Solving the damped pendulum equations of motion 👀 #19
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I found that when only changing the values of damping and stiffness, I found that both values around 0.3 ~ 0.35 gave really quick damping results. When increasing the stiffness by a large margin it has a similar result to that of increasing the damping by a large margin. There is likely more that could be done by changing the mass and the length of the pendulum. Additionally if the mass of the cart was add that could also cause variation in the damping. |
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When testing the model, I found that damping and stiffness values around 0.3-0.35 seemed to stop the pendulum motion the quickest. At that range, the motion dies out quick without too much extra oscillation. Increasing stiffness a lot had a similar effect to increasing the damping a lot because both help to reduce the motion faster. |
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For this system, the optimal stiffness and damping are moderate rather than extremely large. Very high damping or stiffness effectively prevents the base from moving, which reduces its ability to absorb energy from the pendulum and slows the decay of oscillations. If the goal is to stop the pendulum motion quickly, the optimal combination typically lies in a range of 0.3-0.35 The analysis also depends on other parameters such as the pendulum mass, length, gravity, the initial angle and velocities, and how performance is measured. |
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I got similar results to everyone else, stiffness and damping coefficients in the 0.3-0.4 range worked best to stop the motion of the pendulum as soon as possible. When playing with the parameters I found that the mass of the pendulum had a direct relationship with the optimal damping/stiffness needed for slowing the pendulum's motion. This is similar to normal spring-mass-damper system where the amount of damping needed to be critically damped increases with mass. On the other hand the length of the pendulum seems to have an inverse effect with the optimal stiffness/damping. This makes sense as the length of the pendulum determines its frequency, an increase in length would decrease its frequency, decreasing the optimal stiffness, subsequently decreasing the optimal damping coefficient. |
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When I experimented with the model, I also found that moderate values of stiffness and damping in the range of 0.3-0.35 seemed to perform best. In my tests, this range of coefficients seemed to reduce the pendulum motion the fastest without causing excessive oscillation. If the damping or stiffness was increased too much, the base essentially stopped moving, which reduced its ability to absorb energy from the pendulum and actually slowed the overall damping of the motion. In addition to stiffness and damping, I think several other parameters would influence the optimal values. The mass and length of the pendulum affect its natural frequency and inertia, which would likely change the damping required for the quickest decay. The mass of the base could also play an important role since a heavier or lighter base would respond differently to the pendulum’s motion. Initial conditions, such as the starting angle or velocity, may also impact how quickly the system settles. |
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In my analysis, I found that the most effective stabilization occurs when the stiffness and damping are balanced around the 0.3-0.35 range. It was surprising to see that more isn't always better, if the base is too rigid, it stops absorbing energy entirely. I also noticed that the pendulum's length plays a huge role; as L increases, the frequency drops, which seems to inversely affect the optimal stiffness required. This suggests that for a real-world application, like stabilizing a load on a crane or a ship, the damping system has to be precisely tuned to the specific geometry and mass of the object being moved. |
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From the simulations, it seems that the best combination of stiffness and damping is in a moderate range rather than extremely large values. Values of k and b around 0.3–0.35 appear to reduce the pendulum motion the fastest. In this range, the base is still able to move enough to absorb energy from the pendulum while also providing enough damping and restoring force to reduce the oscillations. When the stiffness or damping becomes too large, the base barely moves, which actually reduces its ability to absorb energy and can slow down how quickly the pendulum settles. Other parameters would also influence the optimal values. The mass of the pendulum affects the inertia of the system, so a larger mass would likely require more damping or stiffness to stop the motion quickly. The length of the pendulum also plays an important role because it changes the natural frequency of the system. Initial conditions, such as the starting angle or any initial velocity, could also affect how quickly the motion dies out. If the mass of the base or cart were included in the model, that would likely have a significant effect as well, since the damping behavior depends on how the base responds to the pendulum’s motion. |
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From the simulations I ran, the best combination of stiffness and damping tends to lie in the 0.3-0.35 range, as this was what caused the pendulum to stop moving as fast as possible. Like previously stated, I would not have predicted that if the damping or stiffness is increased too much, the base moves far less and it is actually more difficult to slow the pendulum. Some other parameters that could be changed to affect the analysis are the length of the pendulum, its mass, and initial conditions, as well as the mass of the cart. |
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I found that the optimum combination of stiffness and damping is around 0.3 to 0.45 to stop the pendulum form moving. While I kept going lower at some point the oscillations became erratic then eventually converge to the same plot vice versa if you were to make it very large like in the video. Fooling around a bit I found that l made a significant difference to decrease the number of oscillations and time to stop if the application called for it. |
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@UConn-Dynamics Very cool findings, it sounds like b = 1/3 kg/s was an optimum damping. I'm assuming the other values stayed the same?
