Fast Fourier Transform

Classifications, applications, and design challenges of drones: A review

Vibration Control of Active Structures

Review of State of Art of Smart Structures and Integrated Systems

Chatter is a self-excited vibration that can occur during machining operations and become a common limitation to productivity and part quality. For this reason, it has been a topic of industrial and academic interest in the manufacturing sector for many years. A great deal of research has been carried out since the late 1950s to solve the chatter problem. Researchers have studied how to detect, identify, avoid, prevent, reduce, control, or suppress chatter. This paper reviews the state of research on the chatter problem and classifies the existing methods developed to ensure stable cutting into those that use the lobbing effect, out-of-process or in-process, and those that, passively or actively, modify the system behaviour.

Chatter in machining processes: A review

Preface to the third edition.- Preface to the second edition.- Preface to the first edition.- 1 Introduction.- 1.1 Active versus passive.- 1.2 Vibration suppression.- 1.3 Smart materials and structures.- 1.4 Control strategies.- 1.4.1 Feedback.- 1.4.2 Feedforward.- 1.5 The various steps of the design.- 1.6 Plant description, error and control budget.- 1.7 Readership and Organization of the book.- 1.8 References.- 1.9 Problems.- 2 Some concepts in structural dynamics.- 2.1 Introduction.- 2.2 Equation of motion of a discrete system.- 2.3 Vibration modes.- 2.4 Modal decomposition.- 2.4.1 Structure without rigid body modes.- 2.4.2 Dynamic flexibility matrix.- 2.4.3 Structure with rigid body modes.- 2.4.4 Example.- 2.5 Collocated control system.- 2.5.1 Transmission zeros and constrained system.- 2.6 Continuous structures.- 2.7 Guyan reduction.- 2.8 Craig-Bampton reduction.- 2.9 References.- 2.10 Problems.- 3 Electromagnetic and piezoelectric transducers.- 3.1 Introduction.- 3.2 Voice coil transducer.- 3.2.1 Proof-mass actuator.- 3.2.2 Geophone.- 3.3 General electromechanical transducer.- 3.3.1 Constitutive equations.- 3.3.2 Self-sensing.- 3.4 Reaction wheels and gyrostabilizers.- 3.5 Smart materials.- 3.6 Piezoelectric transducer.- 3.6.1 Constitutive relations of a discrete transducer.- 3.6.2 Interpretation of k2.- 3.6.3 Admittance of the piezoelectric transducer.- 3.7 References.- 3.8 Problems.- 4 Piezoelectric beam, plate and truss.- 4.1 Piezoelectric material.- 4.1.1 Constitutive relations.- 4.1.2 Coenergy density function.- 4.2 Hamilton's principle.- 4.3 Piezoelectric beam actuator.- 4.3.1 Hamilton's principle.- 4.3.2 Piezoelectric loads.- 4.4 Laminar sensor.- 4.4.1 Current and charge amplifiers.- 4.4.2 Distributed sensor output.- 4.4.3 Charge amplifier dynamics.- 4.5 Spatial modalfilters.- 4.5.1 Modal actuator.- 4.5.2 Modal sensor.- 4.6 Active beam with collocated actuator-sensor.- 4.6.1 Frequency response function.- 4.6.2 Pole-zero pattern.- 4.6.3 Modal truncation.- 4.7 Admittance of a beam with a piezoelectric patch.- 4.8 Piezoelectric laminate.- 4.8.1 Two dimensional constitutive equations.- 4.8.2 Kirchhoff theory.- 4.8.3 Stiffness matrix of a multi-layer elastic laminate.- 4.8.4 Multi-layer laminate with a piezoelectric layer.- 4.8.5 Equivalent piezoelectric loads.- 4.8.6 Sensor output.- 4.8.7 Beam model vs. plate model.- 4.8.8 Additional remarks.- 4.9 Active truss.- 4.9.1 Open-loop transfer function.- 4.9.2 Admittance function.- 4.10 Finite element formulation.- 4.11 References.- 4.12 Problems.- 5 Passive damping with piezoelectric transducers.