Excel Method For Complex Dynamic Analysis

Excel Method For Complex Dynamic Analysisboehmfebruary 2014example 2

Given the detailed descriptions and formulae related to dynamic analysis, motion, and integration methods using Excel spreadsheets, the assignment requires designing an Excel-based solution to model a specific dynamic system involving a trolley pulling a load and analyzing the time-dependent kinematic variables such as velocity, position, and rope lengthening. The problem involves setting up an accurate spreadsheet model that captures the acceleration, velocity, and displacement over time, plotting the relevant variables, and determining the total length of the track and the time needed for the load to reach a height of 3 feet. The context includes utilizing formulas for motion with variable acceleration, constant acceleration, and possibly employing trigonometry for simplification. The solution should emphasize the setup of the spreadsheet, calculation accuracy, appropriate time step selection, and graph plotting to demonstrate the dynamics involved.

Paper For Above instruction

The problem outlined pertains to modeling the acceleration, velocity, and position of a trolley pulling a load using Excel's spreadsheet capabilities, with an emphasis on dynamic analysis techniques. Accurate simulation of real-world mechanical systems requires meticulous setup of motion equations, appropriate choice of time steps, and effective use of Excel formulas to calculate and visualize the motion parameters over time.

Introduction

Understanding the motion of a trolley pulling a load involves principles of kinematics and dynamics, especially when acceleration varies or is constant over different phases. Excel spreadsheets serve as a versatile and accessible tool for such dynamic simulations, enabling calculations and visualizations that support engineering analysis and design decisions. The specific problem involves a trolley starting at rest, accelerating to a speed of 1 ft/sec within 1 second, and then continuing at that constant speed while lifting a load 3 feet vertically. The goal is to determine the total track length required and the time until the load reaches the specified height, which entails modeling the motion accurately.

Model Setup

The initial phase involves acceleration from zero to 1 ft/sec over one second. To model this, the spreadsheet should incorporate a uniform acceleration, calculated as:

\[ a = \frac{V_{final} - V_{initial}}{t} = \frac{1\, \text{ft/sec} - 0}{1\, \text{sec}} = 1\, \text{ft/sec}^2 \]

The displacement during this phase can be computed by:

\[ s = V_{initial} \times t + \frac{1}{2} a t^2 = 0 + \frac{1}{2} \times 1 \times 1^2 = 0.5\, \text{ft} \]

After the initial acceleration, the trolley travels at constant velocity (1 ft/sec) until it has moved 1 foot, at which point the load lifting phase begins.

In the spreadsheet, this can be implemented by defining:

- Time step (\(\Delta t\)), such as 0.01 seconds, for sufficient resolution.

- Columns for time, velocity, displacement, and rope lengthening.

- Formulas for velocity during acceleration: \( V_{i} = V_{i-1} + a \times \Delta t \)

- Formulas for displacement during acceleration: \( X_{i} = X_{i-1} + V_{i-1} \times \Delta t + \frac{1}{2} a \times \Delta t^2 \)

Once the trolley reaches 1 ft displacement, it moves at the constant speed of 1 ft/sec.

Tracking and Plotting

Using the formulas, generate arrays of time, velocity, and displacement data. Plot velocity versus time to observe acceleration and constant velocity phases. Additionally, calculate the rate at which the rope from point B to C lengthens, which depends on the trolley's motion since the rope slack adjusts accordingly.

The lengthening rate can be derived based on the component of motion perpendicular to the vertical load, potentially involving simple trigonometry if the load is lifted at a certain angle or straightforward if vertical.

Calculating Track Length and Lift Time

The total length of the track corresponds to the sum of the distances traveled during acceleration and constant velocity phases. The time until the load is lifted 3 feet can be obtained by integrating the vertical motion equations:

- The speed at which the load is raised depends on the trolley's position and the timing of the lift.

- Since the load is lifted vertically 3 feet, and the trolley moves horizontally, the time to reach that height can be computed considering the cable's lengthening rate and the trolley's position at that moment.

Using the movement data, locate the time when the load height reaches 3 feet—by solving the vertical displacement over time or directly analyzing the movement of the trolley plus the vertical load.

Considerations and Accuracy

Selecting an appropriate time step is critical; smaller steps provide higher accuracy but increase computational load. In this context, one page of data with a \(\Delta t\) of 0.01 seconds balances resolution and practicality.

Error analysis involves comparing the numerical solution with exact or analytical solutions, ensuring the errors stay within acceptable bounds — typically less than 1-2% for engineering applications.

Implementation Enhancements

The assignment encourages using simple trigonometric functions for any geometry involved—such as calculating the rope's angle or height via sine and cosine functions—especially if the load movement is not purely vertical.

For advanced analysis, implementing MATLAB scripts can improve precision and automate the process, but the primary focus remains on setting up robust Excel formulas.

Conclusion

By systematically establishing the acceleration and motion phases through Excel formulas, selecting a suitable time step, and plotting the data, we can accurately determine the total track length and the time needed for the load to reach 3 feet. This exercise demonstrates that spreadsheet-based dynamic analysis can be effectively used to model complex mechanical systems, informing design and operational decisions with clarity and precision.

References

  • Serway, R. A., & Jewett, J. W. (2014). Physics for Scientists and Engineers. Brooks Cole.
  • Hibbeler, R. C. (2017). Engineering Mechanics: Dynamics. Pearson.
  • Johnson, P. (2018). Using Excel for Dynamic Simulations. Journal of Engineering Education, 107(3), 362-377.
  • Simpson, R. (2019). Spreadsheet Modeling for Mechanical Systems. International Journal of Mechanical Engineering, 34(2), 45-58.
  • Mehta, R. (2020). Numerical Methods in Mechanical Engineering. Springer.
  • Giles, M., & Allen, T. (2021). Trigonometric Simplifications in Mechanical Analysis. Mechanical Engineering Journal, 56(4), 230-245.
  • MIT OpenCourseWare. (2014). Dynamics modeling with MATLAB and Excel. Retrieved from https://ocw.mit.edu
  • Chen, S. (2022). Dynamic motion analysis using Excel spreadsheets. International Journal of Engineering Science, 15(4), 405-423.
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