Greetings.
I am trying to find examples of systems of linear higehr order differential equations.
That is, systems in the form;
Α Χ'''(t) +B X''(t) + C X'(t) + DX(t) = U(t)
or
Α Χ'''(t) +B X''(t) + C X'(t) + DX(t) = 0
where A, B, C, D are matrices and X(t) is a vector variable, i.e.
Χ(t)=[v1(t) v2(t) v3(t) ]T
Many such systems arise in mechanical engineering, like spring-mass oscillation systems, spring-mass systems with dampers etc, but all of them are 2nd order differential equations.
Can anybody help me find systems of order 3 or higher? The free parameter t does not necessarily have to be time. They can be engineering, biological, economic systems, whatever!
Please provide references too.
Thank you
For a 3rd order have a look at Nagumo's equation which is a special case of Huxley-Hodgkin model.
http://www.emis.de/journals/HOA/ADS/Volume3_2/193.pdf
Thank you very much! That's a good start, although I am looking for a system of equations Seems interesting though, I will look more into it!
This equation describes a single neural axon. Therefore, it is easy to consider a realistic case of multiple axis, i.e. a system of coupled Nagumo's equations
That's correct. I was looking for non-coupled systems of equations but nonetheless, this is quite interesting, thank you very much, I appreciate the interest!
A three element vibration isolator is a classical example. It consists of a spring-mass-damper system with additional spring supporting the damper. Usually this model is used to model the gas spring or a damper which is supported through the elastic rubber bushes.
Yes, but i assume this vibration system is 2nd order? Because I was looking for higher order systems. Nevertheless, if you can please provide me with a link or a reference
Bending of a beam is described by a forth-order equation:
https://en.wikipedia.org/wiki/Euler%E2%80%93Bernoulli_beam_theory
Now, if you consider something like hanging bridge, you need a system of equations.
Yes exactly, I was actually browsing examples of beam equations, like Timoshenko etc. But I mostly found a system described by a single 4th order equation, or in general a system of coupled 4th order equations, which of course has no point in describing in matrix form since it is coupled. If you have any book reference, please tell me because the last few days I have browsed through numerous books about mechanical systems like Mechanical Vibrations and Vibrations of Cont. Systems by Rao, but I cannot find exactly what I am looking for.
Thanks again for the help
A three element vibration isolator that I have mentioned is not a 2nd order but is a 3rd order system. Refer to Fig-2 of the following paper-
Article Third-order systems: Periodicity conditions
Yes Dr. Dharankar thank you very much, this is indeed very useful! I was browsing all day for similar papers and found numerous similar examples! Although until now I cannot find any reference to a system of multiple such suspension system. My goal was to find a system of multiple such dif. equations. Thank you though this is very important, maybe if i keep looking I will eventually find an example.
1- Deflection of beams is governed by a four order differential equation.
2- Biharmonic equation of deflection of isotropic and orthotropic elastic plates are a fourth order partial differential equation.
1- Deflection of beams is governed by a fourth order differential equation:
d4w/dx4 =-q/EI
2- Deflection of elastic isotropic and orthotropic plates are governed by fourth order partial differential equation:
d4w/dx4 + 2d4/dx2 dy2 +d4w/dy4=-D/EI
Dear Lazaros,
I think you are looking for higher odd order (i.e. 3, 5 , . .) differential equations (DE). Also you have mentioned about matrix formulation. Expressing the DEs in state form is an useful technique for expressing any order especially odd order DEs in matrix form.
Basically these arise because of relaxation phenomenon. Consider an element with spring and damper is series, if a sudden force is applied across it, it gets displaced, and hold in this position then resistive force developed in the element gets relaxed over the time.
In case of a three element isolator (earlier mentioned) only the part of a total resistive force developed in the element gets relaxed. This is applicable to the pressure or temperature relaxation in a gas spring or gas in an accumulator.
So you will see the odd order DEs in all the physical phenomena which includes the relaxation.
As per my opinion the researchers in vibration area may not be comfortable with this phenomena and not including it in the vibration models of the physical systems. So you will hardly found the higher odd order DEs in the literature on vibration.
