Dynamic Characteristics on the Dual-Power State of Flow.pdf

Dynamic Characteristics on the Dual-Power State of Flow in Hydro-Mechanical Transmission Jibin Hu and Shihua Yuan Xiaolin Guo School of Mechanical and Vehicular Engineering Department of Automotive Engineering Beijing Institute of Technology Tsinghua University Beijing 100081, China Beijing 100084, China hujibin ijz is the mechanical path ratio. ip is the transmission ratio from gear Z22 to Z3. ihz is the conflux ratio of mechanical path. ihy is the conflux ratio of hydrostatic path. ib is the transmission ratio from gear Z5 to Z7. MTF1 is the variable displacement hydrostatic unit and can be describe as a variable gyrator. The modulus of the gyrator is decided by parameter qp of the signal generator. qm and qml stand for conversion gain coefficient of the fixed displacement hydrostatic unit, furthermore, qm qml=1. 1-junction is a co-flow node in which flow variables is equal. 0-junction is a co-effect node in which effect variables is equal. 10 20 19 18 17 16 15 1413 12 11 29 28 27 26 25 24 2322 21 9 8 7 6 5 4 3 2 1 59 58 57 56 50 55 553 52 51 48 47 46 45 44 43 42 49 41 4039 38 37 36 35 34 33 32 31 30 1 I Io R ?o MTF MTF1 010TF qm Sf no Se Tb R Rp C Cp I Igl C Cm R Rm 1 R ?fm R ?b I Ib I Im 10 C Co 0 C Cb R Rgl 1 TF ihy 1 01TF qm1 R Rdl I Idl TF io R ?fp I Ip 1 Se pdl TF ip 1 TF ib 0 0 1TF ihz R ?jz1 I Ijz1 C Cjz1 011TF ijz R ?jz3 C Cjz2 I Ijz3 I Ijz2 R ?jz2 qp Fig.2 Bond graph model of the HMT system ?0 is coefficient of viscous friction on input shaft (Ns/m). ?fp is coefficient of viscous friction counteracting the rotation of the variable displacement hydrostatic unit. ?fm is coefficient of viscous friction counteracting the rotation of the fixed displacement hydrostatic unit. ?b is coefficient of viscous friction on output shaft. Rgl is leakage fluid resistance of oil in high pressure hydrostatic loop (Ns/m5 ). Rdl is leakage fluid resistance of oil in low pressure hydrostatic loop. Rp is leakage fluid resistance of oil in the variable displacement hydrostatic unit. Rm is leakage fluid resistance of oil in the fixed displacement hydrostatic unit. ?jz1 is coefficient of viscous friction in drive shafting of the mechanical path transmission. ?jz2 is coefficient of viscous friction in driven shafting of the mechanical path transmission. ?jz3 is coefficient of viscous friction in conflux shafting. Co is coefficient of pliability of the input shaft (m/N). Cb is coefficient of pliability of the output shaft. Cp is the fluid capacitance of inner oil in the variable displacement hydrostatic unit (m5/N). Cm is the fluid capacitance of inner oil in the fixed displacement hydrostatic unit. Cjz1 is coefficient of pliability of the drive shafting of the mechanical path transmission. Cjz2 is coefficient of pliability of the driven shafting of the mechanical path transmission. Io is the moment of inertia of the input shaft. Ip is the moment of inertia of the variable displacement hydrostatic unit. Im is the moment of inertia of the fixed displacement hydrostatic unit. Ib is the moment of inertia of the output shaft. Igl is the fluid inductance in high pressure oil loop (Ns/m5). Idl is the fluid inductance in low pressure oil loop. Ijz1 is the moment of inertia of the drive shafting of the mechanical path transmission. Ijz2 is the moment of inertia of the driven shafting of the mechanical path transmission. Ijz3 is the moment of inertia of the conflux shafting. C. State equations of the HMT system Analyzing the dynamic characteristic of system using bond graph methods need to choose state variables of system reasonably and establish state equation of the system according to the known bond graph model of system. In a general way, the generalized momentum p of inertial unit and the generalized displacement of capacitive unit are introduced as state variables of system 510. If causalities of the bond graph are annotated according to principle of priority of the integral causality, some