Dear Mohammadreza Atarodi,

First, by looking at the graph, your system seems okay.  Of course if you choose initial conditions equal zero, depend on the situation, it should remain zero; I suppose you are considering both initial condition zeros, position and speed, but the latter is more significant in the current case. Please, keep in mind that acceleration is created by force, not by speed, you can have a car in constant speed, it is similar to 0, first law of Newton.

Second, I suppose you are using a numerical method. According to my own experience, numerical method, depend on the type, depending on the period of simulation can "blow up;" even for quite good software such as Matlab. It creates fake behaviors. Even the mathematical model can create. Models in general are build to operate in certain conditions, time and parameters, of you leave this domain, the method starts to behave oddly.   

Can we accept that, high acceleration in spring mass of vehicle suspension system is consequence of chaotic dynamic?

I would not say 'chaos,' I would say "instability." According to my view, chaos would make your system to behavior more likely a periodic system, maybe increasing acceleration, but still not complete "chaos", complete chaos according to my view is instability.  See Lorenz System https://www.google.it/search?q=lorenz+system&biw=1366&bih=667&tbm=isch&tbo=u&source=univ&sa=X&ved=0ahUKEwiThuPsnpjMAhVG1hQKHa-hDxEQsAQILw

So what I would suggest you to do before any significant energy is lost: 1) take a break and double check your codes, such as look for other cases that you have an expected behavior, it can happen that the code is correct and you are wrong, it happens often; 2) be sure you know which numerical method you are applying, assuming you are using one. Last but not least, double check your codes, it may escape bugs that creates strange behavior as the one you have pin pointed.  

After that, if the problem persist, I will be glad to sleep over the problem with you. 

Good luck,

All the best,

Jorge Pires,

thanks  mr Jorge Pires for  answer.

at first i used order 4 -runge kutta- method to solve equation and then i used Poincare map and bifurcation diagram and... for recognize chaos in this system and  this equipment  proved that my system is chaotic.

 my input for the system is A*sin(w*t) and my initial conditions are zero you know that my system is a mechanical system......

if in this mechanical system i dont have any input my response must be zero but i receive response for my equation.

i attached my  system response(acceleration)that it was for A==0 and you can see that acceleration isnt zero and i dont know why?

after this step i want design a controller for system but this high acceleration needed high energy  for control. 

thank you

Hi Mohammadreza,

I completely agree with Jorge that this is a case of instability (at least marginal instability) and not chaos.

Other than the precautionary measures he has already suggested, in your position I would have checked for resonance conditions (close to zero damping ratio) in the system model. Other things apart, the fact that your system is showing a significant response with zero initial conditions as well as no excitation input makes one suspect an almost sure case of resonance. 

-Sanjay

thanks mr Sanjoy.i can show you chaotic vibration of this system....

i think this especial treat is caused by strange attractor...but i dont know my idea is true or no??

It looks like your transient period is to short and the system has not yet settled on the attractor.

What is your transient time? Have you waited long enough to start drawing Poincare sections? I recommend using transient t = 1000*T, where T is the excitation period.

If you want to be sure, that you have chaotic dynamics calculate Lyapunov exponents.

The high acceleration is normally seen when natural frequency of the sprung mass matches to that of excitation frequency. ( Resonance frequency.) Check by allowing more time.

Thanks from all