The fastest insight into the large amplitude vibration of a string

This paper recommends a simple and excusive approach to a strongly nonlinear oscillator. Its frequency property can be immediately obtained by the simplest calculation. The results show that the method leads to an approximate solution with relatively high accuracy. Considering the simplest solution process, this paper provides a highly efficient tool for fast determination of the amplitude-frequency relationship of a nonlinear oscillator. The large amplitude vibration of a string is used as an example to illustrate the solution process.


Introduction
Strings/ropes belong to the oldest means of transmitting a force and, therewith, also power over a distance. Their high flexibility offers means to transmit force through tightest and practically even unapproachable places. They are also characterized by high strength-to-weight ratio, quiet, smooth and free running, long life expectancy, capability of 3D movement in various directions and around bends, minimal maintenance costs. Those exquisite properties made them practically an inevitable element in numerous machineries today, including all kinds of cranes, ropeways, pulleys, etc. Being such an important element of various machines, consideration of their mechanical behavior and developing methods for fast assessment of important parameters of their mechanical behavior are worth of effort.
Since a string has negligible flexural, torsional and shear stiffness and practically nearly zero buckling load, it can be idealized as a one-dimensional elastic continuum, which does not transmit bending and torsional moments and neither shear and longitudinal pressure forces. A vibrating string is at the same time one of the simplest example of a distributed parameter system, but also one of the most interesting ones. During vibration, a string deflects transversely, and quite often the achieved amplitudes call for consideration of nonlinear effects in order to reach suitable accuracy of the obtained results.
We consider a string's transverse vibration with a large amplitude as illustrated in Fig.1. The governing equation can be obtained as follows (Mahabadi & Pazhooh, 2018). (1 ( ) ) with the following boundary conditions where w is the transverse displacement, =/ c  is the transverse wave's velocity,  and  are, respectively, the tension and the mass per unit length, respectively. We assume that the solution of Eq.
(1) can be expressed as By the Galerkin technology, we obtain the following nonlinear vibration of strings with large amplitude (Mahabadi & Pazhooh, 2018).
where β is a constant, A is the amplitude. Eq. (4) occurs in various fields, such as the long cable vibration, the bridge vibration, the MEMS vibration (Anjum & He, 2020a, 2020b, 2020c; Lai, et al., 2008;Skrzypacz, et al., 2019). Eq. (4) can be solved by various analytical methods, e.g., the variational iteration method, the homotopy perturbation method (He & Latifizadeh, 2020;He & El-Dib, 2020;He & Jin, 2020;He, 2020aHe, , 2020bHe, 2006). A fast insight into the frequency property is much needed in practical applications, so the solution process should be as simple as possible. In this paper we will apply the simplest method (He, 2019a,b) in all literature to fast elucidate the frequency property of Eq.(4), the method is called He's frequency formulation and various modifications were appeared in literature (He, Wang, Yao, 2019;Ren & Hu, 2019a, 2019b.

He's frequency formulation
Consider the following nonlinear oscillator He's frequency formulation is (He, 2019 a where N is a constant, 01 N  . Consider the Duffing oscillator (He, 2006) Eq. (9) is same as those obtained by the variational iteration method and the homotopy perturbation method (He 2006), so we recommend 0.8 N  for fast insight into the frequency property of a practical problem.
According to Eq.(6), the frequency of Eq.(4) can be written as The approximate solution is We choose 0.8 N  , Table 1 and Figure 1 show the accuracy of the simplest estimation of Eq. (10).

Discussion and conclusion
In this paper we recommend 0.8 N  , an optimal choice can be made by the least square method where 2/ T   , and  is given in Eq. (10). From Eq. (12), the value for N can be optimally identified.