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# Double Pendulum Design Project

### INTRODUCTION

Developing and evaluating a dual pendulum mechanism constitutes a stimulating and demanding undertaking that necessitates a cross-functional methodology. The current implementation concerns the development of a dynamic apparatus that can be subjected to numerical simulation by applying principles acquired in the Basic Mechanics 2 curriculum. The double pendulum system is comprised of two pendulums that are connected at their endpoints. This system displays an unusual dynamic behavior that relies highly on the initial conditions of position and velocity. A method of interdependent ordinary differential equations describes the dynamics of the double pendulum. Under specific energy conditions, the motion of the double pendulum exhibits chaotic behavior.

This endeavor aims to devise and scrutinize a dual pendulum configuration while adhering to predetermined limitations. The presented design scenario requires that the design team operate within specific constraints, including a maximum height of 1.5 meters, a total mass not exceeding 1.2 kilograms, and an initial energy of 1.4 Joules. The undertaking comprises two distinct scenarios, each with disparate objectives. The initial task entails achieving the highest possible amalgamation of the maximum vertical position of m2 and its corresponding vibration within a 10-second time frame. The second scenario necessitates the team’s contemplation of a damping ratio of 0.2. The objective is to attain the utmost combination of the maximum y position of m2 and the full-time vibration of the double pendulum system, starting from the application of an initial energy of 1.4 Joule until the double pendulum reaches a state of rest, where the total energy is less than or equal to 0.05 Joule.

In order to attain these objectives, the design team is required to undergo a systematic design procedure. The initial phase involves acknowledging the necessity and discerning prospects for enhancement. This project aims to develop a double pendulum system that satisfies the designated design criteria and limitations while accomplishing the objectives. The performance standards serve as the design criteria, whereas the constraints imposed on the designer, the final design, or the manufacturing process are called design constraints.

The objective of PD1 was to construct a precise mathematical framework that characterizes the dynamics of a descending entity. Multiple models were posited on varying assumptions, and their predictions were juxtaposed with empirical data. Nonetheless, the impact of air resistance, which can substantially influence the movement of an entity in actuality, was not taken into account.

In the upcoming PD2 session, we will expand upon our prior efforts and integrate the influence of air resistance into our computational models varying assumptions concerning air resistance on the prognostications of our models and subsequently juxtapose them with empirical data. The primary aim of this research is to investigate present a theoretical framework that effectively characterizes the trajectory of a descending entity in the context of aerodynamic drag.

Through the incorporation of air resistance, it is anticipated that a more comprehensive comprehension of the underlying physics governing the descent of objects can be attained, thereby enhancing our capacity to forecast their trajectory in practical applications. Our work’s potential applications span various fields, including engineering, physics, and sports science. Accurate modeling of the motion of falling objects is crucial in these domains for design and analysis purposes ( Xu, Z.et, al 2021).

If modifications or revisions are deemed necessary for the PD1 report, they must be duly attended to in the subsequent PD2 report. The proposed update entails a comprehensive evaluation of the PD1 report, coupled with the assimilation of any novel data or modifications that have been unearthed. This may encompass any identified errors, novel discoveries, or require changes to the initial proposal.

Furthermore, the update may encompass the integration of feedback received regarding the initial report. This feedback can be provided by peers, instructors, or other pertinent stakeholders. Incorporating feedback into the revised account is a crucial step in ensuring the accuracy, feasibility, and achievement of the stated goals of the proposed model. The revised version should be concise and unambiguously delineate the modifications implemented to the initial document. By ensuring that the PD2 report is built upon the work that has already been conducted in PD1, the iterative process of scientific inquiry and research can be demonstrated.( Fu, X., Du, et al 2023).

To achieve a favorable outcome in PD2, the team must establish specific objectives and a comprehensive work strategy that includes a precisely delineated allocation of tasks. It is recommended that the couple engages in a collaborative process to develop a clear objective for the project and establish measurable targets to be attained. This may entail formulating a mathematical or computational model that effectively captures the dynamics of an object subject to air resistance, characterizing the sources of error or uncertainty in the model, or investigating the potential utility of the model across different domains Zhang, R. et al. (2021) post on The implementation of a low-stiffness isolation system for seismic retrofitting of a heritage structure is proposed.

Upon establishing the objectives, the team should devise a comprehensive work plan delineating the requisite undertakings for accomplishing said objectives.

Creating a Gantt chart for PD2 is crucial in maintaining team focus and meeting project deadlines. The Gantt chart is a graphical illustration of the project timetable that displays the initiation and culmination dates of individual tasks and their interdependencies. This approach can assist the team in recognizing probable bottlenecks and making appropriate modifications to the timetable.

EXAMPLE

The analysis and numerical simulation of the models proposed in PD1 are required in PD2, with the additional consideration of the impact of air friction or damping. Consequently, the students must make necessary adjustments to their models by incorporating the effects of air resistance or damping into their respective equations. The influence of air resistance on the kinematics of an object can be substantial, mainly when the thing is in motion at elevated velocities.

The students must conduct numerical simulations to verify the precision of their models and juxtapose them with empirical observations. The report for each design is recommended to include screenshots of the simulation software. It is recommended that the screenshots display pertinent information such as input data, output graphs, time-acceleration graphs, and other relevant details that would facilitate comprehension of the simulations by the readers.

