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Modeling of a Hydraulic Braking System

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Modeling of a Hydraulic Braking System Modeling of a Hydraulic Braking System CHRISTOPHER LUNDIN Master’s Degree Project Stockholm, Sweden 2015 IR-EE-SB 2015:000 Abstract The objective of this thesis is to derive an analytical model representing a re- duced form of a mine hoist hydraulic braking system. Based primarily on fluid mechanical and mechanical physical modeling, along with a number of simplifying assumptions, the analytical model will be derived and expressed in the form of a system of differential equations including a set of static functions. The obtained model will be suitable for basic simulation and analysis of system dynamics, with the aim to capture the fundamentals regarding feedback control of the brake system pressure. The thesis will mainly cover hydraulic servo valve and brake caliper modeling including static modeling of brake lining stress-strain and disc spring deflection- force characteristics. Nonlinearities such as servo valve hysteresis, saturation, effects of under- or overlapping spool geometry, flow forces, velocity limitations and brake caliper frictional forces have intentionally been excluded in order not to make the model overly complex. The hydraulic braking system will be described in detail and basic theory that is needed regarding fluid properties and fluid mechanics will also be covered so as to facilitate the reader in his understanding of the material presented in this work. Overall, the scope of this thesis is broad and more work remains in order to complement the model of the system both qualitatively and quantitatively. Although not complete in its simplified form and with known nonlinearities aside, the validity of the model in the lower frequency domain is confirmed by results given in form of measurements and dynamic simulation. Static analysis of the brake caliper model is also verified to be essentially correct when comparing calculated characteristics against actual measurements, as is also the case for the static models of the brake lining and disc-spring characteristics. Contents 1 Introduction 2 1.1 Mine Hoists . .2 1.2 Hoist Brake System . .3 1.2.1 Brake Control System . .4 1.2.2 Brake Hydraulic System . .5 1.2.3 Brake Mechanical System . .5 1.3 Project . .7 1.3.1 Lab Environment . .7 2 The Hydraulic Braking System 10 2.1 The Hydraulic Power Unit . 10 2.2 The Hydraulic Control Unit . 11 2.3 Transmission Lines . 13 2.4 Disc Brake Calipers . 15 2.5 Reduced System . 16 3 Modeling of the Hydraulic Braking System 18 3.1 Fluid Properties . 18 3.1.1 Fluid Mass Density . 18 3.1.2 Fluid Bulk Modulus . 20 3.1.3 Effective Fluid Bulk Modulus . 25 3.1.4 Fluid Viscosity . 29 3.2 Fluid Mechanics . 32 3.2.1 Navier-Stokes Equations . 32 3.2.2 Bernoulli Equation . 35 3.2.3 Orifice Flow . 35 3.2.4 Pressure Dynamics in a Hydraulic Volume . 39 3.3 Hydraulic Components . 41 3.3.1 Servo Valve . 41 3.4 Mechanical Components . 54 3.4.1 Brake Caliper . 54 3.4.2 Disc Springs . 61 3.4.3 Brake Lining . 64 CONTENTS ii 3.4.4 Simplified Brake Caliper Model . 67 3.5 Brake Stand . 73 3.6 Dynamic Model . 74 4 Model Parameterisation, Simulation and Validation 77 4.1 Model Parameters . 77 4.1.1 Fluid Properties . 77 4.1.2 Brake Calipers . 78 4.1.3 Servo Valve . 80 4.2 Static Characteristics . 85 4.3 Disc Spring Force . 85 4.4 Pressure Dynamics . 88 4.4.1 Simulation . 88 5 Discussion and Conclusions 91 Bibliography 93 List of Figures 1.1 Double-skip friction hoist. .3 1.2 Brake control system. .4 1.3 Brake hydraulic system. .5 1.4 Brake caliper. .5 1.5 Brake stand. .6 1.6 Hydraulic power unit. .7 1.7 Hardware-In-the-Loop Simulation (HILS). .9 2.1 Hydraulic diagram. 11 2.2 Hydraulic diagram. 12 2.3 Valve manifold. 12 2.4 Piping. 14 2.5 Transmission line. 14 2.6 Modal approximation. 15 2.7 Hydraulic cylinder. 16 2.8 Reduced system. 17 3.1 Fluid stress and strain. 22 3.2 Fluid volume and pressure. 23 3.3 Fluid bulk modulus. 24 3.4 Container . 25 3.5 Effective fluid bulk modulus. 29 3.6 Shear stress and strain. 29 3.7 Shear flow. 30 3.8 Kinematic viscosity as a function of temperature. 32 3.9 Flow through an orifice. 36 3.10 Approximated discharge coefficient. 39 3.11 Hydraulic volume. 39 3.12 Servo valve symbol. 42 3.13 Servo valve. 42 3.14 Torque motor. 43 3.15 Spool. 43 3.16 Spool displacement. 44 LIST OF FIGURES iv 3.17 Valve flow. 45 3.18 Wheatstone bridge. 46 3.19 Flow as a function of current. 49 3.20 Saturation. 49 3.21 Blocking both actuator ports. 50 3.22 Pressure Sensitivity. 51 3.23 Load attachment. 52 3.24 Internal valve leakage. 53 3.25 Flow forces. 53 3.26 Caliper halve. 55 3.27 Model of the caliper yoke. 55 3.28 Hydraulic unit. 56 3.29 Model of the hydraulic unit. 57 3.30 Brake shoe. 57 3.31 Brake release. 58 3.32 Gap adjustment. 58 3.33 Model of the brake shoe. 59 3.34 Brake caliper model. 59 3.35 Spring stack. 61 3.36 Disc spring. 61 3.37 Stack deflection and force. 62 3.38 Stack deflection and spring rate. 63 3.39 Friction during compression. 64 3.40 Stress-strain characteristic. 65 3.41 Stress-strain characteristic and Young’s Modulus. 66 3.42 Simplified caliper model. 68 3.43 Brake caliper static characteristics. 72 3.44 Brake calipers. 73 4.1 Measurements and pressure sensitivity. 84 4.2 Measured characteristics. 86 4.3 Pressure response with different signal inputs. 89 4.4 Simulation. 90 List of Tables 4.1 Fluid Properties . 78 4.2 Brake Caliper Parameters . 78 4.3 Brake Lining Parameters . 79 4.4 Disc Spring Parameters . 79 4.5 Valve Rating . 80 4.6 Servo Valve Parameters . 83 4.7 Clamping Force and Air Gap Adjustment . 85 Chapter 1 Introduction 1.1 Mine Hoists Mine hoists are used for the purpose of vertical transportation of ore, personnel and equipment in underground mine shafts. With the use of a mine hoist in an underground mine, production may increase significantly as it is the most efficient way of both elevating ore to the surface and transporting personnel to and from the deep levels of a mine. Hoisting distances typically lie in the range between 500 and 1500 meters with payloads up to about 60 tons. Mechanically, a mine hoist may be configured differently depending on a number of factors, but.
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