Analytical and Dynamic study of Pulled Mass Nonlinear Vibration by Two Cables using Newton's Harmonic Balance Method

In recent years, much research has been done on nonlinear vibrations, and analytical and numerical methods have been used to solve complex nonlinear equations. The behavior of nonlinear oscillating equations is discussed until the second order is approximated. Harmonic balance method, which itself has limitations in application. This method continues to be able to study a wider range of nonlinear differential equations. In general, nonlinear vibration problems are of great importance in physics, mechanical structures, and other engineering research. First, the equation of nonlinear vibrations governing the mass of the particle mass connect to the drawn cable is calculated and then the Newton Harmonic Balance Method is used to study the nonlinear vibrations of the set and obtain the answer and its frequency. The method (NHBM) is done with Maple software and a comparison between the results of this method with the solution methods used by other researchers is shown to be a good match. 
  



Introduction
Nonlinear vibration issues are of great importance in physics, mechanical structures and other engineering research. Vibration response, stability, and frequencies are the main components of a system's vibration check Therefore, investigating the influence of different parameters in these areas can be an important step in the design process. In the 1980s, many researchers used numerical or approximate methods to conduct research. It should be noted that the methods used by these researchers could not have predicted some important nonlinear phenomena such as subharmonic responses and turbulence. At certain intervals of vibration, the system has turbulent vibrations. In this case, the vibrations will be intense and unpredictable. One of the most common of these methods is the perturbation method, which involves expanding the series around a small parameter in a nonlinear system. Newton's harmonic methods are approximate, similar to the February series method used for linear oscillators. And it can be used to investigate the periodic solutions, which are as follows: Harmonic balancing methods are another common method in this field that have their own limitations in application. Due to these limitations in the use of methods, there is always an attempt to develop analytical and semi-analytical methods for the ability to study a wider range of nonlinear differential equations. Therefore, in this paper, Newton's harmonic balance (NHBM), which is a combination of Newton's method and harmonic balance method, is used to solve the problem of nonlinear vibrations of the mass of the particle connected to the drawn cable [1]. The subject of particle mass bound to non-mass cables has also been investigated by researchers using various solutions. One of the most recent studies was that of a senior researcher and Bele´ndez et al [2] who provided an approximate solution using the hypertrophy method of perturbation for the problem. Sehreh et al [3] also used an approximate solution using the maximum minimization method, the frequency amplitude frequency method, and the parameter expansion method to solve this problem. Akbarzadeh et al [4] also used non-linear vibration frequency amplitude frequency formula to investigate the mass connected to the drawn cables and compared the frequencies in different vibration ranges with the harmonic balance method. Khaled et al. [5] also used an approximate homeopathy method to solve the problem of nonlinear vibration of a mass connected to a drawn cable. Betaineh [6] used Homotopic Analysis Method (HAM) to solve the problem and find the periodic answer. This article also uses the NHBM method to facilitate the computational process. Figure 1 shows an L-shaped horizontal elastic beam and an E-shaped elastic module of level I, which are subjected to axial load P, uniform cross-sectional area, and assumed homogenous material. The beams are modeled according to Bernoulli's Euler beam theory. In the Bernoulli beam hypothesis, the cross-sectional plates remain flat after deformation, the midline will remain perpendicular to the middle plate. In this section, we obtain the differential equation of the transverse vibrations of the beam, using the energy method, and in Figure 1, the element shows the initial length of the dx from the beam after deformation and displacement. The geometry of the problem is as shown in Figure 1. Consider the motion of a particle of mass m in the direction of x, which is connected to the cables drawn on both sides, and the cables are fixed at the end. If we apply Hooke's law to each piece of cable drawn, the T-tension in each part is equal to:

Statement of the Problem
In this case, the length L is long, and the length a is x = 0 and k is the hardness factor. The sum of the forces acting in the direction of x on the mass m are as follows: (2)  Where sinϴ=(x/L). Since 2 = 2 + 2 ، equation 2 can be written as follows: If x<<d means the amplitude of the vibrations is low, the relation 3 becomes simpler [7]: The equation of motion is as follows: Where α and β are defined as follows: is the equation governing the vibrations of the set in Figure 1 and the set is in the following initial conditions: In this case, a determines the maximum amplitude of the cross-sectional vibration of the beam, which is shown in Figure 2 of Newton's harmonic balance method.

Newton Harmonic Balance Method
The    By placing relations 10 and 11 in Equation 12 for analytical approximation, the first order is as follows: (12) 1 ( ) = . ∆ 1 = ∆ " = ∆ 1 2 = 0 For our second approximation, we use Equation 13 After placing Equations 10 and 11 in Equation 9, we get the coefficients 3 and by zero and solve the equations of terms C1 and ∆ 1 2 simultaneously. (13) Therefore, the approximation of the second order for frequency and displacement is obtained as follows [10]:

Problem Solving using Newton's Harmonic Balance Method
Using the harmonic balance method to obtain the equation frequency, assume that the answer is z = Acos ωt, by placing the assumption answer in equation 5 to the following equation: To obtain the frequency, coefficients are equated on both sides: ( 1 7 ) = √0.75 2 + Therefore, the first approximation of the equation was obtained in the form of 1 = (0.75 2 + ) 1/2 , to obtain the second approximation of Newton's harmonic equilibrium method as stated by placing τ=ωt in the equation 16 is as follows. , the equation is written as follows: The following equation obtained by linear this equation with respect to ∆ 1 and ∆ 1 2 : (20) Therefore, the approximation of the second order for frequency and displacement by Newton harmonic balance method will be as follows:

Results and Discussion
In this section, first, the accuracy of the obtained relations is compared with the researches and then by drawing the equations obtained by maple software, the effect of coefficients and parameters on the frequency and response is investigated. Table 1 compares the results of the HBM and He's amplitude frequency formulation methods investigated in reference [4] with the method used in the present paper, which shows a good correlation in the results due to the small difference. After verification, the effect of the coefficients on the system frequency is investigated using diagram, which shows the behavior of dimensionless deviation z(A,t) and α = π and β = 0.15 and the maximum displacement in the middle of the beam is equal to five. In Figure 4, the frequency changes with respect to the amplitude by changing β in α = 14.3 are dimensionless and in Figure 5, the displacement obtained by the NHBM method in α = π, β = 0.15 and in Figure 6, the frequency changes with respect to the amplitude by changing α in β = 0.15 are parameters, and in Figure 7, the frequency changes with respect to the α = β = 0.15 amplitude are shown.

Conclusion
This paper investigates the response to nonlinear vibrations of a mass connected to a cable. After obtaining the equation governing the vibrations of the beam, a new Newton harmonic balancing method (NHBM) is used to solve this equation and complex dynamic behavior, such as a drastic change in the output regime, predicts a sudden instability. Using the Newtonian harmonic equilibrium (NHBM) method for quantitative analysis, the frequency response is predictable and it was found that the frequency of nonlinear free vibration decreases with increasing initial tensile ratio and control parameters (pressure and voltage). Comparing the results of this method with the work of other researchers shows that this method has a good accuracy.