I don't smoke, but this portofino was from a project that didn't turn out, and being a bit contemplative today, I brought the portofino out onto the deck and pondered... |

I was reminded today, that as many times as these systems have been worked out, and solutions provided for (systems that have a torsional resonances that are of consequence) -- we just keep on creating new ones. So in an effort to share something I know a lot about here's a short how-to. This will take me a few days and I will scan calcs and spreadsheet formulas for those who want to sing along...

Like any vibration problem, we can identify a mass, or in this case, an inertia. A stiffness and a forcing function. Torsional vibrations and their correct mitigation are sort of everyday stuff for engine manufacturers. I learned most of what I know as an analysis engineer a million years ago at a company called Caterpillar. I did lots and lots of engine to drivetrain analysis and I suppose correctly said, synthesis. So here goes, I hope you enjoy the detour from the normal stuff I like to post.

The torsional pendulum...

The two mass system is discussed below. It's taken me a bit longer to post as I was stuck in the O'Hare vortex for a few days. Luggage is still out there somewhere -it's ok, United knows very well where I live LOL. I actually ended up driving home last night from Wausau, Wisconsin with three other stranded yoopers all trying to get to CMX...

Most systems in the "real world," are never this straight forward. It's a very good approximation though for the first torsional frequency of a more complex system. There is a wrinkle to this I should mention. If the line shafting goes through a gearbox the reflected inertia is either multiplied of divided by the ratio squared depending on if the gearbox is a speed increaser or reducer. If you have a question about that or how to calculate an equivalent stiffness or inertia just jot it down in a comment, I'll be happy to work out an example for you. I've done hundreds of marine compound and single gearboxes with multiple branched kinematics, and used the torsional forced response as the forcing function for the isolation system of large diesel engines, and in a fair amount of vehicle drivetrains too. So just ask, I like this stuff!

Inertia units can vary a lot so be careful. Here's a short list. I am messing around with an app for Anroid and iOS devices for vibes and have an expanded set of conversions for both stiffness and inertia values. Always work out your own dimensional conversions before you grab anything off of the web. I believe these to be correct but ask you to make sure for yourself before using them. The app has a reference calculation for each conversion and they have all been verified. I'll post the availability of the app when I'm done banging on it.

Normal modes of an undamped system. Below find characteristic mode shapes for multi-mass systems. There is another aspect of this that you should consider. Simply knowing the frequency and the mode shape really doesn't give you enough information to change the system to move the mode to a place where you want it. You really also need strain energy to do that confidently. So in the example to be posted here you will see a three mass two spring system with mode shapes and strain energy plots. I'll explain a bit more about this when I get there.

Holzer's method in a nutshell - and spreadsheet formulas using goal seek are below. Holzer came up with a method to balance the distribution of energy in a multi-mass system using an iterative or marching algorithm. It's as accurate as the residual value you are willing to accept and is generally not used for damped systems due to the inability to handle non-linearities. But it's still a good method for systems with very small hysteretic damping. This works well for engines and drivetrains. There are unique energy methods for heat loads on couplings etc., I will not cover those here, but would be happy to return a text if you want to comment on this post. Similarly there are methods to characterize chains, belts, gear teeth, propellers and just about any physical connection. Where Holzer and methods like it have problems are with non-linear stiffness or non-linear damping of significant magnitude...

Here's a simple 4 mass (inertia) 3 spring system. Consider that at the end of the day, any mathematical treatment of this topic is an artifact of simplifications that are ascribed to in order to define the elements of what can then be a "solvable" problem. Let me explain... Inertia values are pretty easy to calculate or to measure. BUT, what is an inertia? A deceptively simple question. In an engine, we easily see crankshaft counter-weights, the large end of the conrod... but what part of the reciprocating piston mass is transiently involved - or is it (in a forced response simulation you have inertia torque and gas torque and they oppose each other)? What about the pin end of the rod? Not too many things are as easy to calculate as a flywheel, gear, coupling or ring gear... those are nice and straight forward. Now I will ask you, what is a torsional spring? Hmmmm, that looks easy too, until you consider for instance that a twisting crankshaft has an offset pin, it has geometry (the flanks or cheeks depending on which side of the Atlantic you live on) that are pretty stiff relative to the journal or the crank pin and they all deform - twist if you will - ... so how do we arrive at a spring rate for that? Which chunk of that system is inertial and where does it go in the 4 mass 3 spring model of this 4 cylinder engine? Then one last idea to ponder... lets consider that the entrained water circulating with a propeller is influenced by the hull shape, the angle of the blades and in heavy seas, when you momentarily uncover a screw... you have no entrained water and a mass-elastic system that is very different than the one you know when everything is as it should be. Then there's ice and the impact forces it can produce.... remember that a polymeric coupling is rate sensitive! IE it can get very stiff at high rates of impact. Lots to noodle on here! But, just take your time and cover all of the conditions.

Are there too many variables to make torsional vibration analysis believable? NO! What you need to do is to consider all of these scenarios, and set banded speed ranges if required, and take data... validate your predictions by measuring the first 2 or 3 modes. Higher modes... are pretty hard to predict with high accuracy because of the way we descritize a system. The real world isn't descritized... and the distribution of strain is far more nuanced and complex than our analysis methods allow us to see. When you test, you are looking for strain, an fft of those strains, and the peak amplitudes of the line shafting. When you acquire this data you need to go back to your model to see if makes sense. If not, perhaps some initial assumptions need to be adjusted.

Later today, (time !) I'll try to post a spread sheet and a few more thoughts about this. As always, if you would like to correspond about this topic just write a comment. -LJO

One other last thought, before I forget! An engine's crankshaft is a progressively loaded spring. Which means that the power delivery side of the crank will be twisted more than the cranknose end. Don't forget that this has implications too. Counter-weighting and progressive unbalance occurs toward that end of an engine and is the reason the power driving end of an engine tends to have higher vibration amplitudes in it's block and through it's mounts - and a phenomena known as flywheel wobble. The last throw of a crank will twist... but it also will bow under the gas load - very slightly, that induces a flywheel wobble consistent with the firing of any cylinder associated with the final crank throw. For this reason, many marine engine manufacturers design in a second 'zero bearing.' Essentially, a double bearing on the crankshaft at the last throw. The torque in the drivetrain is also reacted... and a simple free body diagram reveals that the torsional harmonics get reacted in the engine mounts, which in turn drive those harmonics into the hull structure with some transmissability dependent on the mount system. OK, will revisit later! LJO

Forced response and damping...

Modal synthesis and the common ODE approach...

What recip engine torque looks like...

This is why engine engineers live in the frequency domain and drivetrain engineers live in the time domain...

Forced response of multi-mass recip engine systems...

Energy method handy for heat loads on couplings...

Model validation strategies...

The end!

## 1 comment:

Hello Mr. Oberto,

I am currently trying to educate myself about torsional vibrations in marine engineering. On my way I found this helpful blog post, unfortunately the spreadsheet formulas you posted are no links but just blank text. I would be very grateful if u could provide the links.

Greetings,

H. van Rooijen

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