Amplitude Incremental Method: A Novel Approach to Capture Stable and Unstable Solutions of Harmonically Excited Vibration Response of Functionally Graded Beams under Large Amplitude Motion

B. Panigrahi; G. Pohit

doi:10.1515/ijnsns-2018-0235

Article

Amplitude Incremental Method: A Novel Approach to Capture Stable and Unstable Solutions of Harmonically Excited Vibration Response of Functionally Graded Beams under Large Amplitude Motion

B. Panigrahi and G. Pohit

Published/Copyright: April 16, 2019

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From the journal International Journal of Nonlinear Sciences and Numerical Simulation Volume 20 Issue 5

Abstract

An interesting phenomenon is observed while conducting numerical simulation of non-linear dynamic response of FGM (functionally graded material) beam having large amplitude motion under harmonic excitation. Instead of providing a frequency sweep (forward or backward), if amplitude is incremented and response frequency is searched for a particular amplitude of vibration, solution domain can be enhanced and stable as well as unstable solution can be obtained. In the present work, first non-linear differential equations of motion for large amplitude vibration of a beam, which are obtained using Timoshenko beam theory, are converted into a set of non-linear algebraic equations using harmonic balance method. Subsequently an amplitude incremental iterative technique is imposed in order to obtain steady-state solution in frequency amplitude plane. It is observed that the method not only shows very good agreement with the available research but the domain of applicability of the method is enhanced up to a considerable extent as the stable and unstable solution can be captured. Subsequently forced vibration response of FGM beams are analysed.

Keywords: non-linear vibration; harmonic balance method; amplitude incremental technique; FGM beams; Timoshenko beam theory

MSC 2010: 37M05

PACS: 02.70.-c

Funding statement: This research received no specific grant from any funding agency in the public, commercial or not-for-profit sectors.

Conflict of Interest

None.

Appendix A

Elements of stiffness and mass matrices and load vector for uniformly distributed loading of eq. (8), for i=1, 2, 3, …, N and j=1, 2, 3, …, N are mentioned below:

(19)Mj,i+N=Mj,i+2N=Mj+N,i=Mj+N,i+2N=Mj+2N,i =Mj+2N,i+N=0, Mj,i=∫01ΦujΦuidξ, Mj+N,i+N=∫01ΦwjΦwidξ, Mj+2N,i+2N=m3∫01ΦΨjΦΨidξ, Kj,i+N=Kj,i+2N=Kj+N,i=Kj+N,i+2N=Kj+2N,i =Kj+2N,i+N=0,Kj,i=α∫01δΦujδξδΦuiδξdξ,Kj+N,i+N=k4α2∫01δΦwjδξδΦwiδξdξ,Kj+2N,i+2N=k3α∫01δΦΨjδξδΦΨiδξdξ+k4α3∫01ΦΨjΦΨidξ,Knlj,i=Knlj,i+2N=Knlj+N,i+2N=Knlj+2N,i =Knlj+2N,i+N=Knlj+2N,i+2N=0, Knlj,i+N=α∫01δΦujδξδΦwiδξδwδξdξ, Knlj+N,i =α∫01δΦwjδξδΦuiδξδwδξdξ, Knlj+N,i+N=14α∫01δΦwjδξδΦwiδξδwδξ2dξ,

Non-zero elements of Knl1∗ and Knl2∗ are given below:

(20)Knl1∗j,i+N=α∫01δΦujδξδΦwiδξδwbjδξdξ,Knl1∗j+N,i=α∫01δΦwjδξδΦuiδξδwajδξdξ,Knl2∗j+N,i+N=14α∫01δΦwjδξδΦwiδξδwbjδξ2dξ,

Load vectors F_l for various loading patterns are given as

(21)(F)j=(F)j+2N=0, (F)j+2N=FL2K1h2bΦwj|ξ=ξpfor concentrated loading(F)j=(F)j+2N=0,(F)j+2N=fL2K1h2bL∫01Φwjdξfor UDL loading(F)j=(F)j+2N=0, (F)j+2N=L2K1h2bL∫01f(ξ)Φwjdξfor Triangular loading(F)j=(F)j+2N=0,(F)j+2N=L2K1h2bL∫01f(ξ)Φwjdξfor Hat loading

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Received: 2018-08-09

Accepted: 2019-03-30

Published Online: 2019-04-16

Published in Print: 2019-08-27

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Articles in the same Issue

https://doi.org/10.1515/ijnsns-2018-0235

Keywords for this article

non-linear vibration; harmonic balance method; amplitude incremental technique; FGM beams; Timoshenko beam theory