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## ODE Problem

Valued Contributor III
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There are two forms of measurements in common use on modern bridge monitoring, strain gauges and accelerometers.  The modern accelerometer at 2000 data points per second gives a lot of acceleration data, but successfully converting it to velocity and displacement can be problematic, due to temperature and other variables.

In measuring a damaged bridge recently, a friend used an accelerometer in parallel to another team who used strain gauges.  The bridge was load tested with a three axle truck.

Comparing the results is interesting and a good Fortran problem.

Fryba from Czechoslovakia published a number of papers in the late 1960s on displacement of bridges under moving loads and then published a book in 1972 which has been released as late as 1999. in English.  His ODE solutions are essentially as given in the 1960s.  Fryba solved the general ODE, he provides the equations and the solutions but not the code merely noting it was solved on a URAL computer. Most modern papers refer to Fryba's book, but a lot do not solve the general equation and the ACM ODE solver is not general.  Not general means velocity ignored.

Fryba says he used the RKN method, which is quite commonly used form the ACM ODE solver, but there limited western works on the general solver.  Lear from NASA in 1975 published the specific algorithms, and the guy from the degenerateconic website provided a Fortran subroutine for one ODE solver, but nothing else.

This is the code I have been discussing today in the posts on ALLOC errors.  I am slowly getting to Fryba's solutions, it just takes a while with the complexity of the equations and determining the unknown variable values, he gives you a lot but not all.

In the attached figure you can see Fryba's measured values and computer results are shown for a single span bridge.  In order to upload the figure I had to put it into a word doc.  Fryba's theory is that the deflection returns to zero as soon as the train leaves the bridge, but the experimental measurements show that there is movement after the train leaves the bridge.

Fryba does not provide the velocity and acceleration profiles, it is possible using the LEAR algoithtm to get the velocity and a simple extension provides the acceleration.  The wiggle shape shown by Fryba is what we observe if we isolate the vehicle running over the deck and only do a very short estimate of velocity and displacement.

But I am stumped as to why Fryba insists that the displacement is zero as soon as the vehicle leaves the bridge, I would see the experimental as more meaningful.

I am slowly getting the full equations into the code, but I was seeking any thoughts on whether Fryba's assumption is to simple.

If it comes to zero displacement and stays this means the acceleration and velocity are also exactly zero and this seems improbable.

John

8 Replies
New Contributor II
2,966 Views

John,

I don't have the Fryba paper available, but you are dealing with a time varying load being applied to the bridge ( at the fastener/sleeper spacing ).

As this is a transient load, there would be a transient response after the train leaves the bridge, which is smaller and less significant for bridge deflection design, but significant for train ride performance.

The more significant (equivalent static) response is as the position of the train load changes  along the bridge.

If Fryba suggests the displacement is zero after the train leaves the bridge, then it suggests he is only considering the "equivalent static" varying load as the train position changes and varies the load on each fastener.

There are also a number of other considerations for time varying load effects of varying significance that can affect this deflection estimate. These can result in variations to the load applied to the bridge deck at the base of the fastener points. Some of the main considerations I have identified are fastener spacing and stiffness, rail flexibility, bogie mass and stiffness and wagon mass and velocity, although there could be others in other particular cases. This results in the following types of transient movement effects. that I have considered:

1) The continuous rail will flex between the fastener supports and so can change the load distribution between fasteners.

2) The elastic stiffness of the fasteners can vary the load distribution between fasteners.

3) The vertical movement of the rail can modify the time varying force in the train bogie, when considering the vertical inertia loads of the train bogie and wagons.

In my analysis of train, rail and bridge movement I try to include all 3, but separately consider the 3rd to simplify the calculation.

I should review Fryba's ODE approach. However it is possible using Fortran to include many other identified load varying effects in a time varying/stepping FEA analysis to get a better understanding of the relative significance of these effects, rather than what I would assume to be a very complex analytical solution that ignores a number of these other significant effects.

Valued Contributor III
2,966 Views

John

Fryba had data for trains from the 1960s so the quality of the data - ie resolution etc - is a small problem.

I have a solid FEA analysis using Stability functions - if one tries with normal structural FEA, constant coefficients you run into the problem of matching high frequencies, I have played with this since 2011.

, we can predict these things quite wel with SFl, but the problem is the Stability functions are a pain to converge and they are slow as slow.  But they work but not in near real time on a NUC 3 on a bridge.

Fryba's method however will run in seconds on a NUC Core i3, all of the problems you identify exist, I am slowly stumbling onto them - but the beauty of a simple system that will run in seconds out weighs the pain of the next year as we learn to use it.

I enclose the paper -- I talked to his former Dept Head and he died 3 months ago.

We are trying to match real data at the moment and it is a lot of fun.  - We do road bridges, not rail.  We ahve two very real problem cases I am looking at currently, we shall see.

Thanks for the response.  Our real problem is data richness - not the models - but the capacity of a small computer on a bridge.

John

Valued Contributor III
2,966 Views

PS: A lot of the modern papers copy from Fryba but change the math a little in notation and them delete damping.  However we do not criticize modern papers - not good for the academic image.