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For this, I started at a stiffness of 0.1, and iteratively increased it until the pendulum achieved a damped state (settling at x=0). This is somewhere around 0.3 kg/s. Below is an animation- we can see it dying out about x = 0.
Here is a list of all parameters used for this: begin
k = 2 # N/m
b = 0.3 # kg/s
g = 9.81 # m/s/s
m = 0.1 # kg mass of pendulum bar
L = 0.9 # m length of pendulum bar
end |
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After experimenting with the parameters in the notebook, my analysis also agrees with many of the other results posted here. I found that stiffness and damping values around 0.30–0.35 stop the pendulum motion the fastest. In this range the system dissipates energy without introducing additional oscillation before settling. If the damping is too low, the pendulum continues oscillating for a long time. If stiffness becomes too large, the system can introduce additional dynamic response before settling. Other parameters I considered were the initial angle and angular velocity, since they determine how much energy is initially in the system. Physical parameters like pendulum length and mass also influence the natural frequency, could also affects how the damping interacts with the motion. |
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Testing revealed that damping and stiffness values around 0.3-0.35 stopped the pendulum motion the quickest, with minimal oscillation. Increasing stiffness produced smaller effects to raising damping, as both help dissipate energy faster. Some other parameters like the mass of the pendulum, the length, and cart mass also influence decay rates, this means that optimal damping/stiffness values would shift with a different system configuration. |
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Hello @cooperrc saw the post about the project, is it supposed to be due tomorrow at 11:59 pm or was that accidental. Seems last minute if that is the intended due date. |
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That is accidental. I wanted to open up group enrollment, not add a 24-hour deadline. Thanks for checking. |
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After experimenting with the parameters in the notebook, I also found that moderate values of stiffness and damping seem to stop the pendulum motion the fastest. Values around 0.3–0.35 reduce the oscillations efficiently because the base can still move enough to absorb energy while the damping removes energy from the system. If the damping is too low, the pendulum continues oscillating for a longer time, while very large stiffness or damping can limit the base motion and reduce how effectively the system releases energy. I also considered how other parameters could affect the optimal values, such as the pendulum mass, pendulum length, and the initial angle or velocity. These factors change the energy in the system and the natural frequency, which would likely shift the stiffness and damping needed to stop the motion quickly. |
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I found the optimal combination of stiffness and damping to be roughly 0.34. This does seem to be in line with what everyone else was getting with a value between 0.3 and 0.4. This value of stiffness was the optimum combination as the goal was to stop the pendulum from moving. Other parameters that were considered in my analysis were mass and pendulum length. These are both important to consider in the analysis as they would affect the starting energy and natural frequency of the system. By changing these factors I would need to retest for the optimal combination of stiffness and damping. |
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For this analysis, I tested different combinations of spring stiffness and damping to see which setup slowed the pendulum down the best. Instead of choosing values by guessing, I used a parameter sweep and gave each case a motion score based on how much the pendulum was still moving near the end of the simulation. |
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Hi @ericd3030. |
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With the help of copilot, I parametrized evaluation of various stiffness and damping terms to create trace plots for the pendulum free end, as an extension of the example I discussed in the rigid body motion discussion (#16 (comment)). I've included plots that show the trace for a couple different permutations. For these particular cases, for comparison's sake, I've kept k = 2 N/m. In the first figure, I've plotted the trace for a system with damping equal to 0.3 kg/s--this is optimal value that a lot of people are reporting and I had this observation as well. In the next plot, I've plotted the trace for a system with damping equal to 100 kg/s. As discussed in the video and in the documentation, a higher damping value doesn't nesecarily damp motion in this case due to frequency characteristics of the system. In the same (default) timeframe as in the first case, the free end is almost back to where it started. Another parameter we can investigate changing is the pendulum mass. In figure 3, I plotted the trace of the pendulum free end with the same parameters as in figure except with a mass of m=10 kg instead of 0.10 kg. It is clear that this new mass serves to damp the system much more than the original mass--this again shows how the frequency of the system is tied to a number of parameters.
