 220 The Study on Nonlinear Model of Pantograph-Catenary Coupling System for

40 km/h, the contact forces of nonlinear model and linear model can reflect the lateral excitation of the finger plate. When the speed exceeds 40 km/h, only the nonlinear model can reflect the lateral excitation caused by finger plate. The nonlinear pantograph-catenary coupling dynamics model is more suitable to the straddle-type monorail pantograph-ca-tenary coupling system research.


Introduction
Straddle-type monorail is a new type form of urban rail transit system which has some advantages such as: small turning radius, low manufacturing cost, strong climbing ability and ride comfort [1,2]. Because of the unique driving principle of straddle-type monorail vehicles, the structure of atraddle-type monorail bogie is also special. The running part of the bogie consists of four running wheels, four guiding wheels and two stabilizing wheels, all of which are rubber tires. While running wheels provide the longitudinal movement, guide wheels lead the monorail vehicle along the track. Moreover, stabilization wheels prevent excessive rolling motion of the monorail. The catenary of straddletype monorail is rigid catenary, positive and negative catenary are set on the left and right side of the track beam respectively. Because the rubber tires of running wheel, guide wheel and stabilizer wheel are flexible. Therefore, the bogie is sensitive to lateral external effects such as track alignment change, turnout joint transition and cross wind during operation, which will lead the lateral vibration of the bogie.
Because of the unique driving principle of straddle-type monorail, the interaction between side mounted pantograph and the side catenary is also special. Therefore, the interaction between side mounted pantograph of the monorail and the side catenary is one of the factors that limits the operating safety of monorail. The working environment of straddle-type monorail vehicle pantograph is different from that of metro and high-speed railway, so it is necessary to study the pantograph modeling of straddletype monorail.
Massat [16] establishes three-dimensional rigid body model of pantograph using MSC software. Rauter [17] establishes a multi-rigid-body dynamic model of pantograph, the number of objects and the position of articulation in the model are close to the real body. Ambrosio [18] established a rigid-flexible hybrid model of pantograph. There is no research on pantograph modeling of straddle-type monorail. Therefore, research the accurate pantograph model suit for straddle-type monorail has great significance to improve the operation safety of straddle-type monorail.

Two-dimensional model of pantograph
Straddle-type monorail pantograph is mainly composed of a bottom plate, a lower frame, an upper frame, a connecting rod, a balance bar, a pantograph head seat, a pantograph spring, a pantograph head and a cylinder parts (positive pantograph only) and so on. The pantograph is reduced to the model shown in Fig.1 a. A is the hinge point between the lower frame and the bottom plate, B is the hinge point between the connecting rod and the bottom plate, C is the hinge point between the lower frame and the upper frame, D is the hinge point between connecting rod and the lower frame, E is the hinge point between balance bar and the connecting rod, F is the hinge point between balance bar and the pantograph seat, G is the hinge point upper frame bar and the bow head, K and P are the installation position of pantograph spring. Q1, Q2, Q3, Q4, Q5 are the gravity center of lower frame, upper frame, connecting rod, balance bar, bow head seat respectively. α, β, γ, δ are the angles between AC, BD, CD, AB and X axis respectively, ε is the angle between CD and CG, θ is the angle between FG and X axis. Fc is the contact force between pantograph and catenary. Since the lower frame is the active part, A is selected as the origin point of coordinate system, the direction of the monorail running is defined as the positive direction of the X axis, the yaw direction of the vehicle is defined as the Y axis, the vertical upward direction is the positive direction of the Z axis.

Differential equation of pantograph head
The motion differential equation of pantograph head component is established on the basis of Newton's second law [2][3][4][5] (2) where: T is the total kinetic energy of the frame system; V is the total potential energy of a frame system, V = 0.

