Mathematical Problems in Engineering

Volume 2015, Article ID 928152, 14 pages

http://dx.doi.org/10.1155/2015/928152

## Nonlinear Coupling Characteristics Analysis of Integrated System of Electromagnetic Brake and Frictional Brake of Car

School of Automobile and Traffic Engineering, Jiangsu University, Zhenjiang 212013, China

Received 2 February 2015; Revised 29 April 2015; Accepted 3 May 2015

Academic Editor: Francesco Braghin

Copyright © 2015 Ren He and Donghai Hu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Since theoretical guidance is lacking in the design and control of the integrated system of electromagnetic brake and frictional brake, this paper aims to solve this problem and explores the nonlinear coupling characteristics and dynamic characteristics of the integrated system of electromagnetic brake and frictional brake. This paper uses the power bond graph method to establish nonlinear coupling mathematical model of the integrated system of electromagnetic brake and frictional brake and conducts the contrastive analysis on the dynamic characteristics based on this mathematical model. Meanwhile, the accuracy of the nonlinear coupling mathematical model proposed above is verified on the hardware in the loop simulation platform, and nonlinear coupling characteristics of the integrated system are also analyzed through experiments.

#### 1. Introduction

The function of automotive active braking control system has been constantly improved since the 1990s. In order to eliminate the defects of the function, the active braking control system has developed rapidly in the direction of integration and intelligence [1, 2] while the brake-by-wire system might develop in the direction of active braking control system in the future. As a transitional product between the traditional brake system and brake-by-wire system, electronic hydraulic brake has attracted much attention from the automobile component manufacturers [3, 4]. EHB has eliminated the coupled relationship between the brake pedal and the driver without vacuum booster [5] which makes friction brake more simple and compact [6, 7]. EHB can also control braking force of each wheel separately and realize the maximization of the braking energy recovery easily. As a result, EHB is of special application value for electric vehicles [8].

Though these advantages of EHB are obvious, its study might be just in its infancy and there are still a lot of problems to be remedied. The EHB’s dynamic characteristics are closely related to gas pressure of the accumulator since the EHB applies the accumulator as the main pressure source. The large gas pressure of the accumulator makes the quick brake response possible, but it needs further improvement in precision control [9]. Though the high-speed solenoid valve of traditional ABS can maintain a higher reliability under the emergency brake, EHB’s constant braking under the traditional brake system would inevitably result in lower reliability. Compared with the traditional brake system, EHB has realized electronic control, but its nature has not changed as a friction brake. Friction brake might produce poisonous dust particles and harsh braking noise, because disc brake still uses the frictional force between the friction block and the brake disc to realize braking when the vehicle is running on the crowded road [10].

In order to address these problems, the electromagnetic brake might be an optimal alternative. The EMB is a kind of noncontact brake with faster response and easier control [11]. A calculation method of the magnetic induction intensity and braking torque of electromagnetic brake was put forward and the synovial variable structure controller was designed and antilock control experimental study was carried on in a miniature model of the EMB [12]. However, the EMB also has some disadvantages: EMB requires a larger exciting current, lacks failure safety mechanisms, and has a lower braking efficiency at low speeds. Considering advantages and disadvantages of these two brakes, some experts proposed that EMB and EHB be integrated to establish an integrated system of electromagnetic brake and frictional brake. In the bus equipped with the integrated system (ISEFB), the synovial nonlinear variable structure controller was developed by Sohel Anwar for the ABS/ESP control. Experimental results showed that the ISEFB had good braking performance on roads of low adhesion coefficient [13, 14].

In this paper, both the research purpose and the research significance of this integrated system of electromagnetic brake and frictional brake are explained systematically. Meanwhile, this paper also provides the theoretical foundation for the research on design and control of ISEFB through analyzing its nonlinear coupling characteristics and dynamic characteristics. In the second part, nonlinear coupling mathematical model related to ISEFB is established and the accuracy of the mathematical model and nonlinear coupling characteristics of ISEFB are analyzed and verified on the hardware in the loop simulation platform, respectively. Dynamic characteristics of ISEFB are discussed in the third part. In the case of emergency brake, the influences of the integrated system of electromagnetic brake and frictional brake on the design and the control strategy are analyzed in the last part.

#### 2. Modeling and Analysis of Nonlinear Coupling Characteristics of ISEFB

##### 2.1. Nonlinear Mathematical Model of EHB

###### 2.1.1. Bladder Accumulator

The bladder accumulator stores high-pressure braking fluid from the hydraulic pump and then conveys the fluid to the disc brake. When the bladder accumulator is connected to the hydraulic brake system, a certain length is available at the entrance of connecting pipeline. Because the connecting pipeline has a significant influence on the dynamic characteristics of the bladder accumulator, it is an indispensable part of the bladder accumulator. High-pressure braking fluid flows into the accumulator when the inlet pressure of connecting pipeline is higher than the gas pressure of bladder accumulator. The equation of the fluid pressure balance in the connecting pipeline can be written aswhere , , , , and represent working pressure of the hydraulic pump, gas pressure of the bladder accumulator, actual flow of the bladder accumulator, fluid resistance, and hydraulic inductance of the bladder accumulator, respectively.

