# A Swing of Beauty: Pendulums, Fluids, Forces, and Computers

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## Abstract

**:**

## 1. Introduction

- Under-damped: The pendulum will swing back and forth, although its amplitude of oscillation will steadily decline, until it asymptotically approaches its equilibrium.
- Critically-damped: the pendulum returns to equilibrium as quickly as it can. If the damping parameter were made slightly more or slightly less, it would result in the pendulum returning slower to its equilibrium position.
- Over-damped: the pendulum moves towards its equilibrium position slower than the critically-damped case. There is no oscillation.

## 2. Methods

#### 2.1. Model Geometry

#### 2.2. Model Construction

- Position of Lagrangian Points
- Forces on Each Lagrangian Point (Horizontal/Vertical and Normal/Tangential Forces)
- Fluid Velocity
- Fluid Vorticity
- Forces spread from the Lagrangian mesh onto the Eulerian grid

## 3. Results

- Angular Displacement of the pendulum bob
- Speed of the pendulum bob
- Forces acting on the pendulum bob
- Effect the pendulum bob has onto the fluid
- Comparison between reduced ODE model and FSI model

#### 3.1. Angular Displacement of the Pendulum Bob

#### 3.2. Speed of the Pendulum Bob

#### 3.3. Forces on the Pendulum Bob

#### 3.4. Effect the Pendulum Bob Has onto the Fluid

#### 3.5. Numerical Comparison & Validation

## 4. Discussion and Conclusions

- A connection to where students may have seen fluid drag laws previously, i.e., the Stokes Drag Law and Pendulum Motion. Furthermore, it illustrates for students that famous laws of physics were discovered with systems that seem as “basic” as that of a pendulum.
- The differences that may arise between modeling a system using a reduced-order ODE model and attempting to computationally model all aspects of the system to a higher degree. We hope this shows students that reduced models are valuable in that they are usually easier to solve while (hopefully) capturing a bulk of a system’s dynamics. However, there are clear disadvantages as illustrated by the discrepancies that arise between the reduced order model and computational model—many dynamics are not captured in the reduced-model, e.g., the vortex wake or drafting, that maybe particularly interesting or important to understanding the system as a whole.
- Similarly, the full dynamical richness of a system may only be explored by investigating its explicit fluid mechanics, even in a system as seemingly “simple” as a single pendulum immersed in a fluid. Moreover, to even study systems involving fluids and objects immersed therein, it requires either sophisticated experimental techniques or computational expertise. This work shows that a computer can be an immensely powerful tool for performing science. More than that, programming knowledge is highly sought after in this day and age [83,84].
- The observation that even systems that are routinely studied in some introductory courses, like a pendulum, may still have open, exciting research questions that scientists and engineers actively pursue.

## Supplementary Materials

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

CFD | Computational Fluid Dynamics |

FSI | Fluid-Structure Interaction |

Re | Reynolds Number |

IB | Immersed Boundary Method |

ODE | Ordinary Differential Equation |

## Appendix A. Instructor Resources

**Teaching Resources:**

**Pendulum_Classroom_Supplement.pptx/.pdf**: presentations which may be used in class; slides that tell the story of the paper. Note that the $.pptx$ file has embedded movies in $.mp4$ format.**Movies**: directory containing movies ($.mp4$ format) pertaining to each simulation shown in the manuscript.- Note that an open-source fluid-structure interaction model of a point-mass pendulum can be found at: https://github.com/nickabattista/IB2d in the sub-directory:IB2d → matIB2d → Examples → Examples_Education→ Pendulum.
- Visualization software used: VisIt (https://visit.llnl.gov/) (v. 2.12.3)

## Appendix B. Immersed Boundary Method

**F**(r,t), is specific to the system being explored. For this pendulum model, it takes the following form

#### IB Algorithm

**Step 1:**Calculate the force density, ${\mathbf{F}}^{n}$ on the immersed boundary, from its current boundary configuration at time n, ${\mathbf{X}}^{n}$.

**Step 2:**Use Equation (A3) to spread the force from the Lagrangian boundary to the Eulerian (fluid) mesh to compute ${\mathbf{f}}^{n}$

**Step 3:**Solve the Navier-Stokes equations, A1 and A2, on the Eulerian grid, thus updating ${\mathbf{u}}^{n+1}$ and ${p}^{n+1}$ from ${\mathbf{u}}^{n}$, ${p}^{n}$, and ${\mathbf{f}}^{n}$.

**Step 4:**Update the Lagrangian point positions, ${\mathbf{X}}^{n+1}$, using the local fluid velocities, ${\mathbf{U}}^{n+1}$, computed from ${\mathbf{u}}^{n+1}$ and (A4).

