New video: Kinematic solutions for a piston-crank system 👀

[cooperrc]

Kinematic solutions for a piston-crank system

In this notebook + video, we go through the analysis of a multibody dynamic simulation for piston-crank kinematics. We keep track of more variables (12 instead of 2-3), but we get a glimpse into how MBD systems and kinematic constraints can be handled in a more general sense of interacting rigid bodies.

Can you recreate the graphs in Shabana's prescribed_rotation_of_the_crankshaft? This is a great verification step that shows our solution process is repeatable.

[cjdipietrantonio]

With the results from the code Professor Cooper wrote during the lecture video, we can recreate the graphs shown in Shabana's "Prescribed Rotation of the Crankshaft" example. Below are the results for the connecting rod and slider block. The position level plots show the connecting rod orientation and slider block displacement versus crank angle (as shown in Figures 3.37a and 3.37b of Shabana). The velocity level plots show the angular velocity of the connecting rod and the velocity of the slider block versus crank angle (as shown in Figures 3.38a and 3.38b of Shabana). We can observe that these plots generally match the shapes and values shown in Shabana's figures.

The code from the lecture could also be updated to compute the accelerations for each body by taking the time derivative of the velocity constraint equation and solving the resulting linear system at each time step. Nevertheless, the plots above demonstrate the agreement between our simulation results and those shown in Chapter 3 of the text. In Chapter 3, other plots show the positions, velocities, and accelerations of the crankshaft/connecting rod versus time. These plots could also be recreated by following the same process used to make the plots above!

[jordanbollacke]

These are the plots that julia helped me generate -- they don't match Shabana's, but I'm not sure why they are differing so much.

[JulianPojano]

This is what I was able to get working. Looks pretty much exactly like the first 4 plots in the textbook.

[AuZwick]

Here is what I was able to get.

[danny-kruzick]

I was able to recreate the main Shabana-style verification plots from the piston-crank notebook by starting with the same multibody constraint formulation in the video. The nice part of this setup is that even though the system has 12 coordinates, the solution process stays organized: first solve the nonlinear constraint equations for position, then solve the linearized constraint equation for velocity.

For the verification plots, I focused on the same quantities several others highlighted: the connecting rod orientation $\theta_3$ versus crank angle $\theta_2$, the slider displacement $x_4$ versus $\theta_2$, the connecting rod angular velocity $\dot{\theta}_3$ versus $\theta_2$, and the slider velocity $\dot{x}_4$ versus $\theta_2$. These plots do match the same overall shapes shown in the example. Looks like other posts have similar plots. 

As mentioned, this serves as a strong repeatability check. We are not just solving one closed-form kinematics problem, but using a general multibody constraint approach and still recovering the same trends shown in the textbook. That makes the method feel much more reusable for more complicated systems where a direct hand-derived solution would be much harder.

[whteCobra]

I’ve taken the piston-crank kinematics and recreated them using the standard geometric relation, keeping the crank rotation constant. I then plotted the slider’s position, velocity, and acceleration over one full cycle. The position curve shows the expected asymmetric periodic motion, which is due to the finite length of the connecting rod. The velocity is almost sinusoidal, but it’s slightly skewed between the forward and return strokes. The acceleration has strong nonlinear peaks near the dead-center positions. These trends match what we see in Shabana’s reference behavior. When I compared these results to a constraint-based MBD solution, they were very close (the differences are almost as small as the numerical precision), which really confirms that the formulation is consistent and can be repeated.

[alw14027]

[ktburnash]

These are the plots I got.

[Hlpred]

I was able to recreate the plots from the textbook using the provided notebook.

[matlabkid]

Recreated the plots from the textbook using the provided lecture notebook, the negative in Ctq got me for the acceleration plots at first.

[joshuajimenez1]

The crankshaft rotation is used as the system input, and all kinematic quantities are plotted as functions of crank angle. These results match standard slider crank kinematics and show validity of the modeling approach.

[med17018]

I was able to recreate plots 3.37-3.42 from Shabana's prescribed rotation of the crankshaft:

[jhngms]

See the figure below for the plots I generated.

[ojmoney00]

Here are the graphs from the example given in the textbook. The only thing I had to change was the initial angle to 30 degrees at pi/6 radians. "Initially, the crankshaft is assumed to make an angle θ20 = 30°." This shifts the graphs over slightly, so they exactly match with the ones given in textbook. I would say the main difficulty with this is figuring out the correct q_sol[:, _] to use for each value, but other than that the graphs to match the ones exactly from the textbook, proving that our solution in the Julia code is repeatable.

[avagabryshak]

these are what i did.

