An Introduction to Nonlinearity Mike Brennan (UNESP) Bin Tang (Dalian University of Technology, China) Gianluca Gatti (University of Calabria, Italy) An Introduction to Nonlinearity • Introduction • Historical perspective • Nonlinear stiffness • Symmetry • Asymmetry • Nonlinear damping • Conclusions Dynamical System Inertia force m + Damping force c + Stiffness force k = Excitation force Common stiffness nonlinearities Common damping nonlinearities Historical Perspective Historical Perspective Galileo Galilei, 1564–1642, studied the pendulum. He noticed that the natural frequency of oscillation was roughly independent of the amplitude of oscillation Christiaan Huygens, 1629–1695, patented the pendulum clock in 1657. He discovered that wide swings made the pendulum inaccurate because he observed that the natural period was dependent on the amplitude of motion, i.e. it was a nonlinear system. Historical Perspective Robert Hooke, 1635–1703, is famous for his law which gives the linear relationship between the applied force and resulting displacement of a linear spring. Isaac Newton, 1643–1727, is of course, famous for his three laws of motion. Historical Perspective John Bernoulli, 1667–1748, studied a string in tension loaded with weights. He determined that the natural frequency of a system is equal to the square root of its stiffness divided by its mass, n km Leonhard Euler, 1707–1783, was the first person to write down the equation of motion of a harmonically forced, undamped oscillator, my ky F sin t . He was also the first person to discover the phenomenon of resonance Historical Perspective Robert Hooke, 1678 F ky 9 years Isaac Newton, 1687 F my 63 years Leonhard Euler, 1750 my ky Fsin t Historical Perspective Hermann Von Helmholtz, 1821–1894, was the first person to include nonlinearity into the equation of motion for a harmonically forced undamped single degree-of-freedom oscillator. He postulated that the eardrum behaved as an asymmetric oscillator, such that the restoring force 2 was f k 12 y k y which gave rise to additional harmonics in the response for a tonal input. John William Strutt, Third Baron Rayleigh, 1842–1919, considered the free vibration of a nonlinear single-degree-of-freedom in which the force-deflection characteristic was symmetrical, given by 3 f k13 y k y The Duffing Equation 3 my cy k13 y k y Fcos t Georg Duffing 1861-1944 The Duffing Equation: Nonlinear Oscillators and their Behaviour Editors: Ivana Kovacic and Michael J. Brennan Published in 2011 Nonlinear stiffness Nonlinear Spring f f k stiffness x hardening spring x linear force f softening For a linear system spring f kx displacement x Symmetric Nonlinear Spring Force-deflection 3 Fs k13 y k y Non-dimensional 3 Fs y y 2 Fss F k1 l y y d0 k3 d 0 k 1 d0 =original length of spring Symmetric Nonlinear Spring 3 Fs y y Fys 3 Fs y y Note symmetry Asymmetric Nonlinear Spring Force-deflection 2 3 Fs k13 y ky2 k y Non-dimensional 2 3 Fs yy y k2 d 0 k 1 d0 =original length of spring Asymmetric Nonlinear Spring 23 23 Fs y y y Fs y y y linear 0 23 23 Fs y y y Fs y y y Asymmetric Nonlinear Spring 23 23 Fs y y y Fs y y y linear 0 23 23 Fs y y y Fs y y y Asymmetric Nonlinear Spring 23 23 Fs y y y Fs y y y linear 0 23 23 Fs y y y Fs y y y Asymmetric Nonlinear Spring 23 23 Fs y y y Fs y y y linear 0 Note Asymmetry 23 23 Fs y y y Fs y y y Pendulum – softening stiffness Stiffness Moment d 2 ml2 mglsin M cos t dt 2 35 sin .... 