Linear and Nonlinear Instabilities in Mechanical Systems

HIROSHI YABUNO is Professor of Mechanical Engineering at University of Tsukuba in Japan. In 1990, he attained his Ph.D. in Engineering from Keio University, Japan and was appointed Professor at Keio University. He was a Visiting Scholar at the Virginia Polytechnic Institute and State University in 1997 and, in 2002 and 2008, he was a Visiting Professor at the University of Rome “La Sapienza”. He was also Chair of Working Party II (Dynamical Systems and Mechatronics) of IUTAM. He is an associate editor of international journals including Journal of Computational and Nonlinear Dynamics (ASME), Nonlinear Dynamics, Journal of Vibration and Control, and International Journal of Dynamics in Various Mechanical Systems and Control. His research interests include analysis and control of nonlinear dynamics and positive utilization of the nonlinear instability phenomena to mechanical systems in particular, micro/nano resonators.

• Preface 1References 81 Equilibrium States and their Stability 111.1 Equilibrium states 111.1.1 Spring-mass system 121.1.2 Magnetically levitated system 161.1.3 Simple pendulum 201.2 Work and potential energy 231.3 Stability of the equilibrium state in conservative systems 271.4 Stability of mechanical systems 291.4.1 Stability of spring-mass system 291.4.2 Stability of magnetically levitated system 311.4.3 Pendulum 321.4.4 Stabilization control of magnetically levitated system 32References 342 Linear Dynamical Systems 352.1 Vector field and phase space 352.2 Stability of equilibrium states 402.3 Linearization and local stability 412.4 Eigenvalues of linear operators and phase portraits in a single-degree-offreedom system 442.4.1 Description of the solution by matrix exponential function 442.4.2 Case with distinct eigenvalues 452.4.3 Case with repeated eigenvalues 492.4.4 Case with complex eigenvalues 542.5 Invariant subspaces 602.6 Change of stability due to the variation of system parameters 61References 673 Dynamic Instability of Two-Degree-of-Freedom-Systems 693.1 Positional forces and velocity-dependent forces 693.2 Total energy and its time-variation 713.2.1 Kinetic energy 713.2.2 Potential energy due to conservative force FK 723.2.3 Effect of velocity dependent damping force FD 763.2.4 Effect of circulatory force FN 783.2.5 Effect of gyroscopic force FG 81References 834 Modal Analysis of Systems Subject to Conservative and Circulatory Forces 854.1 Decomposition of the matrix M 864.2 Characteristic equation and modal vector 894.3 Modal analysis in case without circulatory force 904.4 Modal analysis in case with circulatory force 974.4.1 Case study 1: _i are real 1004.4.2 Case study 2: _i are complex 1034.5 Synchronous and nonsynchronous motions in a fluid-conveying pipe (video) 114References 1155 Static Instability and Practical Examples 1175.1 Two-link model for a slender straight elastic rod subject to compressive forces 1175.1.1 Static instability due to compressive forces 1175.1.2 Effect of a spring attached in the longitudinal direction 1225.2 Spring-mass-damper models in MEMS 1255.2.1 Comb-type MEMS actuator devices 1255.2.2 Cantilever-type MEMS switch 129References 1316 Dynamic Instability and Practical Examples 1356.1 Self-excited oscillation of belt-driven mass-spring-damper system 1356.2 Flutter of wing 1396.2.1 Static destabilization in case when the mass center is located in front of the elastic center 1456.2.2 Static and dynamic destabilization in case when the mass center is located behind the elastic center 1466.3 Hunting motion in a railway vehicle 1496.4 Dynamic instability in Jeffcott rotor due to internal damping 1616.4.1 Fundamental rotor dynamics 1616.4.2 Effects of the centrifugal force and the Coriolis force on static stability 1666.4.3 Effect of external damping 1706.4.4 Dynamics instability due to internal damping 1746.5 Dynamic instability in fluid-conveying pipe due to follower force 178References 1807 Local Bifurcations 1837.1 Nonlinear analysis of a two-link-model subjected to