Friday, October 24, 2014

A matrix solution to a simple torsional vibration problem

It's been a while since I thought about this, and because I still get a lot of inquiry on torsional vibrations from my post on the subject months ago and continue to get good questions,  I'm revisiting it again.

I've worked a lot of these problems in lower drivetrains, hybrid vehicles with inertially massive rotors, dual mass flywheels that make stuff go clunk... and ship propulsion... so I thought why not set up a simple 4 mass system as I started off doing, but never quite finished and show how to solve for the natural frequencies of the various modes and look at the mode shapes and discuss what they may influence in any design...

As always, this is an area I've worked on as an engineer for around thirty years.  I've done a lot of work in transmission design - both gearboxes and cases and both gear and bearing life and noise radiated analysis, diesel engine design and lots of structures too... but this torsional stuff is sort of a favorite area of mine!

So let's use the example of a 4 stroke 4 cylinder engine with a flywheel, first attached to a propellor, then the same system used to drive a rear wheel drive tractor... We'll have case A and case B.

The first step is to discritize the components into their equivalent mass elastic properties.  Here error is introduced as in the real world nothing is ever a cylinder or a prismatic shape and when things twist they are generally NOT linear in their stiffness, but they are reasonably so in a defined range of interest.  We can approximate the pieces of a system with these understood inertial and physical characteristics, but in fact we have more complicated geometry like fillets, oil holes, tapered flanges, splines and on and on and on... Where we see this error having a more profound effect is in the accuracy of prediction of the higher modes - modes where the mode shape and frequency accuracy are more and more dependent on several nodal crossings and the distribution of energy across many elements of the system.  Fortunately the higher modes of excitation are USUALLY, but not ALWAYS, of little or no concern to us as hysteretic damping and the lack coupled with the lack of a high energy forcing function usually keep things from finding a way to get to amplitude magnification.  Happily it's the way things usually work out to be, but not always! So ignoring modes of higher order, say the 4th or 5th mode and beyond are commonly done once we build a prototype and have test data to look at.  So the first 3 modes are usually what you want to be aware of, and a good rule of thumb is to understand what excitation exists in your system and to understand that dwelling on a mode could be a problem.  There are ways to move modes (soft couplings or even rigid couplings... adding inertia or reducing inertia... lots of stuff can be done to synthesize any system).

The first step in a normal modes analysis is just to solve the set of ODE's which is usually done without damping.  Sort of a quick look at what the system is tuned for...  We can solve these equations a number of ways.  A popular method is to use the Holzer Method, another is by direct matrix algebra.  Lets take a look at these two.

General Approach Using Transfer Matrices with a sweeping Omega term in Goalseek:


Mass Elastic System for Case A:

The Transfer matrices Method uses Point and Field Matrices to characterize the system, then the Omega variable is swept to find the zero crossings, which define the Mode Shapes...

Holzer's Method is a good alternative, both techniques are easy and fast
As time allows I'll add more detail to the calculation steps I'll post the cell equations if anyone wants them...


Mass Elastic System for Case B:

Wednesday, October 22, 2014

Roll out from MSP

Just a quick post to share the last departure from MSP.   I noticed this little guy out of my excellent peripheral vision...

What's that you ask?
Seems like we took off with one of our nations finest aircrews...

Why it's a very cool C-17.

Friday, October 17, 2014

Cinque Terra Vacation

Sorry I haven't posted in some time, it has been a very busy summer for me but I did manage to get in a very nice vacation hiking the Cinque Terre on the very rugged coast of Italy. 
Here's a few pictures to share.  I'll post a few more pics over the weekend. I did make it to Firenze, Como, Milan and Mandello del Lario and toured the Moto Guzzi museum...

The best vacation EVER...

Monterosso al Mare as seen from the trail
The Cinque Terre are five villages that dot the Italian Riviera and are part of a National Park.  The trails are made of macadam and cut stone with hundreds and hundreds of steps that wind there way through the steep cliffs that plunge to the sea.  The views are truly awesome with treks through vineyards and olive tree groves all part of a days hike.  The first town in the sequence of the five going South is Monterosso al Mare, shown above.  There's this huge sculpture in the old part of the city...

Neptune - a concrete and iron sculpture created in 1910, during WW2 it was damaged. Originally the statue had a trident and huge shell...
Typical trail cut into the steep terrain with lots and lots of...
STEPS and more steps!
The trail follows the land features in such a way that you climb and serpentine your way though the dense vegetation from town to town.

Trail side view of Vernazza, I recommend eating at Batti Batti

There are many wonderful little places to explore.  The restaurants and shops are small and unique with a lot of local art on display.  Seafood is a common staple of the Cinque Terre and I dont think you can go wrong with ordering any seafood.
Artist on the street

So after hiking the Cinque Terre the next stop was Florence or Firenze in Italian...  An amazing old world city with many treasures to experience.

View from Michelangelo Piazza
The Uffizi Gallery was wonderful as was the Galileo Museum.

Sculpture of Christ
 The Ponte Vecchio is a fascinating place... very sparkely

Wall to wall jewelery of the most exquisite designs

A most cooperative mime at the Uffizi

Around the corner from the Uffizi...

Moving around Italy is pretty easy and efficient on the TrenItalia rail system.

Train station at Milan Airport (Malpensa)
Lake Como - a very beautiful place!
 On the last day - Moto Guzzi!

The train station is right across the street from the plant!

Carlo Guzzi and I hanging out in downtown Mandello del Lario...

It is I knocking at the big Red Door at the Moto Guzzi Factory...

 You can hear the famous 1957 Guzzi V-8 here...  The factory rep explained to me that in the second year of racing they discovered that they had to add an extra fuel chamber on the carbs just so the engine wouldn't starve for fuel in corners... More art from Italy - desmodronic valves too!

Ok, not to diminish Honda's break out CB 750 I-4 "Dream Machine." But this Magnum Opus of Carcano's rolled out of  the Mandello plant 12 years before... AND this mill ran the Monza 500cc Grand Prix Circuit -- I'd doff my fedora (if I had one) to Giulio Carcano Ing extraordinaire. 

Carcano was one heck of an engineer!
I really think ├╝ber creative people tend to have crazy hair...

Hope you enjoyed the pics.  I have a 100's more, if anyone is inetersted in finding out more about Cinque Terre just ask.