- 5.1 Introduction.- 5.2 Resistive shunting.- 5.3 Inductive shunting.- 5.4 Switched shunt.- 5.4.1 Equivalent damping ratio.- 5.5 References.- 5.6 Problems.- 6 Collocated versus non-collocated control.- 6.1 Introduction.- 6.2 Pole-zero flipping.- 6.3 The two-mass problem.- 6.3.1 Collocated control.- 6.3.2 Non-collocated control.- 6.4 Notch filter.- 6.5 Effect of pole-zero flipping on the Bode plots.- 6.6 Nearly collocated control system.- 6.7 Non-collocated control systems.- 6.8 The role of damping.- 6.9 References.- 6.10 Problems ..- 7 Active damping with collocated system.- 7.1 Introduction.- 7.2 Lead control.- 7.3 Direct velocity feedback (DVF).- 7.4 Positive Position Feedback (PPF).- 7.5 Integral Force Feedback(IFF).- 7.6 Duality between the Lead and the IFF controllers.- 7.6.1 Root-locus of a single mode.- 7.6.2 Open-loop poles and zeros.- 7.7 Actuator and sensor dynamics.- 7.8 Decentralized control with collocated pairs.- 7.8.1 Cross talk.- 7.8.2 Force actuator and displacement sensor.- 7.8.3 Displacement actuator and force sensor.- 7.9 References.- 7.10 Problems.- 8 Vibration isolation.- 8.1 Introduction.- 8.2 Relaxation isolator.- 8.2.1 Electromagnetic realization.- 8.3 Active isolation.- 8.3.1 Sky-hook damper.- 8.3.2 Integral Force Feedback.- 8.4 Flexible body.- 8.4.1 Free-free beam with isolator.- 8.5 Payload isolation in spacecraft.- 8.5.1 Interaction isolator/attitude control.- 8.5.2 Gough-Stewart platform.- 8.6 Six-axis isolator.- 8.6.1 Relaxation isolator.- 8.6.2 Integral Force Feedback.- 8.6.3 Spherical joints, modal spread.- 8.7 Active vs. passive.- 8.8 Car suspension.- 8.9 References.- 8.10 Problems.- 9 State space approach.- 9.1 Introduction.- 9.2 State space description.- 9.2.1 Single degree of freedom oscillator.- 9.2.2 Flexible structure.- 9.2.3 Inverted pendulum.- 9.3 System transfer function.- 9.3.1 Poles and zeros.- 9.4 Pole placement by state feedback.- 9.4.1 Example: oscillator.- 9.5 Linear Quadratic Regulator.- 9.5.1 Symmetric root locus.- 9.5.2 Inverted pendulum.- 9.6 Observer design.- 9.7 Kalman Filter.- 9.7.1 Inverted pendulum.- 9.8 Reduced order observer.- 9.8.1 Oscillator.- 9.8.2 Inverted pendulum.- 9.9 Separation principle.- 9.10 Transfer function of the compensator.- 9.10.1 The two-mass problem.- 9.11 References.- 9.12 Problems.- 10 Analysis and synthesis in the frequency domain.- 10.1 Gain and phase margins.- 10.2 Nyquist criterion.- 10.2.1 Cauchy's principle.- 10.2.2 Nyquist stability criterion.- 10.3 Nichols chart.- 10.4 Feedback specification for SISO systems.- 10.4.1 Sensitivity.- 10.4.2 Tracking error.- 10.4.3 Performance specification.- 10.4.4 Unstructured uncertainty.- 10.4.5 Robust performance and robust stability.- 10.5 Bode gain-phase relationships.- 10.6 The Bode Ideal Cutoff.- 10.7 Non-minimum phase systems.- 10.8 Usual compensators.- 10.8.1 System type.- 10.8.2 Lead compensator.- 10.8.3 PI compensator.- 10.8.4 Lag compensator.- 10.8.5 PID compensator.- 10.9 Multivariable systems.- 10.9.1 Performance specification.- 10.9.2 Small gain theorem.- 10.9.3 Stability robustness tests.- 10.9.4 Residual dynamics.- 10.10References.- 10.11Problems.- 11 Optimal control.- 11.1 Introduction.- 11.2 Quadratic integral.- 11.3 Deterministic LQR.- 11.4 Stochastic response to a white noise.- 11.4.1 Remark.- 11.5 Stochastic LQR.- 11.6 Asymptotic behavior of the closed-loop.- 11.7 Prescribed degree of stability.- 11.8 Gain and phase margins of the LQR.- 11.9 Full state observer.- 11.9.1 Covariance of the reconstruction error.- 11.10Kalman-Bucy Filter (KBF).