XXVI. ASR '2001 Seminar, Instruments and Control, Ostrava, April 26-27, 2001 Paper 77 Nonlinear Identification of Global Characteristic of Damping Car System VOTRUBEC, Radek Ing., TU Liberec, Hálkova 6, Liberec 1, 461 17, radek.votrubec@vslib.cz Abstract: This article deals with identification of absorption system of car. It s a springdamper system, which is consequently used to build up a quarter car model. Given method is possible to use for identification of general dynamic system. In the paper there is projected a new method of an identification of a damper force. This method will be a global characteristic that expresses a damper force as a function of a velocity and an acceleration of the relative motion of a piston rod and working cylinder instead of so called velocity characteristics with a hysteresis loop, that depends on a frequency. Keywords: damper, identification 1 Description of spring-damper system Absorption system consists of spring (3) attached by means of metal holders (2) to damper (4). Whole system is fixed to vehicle body by means of rubber silent blocks (1), Fig 1. Figure 1 - Spring-damper system Identification of stand alone parts of such system (separately spring and damper ) would require toilsome disassembly thus better way is to consider this system as whole and find general dependence of force F on length of system L and its derivations L, L, it means F = f ( L, L', L",...) (1) - 1 -
2 Identification method Given measured set of data which consists of output signal (force) and position (length of spring-damper system) as input signal we have to find dependence of force on length and it s derivations. If we find some function describing dependence of force on certain derivation (including zero derivation), we subsequently substract this force contribution from total force. This way we obtain suitable data for looking for dependence on other derivations. If we again find some dependence, this contribution is again substracted and we can continue in finding dependence on other derivations. This way we continue until remaining portion of force is negligible. Note that function describing particular dependences can be two or more variable function. 3 Identification of spring-damper system Identification consists in searching some dependence of force on position and it s derivations. Figure 2 shows measured data. On the top there is a position as a function of time. Below is force as a function of position, velocity and acceleration. Figure 2 Measured data Nonlinear characteristic of dependence of force on position can be determined from signals of very low frequencies and from step shaped signals. For results see Figure 3. - 2 -
Figure 3 Dependence of force on position Using found dependence we consequently substract position contribution (coming mainly from metal spring) from total force. See Figure 4 F(x ) and F(x ). Figure 4 - Measured data after position contribution substracted - 3 -
Now we can determine nonlinear function describing dependence of force on velocity, Fig. 5. Figure 5 - Dependence of force on velocity Further we substract force contribution coming from velocity and this way we get data as in Figure 6, F(x ). Figure 6 - Measured data after position and velocity contributions substracted - 4 -
Remaining part of force can be neglected. To build up a quarter car model found velocity and position dependencies of force are sufficient, because the acceleration on the car model doesn t too much usually. 4 Force as a two variable function of velocity and acceleration When examining signals of high frequencies or signals with no continuities in position or velocity where acceleration achieves high levels ( input signal - hydraulic drive movement changes rapidly ) raise of force can be observed, Fig.7. These hysteresis loops are caused with so called inertial part of force of damper. Note that position contribution is substracted in characteristic F(x ) and velocity contribution is substracted in F(x ) according to previous chapters. Figure 7 Signals with no continuities in velocity In these cases the fit could be rated as inaccurate, see Fig.8. Figure 9 shows corresponding parts of input and output signals. Extending of spring-damper system in blue, contraction in red. Deviations of polynomial fit to measured data correspond to situation when direction of movement changes and high levels of acceleration are achieved ( green and yellow parts of curve ). - 5 -
Figure 8 Polynomial fit of velocity dependence - detail Figure 9 Corresponding parts of input and output signal Attempts to find some dependence of force on acceleration failed. In such cases dependence of force can be considered as two variable functions of velocity and acceleration. - 6 -
It means that we can not substitute general dependence (1) with F = f ( L) + f 2 ( L' ) + f3 ( 1 L ") (2) but we have to consider dependence F = f1( L) + f 2 ( L', L") (3) Measured data after position contribution of force substracted are shown in Figure 10. Figure 10 Measured data after position contribution of force substracted So we try to describe this dependence by quadratic plain using least square error method. 2 2 Fi = a0 i L' + a1 i L" + a2i L' L" + a3i L' + a4i L" + a5i (4) Found plain is shown in Figure 11. - 7 -
Figure 11 Found quadratic plain Results of identification are shown in Figure 12 and Figure 13. Measured data in white, simulated data using 1D description in yellow and 2D simulated data in red. Figure 12 - Comparison of 1D and 2D description of spring-damper system - 8 -
Figure 13 Comparison of 1D and 2D description, detail after position contribution substracted 5 Conclusions There is shown a new method of an identification of a car damping system as a global dependence of force on a position, velocity and acceleration. Its result improves actual description of quarter car model. 6 References ŠKLÍBA, J. 2000. Some problem of hydraulic damper modeling, Colloquium Dynamics of Machines 2000, Praque 2000-9 -