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I worked through the notebook and played around with the stiffness and damping a bit to see what actually stops the motion fastest. What stood out is that there is not one single “perfect” value, it is more about the balance between stiffness and damping. If the damping is too low, the system just oscillates for a long time. If it is too high, the motion becomes really sluggish and takes longer to settle. The fastest way to kill the motion is somewhere in between, basically a critically damped type response. For this setup, that happens when the damping is matched to the stiffness so the base motion absorbs energy without introducing long transients. When I varied both, increasing stiffness generally helps limit how much the base moves, but if it is too stiff, you lose the benefit of that energy dissipation through motion. So the best case ends up being moderate stiffness with damping tuned just high enough to remove oscillations without slowing everything down too much. Besides stiffness and damping, a few other parameters matter a lot. The mass and length of the pendulum change the natural frequency, which shifts what “optimal” damping looks like. Initial conditions also matter more than you might expect, since larger initial angles can push the system into more nonlinear behavior. Gravity is fixed here, but it still sets the time scale of the motion. |
I have plotted the X displacement for the cart as well as the pendulum angle for the system with no displacement showing the weird mechanics you get with an undampened 3 DOF system |
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The results show that the optimum damping value is not the largest damping value. Very large damping makes the cart almost fixed, which reduces energy transfer from the pendulum to the damper. Moderate damping allows the cart to move and absorb more energy, causing the oscillations to decay faster. The best combination of stiffness k and damping b would likely use moderate damping and enough stiffness to stabilize the cart without completely restricting motion. If the system is too stiff or overdamped, the pendulum can continue oscillating longer. Other important parameters include the pendulum length L, mass m, gravity g, and the initial angle and velocity, since they all affect the system’s natural frequency and energy response. |
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For the goal of stopping the pendulum from moving, I found that damping and stiffness values around 0.3–0.35 seemed to stop the pendulum motion the quickest. At that range, the motion dies out fairly quickly without creating too much extra oscillation. Based on the notebook, the best option is not simply using the largest damping or stiffness value, because too much of either can make the system respond less effectively or create unwanted behavior. Increasing stiffness a lot can have a similar effect to increasing damping a lot because both help reduce motion faster, but the best combination seems to be a balanced, moderate range. Other parameters I would consider are the mass and length of the pendulum, the mass of the cart/base, gravity, the initial release angle, base displacement, settling time, and remaining oscillation. Changing the mass, length, or cart mass would likely change what the best damping and stiffness values are. |
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I found that the best stiffness and damping values were around 0.3–0.35. In that range, the pendulum motion slowed down quickly without adding too much extra oscillation. If the damping is too low, the pendulum keeps swinging, but if the damping or stiffness is too high the base does not move enough to absorb energy from the pendulum. I also think the pendulum mass, length, base mass, and initial angle would matter because they change the system’s natural frequency. |
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This discussion helped clarify how equilibrium points and linearization are used to analyze system behavior without solving the full nonlinear equations. Identifying equilibrium configurations first makes it much easier to reason about stability especially for systems like pendulums where intuition alone can be misleading. Linearizing the equations of motion about an equilibrium showed how small angle or small perturbation assumptions simplify the dynamics while still capturing the local behavior accurately. From the examples it was clear that some equilibria are stable while others are unstable even though they may look physically similar. This distinction becomes obvious once the linearized system and eigenvalues are examined. Overall, this was a useful bridge between nonlinear dynamics and practical analysis techniques and it reinforced why linear models are so widely used for stability analysis, control design, and understanding local system behavior near operating points. |
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im not sure if I did something wrong as I am comparing my work to everyone else, but I am finding that the damping value that is best is 0.5. I am not sure if i made a mistake with my coding but the optimum I found was a b of 0.5 and a k of 2. all the factors of the pendulum could affect this, and changing the pendulum could definitely change these values |