compressive forces 1847.1.1 Nonlinearity of equivalent spring stiffness 1847.1.2 Equilibrium states and their stability 1867.2 Reduction of dynamics near a critical point 1907.3 Pitchfork bifurcation 1967.4 Other codimension one bifurcations 1977.4.1 Saddle-node bifurcation 1977.4.2 Transcritical bifurcation 1997.4.3 Hopf bifurcation 2007.5 Perturbation of pitchfork bifurcation 2047.5.1 Bifurcation diagram 2047.5.2 Analysis of bifurcation point 2077.5.3 Equilibrium surface and bifurcation diagrams 2097.6 Effect of Coulomb friction on pitchfork bifurcation 2117.6.1 Linear analysis 2127.6.2 Nonlinear analysis 2147.7 Nonlinear characteristics of static Instability in spring-mass-damper models of MEMS 2177.7.1 Pitchfork bifurcation in comb-type MEMS actuator device 2187.7.2 Saddle-node bifurcation in MEMS switch 220References 2228 Reduction Methods of Nonlinear Dynamical Systems 2258.1 Reduction of the dimension of state space by center manifold theory 2268.1.1 Nonlinear stability analysis at pitchfork bifurcation point 2268.1.2 Reduction of nonlinear dynamics near bifurcation point 2298.2 Reduction of degree of nonlinear terms by the method of normal forms 2338.2.1 Reduction by nonlinear coordinate transformation: Method of normal forms 2338.2.2 Case in which the linear part has distinct real eigenvaules 2358.2.3 Nonlinear term remaining in normal form 2388.2.4 Reduction in the neighborhood of Hopf bifurcation point 240References 2469 Method of Multiple Scales 2479.1 Spring-mass system with small damping 2489.2 Introduction of multiple time scales 2519.3 Method of multiple scales 2539.4 Slow time scale variation of amplitude and stability of periodic solutions 256References 25610 Nonlinear Characteristics of Dynamic Instability 25910.1 Effect of nonlinearity on dynamic instability due to negative damping force 26010.1.1 Cubic nonlinear damping (Rayleigh type and van der Pol type) 26010.1.2 Self-excited oscillation produced through Hopf bifurcation 26110.1.3 Self-excited oscillation by linear feedback and its amplitude control by nonlinear feedback 26910.2 Effect of nonlinearity on dynamic instability due to circulatory force 27110.2.1 Derivation of amplitude equations by solvability condition 27210.2.2 Effect of cubic nonlinear stiffness on steady state response 278References 28111 Parametric Resonance and Pitchfork Bifurcation 28311.1 Parametric resonance of vertically-excited inverted pendulum 28411.1.1 Equation of motion 28411.2 Dynamics in case without excitation 28511.2.1 Dimensionless equation of motion subject to vertical excitation 28611.2.2 Trivial equilibrium state and its stability 29011.2.3 Nontrivial steady state amplitude and its stability 291References 29512 Stabilization of Inverted Pendulum under High-Frequency Excitation 29712.1 Equation of motion 29812.2 Analysis by the method of multiple scales 29912.2.1 Scaling of some parameters 29912.2.2 Averaging by the method of multiple scales 30012.3 Bifurcation analysis of inverted pendulum under high-frequency excitation 30212.3.1 Subcritical pitchfork bifurcation and stabilization of inverted pendulum 30212.3.2 Global stability of equilibrium states 30512.4 Experiments 30712.5 Effects of the excitation direction on the bifurcation 30812.5.1 Averaging by the method of multiple scales 30912.5.2 Excitation inclined from the vertical direction and perturbed subcritical pitchfork bifurcation 31012.5.3 Supercritical pitchfork bifurcation in horizontal excitation and its perturbation due to inclination of the excitation direction 31112.6 Stabilization of statically unstable equilibrium states by high-frequency excitation 311References 31213 Self-excited Resonator in Atomic Force Microscopy (Utilization of Dynamic Instability) 31513.1 Principle of frequency modulation atomic force microscope (FM-AFM) 31613.2 Detection of frequency shift based on external excitation 32213.3 Detection of frequency shift based on self-excitation 32513.4 Amplitude control for self-excited microcantilever probe 327References 32814 High-Sensitive Mass Sensing