Flow continuity equation in the connecting pipeline can be described aswhere is hydraulic capacitance of the bladder accumulator.

If the liquid flow direction is positive in flowing into the bladder accumulator, the hydraulic capacitance of the bladder accumulator can be expressed aswhere and are the initial stress and volume of the accumulator, respectively. is the polytropic process index of the gas of which in isothermal process and in the adiabatic process.

###### 2.1.2. High-Speed Solenoid Valve

High-speed solenoid valve, a primary control component of EHB, has two working conditions: open (oil outlet connected to drain port) and shut (oil outlet connected to oil inlet) [15]. The flow continuity equation of oil inlet and oil outlet can be expressed aswhere is discharge coefficient and in the turbulent state . , , and represent hydraulic pressure at oil inlet, outlet, and hydraulic chamber of high-speed solenoid valve, respectively. and are sectional area at oil inlet and outlet.

Since oil pressure is affected by oil compressibility in hydraulic cavity, the pressure balance equation can be obtained aswhere and represent bulk modulus of elasticity of the brake oil and hydraulic chamber volume of high-speed solenoid valve, respectively.

###### 2.1.3. Braking Pipe

The pressure of brake cylinder changes during braking as the pressure of braking pipe changes. Lumped parameter method is a main method used for establishing the mathematical model for dynamic characteristics of braking pipe. In this method, the braking fluid in braking pipe is treated as an integral whole, where pressure, velocity, and other physical quantities are considered to be the same everywhere in the whole line and the influence of liquid viscosity is neglected. In the case of laminar flow in braking pipe, the pressure balance equation and flow continuity equation for the braking pipe can be expressed aswhere , , and are fluid resistance, hydraulic inductance, and hydraulic capacitance of the braking pipe, respectively. and are pressure at oil inlet and outlet of the braking pipe. and are flux at oil inlet and outlet of the braking pipe.

###### 2.1.4. Brake Cylinder

The pressurization process of brake cylinder can be divided into two phases. One is the idle motion stage in which the air gap is eliminated between braking block and brake disc. And the other is the stable pressurization stage in which the braking lining and brake disc deformation work [16]. The piston of brake cylinder first overcomes the frictional force before moving up and down during the idle motion stage. In this stage, high-pressure braking fluid from the bladder accumulator is only employed to balance all kinds of resistance in the process of piston movement until air gap disappears. Without considering the compressibility of brake fluid, the flow continuity equation of brake cylinder during idle motion stage can be expressed aswhere , , and represent sectional area and displacement of the piston and flux of the braking cylinder, respectively.

The force situation and the piston movement of disc brake can be expressed approximately by quality-spring-damper system. If the initial displacement of piston is zero, there arewhere , , , , and represent weight of the piston, the damping coefficient of the piston moving in the braking cylinder, stiffness of the return spring, stiffness of the spring of brake disc, and pressure in the braking cylinder, respectively.

Oil compressibility and elastic deformation of brake disc and brake cylinder are treated as main factors in analyzing the flow characteristics of brake cylinder because the motion displacement of piston is less after the idle motion stage. If the flow direction of brake cylinder fluid is positive, the flow continuity equation of brake cylinder can be expressed aswhere is volume of the braking cylinder. is the equivalent bulk modulus of elasticity the braking cylinder.

Considering the stiffness of brake disc, the formula of can be expressed as

According to the pressure of brake cylinder, the braking torque of disc brake can be calculated as follows:where is the friction coefficient between the braking block and brake disc. is the effective radius of brake disc.

##### 2.2. Nonlinear Mathematical Model of EMB

Magnetomotive force (MMF) of EMB is composed of eddy current MMFs and excitation MMFs. Because both eddy current MMFs and excitation MMFs are AC excited, the analysis of EMB’s magnetic circuit becomes complicated [17]. Therefore, the magnetic circuit of electromagnetic brake in this paper is assumed as static magnetic circuit and an equivalent DC magnetic circuit. The simplified equation of magnetic circuit could be expressed aswhere is the excitation MMFs and . is the eddy current MMFs and . is magnetic field intensity in the air gap and . is the depth of the skin and . , , , , , , , , , , , and represent length of the air gap, magnetic field intensity of the iron core, length of the iron core, magnetic field intensity in the brake disc, coil turns, the exciting current, the magnetic flux, sectional area of the magnetic circuit, vacuum magnetic permeability of brake disc, relative permeability of brake disc, rotation speed, and resistivity of the brake disc, respectively.

The brake disc is mainly made of gray cast iron, and the main material of iron core is industrial pure iron. Both materials are soft magnetic materials. The relationship between magnetic induction intensity and applied magnetic field strength is saturation nonlinear as shown in Figure 1. Because saturation nonlinearity of soft magnetic materials is important to the performance of brake disc, it cannot simply be linearized during building the mathematical model of EMB.