## Appendix C. Additional Pendulum Data

**Figure A1.**Depicting the angular displacement (radians) vs. time (s) for pendulums with the same mass but different radii. (

**a**–

**c**) give data for a specific mass, either $m=1\times {10}^{4}\mathrm{kg},2\times {10}^{3}\mathrm{kg}$, or $2\times {10}^{2}\mathrm{kg}$, respectively, and a variety of radii in each.

**Figure A2.**(

**a**) Plot illustrating the decay of the height (m) that the pendulum bob reaches as the pendulum continues to swing for the case of $m=1\times {10}^{4}\mathrm{kg}$ for a variety of radii. The peak amplitude decays exponentially as illustrated by the linear relationship between the logarithm of the amplitude against peak number, as shown in (

**b**).

**Figure A3.**(

**a**) Plot depicting the linear speed of the pendulum bob against non-dimensional time given as the # of swings (half a full displacement cycle) for the case of $m=1\times {10}^{3}\mathrm{kg}$ for a variety of radii. Speed peaks in the middle of a swing corresponding to when the pendulum has zero angular displacement from the vertical and the peak speed appears to decay exponentially, given by the linear relationship in (

**b**).

**Figure A4.**(

**a**) Phase space of linear speed of the pendulum bob vs. angular displacement (radians) for a variety of radii in the case of $m=5\times {10}^{3}\mathrm{kg}$. (

**b**) A closer look at the last simulated cycle’s phase space for each case.

**Figure A5.**Phase space of linear speed of the pendulum bob vs. angular displacement (radians) for a variety of masses for cases: (

**a**) $r=0.001\mathrm{m}$, (

**b**) $r=0.005\mathrm{m}$, (

**c**) $r=0.015\mathrm{m}$, and (

**d**) $r=0.025\mathrm{m}$.

**Figure A6.**Drag forces (N) over time in seconds for a variety of masses for cases with (

**a**,

**d**) $r=0.015\mathrm{m}$, (

**b**,

**e**) $r=0.020\mathrm{m}$, and (

**c**,

**f**) $r=0.025\mathrm{m}$. The semi-log data is provided in (

**d**–

**f**) to highlight a linear relationship between the logarithm of the drag force and time. This linear relationship suggests an exponential decay in drag force over time.

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**Figure 1.**(

**a**) A pendulum of length L with circular bob of radius, r, and mass, m. (

**b**) Angular displacement (in radians) over time for various gravity pendulums of differing radii. The non-dimensional time is given in terms of the number of periods of the case with radius, r.

**Figure 2.**Angular displacements against non-dimensional time for damped physical pendulums in the case of (

**a**) constant radius and varied damping, b, and (

**b**) constant b and varied radii.

**Figure 3.**(

**a**) Model of an immersed pendulum with circular bob of radius r in a viscous, incompressible fluid. The fluid has density and viscosity of $\rho $ and $\mu $, respectively. The pendulum has length L and the bob has mass m, concentrated at its center. (

**b**) The computational geometry illustrating the fiber model construction of the discretized Lagrangian mesh.

**Figure 4.**Snapshots of a single moment in time for a pendulum with mass, $m=5\times {10}^{2}\mathrm{kg}$, and radius, $r=0.0175\mathrm{m}$, providing some of the data stored during the time-step in which the simulation time reached $t=0.70\mathrm{s}$, i.e., positions of Lagrangian points (pendulum), the velocity vector field, magnitude of velocity, and vorticity. Note that data from giving the force spread from the Lagrangian grid (pendulum) onto the Eulerian (fluid) grid is not shown. Lagrangian Coherent Structures (LCS) via finite time Lyapunov exponents (FTLE) are also illustrated, although they were computed during the post-processing stage, after the data was collected.

**Figure 5.**Snapshots of multiple pendulums’ (of differing radius) angular displacement over time in the case of $m=1\times {10}^{3}\mathrm{kg}$.

**Figure 6.**Depicting the angular displacement (radians) vs. time (s) for pendulums with the same radius but different masses. (

**a**–

**c**) give data for a specific radius, either $r=0.005\mathrm{m},0.015\mathrm{m}$, or $0.025\mathrm{m}$, respectively, for 4 orders of magnitudes in mass in each.