energy storage elements in bond graph maybe have differential causality on occasion. Under the circumstances, the amount of state variables of the system is equal to the counterpart of energy storage elements which have the integral causality. Energy variables of the energy storage elements which have the differential causality depend upon state variables of the system. These variables are dependent variables. Algebraic loop problem will occur while establishing state equation of 891 these kinds of bond graph. The bond graph model of the HMT system established as above belongs to these kinds. In Fig.2, energy variables in inertial elements Io, Ijz2 and Im have differential causalities. The resolution is to express the generalized momentum and the generalized displacement of energy storage elements which have the differential causality with involved state variables and to work out the first derivative of these equations toward time. The expressions of the inertial elements Io, Ijz2 and Im are derived as follows: 274 p I Iii p p opo ? ? (1) 11 1 2 15 p Ii I p jzjz jz ? (2) 43 1 49 p I qI p dl mm ? (3) Therefore, the amount of state variables of the HMT system is just 12: )( 2 tq , )( 9 tq , )( 11 tp , )( 18 tq , )( 20 tp , )( 27 tp , )( 31t q , )( 34 tp , )( 37 tq , )( 43 tp , )( 55 tq, )( 58 tp . The input state vector: U ?T bdlo Tpn?. According to the structural characteristics of the system shown by bond graph, the differentials of state variables can be describe as functions of state variables related to input variables. 12 state equations can be formulated as follows: 272 p I ii nq p po o? ? (4) 2711 1 9 1 p I i p I q p p jz ? (5) 18 21 11 2 11 21 2 9 11 11 11 q CiC p iIC i q CC p jzjzjzjz jzjzjz jz ? ? ? ? ? (6) 20 3 11 1 18 11 p I p Ii q jzjzjz ? (7) 5520 3 3 18 2 20 11 q Cii p I q C p bbhzjz jz jz ? ? ? (8) 27 2 22 9 12 2 2 27 p IC li q CC i q CC ii p p opofp jz p o po ? ? dl p p p p C qt q CC qt 2 31 2 )()(? ? (9) 34312731 11 )( p I q CR p I qt q glppp p ? ? ? (10) 37343134 11 q C p I R q C p mgl gl p ? (11) 43373437 111 p I q CR p I q dlmmgl ? (12) 43 2 3 2 37 3 43 1 p IqC Rq q CC p dlm dlmfm m ? ? ? ? dl bbhym p C q CiiqC 3 55 3 11 ? (13) 584320 3 55 111 p I p Iqii p Iii q bdlmbhyjzbhz ? (14) b b b b Tp I q C p? 585558 1? ? (15) Where, 1 2 1 2 1 1 jzjz jz Ii I C? , p opo I Iii C 22 2 1? , 2 3 1 mdl m qI I C? . III. SIMULATION RESULTS In these equations above, with the structural and calculative parameters of the known HMT system, dynamic simulation can be done in computer. In the process of simulation, initial values are given primarily. After the system stabilized, input signal is stimulated. Meanwhile, the results of dynamic response of the system are recorded. The response curves of the output speed of system and the oil pressure in main pipe of the bump-motor system under varied input signals are shown from Fig.3 to Fig.8. Fig.3 shows the pulsed response curves of the output speed and the oil pressure of the system as the load change instantaneously. The rising time of the oil pressure response is 22ms. The control time is 445ms. The overshoot is equal to 86%. Times (s) Fig. 3 Pulsed response of the system as load changing Pressure Output speed Speed response (rpm) Pressure response (MPa) 892 Fig.4 shows the pulsed response curves of the output speed and the oil pressure of the system as the speed changes instantaneously. The rising time of the speed response is 17ms. The control time is 479ms. The overshoot is equal to 65%. Times (s) Fig. 4 Pulsed response of the system as speed changing Fig.5 shows a group of slope response curves as the angle of swing plate of the variable displacement bump is a ramp excitation. In this figure, the ascending gradients of the angle of swing plate whose range is from 0 to its maximum (correspondingly, relative rate of changing displacement is from 0 to 1, i.e.10?) are assigned some values respectively, such as 50, 20, 8, 4 (corresponding rising time for ramp excitation are 0.04, 0.10, 0.25, 