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I tried playing around with the stiffness and damping values to see how that affects how quick;y the pendulum settles. At first, increasing the damping had a huge affect, but as I continued I reached a certain point where the difference was not really noticeable. I guess too much damping just makes it slower to return to equilibrium versus killing the oscillation right away, which was interesting. When stiffness and damping were in the middle range, this seemed to work best. I got my best results around k = 0.3-0.35 and a damping value around the same range. I'm not 100% if this is the actual best values, but these were the ones that worked when I gave it a go. I think changing the pendulum length and mass would definitely affect what the most "successful" damping value actually is. So, the values I used that work well aren't the only best answer I believe it varies based on the setup, of course. |
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I thought it was interesting that increasing the damping does not automatically make the pendulum stop faster. From the plots, the very large damping case did not look like the best choice because it changed the response a lot and still allowed large oscillations. The smaller damping values seemed more useful, especially around the middle range, because they reduced the motion without making the system behave strangely. For an optimum combination, I would probably compare stiffness and damping based on settling time, maximum angle, and how quickly both θ and θ approach zero. With the stiffness fixed at k=2, the damping values around b=1 to 1.5 looked like a better starting point than either very low damping or extremely high damping. Too little damping lets the pendulum keep oscillating, but too much damping can over-constrain the support motion and create a poor coupled response. I would also consider the pendulum mass, length, and initial angle because those affect the natural frequency of the system. Since the support motion and pendulum rotation are coupled, the best damping value depends on more than just slowing the cart down. The goal would be to tune k and b so the support absorbs energy from the pendulum without creating extra motion in the base. |
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I found that the optimum combination of stiffness and damping was between 0.3-0.4. Similar to my peers, I found that by greatly increasing the stiffness, the model acted similarly to when I increased the damping greatly. This shows there is a relationship between optimal stiffness and damping and mass. High amounts of damping and stiffness keep the model stagnant, slowing its decay of oscillations. My senior design has much to do about stiffness and damping but on cantilever beams so it was interesting to see how stiffness and damping influenced the oscillations of the pendulum. Further steps would be to see the effects of the mass and length of the pendulum to see how it influences the oscillations. |
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From testing the system, it seems like the optimal stiffness and damping values are in the range of about 0.3–0.35 for stopping the pendulum motion the fastest. At this range, the oscillations decay quickly without creating too much extra motion or overshoot. If the damping or stiffness becomes too large, the base does not move enough to effectively absorb energy from the pendulum, so the motion can actually take longer to die out. |
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While testing the notebook, I found that the moderate values of stiffness and damping seemed to work best, and the quickest reduction in the pendulums motion happening around the 0.3–0.4 range. What I found interesting was that increasing either the damping or stiffness too much made the system less effective. When those values are very large, the base barely moves, so it can’t absorb energy well from the pendulum. At that point, the system starts acting more like a fixed support, and the pendulum takes longer to settle. The mass and length of the pendulum affect the system’s inertia and natural frequency, which changes where the optimal damping ends up. The initial angle matters too since starting at a bigger angle just means there’s more energy in the system to get rid of. If the mass of the cart or base were included, I think it would also make a noticeable difference, since a heavier or lighter base can change how easily it can move. |
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Solving the damped pendulum equations of motion
In this notebook and 2 videos, I go through the troubleshooting to get a set of coupled differential equations into a solveable form and then visualize the results to see the predicted motion. Two things surprised me in this analysis:
What do you find as the optimum combination of stiffness and damping if the goal is to stop the pendulum from moving? What other parameters are you considering in the analysis?
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