Appendix 1.2 – Effect of damping ration on the response of 2nd order system
Summary
The 1st order systems consisted of Resistor-Capacitor (RC) Network, Flow Restrictor - Volume/Vessel Network which had a time constant value of 64% of the output. A Series Inductor Capacitor and Resistor (LCR) Network and Spring – Mass Damper system were used as 2nd order systems to determine the effect of damping ratio of on the output.
1. Results
The following figures and graphs represent the theoretical analysis and the experimental results obtained from the Resistor-Capacitor (RC) network, the Flow Restrictor – Volume/Vessel network, the Spring – Mass Damper system and Series LCR network respectively. The numbering of graphs are in respect to the numbering of figures as shown below.
2. Discussion
The experimental results of the RC showed some differences from the theoretical results. This was caused by the assumption that the system is perfect in theoretical calculation but in reality, there were some extra internal resistance from the oscilloscope which was not considered during the calculation.
With the aid of the LCR series network simulator, it was established that the response of the 2nd order system was the same as recorded in the Spring – Mass Damper system experiment.
The responses of these 1st and 2nd order systems were proven to be the same with additional aid from the solutions of their mathematical models. It can be concluded that all 1st and 2nd order systems will have such responses.
3. Conclusion
From the experiments, the period of 1 time constant for the 1st order system is deduced to be 64.1% of the final output. It is also illustrated that any 1st order systems irregardless of measuring parameters will have the same response, which applies to the 2nd order systems as well.
The stability of the 2nd order systems relies heavily on the damping ratio. If the ratio is set to be more than one, the system will be over damped but stable. It will take a longer time for the system to reach its steady state. But if the ratio is set to be lesser than one, the system will become under damped and unstable which may not also reach its steady state. In fundamental nature, it is important to set the damping ratio to a suitable value which makes the response of the system fast and stable. The ideal or perfect value of the damping ratio in most common cases is 0.699.
4. References
Dr C Lekakou, University of Surrey Engineering experiments Methods 2008-09
Dr Franjo Cecelja University of surrey 1st Semester Note
Franklin, G.F, Powell, J.D, Emami-Naeini, A, 2002, Feedback Control of Dynamic Systems 4th edition, Prentice Hall International
Appendix 1.1
Theoretical calculations of 2nd order system.
From Newton’s 2nd law:
M + C+ Kx(t) = f(t)
Taking Laplace Transform with initial conditions of zero gives:
Msx(s) + Csx(s) +Kx(s) = f(s)
Rearranging gives the Transfer function:
=
for =
which gives the general soulution:
Real distinct poles (>1), poles are and
x(t) = 1 + + ,
Real equal poles (=1), both poles are equal to -
x(t) = 1 – ,
Complex poles (0< < 1), poles are equal to -
x(t) = 1-,
where = and =
Appendix 1.2
Effect of damping ratio on the response of 2nd order system.