- 11.11Linear Quadratic Gaussian (LQG).- 11.12Duality.- 11.13Spillover.- 11.13.1Spillover reduction.- 11.14Loop Transfer Recovery (LTR).- 11.15Integral control with state feedback.- 11.16Frequency shaping.- 11.16.1Frequency-shaped cost functionals.- 11.16.2Noise model ..- 11.17References.- 11.18Problems.- 12 Controllability and Observability.- 12.1 Introduction.- 12.1.1 Definitions.- 12.2 Controllability and observability matrices.- 12.3 Examples.- 12.3.1 Cart with two inverted pendulums.- 12.3.2 Double inverted pendulum.- 12.3.3 Two d.o.f. oscillator.- 12.4 State transformation.- 12.4.1 Control canonical form.- 12.4.2 Left and right eigenvectors.- 12.4.3 Diagonal form.- 12.5 PBH test.- 12.6 Residues.- 12.7 Example.- 12.8 Sensitivity.- 12.9 Controllability and observability Gramians.- 12.10Internally balanced coordinates.- 12.11Model reduction.- 12.11.1Transfer equivalent realization.- 12.11.2Internally balanced realization.- 12.11.3Example.- 12.12References.- 12.13Problems.- 13 Stability.- 13.1 Introduction.- 13.1.1 Phase portrait.- 13.2 Linear systems.- 13.2.1 Routh-Hurwitz criterion.- 13.3 Lyapunov's direct method.- 13.3.1 Introductory example.- 13.3.2 Stability theorem.- 13.3.3 Asymptotic stability theorem.- 13.3.4 Lasalle's theorem.- 13.3.5 Geometric interpretation.- 13.3.6 Instability theorem.- 13.4 Lyapunov functions for linear systems.- 13.5 Lyapunov's indirect method ..- 13.6 An application to controller design.- 13.7 Energy absorbing controls.- 13.8 References.- 13.9 Problems.- 14 Applications.- 14.1 Digital implementation.- 14.1.1 Sampling, aliasing and prefiltering.- 14.1.2 Zero-order hold, computational delay.- 14.1.3 Quantization.- 14.1.4 Discretization of a continuous controller.- 14.2 Active damping of a truss structure.- 14.2.1 Actuator placement.- 14.2.2 Implementation, experimental results.- 14.3 Active damping generic interface.- 14.3.1 Active damping.- 14.3.2 Experiment.- 14.3.3 Pointing and position control.- 14.4 Active damping of a plate.- 14.4.1 Control design.- 14.5 Active damping of a stiff beam.- 14.5.1 System design.- 14.6 The HAC/LAC strategy.- 14.6.1 Wide-band position control.- 14.6.2 Compensator design.- 14.6.3 Results.- 14.7 Vibroacoustics: Volume displacement sensors.- 14.7.1 QWSIS sensor.- 14.7.2 Discrete array sensor.- 14.7.3 Spatial aliasing.- 14.7.4 Distributed sensor.- 14.8 References.- 14.9 Problems.- 5 Tendon Control of Cable Structures.- 15.1 Introduction.- 15.2 Tendon control of strings and cables.- 15.3 Active damping strategy.- 15.4 Basic Experiment.- 15.5 Linear theory of decentralized active damping.- 15.6 Guyed truss experiment.- 15.7 Micro Precision Interferometer testbed.- 15.8 Free floating truss experiment.- 15.9 Application to cable-stayed bridges.- 15.10Laboratory experiment.- 15.11Control of parametric resonance.- 15.12Large scale experiment.- 15.13 References.- 16 Active Control of Large Telescopes.- 16.1 Introduction.- 16.2 Adaptive optics.- 16.3 Active optics.- 16.3.1 Monolithic primary mirror.- 16.3.2 Segmented primary mirror.- 16.4 SVD controller.- 16.4.1 Loop shaping of the SVD controller.- 16.5 Dynamics of a segmented mirror.- 16.6 Control-structure interaction.- 16.6.1 Multiplicative uncertainty.- 16.6.2 Additive uncertainty.- 16.6.3 Discussion.- 16.7 References.- 17 Semi-active control.- 17.1 Introduction.- 17.2 Magneto-rheological fluids.- 17.3 MR devices.- 17.4 Semi-active suspension.- 17.4.1 Semi-active devices.- 17.5 Narrow-band disturbance.- 17.5.1 Quarter-car semi-active suspension.- 17.6 References.- 17.7 Problems.- Bibliography.- Index.