The utilization of simulations is expected to yield significant insights into the performance of the proposed models, thereby facilitating the identification of any potential limitations or shortcomings. Through the process of analysis and simulation, the students will have the capacity to ascertain the model that is most appropriate for their objectives. In addition, individuals will be able to enhance the model’s design and adjust its parameters to attain the intended level of performance.

The report should furnish an exhaustive examination of the simulation outcomes, encompassing any discernible trends or patterns that surfaced throughout the simulations. The students are expected to elucidate their utilization of simulation data in enhancing their models and accomplishing their objectives. Including a discourse on the constraints of the simulations and the presumptions undertaken during the modeling procedure is imperative in the report.

The evaluation of the proposed models from PD1 is a mandatory task for students in PD2, which involves considering the influence of air friction, also known as damping. The assessment is crucial for determining the optimal model that aligns with the team’s objectives. The students must comprehensively evaluate each model to ascertain its efficacy in practical situations. ( Zhuang P. et al., 2022).

To assess the suggested models, the students must consider multiple factors, such as the damping impact on the system, the system’s stability, and the precision of the model in forecasting the system’s behavior. Conducting a comparative analysis of the model’s performance is imperative to ascertain the optimal model that aligns with the team’s objectives.

The assessment should encompass simulations and analyses utilizing pertinent software applications. It is recommended that students furnish screenshots of the software outputs about each design, containing input data, output graphs, and time acceleration graphs. In addition, the team must conduct a comparative analysis between the simulation outcomes and empirical data to authenticate the precision and reliability of the models.

EXAMPLES

The students must furnish a comprehensive assessment document that encapsulates the merits and demerits of every proposed model and suggests the most appropriate model that accomplishes the team’s objectives. The assessment should be grounded on robust engineering principles and pertinent data analysis.

Several models were suggested during PD1 to accomplish the team’s objective of devising a projectile motion mechanism. Nevertheless, the PD2 analysis incorporated the influence of air resistance (damping) on said models. The anticipation was that this deliberation would result in recognition of an enhanced model that could exhibit greater precision in practical scenarios.

Upon analyzing the proposed models in PD1, which incorporated the factor of air friction (damping), it was noted that specific models needed to exhibit the anticipated level of performance. The models that showed superior performance in accuracy and precision were discovered to be comparatively less productive in air resistance (damping). Conversely, specific models that exhibited lower efficacy initially demonstrated improved performance in the presence of air resistance (damping).

A comprehensive analysis was conducted to compare the proposed models in PD1 and PD2 to determine the optimal model for attaining the team’s objectives. The study involved a comparative analysis of the model’s efficacy, precision, accuracy, and computational intricacy. The study conducted a comparative analysis and subsequently proposed a novel model that integrated the impact of air friction (damping) and exhibited superior performance in practical scenarios.

The influence of air resistance, also known as damping, on the suggested models, was noteworthy and must not be disregarded in the development of a projectile motion mechanism. Hence, it is imperative to consider the impact of air resistance (damping) in the developmental stage to attain the intended outcomes.

The process of identifying the most suitable model in PD2 involves a thorough examination of the models that were put forth in PD1 and PD2. The evaluation of the models should be conducted considering their efficacy in fulfilling the primary objectives of PD2 while also accounting for the influence of air resistance (damping).

The methodology to select the most suitable model is clearly defined and based on objective criteria. Evaluating a method or technique may encompass various factors, including but not limited to accuracy, precision, computational efficiency, simplicity, and ease of implementation. The weighting of each criterion should be determined based on its significance in attaining the primary objectives of PD2.

A potential method for determining the most suitable model is to employ a decision matrix. The decision-making process involves utilizing a decision matrix, which entails allocating a numerical value to each model based on its corresponding weight and criterion. Subsequently, the scores are aggregated, and the model exhibiting the maximum score is designated as optimal.

An alternative methodology involves a multi-criteria decision analysis (MCDA) approach, such as the analytic hierarchy process (AHP), which entails decomposing a multifaceted decision problem into a hierarchical structure of criteria and sub-criteria and subsequently evaluating the alternatives against each measure. The Analytic Hierarchy Process (AHP) offers a systematic approach for determining the comparative significance of individual criteria and for allotting ratings to each model on the basis of their respective achievements concerning each standard. In general, selecting the most suitable model necessitates a comprehensive evaluation of the proposed models and the appropriate methodology to ascertain that the chosen model aligns with the primary objectives of PD2 and is ideal for practical application.

### Reference

Zhang, R., Wu, M., Lu, W., Li, X., & Lu, X. (2021). Seismic retrofitting of a historic building using an isolation system with a weak restoring force. Soil Dynamics and Earthquake Engineering, 148, 106836.

Fu, X., Du, W. L., Li, H. N., Li, G., & Gan, S. (2023). Wind-Induced Response and Equivalent Static Wind Load of Transmission Lines Considering the Location Updating Effect. IEEE Transactions on Power Delivery.

Zhuang, P., Wei, L., Wang, W., & Han, M. (2022). Feasibility evaluation of pre-pressed spring devices for vertical isolation of single-layer spherical lattice shell structures. Soil Dynamics and Earthquake Engineering, 158, 107308.

Xu, Z., Zhao, Y., Deng, C., & Li, Q. (2021, October). Research on seismic isolation of truss string structure with rubber bearings considering relative rotation. In Structures (Vol. 33, pp. 1428-1438). Elsevier.

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