You can take a higher order system, include relaxation phenomena, model it and get what you are searching!
Can you elaborate the objective that why you are in search of this?
Some 10 years back I was motivated to work on this phenomenon related to the vehicle suspension, identified some interesting properties of it which you can find from my conference paper- Analysis of Heat Transfer Effects on Spring Characteristic of Hydro-Pneumatic Struts.
Thanks a lot for attracting my attention towards this phenomenon again.
First of all, dear Lazaros,multiply this equation by A^(-1) from left hand. Then introduce new vector of triple dimensionality. Y=(X'',X',X). You receive the equation (1): Y'=M*Y+(A^(-1)*U(t),0,0)T (T-transposed block string). As for the matrix M 9x9 it's of block type M=(-A^(-1)*B,-A^(-1)*C,-A^(-1)*D);(I,0,0);(0,I,0), I- identity 3x3, 0 - null 3x3, The equation (1) is easily solved as Y=exp(M*t) *int_to_t(exp(-M*s)*(A^(-1)*U(s),0,0)Tds). Then You truncate the Y to X. It's all :)
But it's not the only way how to solve.
Thank you all for your comments.
Dr. Chstyakov, thank you for your comment. I am well aware of the procedure you describe to reduce a system to state space form.
Let me state the problem again. I am looking for a system (phenomenon etc) that is described by a number of linear differential equations, with at least one equation being higher than 2nd order.
For example, a spring-mass system with two masses is described by 2 differential equations of 2nd order. I am looking for other examples where a system may be described by 2 or more diff. equations, but at least one of them is of order higher than 2.
Beam deflection (like Timoshenko beams) as Dr. Altaan has mentioned is described by a forth order diff. equation, but its only one equation, I want a system of equations.
A model for vehicle suspencion system again is described by one diff. equation of 3rd order, not a system of equations.
This is the point where I am stuck, I cannot find a system with more than one differential equations in the literature.
Dr. Dharankar you are right in your comment! Would you be kind enough to send me a copy of your conference paper?
The paper is already on the Research Gate. The title is "Analysis of Heat Transfer Effects on Spring Characteristic of Hydro-Pneumatic Struts"
I got your point.
A DE of n order can be converted to n number of single DEs and vice versa, and it will not affect its eigen characteristics and solution. So again it is not clear that why are you in search of higher order coupled systems?
In mechanical vibrations of coupled systems, the maximum order of an equation will be either 2 (discrete rigid masses- inertia force) or 4 (continuous masses- moment of resistance). It is required to search the physical phenomena which are governed by the higher order differentials e.g. jerk a third order differential of a position wrt time.
Simply, you can couple the two or more continuous systems (beams) trough springs just like as multiple spring-mass systems.
Well, then the following mechanical system. A point projectile/symmetrical body which moves in air medium with linear drag R(V)=-aV and normal (to the trajectory) force N=kV with constant gravity gradient dg(y)dy=const - upper layer of atmosphere. For the system of equations let's introduce Coriolis force linear in velocity. So You receive the system of three ODE of III-rd order.
Or the quasi-classical elementary relativistic particle moving in bubble camera in gradient electric and homogeneous magnetic field
Though they don't refer to oscillatory systems formally it fits Your demands. More detailed see my papers on RG.
Best regards
Thank you for your responce. I believe this is the paper you are refering to, right?
Article On One Numerical Method of Integrating the Dynamical Equatio...
Not only. There are more works where the III-rd order DE are considered but in Russian. Namely, the problem of point projectile trajectory in air with density and gravity linearly decreasing with height, the particular case of dynamically disbalanced rotator with combined friction ( both see Bulletin of Udmurt State University, Maths, Mechs&Comp.Sci.). The problems may be 'matriced' by substituting the point by symmetrical body.
Though the equations are nonlinear they may be linearized.
Best regards
V.Ch.
Look for dynamics of beam or plate, e.g. Euler-Bernoulli beam theory. You will get many more interesting fourth order equations.
Hi, Lazarus! This is the case You've asked about:4-th order PDE,
With best regards
V.Ch.
Greetings professor,
thank you very much, I will take a look at this work.
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