Getting a RKN solver to do the general equation is another challenge - thank god, she created NASA.

New Contributor II
2,966 Views

John,

Getting a practical solution to a moving train on a model of a rail supported by fasteners on a bridge has a number of practical problems. Although the rail is much stiffer than the fasteners, the flexing of the rail induces significant higher frequency vibration.

For vibration isolation problems, the rail fasteners and bogie suspension are much more flexible than the rail, so accurate estimations of the rail deflection under the moving axle becomes very important. If modelling the rail as a line element, it becomes very important to get an accurate estimation of the lateral deflection of the rail as the axle moves between nodes, so that a suitable estimation of the axle vertical movement between fasteners is obtained. This relates the time varying fastener forces and bogie inertia forces. Simplifying assumptions of applying time varying loads to each node may not adequately describe the axle vertical movement and a more accurate contact description is required. This can be a difficult contact problem to define in a general FEA program, but is required to describe the transient vibration of the bridge, rail and axle, evident from your experimental measurements. Accurate estimation of this higher frequency movement is difficult and problems in the modelling approach can be identified by changing the node spacing along the rail and comparing the calculated the time varying vertical acceleration of the axle.

I shall read the Fryba paper to see how this is addressed, although dismissing transient vibration after the train passes does not appear adequate. A train is a set of many axles that combine for a total vibration effect.

Valued Contributor III
2,966 Views

John Campbell wrote:

John,

Getting a practical solution to a moving train on a model of a rail supported by fasteners on a bridge has a number of practical problems. Although the rail is much stiffer than the fasteners, the flexing of the rail induces significant higher frequency vibration.

For vibration isolation problems, the rail fasteners and bogie suspension are much more flexible than the rail, so accurate estimations of the rail deflection under the moving axle becomes very important. If modelling the rail as a line element, it becomes very important to get an accurate estimation of the lateral deflection of the rail as the axle moves between nodes, so that a suitable estimation of the axle vertical movement between fasteners is obtained. This relates the time varying fastener forces and bogie inertia forces. Simplifying assumptions of applying time varying loads to each node may not adequately describe the axle vertical movement and a more accurate contact description is required. This can be a difficult contact problem to define in a general FEA program, but is required to describe the transient vibration of the bridge, rail and axle, evident from your experimental measurements. Accurate estimation of this higher frequency movement is difficult and problems in the modelling approach can be identified by changing the node spacing along the rail and comparing the calculated the time varying vertical acceleration of the axle.

I shall read the Fryba paper to see how this is addressed, although dismissing transient vibration after the train passes does not appear adequate. A train is a set of many axles that combine for a total vibration effect.

John

you are absolutely correct. Fryba is a good paper from 1968, given the quality of the instruments and the quality of the computers he did a pretty good job.

He did not ignore the transients - his main interest was peak deflection, he made starting allowances for some of the things you mention.

We have 2000 cycles per second data in three directions and two tilt angles,  we monitor to 1000 Hz and our problem is not the low frequencies that is trivial, the problem is you match the low and then matching the high requires quite sophisticated tools, way past Strand 7 and the ilk.

Our data is uploaded from the bridge to the cloud so we have a lot of data- we pick up the magnetic field from the electrical wires and can tell you if the town's power plant is having problems.  We have so much data that it is easier to process and graph in the cloud.

Yes I agree we have a long way to go and a lot of different problems,

But thankfully there are a few people on this board who are really helpful and interesting.

John

Honored Contributor III
2,966 Views

John Nichols,

The Fryba paper describes a solution using an overly simplified model. One that does not represent a (near) complete physical model. IOW two axles are simulated whereas a modern diesel locomotive has two carriages of three axles. Additionally, the simulation is limited to a single span bridge (of beam design).

Have you experimented with using Finite Element modeling, with which you can describe a more complete model?

Jim Dempsey

Valued Contributor III
2,966 Views

jimdempseyatthecove (Blackbelt) wrote:

John Nichols,

The Fryba paper describes a solution using an overly simplified model. One that does not represent a (near) complete physical model. IOW two axles are simulated whereas a modern diesel locomotive has two carriages of three axles. Additionally, the simulation is limited to a single span bridge (of beam design).

Have you experimented with using Finite Element modeling, with which you can describe a more complete model?

Jim Dempsey

Jim:

I agree with your comments, my main aim was to get the RKN method working.  Fryba publishes a book with a more complete solution for multiple wheels.  Remember his solution is 1968.  We can improve it, but getting a solver first was the important step.

We have huge data sets that are located on the bridge for economic reasons you want to do any analysis on the bridge computer, I know we have played with this since 2007.

I have FEM programs that solve this problem regularily but they take a lot of resources, what I have are the left over resources on a small computer on the bridge after data collection FFT and statistical analysis.  So we will use both methods.

The issue is now data quantity and analysis time.

John

Valued Contributor III
2,966 Views

Plus - I am initially interested in Road Bridges, Fryba's stuff is a nice addition to the toolkit.