0.50s). The rising times of response of the output speed are 43, 108, 255, 505 ms. Overshoot are respectively 47%, 12%, 4%, 2%. Times (s) Fig. 5 Slope response of the system as angle of swing plate changing Fig.6 shows the pulsed response curves of the output speed and the main oil pressure of the system as the angle of swing plate changes instantaneously. The rising time of the speed response is 22ms. The control time is 420ms. The overshoot is equal to 73%. The bond graph model of the two range HMT system established by the author is a linear system. The results of simulation demonstrate that the speed of response of the system is quite fast and the stability is satisfactory, but the overshoot of step response is too large. On condition that the input signal is ramp type and the gradients is greater than 8 (the time interval in which the angle of swing plate changed from 0 to the maximum is not less than 0.25s), the transition process of the system whose overshoot will not exceed 5% will approach steady state. Times (s) Fig. 6 Pulsed response of the system as angle of swing plate changing The results of simulation indicated by Fig.3 Fig.6 is acquired on condition that the fluid capacitances Cm and Cp in the model denoted in Fig.2 are set to 0.0085. As other conditions are invariable, response curves indicated by Fig.7 and Fig.8 can be obtained for Cm and Cp are set to 0.0850. Fig.7 shows the slope response curves of the rotation speed and the pressure as the angle of swing plate changes on the principle of ramp excitation. The rising times of response of the output speed are 87, 121, 204, 519 ms. Overshoot are respectively 52%, 38%, 11%, 5%. Times (s) Fig. 7 Slope response of the system when Cm and Cp are set to 0.0850 Pressure Output speed Speed response (rpm) Pressure response (MPa) Pressure Output speed Speed response (rpm) Pressure response (MPa) Pressure Output speed Speed response (rpm) Pressure response (MPa) Pressure Output speed Speed response (rpm) Pressure response (MPa) 893 Fig.8 shows the pulsed response curves of the rotation speed and the pressure. The rising time of response of the output speed is 68ms. Overshoot is 57%. Compared with the results of simulation indicated in Fig.5 and Fig.6, the speed of response of the system is slowing down and the time interval needed to reach the steady state is delayed. At the same time, the number of oscillations of the response and fluctuating quantity of the pressure is decreasing. The overshoot of the pulsed response increased a little, but the overshoot of the slope response increased a bit as well. Times (s) Fig. 8 Pulsed response of the system when Cm and Cp are set to 0.0850 IV. CONCLUSIONS A bond graph model of the dual-power state of flow of the two ranges HMT system is established based on the bond graph theory. The model can be applied to simulate and study the dynamic characteristics of a hydro-mechanical transmission (HMT) system. On conditions that the displacement of the hydrostatic bump is constant, the system focused in this article can be simplified to a linear stationary system. On conditions that the displacement of the hydrostatic bump changes along with time, the system is a linear time varying system; the transition of the system approaches to stable state while the ramp input signal draws 8s. The value of the fluid capacitance in the hydrostatic system affects the dynamic response performance of the system. A further study on the influence of the fluid capacitance and the fluid resistance will be done. REFERENCES 1 X. Liu, Analysis of Vehicular Transmission System, Beijing: National Defense Industry Press, 1998, pp. 255-310. 2 D. Margolis, T. Shim, “ A Bond Graph Model Incorporating Sensors, Actuators, and Vehicle Dynamics for Developing Controllers for Vehicle Safety,” Journal of the Franklin Institute, Vol. 338, pp. 21-34, 2001. 3 M. Cichy, M. Konczakowski, “ Bond Graph Model of the IC Engine as an Element of Energetic Systems,” Mechanism and Machine Theor