Vibration control of active structures : an introduction

Preface to the third edition- Preface to the second edition- Preface to the first edition- 1 Introduction- 11 Active versus passive- 12 Vibration suppression- 13 Smart materials and structures- 14 Control strategies- 141 Feedback- 142 Feedforward- 15 The various steps of the design- 16 Plant description, error and control budget- 17 Readership and Organization of the book- 18 References- 19 Problems- 2 Some concepts in structural dynamics- 21 Introduction- 22 Equation of motion of a discrete system- 23 Vibration modes- 24 Modal decomposition- 241 Structure without rigid body modes- 242 Dynamic flexibility matrix- 243 Structure with rigid body modes- 244 Example- 25 Collocated control system- 251 Transmission zeros and constrained system- 26 Continuous structures- 27 Guyan reduction- 28 Craig-Bampton reduction- 29 References- 210 Problems- 3 Electromagnetic and piezoelectric transducers- 31 Introduction- 32 Voice coil transducer- 321 Proof-mass actuator- 322 Geophone- 33 General electromechanical transducer- 331 Constitutive equations- 332 Self-sensing- 34 Reaction wheels and gyrostabilizers- 35 Smart materials- 36 Piezoelectric transducer- 361 Constitutive relations of a discrete transducer- 362 Interpretation of k2- 363 Admittance of the piezoelectric transducer- 37 References- 38 Problems- 4 Piezoelectric beam, plate and truss- 41 Piezoelectric material- 411 Constitutive relations- 412 Coenergy density function- 42 Hamilton's principle- 43 Piezoelectric beam actuator- 431 Hamilton's principle- 432 Piezoelectric loads- 44 Laminar sensor- 441 Current and charge amplifiers- 442 Distributed sensor output- 443 Charge amplifier dynamics- 45 Spatial modalfilters- 451 Modal actuator- 452 Modal sensor- 46 Active beam with collocated actuator-sensor- 461 Frequency response function- 462 Pole-zero pattern- 463 Modal truncation- 47 Admittance of a beam with a piezoelectric patch- 48 Piezoelectric laminate- 481 Two dimensional constitutive equations- 482 Kirchhoff theory- 483 Stiffness matrix of a multi-layer elastic laminate- 484 Multi-layer laminate with a piezoelectric layer- 485 Equivalent piezoelectric loads- 486 Sensor output- 487 Beam model vs plate model- 488 Additional remarks- 49 Active truss- 491 Open-loop transfer function- 492 Admittance function- 410 Finite element formulation- 411 References- 412 Problems- 5 Passive damping with piezoelectric transducers- 51 Introduction- 52 Resistive shunting- 53 Inductive shunting- 54 Switched shunt- 541 Equivalent damping ratio- 55 References- 56 Problems- 6 Collocated versus non-collocated control- 61 Introduction- 62 Pole-zero flipping- 63 The two-mass problem- 631 Collocated control- 632 Non-collocated control- 64 Notch filter- 65 Effect of pole-zero flipping on the Bode plots- 66 Nearly collocated control system- 67 Non-collocated control systems- 68 The role of damping- 69 References- 610 Problems - 7 Active damping with collocated system- 71 Introduction- 72 Lead control- 73 Direct velocity feedback (DVF)- 74 Positive Position Feedback (PPF)- 75 Integral Force Feedback(IFF)- 76 Duality between the Lead and the IFF controllers- 761 Root-locus of a single mode- 762 Open-loop poles and zeros- 77 Actuator and sensor dynamics- 78 Decentralized control with collocated pairs- 781 Cross talk- 782 Force actuator and displacement sensor- 783 Displacement actuator and force sensor- 79 References- 710 Problems- 8 Vibration isolation- 81 Introduction- 82 Relaxation isolator- 821 Electromagnetic realization- 83 Active isolation- 831 Sky-hook damper- 832 Integral Force Feedback- 84 Flexible body- 841 Free-free beam with isolator- 85 Payload isolation in spacecraft- 851 Interaction isolator/attitude control- 852 Gough-Stewart platform- 86 Six-axis isolator- 861 Relaxation isolator- 862 Integral Force Feedback- 863 Spherical joints, modal spread- 87 Active vs passive- 88 Car suspension- 89 References- 810 Problems- 9 State space approach- 91 Introduction- 92 State space description- 921 Single degree of freedom oscillator- 922 Flexible structure- 923 Inverted pendulum- 93 System transfer function- 931 Poles and zeros- 94 Pole placement by state feedback- 941 Example: oscillator- 95 Linear Quadratic Regulator- 951 Symmetric root locus- 952 Inverted pendulum- 96 Observer design- 97 Kalman Filter- 971 Inverted pendulum- 98 Reduced order observer- 981 Oscillator- 982 Inverted pendulum- 99 Separation principle- 910 Transfer function of the