**Figure 7.**(

**a**) Plot illustrating the decay of the height (m) that the pendulum bob reaches as the pendulum continues to swing for the case of $r=0.005\mathrm{m}$ for a spectrum of masses. The peak amplitude decays exponentially as illustrated by the linear relationship between the logarithm of the amplitude against peak number, as shown in (

**b**).

**Figure 8.**Plots illustrating the time of the peak in angular displacement against the pendulum bob’s radius for its 1st through 6th peak (

**a**–

**c**) and the time difference between the peaks (

**d**–

**f**) against the pendulum bob’s radius for three different masses: (

**a**,

**d**) $m=2\times {10}^{2}\mathrm{kg}$, (

**b**,

**e**) $m=1\times {10}^{3}\mathrm{kg}$, and (

**c**,

**f**) $m=5\times {10}^{3}\mathrm{kg}$.

**Figure 9.**(

**a**) The period given as a function of a pendulum bob’s radius for a variety of masses. (

**b**) A contour map showing the period as a function of both the pendulum bob’s radius and mass. The highest periods occur for small masses and large pendulum bobs.

**Figure 10.**(

**a**) Plot depicting the linear speed of the pendulum bob against non-dimensional time given as the # of swings (half a full displacement cycle) for the case of $r=0.015\mathrm{m}$ and a variety of masses. Speed peaks near the center of each swing. This corresponds to when the pendulum has approximately zero angular displacement from the vertical. The peak speed appears to begin decaying exponentially, starting on the second or third swing in most cases. This can be seen from linear relationships between peak speed and swings in (

**b**).

**Figure 11.**Plot illustrating that exponential decay appears in peak speed starting with the second swing. There is significantly more decay in peak speed between the first and second swings, than successive swings thereafter.

**Figure 12.**The percentage decrease in speed when comparing pendulum bob speed once it reaches 0 degree angular displacement on the first swing compared between simulated cases in fluid and theoretical value outside of a viscous fluid environment.

**Figure 13.**(

**a**) Phase space of linear speed of the pendulum bob vs. angular displacement (radians) for a variety of masses in the case of $r=0.001\mathrm{m}$. (

**b**) A closer look at the last simulated cycle’s phase space in each case.

**Figure 14.**Drag forces (N) over time in seconds for multiple radii for cases with (

**a**,

**d**) $m=5\times {10}^{2}\mathrm{kg}$, (

**b**,

**e**) $m=1\times {10}^{3}\mathrm{kg}$, and (

**c**,

**f**) $m=5\times {10}^{3}\mathrm{kg}$. The semi-log data is provided in (

**d**–

**f**) to highlight a linear relationship between the logarithm of the drag force and time. This linear relationship suggests an exponential decay in drag force over time.

**Figure 15.**Phase space of drag force (N) versus angular displacement (radians) for a variety of masses in cases of (

**a**) $r=0.005\mathrm{m}$, (

**b**) $r=0.015\mathrm{m}$, and (

**c**) $r=0.025\mathrm{m}$. The data for each case of a specific radius appears to overlap as well as suggesting that as the peaks in angular displacement decay exponentially (see Figure 7), the drag forces also decay exponentially as well.

**Figure 16.**(

**a**) Re vs. Time for $m=2\times {10}^{3}\mathrm{kg}$ and (

**b**) a colormap depicting the temporally-averaged Reynolds number during the first swing for different masses and radii. Note that over time as the pendulum slows down, the average Reynolds number will decrease.

**Figure 17.**The drag coefficient, ${C}_{D}$, during the first pendulum’s first swing (

**a**) and first 4-swings (

**b**) for a variety of radius in the case of $m=1e3\mathrm{kg}$. Note that the drag coefficient maximizes when the pendulum reaches near zero speed at the end of a swing. (

**c**) The temporally-averaged drag coefficients across the first swing for all mass and radius cases considered. (

**d**) A contour map of the temporally-averaged drag coefficients over the first swing from (

**c**) as a function of both the pendulum bob’s mass and radius. Generally higher drag coefficients are seen for larger mass and size pendulum bobs.

**Figure 18.**The average drag coefficient, ${C}_{D}$, vs. average Reynolds number for a variety of masses and radii. The averages were computing over the first swing of the pendulum bob.

**Figure 19.**Colormaps (and its contours) illustrating the time evolution of the fluid’s vorticity, magnitude of velocity, and finite-time Lyapunov Exponent (FTLE), as well as the velocity field (and its streamlines) resulting from the pendulum bob’s first swing in the case of $m=5\times {10}^{2}\mathrm{kg}$ and $r=0.0175\mathrm{m}$.

**Figure 20.**Comparing vortex dynamics among pendulum bob of different radii for a mass of $m=5\times {10}^{2}\mathrm{kg}$.

**Figure 21.**The vortex dynamics of the case ($m,r$) = ($5\times {10}^{2}\mathrm{kg},0.0175\mathrm{m}$) within the first $2\mathrm{s}$ of oscillation.

**Figure 22.**The vortex dynamics of the case ($m,r$) = ($1\times {10}^{4}\mathrm{kg},0.005\mathrm{m}$) on the return swing during its first oscillatory cycle.

