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Time domain analysis of the weighted distributed order rheological model

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Abstract

This paper presents the fundamental solution and relevant properties of the weighted distributed order rheological model in the time domain. Based on the construction of distributed order damper and the idea of distributed order element networks, this paper studies the weighted distributed order operator of the rheological model, a generalization of distributed order linear rheological model. The inverse Laplace transform on weighted distributed order operators of rheological model has been obtained by cutting the complex plane and computing the complex path integral along the Hankel path, which leads to the asymptotic property and boundary discussions. The relaxation response to weighted distributed order rheological model is analyzed, and it is closely related to many physical phenomena. A number of novel characteristics of weighted distributed order rheological model, such as power-law decay and intermediate phenomenon, have been discovered as well. And meanwhile several illustrated examples play important role in validating these results.

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Notes

  1. Since the residue of \(\frac{e^{st}}{\int _{a}^{b} s^{\alpha }\,{\mathrm{d}}\alpha +\lambda }\) equals zero at \(s=\infty \), the path integral of it along \(s\to \infty \) vanishes.

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Acknowledgements

Financial support for this work provided by the National Basic Research Program of China (Nos. 2015CB251601, 2013CB227900), National Natural Science Foundation (Nos. 51322401, 51421003, U1261201), the Fundamental Research Funds for the Central Universities (Nos. 2014YC09, 2014ZDPY08) (China University of Mining and Technology) and the 111 Project (No. B07028).

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Authors and Affiliations

  1. State Key Laboratory for Geomechanics and Deep Underground Engineering, China University of Mining and Technology, Xuzhou, 221116, Jiangsu, P.R. China

    Lili Cao, Hai Pu & Ming Li

  2. School of Mechanics and Civil Engineering, China University of Mining & Technology, Xuzhou, 221116, Jiangsu, P.R. China

    Hai Pu

  3. Service Robots Laboratory of Shandong University, School of Control Science and Engineering Shandong University, Qian Fo Mountain Campus, Jing Shi Road 17923, Jinan, 250061, Shandong, P.R. China

    Yan Li

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  1. Lili Cao
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Correspondence to Hai Pu.

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Cao, L., Pu, H., Li, Y. et al. Time domain analysis of the weighted distributed order rheological model. Mech Time-Depend Mater 20, 601–619 (2016). https://doi.org/10.1007/s11043-016-9314-z

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