compensator- 9101 The two-mass problem- 911 References- 912 Problems- 10 Analysis and synthesis in the frequency domain- 101 Gain and phase margins- 102 Nyquist criterion- 1021 Cauchy's principle- 1022 Nyquist stability criterion- 103 Nichols chart- 104 Feedback specification for SISO systems- 1041 Sensitivity- 1042 Tracking error- 1043 Performance specification- 1044 Unstructured uncertainty- 1045 Robust performance and robust stability- 105 Bode gain-phase relationships- 106 The Bode Ideal Cutoff- 107 Non-minimum phase systems- 108 Usual compensators- 1081 System type- 1082 Lead compensator- 1083 PI compensator- 1084 Lag compensator- 1085 PID compensator- 109 Multivariable systems- 1091 Performance specification- 1092 Small gain theorem- 1093 Stability robustness tests- 1094 Residual dynamics- 1010References- 1011Problems- 11 Optimal control- 111 Introduction- 112 Quadratic integral- 113 Deterministic LQR- 114 Stochastic response to a white noise- 1141 Remark- 115 Stochastic LQR- 116 Asymptotic behavior of the closed-loop- 117 Prescribed degree of stability- 118 Gain and phase margins of the LQR- 119 Full state observer- 1191 Covariance of the reconstruction error- 1110Kalman-Bucy Filter (KBF)- 1111Linear Quadratic Gaussian (LQG)- 1112Duality- 1113Spillover- 11131Spillover reduction- 1114Loop Transfer Recovery (LTR)- 1115Integral control with state feedback- 1116Frequency shaping- 11161Frequency-shaped cost functionals- 11162Noise model - 1117References- 1118Problems- 12 Controllability and Observability- 121 Introduction- 1211 Definitions- 122 Controllability and observability matrices- 123 Examples- 1231 Cart with two inverted pendulums- 1232 Double inverted pendulum- 1233 Two dof oscillator- 124 State transformation- 1241 Control canonical form- 1242 Left and right eigenvectors- 1243 Diagonal form- 125 PBH test- 126 Residues- 127 Example- 128 Sensitivity- 129 Controllability and observability Gramians- 1210Internally balanced coordinates- 1211Model reduction- 12111Transfer equivalent realization- 12112Internally balanced realization- 12113Example- 1212References- 1213Problems- 13 Stability- 131 Introduction- 1311 Phase portrait- 132 Linear systems- 1321 Routh-Hurwitz criterion- 133 Lyapunov's direct method- 1331 Introductory example- 1332 Stability theorem- 1333 Asymptotic stability theorem- 1334 Lasalle's theorem- 1335 Geometric interpretation- 1336 Instability theorem- 134 Lyapunov functions for linear systems- 135 Lyapunov's indirect method - 136 An application to controller design- 137 Energy absorbing controls- 138 References- 139 Problems- 14 Applications- 141 Digital implementation- 1411 Sampling, aliasing and prefiltering- 1412 Zero-order hold, computational delay- 1413 Quantization- 1414 Discretization of a continuous controller- 142 Active damping of a truss structure- 1421 Actuator placement- 1422 Implementation, experimental results- 143 Active damping generic interface- 1431 Active damping- 1432 Experiment- 1433 Pointing and position control- 144 Active damping of a plate- 1441 Control design- 145 Active damping of a stiff beam- 1451 System design- 146 The HAC/LAC strategy- 1461 Wide-band position control- 1462 Compensator design- 1463 Results- 147 Vibroacoustics: Volume displacement sensors- 1471 QWSIS sensor- 1472 Discrete array sensor- 1473 Spatial aliasing- 1474 Distributed sensor- 148 References- 149 Problems- 5 Tendon Control of Cable Structures- 151 Introduction- 152 Tendon control of strings and cables- 153 Active damping strategy- 154 Basic Experiment- 155 Linear theory of decentralized active damping- 156 Guyed truss experiment- 157 Micro Precision Interferometer testbed- 158 Free floating truss experiment- 159 Application to cable-stayed bridges- 1510Laboratory experiment- 1511Control of parametric resonance- 1512Large scale experiment- 1513 References- 16 Active Control of Large Telescopes- 161 Introduction- 162 Adaptive optics- 163 Active optics- 1631 Monolithic primary mirror- 1632 Segmented primary mirror- 164 SVD controller- 1641 Loop shaping of the SVD controller- 165 Dynamics of a segmented mirror- 166 Control-structure interaction- 1661 Multiplicative uncertainty- 1662 Additive uncertainty- 1663 Discussion- 167 References- 17 Semi-active control- 171 Introduction- 172 Magneto-rheological fluids- 173 MR devices- 174 Semi-active suspension- 1741 Semi-active devices- 175 Narrow-band disturbance- 1751 Quarter-car semi-active suspension- 176 References- 177 Problems- Bibliography- Index

