contribution coefficient, inertia contribution coefficient, amplitude-frequency relation, phase-frequency relation, and influence of the order are discussed in detail. The results show that fractional derivatives are applicable for characterizing the viscoelasticity and viscous inertia of materials.

Steady periodic response for a vibration system with distributed order derivatives to periodic excitation

2017 ◽  
Vol 24 (14) ◽  
pp. 3124-3131 ◽  
Author(s):  
Jun-Sheng Duan ◽  
Dumitru Baleanu
Keyword(s):  
Fourier Series ◽  
Steady State ◽  
Fractional Derivative ◽  
Harmonic Excitation ◽  
Periodic Excitation ◽  
Vibration System ◽  
Principle Of Superposition ◽  
Derivative Operator ◽  
Numerical Examples ◽  
Distributed Order

Steady-state periodic responses for a vibration system with distributed order derivatives are investigated, where the fractional derivative operator [Formula: see text] is utilized. The response to complex harmonic excitation is derived and the amplitude–frequency and phase–frequency relations are obtained. For a periodic excitation, we decompose it into the Fourier series, and then make use of the principle of superposition and the results of harmonic excitations to obtain the response. Finally, we examine three numerical examples by using the proposed method.

Dynamic Interaction of Viscoelastic Soil and Fractional Derivative Type Lining with a Circular Tunnel

2012 ◽  
Vol 170-173 ◽  
pp. 1542-1545
Author(s):  
Min Jie Wen ◽  
Zi Ping Su ◽  
Hui Tuan He
Keyword(s):  
Steady State ◽  
Fractional Derivative ◽  
Dynamic Interaction ◽  
Constitutive Behavior ◽  
Steady State Response ◽  
Circular Tunnel ◽  
Resonance Effects ◽  
Material Parameters ◽  
State Response ◽  
Derivative Type

Coupled harmonic vibration of viscoelastic soil and fractional derivative type lining system with a deeply buried circular tunnel is investigated in the frequency domain. Based on theory of elastic and fractional derivative, steady state response of the viscoelastic soil and lining system is studied. Regarding the lining as a medium with fractional derivative constitutive behavior, and the analytical expressins of the displacement and stress of the soil and lining are respectively obtained by the continuity conditions on the inner boundary of lining and the interface between the soil and the lining. The order of fractional derivative model has a greater influence on system dynamic response, and it dependent on the material parameters of lining. With the frequency increasing, the resonance effects of system decrease.

Steady-State Response of a Piping System Under Harmonic Excitations Considering Pipe-Support Friction With Variable Normal Loads

2015 ◽  
Vol 137 (5) ◽  
Author(s):  
José Argüelles ◽  
Euro Casanova
Keyword(s):  
Steady State ◽  
Operating Conditions ◽  
Dynamic Loads ◽  
Computational Time ◽  
Piping System ◽  
Steady State Response ◽  
State Response ◽  
Harmonic Excitations ◽  
Piping Systems ◽  
Support Surfaces

Dynamic loads in piping systems are mainly caused by transient phenomena generated by operating conditions or installed equipment. In most cases, these dynamic loads may be modeled as harmonic excitations, e.g., pulsating flow. On the other hand, when designing piping systems under dynamic loads, it is a common practice to neglect strong nonlinearities such as shocks and friction between pipe and support surfaces, mainly because of the excessive cost in terms of computational time and the complexity associated with the integration of the nonlinear equations of motion. However, disregarding these nonlinearities for some systems may result in overestimated dynamic amplitudes leading to incorrect analysis and designs. This paper presents a numerical approach to calculate the steady-state response amplitudes of a piping system subjected to harmonic excitations and considering dry friction between the pipe and the support surfaces, without performing a numerical integration. The proposed approach permits the analysis of three dimensional piping systems, where the normal forces may vary in time and is based in the hybrid frequency–time domain method (HFT). Results of the proposed approach are compared and discussed with those of a full integration scheme, confirming that HFT is a valid and computationally feasible option.

Steady State Response of Compressible Fractional Derivative Viscoelastic Thick-Walled Cylinder

2012 ◽  
Vol 166-169 ◽  
pp. 1510-1513
Author(s):  
Lin Chao Liu ◽  
Lie Yu ◽  
Huan Xin Yu
Keyword(s):  
Steady State ◽  
Fractional Derivative ◽  
Radial Displacement ◽  
Viscoelastic Model ◽  
Constitutive Relationship ◽  
Steady State Response ◽  
Viscoelastic Problem ◽  
State Response ◽  
Walled Cylinder

Many materials show viscoelastic properties under long term load, because of the complexity of viscoelastic problem, it is not enough for describing the characteristics of material and structure with classic viscoelastic model. The stress-strain constitutive relationship is described by fractional derivative viscoelastic model, the radial displacement and stress of thick-walled cylinder under internal pressure are obtained by using Fourier transform and the properties of fractional derivative, and we also investigated the steady state response of compressible fractional derivative viscoelastic thick-walled cylinder.

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