You will likely receive many responses with high variation in concepts. To offer meaningful information will be risky. Many such attempts at Dynamic Analysis vary in method and results. Thus, to give good guidelines will show the errors of many and sacrifice the diligence of the few.

Some common mistakes have been made by the early academics in choosing the right analytic procedure such as:

1. closed form ordinary linear differential equations vs time step numerical integration with non-linear solutions -- why must we resort to non-linear solutions?

2. choosing the frame of reference between the moving belt and static world on defining motion, displacement, and properly defining dynamic vs static behavior

3. solving the 2-dimensional belt geometric wave equation format vs the linear modulus rod (1-dimensional) with differing reflections and differing apparent elatic modulii

4. how to select the starting point for the solution - to start or not to start - what are the solvable boundary conditions.

5. finding meaningful problems where measurement and theory demonstrate levels of accuracy -- do not apply fudge factors to force the theory to fit the measurements as is currently practiced. The next problem will not easily fit the current fudge factors.

6. You should find a problem that exhibits a high degree of difficulty and what solvers produce an accurate response and what solvers fail - find or define a few key problems that will show the fundamentals with accuracy or not -- there are geomtric complexities we see in many conveyors that will show the complexities. We know them, some our competitive practioners know them, and some do not. I offer one case: If you find a result with time vs a negative belt tension in the plot sequence, you will find, a failed solution in +99.9% of the cases.

7. Not all dynamic analysis codes are accurate and reasonable. In fact, most are not.

8. What starting and stopping algorithms are proposed and how are belt tensions, power, and belt displacements controlled? How is load sharing between multiple pulleys solved? Does the solution use fixed or variable time-integration stepping? What type of forward predictor is being used?

9. You will find difficulties among some hydraulic coupling suppliers that do not want you to model their products -- why?

10. We see others noting how their model results track our published field measurements. Interesting point, since they do not have critical features of the belt such as: a) rubber's viscoelastic rheology, b) idler properties, c) starting and stopping regulation and most critically d) take-up motion with its malfunctioning capstan. We published the results without noting these critical conditions to illustrate the kind of dynamic difficulty one can experience. My question is how did they fit the details of this condition?

I wish you luck with the project. I wonder why you would take on such a thankless task? We invented the most common solution in 1980. We published the first findings in 1982. In 1984 we published the rheological building blocks in Bulk Solids Handling and demosntrated the difference between the 2-D elastic wave equation and a 1-D solution. Some profess today to have a 3-Dimensional solution. Power to them. Does it show advantages over 2-D.

We were told, in a BSH publication 1985-1986 (by memory), various Polish Academics had already solved similar rheological bases in point-by-point fashion. Hmmmm? However, no proof was put forward. We see little of their advancements.

What has not been published in Dynamic Analysis modeling? Where can the modeling be improved? ■

Right, to the question:

1. Alex Harrison's many publications and thesis I believe in about 1982- started with closed form ODE's and more recently saw the angels and switched to non-linear time integration PDE solution:

http://www.conveyorscience.com/dynamicanalysis.html

You can find his thesis under Prof. Alan Roberts guidance at University of Newcastle, Australia.

2. Prof. Gabriel Lodewijks - Delft University of Technology

Lodewijks, G. (1996), Dynamics of Belt Systems, Ph.D. Thesis TU Delft, ISBN 90-370- 0145-9.

3. BSH forum: https://forum.bulk-online.com/showthread.php?t=17908

4. BSH has many publications on Dynamics over the years - you find them.

5. Dr. Funke's 1974 thesis referred to in Dr. Harrison's expository above.

6. My BSH publication, in 1984, describing the need for 2-D - I don't remember which issue

It's a start. ■

Dear Mr. Nordell,

I skimmed through Mr. Lodewijks' thesis "Dynamics of belt systems" and noticed he considered linear viscoelastic 3-component maxwell model as belt elements. Is there any specific reason of considering linear viscoelastic model except for non-linear models. The linear models work accurately for small strains as I read in some literature. Moreover, in normal practice the max belt stretch allowed is 2% ( to the best of my knowledge). Does this affect the choice of constitutive model used ? I have referred to another thesis "Analysis & simulation on the Dynamic process of belt conveyor" by Hong-Yan Li from Northeastern University wherein he used a Standard linear solid model ( Hookean spring in series with a linear kelvin model ). In addition to this, I also referred to your pioneering work where you used a 5 component model. My major concern here is - How is the constitutive model chosen? The current belt manufacturers use NR/SBR/Polybutadiene covers. These cover grades can even reflect linear as well as non-linear behaviour. Sir, I'll be highly grateful if you can throw some light in this matter.

Thanks in advance ■

Dear Mr. Nordell,

I skimmed through Mr. Lodewijks' Thesis "Dynamics of Belt Systems" and found out that he used a Standard Linear Solid Model (hookean spring in parallel with maxwell model) for his rubber cover. Even another thesis " Analysis & Simulation on the Dynamics Process of Belt Conveyor" by Hang-Yan Li incorporated a Standard Linear Solid Model (hookean spring in series with a Kelvin Voigt Model). In your paper at CKIT website of Channar overland conveyor, you suggested idler compressive strain less than 5%, but also mentioned that the rubber cover exhibits a non-linear behavior in large strain range. In practice, the maximum belt stretch is considered to be max 2% (to be best of my knowledge). So, my question is - How to decide the constitutive model for rubber cover, linear or non-linear. On what factors does the choice depend ?"