Vibration Control of Active Structures: An Introduction

Lagrangian dynamics of mechanical systems 1.1 Introduction 1.2 Kinetic state functions 1.3 Generalized coordinates, kinematic constraints 1.4 The principle of virtual work 1.5 D'Alembert's principle 1.6 Hamilton's principle 1.7 Lagrange's equations 1.8 Lagrange's equations with constraints 1.9 Conservation laws 1.10 More on continuous systems 1.11 References 2 Dynamics of electrical networks 2.1 Introduction 2.2 Constitutive equations for circuit elements 2.3 Kirchhoff's laws 2.4 Hamilton's principle for electrical networks 2.5 Lagrange's equations 2.6 References 3 Electromechanical Systems 3.1 Introduction 3.2 Constitutive relations for transducers 3.3 Hamilton's Principle 3.4 Lagrange's equations 3.5 Examples 3.6 General electromechanical transducer 3.7 References 4 Piezoelectric Systems 4.1 Introduction 4.2 Piezoelectric transducer 4.3 Constitutive relations of a discrete transducer 4.4 Structure with a discrete piezoelectric transducer 4.5 Multiple transducer systems 4.6 General piezoelectric structure 4.7 Piezoelectric material 4.8 Hamilton's principle 4.9 Rosen's piezoelectric transformer 4. 10 References 5 Piezoelectric laminates 5.1 Piezoelectric beam actuator 5.2 Laminar sensor 5.3 Spatial modal filters 5.4 Active beam with collocated actuator-sensor 5.5 Piezoelectric laminates 5.6 References 6 Active and Passive Damping with Piezoelectric Transducers 6.1 Introduction 6.2 Active strut, open-loop FRF 6.3 Active damping via 1FF 6.4 Admittance of the piezoelectric transducer 6.5 Damping via resistive shunting 6.6 Inductive shunting 6.7 Decentralized control 6.8 General piezoelectric structure 6.9 Self-sensing 6.10 Other active damping strategies 6.11 Remark 6.12 References Bibliography Index

André Preumont

Papers

Vibration control of active structures : an introduction

Vibration Control of Active Structures

Vibration Control of Active Structures: An Introduction

Mechatronics: Dynamics of Electromechanical and Piezoelectric Systems

A six-axis single-stage active vibration isolator based on Stewart platform