[LaythDuval]

This is what I found:

[augustinet70]

I attempted in recreating the piston-crank kinematic plots in Pluto by modeling the slider-crank geometry over one full crank revolution. I computed the connecting rod angle, slider displacement, connecting rod angular velocity, and slider velocity as functions of crank angle. The resulting plots the expected periodic behavior of the mechanism and verify the piston-crank motion.

[jaydawg-young]

These are the results that I got! Check them out! It seems like it aligns with mostly what everyone else got.

[Wesfrank54]

This is my attempt at recreating the Shabana piston-crank clots using the multibody constraint approach as outlined in the notebook. The plots are the Orientation of the connecting rod, Displacements of the slider block, Angular velocity of hte connecting rod, and the velocity of the slider block. I found that the process was pretty straightforward when constructing these plots.

[jadongs]

I was able to follow the piston-crank kinematic setup and understand how the generalized coordinates and constraint equations work together to describe the motion of the system. One thing that stood out to me was how MBD formulations track many more variables than what may seem necessary at first, but doing this makes the system much more flexible and easier to expand for more complicated mechanisms later on. It was clear that the prescribed crankshaft rotation drives the entire motion of the connecting rod and piston through the constraints. This phenomenon is expressed throughout the consistent graphical expressions of the crankshaft's behavior in each plot. I also noticed that some variables appear redundant in simpler cases, but keeping them in the formulation helps maintain a more general solution approach that could be adapted to larger dynamic systems more easily.

[juliannnatran]

Here are the plots I was able to generate.

[gwgezas123]

Here are the plots I was able to generate. they seem similar to what professor cooper has done. The code was already done so it was pretty easy to get the plots.

[tjs21014]

This discussion helped emphasize how the ideas from this course extend beyond theory into practical engineering applications. Using Lagrangian mechanics, generalized coordinates, and energy methods provides a structured way to model complex systems that would be difficult to handle with force‑based approaches alone. The combination of analytical derivations and simulations made it easier to build intuition about system behavior, stability, and parameter sensitivity. Overall, this course provided a strong foundation for future work in dynamics, controls, and mechanical system design.

[shainaerrico]

[dugganjake]

I did my best to recreate the piston-crank plots from Shabana's prescribed crankshaft rotation example. I focused on slider position, velocity, and the connecting rod's angle and velocity.

The crank angle is defined, so the rest can be solved from its geometry. Once that angle is known, the slider position and connecting rod angle can be calculated using the constraints.

My plots match the same behavior as Shabana's figures. The slider displacement starts off high, decreases to a minimum around the middle of the crank rotation, then increases again. Here is the code I used to generate the plots as well as the plots themselves, they look quite similar to quite a few others in this discussion.

[dugganjake]

The code is out of order for some reason when I added the pictures

[vedantsingh28]

[cbull0]

[honczjoseph]

Verification plots are a good way to confirm the multibody constraint setup works correctly. By plotting the connecting rod angle, slider displacement and their velocities versus crank angle. These are the graphs I was able to output.

[HerminioV]

I recreated the piston-crank graphs by looking at how each part moves as the crank angle changes through one full rotation. The main thing I noticed is that the slider motion is not constant even though the crank rotates at a steady rate. The slider speeds up and slows down because its position depends on the geometry between the crank and connecting rod. For the position part, the constraints keep the links connected at the joints and force the slider to stay on its horizontal path. Once those positions are known, the speed of each part can be found from the velocity constraint equation:

Cqq˙=−Ct
​
That is where the Jacobian becomes useful because it relates the known crank motion to the unknown speeds of the connecting rod and slider. The graphs make sense physically. The connecting rod angle changes smoothly as the crank rotates, while the slider displacement moves back and forth between its minimum and maximum positions. The velocity plots also show that the slider reaches higher speeds during certain parts of the rotation and slows down near the turning points. Overall, the plots helped me see how the constraint equations control both the motion and the speeds of the piston crank system.

[JimInput]

I like to think another way to confirm your slider-crank system is correct is just to look at it and make sure it looks right (:

Here is the one I put together as practice in julia.

[LRand7]

I understood the piston–crank example and how the generalized coordinates and constraints work together to define the motion, and it was interesting to see how multibody‑dynamics models keep extra variables to stay general and scalable for more complex mechanisms. The prescribed crank rotation clearly drives the connecting rod and piston through the constraints, which shows up consistently in the plots. Even though some coordinates seem unnecessary in simple cases, keeping them makes the formulation easier to extend to larger systems. Overall, the discussion highlighted how tools like Lagrangian mechanics, generalized coordinates, and energy methods translate directly into practical engineering modeling, and how combining analysis with simulation builds intuition for system behavior, stability, and parameter sensitivity forming a solid foundation for future work in dynamics.