by Eigenmode Shift 33114.1 Conventional mass sensing by frequency shift of resonator 33214.2 High-sensitive mass sensing by coupled resonators 33314.3 Solution of equations of motion 33514.4 Mode shift due to measured mass 33614.5 Experimental detection methods for mode shift 33714.5.1 Use of eternal excitation 33814.6 Use of self-excitation 339References 34415 Motion Control of Underactuated Manipulator without State Feedback Control 34515.1 What is an underactuated manipulator 34515.2 Equation of motion 34615.3 Averaging by the method of multiple scales and bifurcation analysis 34815.4 Motion control of free link 35215.5 Experimental results 354References 35516 Experimental Observations 35916.1 Experiments of a single degree-of-freedom system (Chapters 2 and 6) 35916.1.1 Stability of spring-mass-damper system depending on the stiffness k and the damping c 35916.1.2 Self-excited oscillation of a window shield wiper blade around the reversal 36216.2 Buckling of a slender beam under a compressive force 36216.2.1 Observation of pitchfork bifurcation (sections 5.1 and 7.1) 36216.2.2 Observation of perturbed pitchfork bifurcation (section 7.5) 36316.2.3 Effect of Coulomb friction on pitchfork bifurcation (section 7.6) 36416.3 Hunting motion of a railway vehicle wheelset (section 6.3) 36516.4 Stabilization of hunting motion by gyroscopic damper (section 6.3) 36716.5 Self-excited oscillation of fluid-conveying pipe (section 6.5) 36816.6 Realization of self-excited oscillation in a practical cantilever (section 10.1.3) 36916.7 Parametric resonance (Chapter 11) 37316.8 Stabilization of an inverted pendulum under high-frequency vertical excitation (Chapter 12) 37416.9 Self-excited coupled cantilever beams for ultrasensitive mass sensing (section 14.6) 37516.10Motion control of an underactuated manipulator by bifurcation control (Chapter 15) 375References 376A Cubic Nonlinear Characteristics 379A.1 Symmetric and nonsymmetric nonlinearities 380A.2 Nonsymmetric nonlinearity due to the shift of the equilibrium state 381A.3 Effect of harmonic external excitation 383B Nondimensionalization and Scaling Nonlinearity 385B.1 Nondimensionalization of equations of motion 385B.2 Scaling of nonlinearity 389B.3 Nondimensionalization of the governing equation of a nonlinear oscillator 391B.4 Effect of harmonic external excitation 392References 394C Occurrence Prediction for Some Types of Resonances 395C.1 Dynamics of a linear spring-mass-damper system subject to harmonic external excitation 396C.1.1 Case with viscous damping 396C.1.2 Case under no viscous damping 399C.2 Occurrence prediction of some types of resonances in a nonlinear springmass-damper system 401References 405D Order Estimation of Responses 407D.1 Order symbol 407D.2 Asymptotic expression of solution 408D.3 Linear oscillator under harmonic external excitation 409D.3.1 Non-resonant case 410D.3.2 Resonant case 411D.3.3 Near-resonant case 411D.4 Cubic nonlinear oscillator under external harmonic excitation 412D.4.1 Large damping case ( = O(1)) 412D.4.2 Relatively small damping case ( = O(_2=3)) 413D.4.3 Small damping case ( = O(_)) 414D.5 Linear oscillator with negative damping 415D.6 Van der Pol oscillator 416D.6.1 Large response case (_0(_) = 1) 417D.6.2 Small but finite response case (_0(_) = o(1)) 417D.7 Parametrically excited oscillator 418D.7.1 Large damping case ( = O(1)) 419D.7.2 Small damping case ( = O(_)) 420D.7.3 Case with cubic nonlinear component of restoring force 422D.7.4 Near-resonant case 423References 425E Free Oscillation of Spring-Mass System under Coulomb Friction and its Dead Zone 427E.1 Characteristics of friction 427E.2 Free oscillation under Coulomb friction 429E.3 Variation of the final rest position with decrease in the stiffness 434References 436F Projection by Adjoint Vector 439G Solvability Condition 441G.1 Kernel and image of linear transformation 441G.2 Solvability condition 443H Effect of Contact Force on the Dynamics of Railway Vehicle Wheelset 451H.1 A slip at the contact point of rolling disk on a plane 452