Vibrations Model for Highway work 1

[Dinosaur]

Hey fellows,

I am a bridge engineer so I infrequently get involved in doing things with differential equations. However, I have been asked to collaborate with a fellow from the concrete pavement arena to analyze/predict the effects of having a transition in the pavement thickness. We agree that our new policy should take into account the traffic speed and the change in pavement thickness, but there will be some resistance in the pavement community because they would like to keep things simple. I don't want to bore you with the politics of roadway work anymore.

To incorporate a rational answer to this question, I want to model a vehicle as a mass supported by a spring (and later a dash-pot shock absorber) riding on a wheel which is pushed up or down by the pavement transition. This will cause the mass to oscillate and we can compute the forces from this oscillation. I would appreciate your help with a number of questions on this problem.

The governing differential equation (as I expect everyone here already knows) is:

[K] x S(t) + [C] x S'(t) + [M] x S''(t) = f(t)

Right now, I am using only a SDOF model, and C=0. I can solve this equation if f(t) = 0, because that is the fundamental free-vibration problem and the solution is both easy to obtain and already well known. However, I am not so well prepared to handle the problem with f(t) being non-zero. To overcome this, I wrote a quick Excel spreadsheet to handle this as a time-step problem. I got very satisfactory results from that.

Now I would like to calibrate my model. I need a reasonable idea of the spring constant for a 2000# vehicle, and I would like the same for a 3000# vehicle and a 4000# vehicle. It is my belief that heavier vehicles will have a combination of mass and suspension to give them a natural frequency near that of a 4000# vehicle.

Is there anyone here that knows something about automobile suspensions that would like to lend me a hand? Thanks for your help.

 

[GregLocock]

Assuming that octoslash means lb, then the resonant frequency will be between 1.2 and 2 Hz (smaller vehicles tending to be higher).

Damping is around 30% critical.

You are ignoring tires, stiffness is 220 N/mm, damping is 7% critical, unsprung mass is 60-100 lb

Whay are you ignoring trucks - they are the ones that'll do real damage, as I am sure you know, damage is proportional to axle weight ^6 (from memory)

Cheers

Greg Locock

SIGlease see FAQ731-376 for tips on how to make the best use of Eng-Tips.

 

[Dinosaur]

Greg,

Thanks for the information. I will be able to use all of this as the project matures. Right now, I am doing only a SDOF model. This precludes me from identifying the tire as a seperate component for the analysis model. But I should be able to pick that up in a later MDOF model.

Regarding the trucks, I believe I will be able to show that the load is proportional to the mass. If the suspension systems are all in the same range of natural frequency, then predicting the damage from a truck would be as simple as converting the mass and magnifying the loads by that ratio. Once I get it to a certain point, I should have resources to verify that. I am actually concerned that once someone at a nearby University hears about what I am doing, they will have more resources and they will pursue my study without me at a quicker pace.

Currently I am calibrating my SDOF model. Once this is done, I will run a large number of cases with varying mass, stiffness (to change the natural frequency over a resonable range), transition length and height. I will compare the answers to the policies used in several states and make a recommendation for my own state and share the results with other states through the FHWA.

Then, if I have some interest in my state, I will make a 2-DOF model, with each axle of the car moving vertically and trying to capture some of the rotational inertia of the car body in that model. The next model, 3-DOF, will include a horizontal component of the car's mass to simulate the feel of a braking type inertia as the car hits the ramp-like pavement transition.

After that, I believe I will model trucks, but that may be well after my University neighbors take the ball and run away with it.

Thanks again for your help.