**Figure 23.**(

**a**) Slopes of the least squares (linear regression) fits through the peaks of angular displacement over time to compute the exponential decay, $\gamma =-\frac{b}{2I}$, for a variety of radii in the $m=5\times {10}^{3}$ kg case. (

**b**–

**f**) Comparison of the FSI and ODE models’ angular displacement over time for a variety of masses and radii.

**Figure 24.**Depicting the dynamics if the ODE model started from the original angular displacement of the FSI pendulum rather than the the 5th peak for the case $(m,r)=(5\times {10}^{2}\mathrm{kg},0.005\mathrm{m})$ and $(m,r)=(5\times {10}^{3}\mathrm{kg},0.015\mathrm{m})$ for (

**a**,

**b**), respectively. A visualization of the exponential decay is also provided with the coefficient either being ${A}_{0}$, the original angular angular displacement, or ${A}_{5}$, the displacement of the 5th peak.

**Figure 25.**Values of the damping parameter, b, as a function of the mass and radius of the pendulum bob.

Parameter | Description | Value |
---|---|---|

L | Pendulum Length | 0.2$\mathrm{m}$ |

r | Pendulum Bob’s Radius | $r\in [0.001,0.025]\mathrm{m}$ |

m | Mass | $m\in [2\times {10}^{2},1\times {10}^{4}]\mathrm{kg}$ |

$\rho $ | Fluid Density | 1000 $\mathrm{kg}/{\mathrm{m}}^{3}$ |

$\mu $ | Fluid (dynamic) Viscosity | 0.01 $\mathrm{kg}/(\mathrm{m}\xb7\mathrm{s})$ |

g | Gravitational Acceleration | 9.81 $\mathrm{m}/{\mathrm{s}}^{2}$ |

${\theta}_{0}$ | Initial Angular Displacement | $-\frac{3\pi}{10}$ radians |

**Table 2.**Table providing number of Lagrangian Points in the circular shell for a particular radius, r.

Radius ($\mathbf{m}$) | 0.001 | 0.0025 | 0.005 | 0.0075 | 0.01 | 0.0125 | 0.015 | 0.0175 | 0.02 | 0.0225 | 0.025 |
---|---|---|---|---|---|---|---|---|---|---|---|

# Lag. Pts in Shell | 12 | 32 | 64 | 96 | 128 | 160 | 194 | 226 | 258 | 290 | 320 |

**Table 3.**Table of numerical temporal, spatial, and fiber model parameters used in our pendulum study.

Parameter | Description | Value |
---|---|---|

$dt$ | time-step | $2.5\times {10}^{-5}\mathrm{s}$ |

${L}_{x}\times {L}_{y}$ | Grid Size | $1\mathrm{m}\times 1\mathrm{m}$ |

$({N}_{x},{N}_{y})$ | Grid Resolution | (1024, 1024) |

$dx=dy$ | Spatial Step | ${L}_{x}/{N}_{x}={L}_{y}/{N}_{y}=0.0009765625\mathrm{m}$ |

$ds$ | Lagrangian Point Spacing | ∼$\frac{{L}_{x}}{2{N}_{x}}$ |

${k}_{sp{r}_{L}}$ | Spring Stiffness Coefficient (Mass to Hinge) | $1.25\times {10}^{8}$$\mathrm{kg}\xb7\mathrm{m}/{\mathrm{s}}^{2}$ |

${k}_{sp{r}_{B}}$ | Spring Stiffness Coefficient (Pendulum Bob) | $2.5\times {10}^{8}$$\mathrm{kg}\xb7\mathrm{m}/{\mathrm{s}}^{2}$ |

${k}_{target}$ | Target Point Stiffness Coefficient | $5\times {10}^{7}$$\mathrm{kg}\xb7\mathrm{m}/{\mathrm{s}}^{2}$ |

${k}_{mass}$ | Massive Point Stiffness Coefficient | $2.5\times {10}^{6}$$\mathrm{kg}\xb7\mathrm{m}/{\mathrm{s}}^{2}$ |

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Mongelli, M.; Battista, N.A.
A Swing of Beauty: Pendulums, Fluids, Forces, and Computers. *Fluids* **2020**, *5*, 48.
https://doi.org/10.3390/fluids5020048

**AMA Style**

Mongelli M, Battista NA.
A Swing of Beauty: Pendulums, Fluids, Forces, and Computers. *Fluids*. 2020; 5(2):48.
https://doi.org/10.3390/fluids5020048

**Chicago/Turabian Style**

Mongelli, Michael, and Nicholas A. Battista.
2020. "A Swing of Beauty: Pendulums, Fluids, Forces, and Computers" *Fluids* 5, no. 2: 48.
https://doi.org/10.3390/fluids5020048