梁远 张 ^*

• 西安建筑科技大学, 陕西西安, China

The Old Babylonian Algorithm, a remarkable mathematical gem from ancient Mesopotamia (around 1800-1600 BC), has long been a subject of fascination for scholars. This ancient algorithm not only represents the advanced intellectual capabilities of the Babylonians but also holds great relevance in modern times. It is particularly renowned for its ingenious method of approximating square roots [1].In recent research, an exciting transformation has occurred. The traditional Old Babylonian Algorithm has been ingeniously modified to solve differential equations, breathing new life into this ancient technique [1]. In this paper, we aim to reveal how this remarkable revival has led to a significant role within the complex and captivating realm of Non-linear Vibration theory, thereby bridging the ancient and the modern and demonstrating the timelessness of mathematical concepts.Nonlinear vibration analysis presents several specific challenges. Strongly nonlinear oscillators, such as the one considered in this study (in the form shown in equation ( 1)), are characterized by the presence of higher-order nonlinear terms. These nonlinearities can lead to complex dynamic behaviors that are difficult to analyze using traditional methods [3][4][5]. For instance, the homotopy perturbation method, while widely used, attempts to convert the original nonlinear problem into a sequence of linear subproblems through the introduction of a homotopy parameter, yet it may still face difficulties in accurately handling highly nonlinear systems [2][3][4]. The variational iteration method, another commonly employed approach, also has its limitations in dealing with the intricacies of nonlinear vibrations [5]. He's frequency formulation, originating from an ancient Chinese algorithm, offers a relatively simple and effective means for treating nonlinear vibration systems and has demonstrated unique value in the field of nonlinear dynamics with its straightforward and efficient nature [6,7,8]. However, the modified Old Babylonian algorithm proposed in Ref. [1] provides a novel perspective and potentially a more efficient solution to the strongly nonlinear oscillator problem in comparison to these traditional methods. It combines the ancient wisdom of the Babylonian algorithm with modern adaptations to better suit the context of nonlinear dynamics analysis.By exploring the application of the modified Old Babylonian algorithm in nonlinear vibration, we hope to contribute to a deeper understanding of both the ancient mathematical heritage and its practical implications in contemporary scientific research. This research not only sheds light on the historical significance of the Old Babylonian Algorithm but also paves the way for further advancements in the field of nonlinear dynamics. We consider a strongly nonlinear oscillator in the form 35 0 u u au bu + + + =(1)This oscillator is distinguished by the existence of the high-order nonlinear terms, where a and b are constants. The initial conditions are set as ( )0 uA = and ( ) 00 u= , where A is the amplitude.Traditionally, numerous methods have been adopted to address such nonlinear oscillator problems.The homotopy perturbation method has gained extensive application. It endeavors to convert the original nonlinear problem into a sequence of linear sub-problems by introducing a homotopy parameter [2][3][4]. The variational iteration method is also a commonly -used approach [5]. Moreover, He's frequency formulation [6,7,8], which stems from an ancient Chinese algorithm, presents a simple yet effective means for dealing with nonlinear vibration systems. It has demonstrated its unique value in the field of nonlinear dynamics with its straightforward and efficient characteristics [9][10][11][12][13].However, in this research, we apply the modified old Babylonian algorithm suggested in Ref. [1]:( ) ( ) 2 35 1 1 n n n n n nn n u u au bu u uu u  +  + + + + = + -(2)Where  is a weighting factor. This modified algorithm [1] provides a novel viewpoint and potentially a more efficient solution to the strongly nonlinear oscillator problem in contrast to the traditional methods.To better understand the derivation of this modified algorithm, let's consider the following step -bystep explanation.Suppose the initial solution is ( )0 cos u A t  = (3)where  is the frequency. Utilizing the modified old Babylonian algorithm, we obtain ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )u u au bu u uu u A t A t aA t bA t A t At At              + + + + = + - - + + + + = + -(4)By setting 31 t  = ， Figure 1 provides an elaborate and in -depth comparison between the approximate solution and the exact one for a diverse range of values of parameters a and b. This comparison is of great significance as it vividly showcases the remarkable performance of the modified old Babylonian algorithm within the scope of this complex problem domain. The excellent agreement between the approximate and exact solutions effectively validates the effectiveness of the modified algorithm. It provides solid evidence that the modified old Babylonian algorithm is not only theoretically sound but also practically applicable.( In conclusion, the modified Old Babylonian algorithm has emerged as a highly promising approach in the analysis of strongly nonlinear oscillators. Through its unique combination of ancient mathematical wisdom and modern adaptations, it offers an alternative and efficient solution method compared to traditional techniques.The successful application of this algorithm in the context of nonlinear vibration not only validates its theoretical soundness but also showcases its practical applicability. The excellent agreement between the approximate and exact solutions, as demonstrated in the comparison presented in this study, provides compelling evidence of its effectiveness.Looking ahead, the findings of this research have significant implications for future studies and applications. In the field of MEMS (Micro-Electro-Mechanical Systems), for instance, the algorithm could potentially be employed to enhance the understanding and optimization of the dynamic behavior of MEMS devices. By accurately modeling and analyzing the nonlinear vibrations in these miniature systems, it may be possible to improve their performance, reliability, and lifespan.Furthermore, this research paves the way for further exploration of the algorithm's capabilities. Future investigations could focus on extending its application to more complex nonlinear systems, such as those with time -varying parameters or multiple degrees of freedom. Additionally, there is potential for integrating the modified Old Babylonian algorithm with other advanced computational and analytical tools, such as machine learning algorithms for more efficient parameter identification and system optimization.Overall, the resurgence of the Old Babylonian Algorithm in the modern context of nonlinear dynamics holds great promise for advancing our understanding and ability to handle complex nonlinear systems, opening up new avenues for research and innovation in various scientific and engineering disciplines.13.