List of figures

I.1 Motion of a particle 3

I.2 Circular motion 6

I.3 A disk rolling on a plane 8

I.4 Phase diagram from a linear harmonic motion 16

1.1 Simplified model of a machine 21

1.2 Harmonic motion of a single degree system and phase angle 23

1.3 Transfer function: forced vibration response 25

1.4 Phase angle between the input force and the mass displacement: forced vibration 26

1.5 Design chart for a soft support 28

1.6 A pair of gears represented as a system with four degrees of freedom 29

2.1 Phase diagram of a conservative nonlinear system 39

2.2 Sketch of a nonlinear pendulum 42

2.3 Phase diagram of a nonlinear pendulum =gl 43

2.4 Phase diagram of the Van der Pol equation with initial conditions within the limit cycle 44

2.5 Phase diagram of the Van der Pol equation with initial conditions outside the limit cycle 45

2.6 Wave form of the Van der Pol equation with initial conditions outside the limit cycle 45

2.7 Frequency spectrum representing the jump phenomenon 48

2.8 Acceleration function due to the friction force 52

2.9 Phase diagram of a self-excited vibration system 52

3.1 Time response of the example equation 3.15 61

3.2 Frequency spectrum of a linear system obtained with the Fourier Transform 61

3.3 The aliasing effect 62

3.4 The overlap concept in the STFT 66

3.5 Time-frequency map constructed with the Hamming function 67

3.6 Time-frequency map constructed with the Gaussian function (50% overlap) 67

3.7 Time-frequency map constructed with the Gaussian function (0% overlap) 68

3.8 Time-frequency map constructed with the Gaussian function (50% overlap) with a high frequency resolution 68

3.9 Time-frequency map produced with the Kaiser-Bessel function (0% ovelap) 69

3.10 Time-frequency map constructed with the Kaiser-Bessel function (50% overlap) 69

3.11 A Morlet mother function for a specific frequency for ß = 8. 73

3.12 Gaussian derivative for N = 2 74

3.13 Paul's function for N = 10 75

3.14 Flow diagram of the Wavelet Transform using the convolution theorem 76

3.15 Time-frequency map using Morlet's mother wavelet 77

3.16 Paul's mother function 78

3.17 Gaussian Derivative mother function with N = 20 78

3.18 Phase diagram of the transient response 82

3.19 Phase diagram of the steady state response 83

4.1 Schematic representation of a lumped-mass system 90

4.2 Actual measurements: mass supported by a rubber strip 91

4.3 Instrumented hammer 92

4.4 Frequency spectrum of an impact response: example of a gear transmission 94

4.5 Time-frequency map of the impact response: example of a gear transmission 95

4.6 Time variation of frequency response 1 96

4.7 Time variation of frequency response 2 96

4.8 Time variation of frequency of the coupling element 97

4.9 Frequency spectrum of the example 102

4.10 Variation of matrix H as a function of O (element H11) 104

4.11 Variation of matrix H as a function of O (element H22) 104

4.12 Variation of matrix H as a function of O (element H12) 105

5.1 Elements of a roller bearing 110

5.2 Schematic representation of the nonlinear stiffness of the rollers 110

5.3 Kinematics of a roller inside the bearing 111

5.4 Axial angle 112

5.5 Radial displacement of a shaft mounted on roller bearings 113

5.6 Mass-spring system for representing a shaft mounted on roller bearings 116

5.7 Free vibration response of a bearing system 118

5.8 Vibration response of a bearing system under an imbalance excitation (10 Hz) 119

5.9 Vibration response of a bearing system under an imbalance excitation (30 Hz) 121

5.10 Vibration response of a bearing system under an imbalance excitation (larger amplitude) 123

5.11 Vibration response of a bearing system with a failed roller 124

5.12 Vibration response of a bearing system with a cage failure 126

5.13 Vibration response of a bearing system with an inner ring failure 127

5.14 Vibration response of a bearing system with an outer ring failure 128

5.15 Vibration response of a bearing system with all excitation forces acting simultaneously 130

5.16 Contact ratio notation 134

5.17 Gear tooth geometry and stiffness parameters 135

5.18 Gear teeth stiffness as a function of the angular rotation 136

5.19 Four degrees of freedom lumped-mass model 138

5.20 Free vibration response of a gear system 141

5.21 Vibration response of a gear system under an unbalanced force rotating at 30 Hz 142

5.22 Vibration response of a gear system under an unbalanced force rotating at 60 Hz 143

5.23 Vibration response of a gear system under an unbalanced force rotating at 300 Hz 144

5.24 Vibration response of a gear system simulating a teeth defect (30 Hz) 145

5.25 Vibration response of a gear system under an unbalanced force rotating at 30 Hz 146

5.26 Stick and slip friction force scheme 147

5.27 Phase diagram of a one DOF friction model 151

5.28 Acceleration as a function of time 152

5.29 Frequency spectrum of a one DOF friction model 152

5.30 Time-frequency map of a one DOF friction model 153

5.31 Schematic representation of a rotor rubbing the casing 154

5.32 Frequency spectrum of a rotor rubbing the casing (measured data) 155

5.33 Time-frequency map of a rotor rubbing the casing 156

5.34 Time-frequency map of a rotor rubbing the casing (higher frequencies) 156

6.1 Accelerometers arrange for torsional...