Sir, you have immense knowledge in this field. I will be highly grateful if you can throw some light in this regard.

Thanks in advance. ■

Originally Posted by Himanshu

Dear Mr. Nordell,

I skimmed through Mr. Lodewijks' Thesis "Dynamics of Belt Systems" and found out that he used a Standard Linear Solid Model (hookean spring in parallel with maxwell model) for his rubber cover. Even another thesis " Analysis & Simulation on the Dynamics Process of Belt Conveyor" by Hang-Yan Li incorporated a Standard Linear Solid Model (hookean spring in series with a Kelvin Voigt Model). In your paper at CKIT website of Channar overland conveyor, you suggested idler compressive strain less than 5%, but also mentioned that the rubber cover exhibits a non-linear behavior in large strain range. In practice, the maximum belt stretch is considered to be max 2% (to be best of my knowledge). So, my question is - How to decide the constitutive model for rubber cover, linear or non-linear. On what factors does the choice depend ?"

Sir, you have immense knowledge in this field. I will be highly grateful if you can throw some light in this regard.

Thanks in advance.

------------------------------------------

The Channar paper described the Belt-Idler interface rubber compressive strain. Belt Stretch strain is an axial tensile strain that occurs within the steel cord or fabric tensile. They are quite different and have different rheology models.

Both types are non-linear. Rubber more so. However, within the modeled visco-elastic range we consider it to be linear.

For steel cord, most use a linear model. However, pull any steel cord in a tensile tester and see the non-linear response. Depends on the accuracy you seek: a) 1st order or, b) 2nd order of accuracy. ■

Originally Posted by Himanshu

Dear Mr. Nordell,

I skimmed through Mr. Lodewijks' thesis "Dynamics of belt systems" and noticed he considered linear viscoelastic 3-component maxwell model as belt elements. Is there any specific reason of considering linear viscoelastic model except for non-linear models. The linear models work accurately for small strains as I read in some literature. Moreover, in normal practice the max belt stretch allowed is 2% ( to the best of my knowledge). Does this affect the choice of constitutive model used ? I have referred to another thesis "Analysis & simulation on the Dynamic process of belt conveyor" by Hong-Yan Li from Northeastern University wherein he used a Standard linear solid model ( Hookean spring in series with a linear kelvin model ). In addition to this, I also referred to your pioneering work where you used a 5 component model. My major concern here is - How is the constitutive model chosen? The current belt manufacturers use NR/SBR/Polybutadiene covers. These cover grades can even reflect linear as well as non-linear behaviour. Sir, I'll be highly grateful if you can throw some light in this matter.

Thanks in advance

----------------------------------------------------------------------------

Dear Himanshu,

You must understand the fundamental differences between the tensile elastic response of a steel spring or polymer fabric, as with belt tension, and the visco-elastic response of rubber under a compressive and shear loading, such as at the belt idler interface.

Choice of models depends on how the modeler wishes to describe the energy loss/belt tension change phenomenon for steady-state and for dynamic operations.

For steel cord response, the essential condition is the frictional loss due to loading the steel cord in tension, which also causes a torsional response due to the helix construction. The steel cord wires try to migrate to the cord center due to the radial action, which causes friction between wires in the cord. This friction is similar to a non-linear coulomb loss, depending on the 1) wire construction and 2) loading. This friction is also dependent on the rubber's restraint of the twist and axial extension. At low belt tensions, below 0.5% sag, the sag geometry will influence the tension and belt hysteresis loss.

For rubber deformation = tension or power loss along the belt is dependent on a rubber temperature, strain, and frequency or strain rate properties for each polymer type. The rubber hysteresis or energy loss at each idler is effected by idler indention into the belt cover, belt flexing between idlers, and rock trampling loss between idlers. Indention loss is due to the compressive and torsional hysteresis loss from deformation and slip of the belt bottom cover over the idler dependent on: 1) belt speed, 2) ambient temperature, 3) idler diameter, 4) idler spacing, 5) idler trough angle, 6) idler surface material, 7) Product(material) loading profile, 8) belt cover thickness, 9) belt cover rubber visco-elastic properties (E';E", G';G"), 10) belt composition and reinforcements, 11) belt tension, 12) Product properties, etc.

I do not believe there is need to go deeper on the subject or on flexure and trampling losses. You can glean understanding of the differences between steel cord modulus and rubber properties and how they influence belt stretch vs. power from the above.

You need to read what is published on these subjects. No knowledgeable individual can give adequate insight in the space allotted in this forum. ■

Sorry for not addressing your request of rheology model selection:

1. Maxwell Model: uses spring (elasticity) and dashpot (viscous loss) in series. This defines a deformation where there is no recovery from spring displacement.

Where is this relevant in our conveyor model? By itself, it is not, unless you are describing pseudo plastic flow.

2. Kelvin-Voigt Model: uses a spring in parallel with the dashpot. Here the displacement is recovered.

How to determine which model to use, best to read about rheology from authors such as:

1. Prof. Nicolas W. Tschoegl "The Phenomenological Theory of Linear Viscoelastic Behavior"

2. Dr.-Ing. Wilhelm Flügge "Viscoelasticity"

These are old texts and show my age. They do give fundamentals of rheology pertaining to rubber. Good reading for the analytical. ■