Lower
LG where: TLower, TLG, TUpper, TUG and TH are the kinetic energy of the lower frame, the connecting rod, the upper frame, the balance bar and the pantograph head seat respectively, which be found in reference [19]. Because the pantograph of monorail vehicle is enveloped by the whole car body, the influence of aerodynamic force on Pantograph can be ignored. Considering the damping of the hinge, the dry friction moment, the damping force of the pantograph spring, the lifting bow moment and the force between the bow head and the frame, the virtual work done on the frame is as follows: In summary, the motion differential equation for the whole pantograph frame of straddle-type monorail can be obtained: The relevant variables in the formula can be found in the reference [19].
The pantograph can be simplified to a linear model with two degrees of freedom by expanding the differential equation of the pantograph frame at the equilibrium position and ignoring the influence of the higher order terms (Fig. 2).

Fig. 2 Linear model of pantograph
where: M is the torque of pantograph raising, which can be found in reference [19].

Differential equation of pantograph frame
According to the displaying center difference formula, the displacement of pantograph head: where: ZG is the displacement of point G at the top of frame. According to the geometric relationship, ZG can be represented by angle α and where: 0  is the angle between 1 L and the pantograph support for the initial position. 0  is the angle between 3 L and the pantograph support for the initial position.
According to the displaying center difference formula, the pantograph lift angle is: According to formulas (1) and (7)-(9), the pantograph head acceleration ZH(n) at n time can be obtained.
According to formulas (1) and (7)-(10), the force FH between bow head and frame can be expressed by n а . By introducing FH into Eq. (6), we can get a one-dimensional equation about n а . This is the differential equation of whole pantograph. where:

. Differential equation of catenary
The catenary and aluminum profile base are regarded as Euler beam; electric insulator is regarded as lumped mass spring model. The stiffness within the span of the catenary is obtained by using the finite element method. The stiffness of the catenary in one span is obtained by using the finite element method. In order to simplify the calculation, the continuous equivalent stiffness of the catenary is used to replace by the stiffness K (t): where: K0 is the average stiffness;  is nonuniformity coefficient of the stiffness; L is the span of the catenary;V is the speed of the monorail, t is time.
where: Kmax, Kmin are the maximum and minimum stiffness of the catenary in one span.

Dynamic model of pantograph-catenary system
The contact force between pantograph and catenary is provided by the pantograph spring. In this paper, penalty function is used to simulate the coupling between pantograph and catenary. When no offline occurs, the contact force is the product of catenary stiffness and pantograph-head displacement. combined the Differential equation of pantograph and Differential equation of catenary, the lateral coupling dynamic model of pantograph and catenary is obtained.
In the formula, () at and () at are acceleration vectors and velocity vectors of system nodes respectively, M is the mass matrix of the system; C is the damping matrix of the system; K is the stiffness matrix of the system; Q(t) is the load vectors of the system.

Lateral vibration model of bogie
When the straddle monorail is running on the track, the excitation of the finger plate on the side of the track will be caused the lateral vibration of the bogie, which will affect the quality of power collection. The lateral vibration model of the monorail can be seen in Fig.3.
where: Mb is the mass of bogie; Zb(t) is the installation tolerances of lateral finger plate; Kb is the equivalent lateral stiffness of Bogie, Cb is the equivalent lateral damping of Bogie.
where: KAS is the lateral stiffness of air spring; KOD is the lateral stiffness of oil-pressure damper; KRT is the lateral stiffness of running tire; KGT is the radial stiffness of guide tire; KST is the radial stiffness of steady tire.
where: CAS is the lateral damping of air spring; COD is the lateral damping of oil-pressure damper; CRT is the lateral damping of running tire; CGT is the radial damping of guide tire; CST is the radial damping of steady tire. Therefore, the lateral vibration acceleration of the monorail is as follow:

Analysis of the contact force
The maximum design speed of straddle monorail vehicle is 80 km/h. In order to evaluate the applicability of the two models, this paper analyzed the contact force response of two model with four different speeds of 20, 40, 60 and 80 km/h respectively.
The track of straddle-type monorail is PC beam which is 22 m long each beam and set finger plate at the joint of two beam. The lateral installation tolerances of finger plate are 3 mm. Therefore, when the vehicle passes through joint of two beam, the lateral vibration of the bogie will affect the contact force between pantograph and catenary. As can be seen from Figs. 4 and 5, two kinds of pantograph-catenary coupling models do not appear offline at all four speeds. At the speed of 20 and 40 km/h, the contact forces of the two models have periodic fluctuations and the fluctuation period is 22 m, which reflects the transverse excitation of the finger plate. As can be seen from Figs. 6 and 7, when the speed exceeds 40 km/h, only the nonlinear model can still reflect the lateral excitation of finger plate.
In order to more intuitively reflect the change of contact force between the two models with train running mileage, Fig. 8 shows the power collection quality evaluation index curve of the two models.    Table 1 The statistics of the power collection quality indexes of the pantograph-catenary system

Frequency response analysis of contact force
The spectrum analysis results of pantograph-catenary system are shown in Figs When the running speed is 80 km/h, the spectrograms of the nonlinear models have three obvious peaks which are 1.07, 3.92 and 11.93 Hz, the spectrograms of the linear models have just one obvious peak which is 4.05 Hz.
When the running speed is below 60 km/h, the peak frequency of linear model is almost the same as that of non-linear model. When the running is exceeding 60 km/h, the spectrograms of the linear model has deviated greatly from the nonlinear model. When the running speed is 80km/h, the spectrograms of the linear models just have one obvious peak, which can't well reflect the low-frequency vibration of the pantograph-catenary system. Therefore, when considering the lateral excitation of the finger plate, only the nonlinear model can better reflect the low frequency vibration characteristics of the pantograph-catenary system. Using M + P modal test system to test the modal of pantograph is seen in Fig.13 a and the test result is shown in the Fig.13 b. The first main frequency of the pantograph is 20.5 Hz, the second main frequency is 48.5 Hz, and the third main frequency is 13.5 Hz.
Analysis show that both models have the worst power collection quality when the running speed is 40 km/h. As can be seen from the test results of Fig. 13, b, the third main frequency of the pantograph is 13.5 Hz. It can be seen from Fig. 10 that when the running speed is 40 km/h, the nonlinear model exhibits a vibration with a frequency of 14.55 Hz, the linear model exhibits a vibration with a frequency of 13.11 Hz, which is closed to the third main frequency of the pantograph. The pantograph system resonates with the excitation of the bow head, which leads to the decrease of power collection quality. a b Fig. 13 Modal test of pantograph: a) model test process of pantograph; b) modal test result of pantograph

Conclusion
The linear and nonlinear dynamic models of straddle monorail pantograph considering the lateral vibration of bogie are derived based on the Lagrange equation. On this basis, the lateral coupling dynamic model of pantograph and catenary is established.
In order to evaluate the applicability of the two models, this paper analyzed the contact force response of two models with different speeds.
1. When the speed is below 40 km/h, the contact forces of the two models are in good agreement, which reflects the transverse excitation of the finger plate. When the speed exceeds 40 km/h, only the nonlinear model can reflect the excitation caused by finger profiles, while the linear model can't clearly reflect the excitation.
2. When the speed is below 60 km/h, both the nonlinear model and linear model can reflect the low-frequency vibration of pantograph-catenary system. When the speed exceeds 60 km/h, only the nonlinear model can reflect the low-frequency vibration of pantograph-catenary system.
3. When considering the lateral excitation of the finger plate, the pantograph system resonates with the excitation of the bow head, the pantograph-catenary system has the worst power collection quality when the running speed is 40 km/h. Therefore, when considering the lateral vibration of the bogie, the nonlinear pantograph-catenary coupling dynamics model is more suitable for straddle-type monorail pantograph-catenary coupling research.