3! 5! 3 stiffness moment mgl 6 Softening Pendulum – softening stiffness 1 0.8 k 2 1 klin 2 k 0.6 klin 0.4 k 24 1 0.2 klin 2 24 0 0 20 40 60 80 90 angle (degrees) Pendulum – softening stiffness 30 60 120 170 Geometrically nonlinear spring Force-deflection d F21 ky 0 s 22 yd Non-dimensional d Fy1, s 2 y 1 Fss F2 kd y y d d do d d0 =original length of spring Geometrically nonlinear spring d F21 ky 0 s 22 yd which approximates to 3 Fs k13 y k y 3 k1021 k d d k30 k d d Geometrically nonlinear spring – non-dimensional which approximates to 3 Fs y y d Fy1, s 2 y 1 1 d d 2 Geometrically nonlinear spring – non-dimensional approximation actual Geometrically nonlinear spring – snap through Fs Geometrically nonlinear spring – snap through Potential Energy 2 V V2 kd 11 V1, d y24 dy 28 V Example of a snap-through system . Nonlinear Energy Harvesting Device – snap through . Nonlinear Energy Harvesting Device Nonlinear Energy Harvesting Device – snap through Positive Stiffness Negative Stiffness Beam Magnets Nonlinear Energy Harvesting Device – snap through 2 Stiff vertical spring 1.8 Optimal vertical spring Soft vertical spring 1.6 1.4 1.2 1 Force 0.8 0.6 0.4 0.2 0 0 0.5 1 1.5 2 Deflection Nonlinear Energy Harvesting Device – snap through 5 4 3 potential energy 2 1 0 -1 -2 1.05 1 0.5 0.95 0 0.9 -0.5 0.85 d displacment Vibration Isolation fe xt m kv c xe ft Vibration Isolation f e k kh h xt m l kv c xe ft An Achievable Stiffness Characteristic 0.3 0.2 0.1 0 stiffness force stiffness -0.1 -0.2 Low stiffness Normalised -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 Normalised displacement Bubble Mount Very low stiffness Static(natural equilibrium frequency) position Large Deflection of Beams Large deflection of beams 2w x2 M EI 3 , 2 w 2 1 x Causes nonlinear effect Softening 2 4w s 2 w 3 22 w w EI4 A 2 EI 2 2 f x, t x x t 2 x x x Large deflection of beams -in-plane stiffness 2 2w 42 wEAl w w A2 EI4 Ts dx 2 f(,) x t t x2l x x 0 Causes nonlinear effect Hardening Large deflection of beams -in-plane stiffness 2 2w 42 wEAl w w A2 EI4 Ts dx 2 f(,) x t t x2l x x Assume simple supports and first0 mode only x x sin w x,() t x q t f x, t F x l 2 cos t l dq2 so m k q k q3 Fcos t dt 2 13 k= 1 T l2 EI 2 EI 4 2 l 3 k 43 EA8 l 1 s 3 Asymmetrical Systems Bubble Mount Bubble Mount Asymmetry stiffnessk 3 k x2 stiffness 13 5 4.5 4 3.5 softening hardening 3 2.5 2 1.5 1 0 0.2 hardening0.4 0.6 0.8 1 hardening1.2 1.4 1.6 1.8 2 displacement Asymmetry 10 Symmetric system 3 Fs k13 y k y 5 F s 0 -5 -10 -2 -1 0 1y 2 Asymmetry 10 Asymmetric system 3 5 Fs F0 k 1 y k 3 y F s 0 -5 -10 -2 -1 0 1 y 2 Asymmetry 10 Asymmetric system 3 5 Fs F0 k 1 y k 3 y F s 0 ˆ ˆ23 ˆ -5 Fs k1 z k 2 z k 3 z -10 -2 -1 0 1 y 2 Asymmetric System - Cable Mode shapes Increasing tension dq2 m k q k q23 k q Fcos t dx2 1 2 3 Due to asymmetry -softening effect Nonlinear damping Nonlinear damping Nonlinear damping Nonlinear damping (a) Damping force fe cx2 m fx h d 22 O ax c 1 x ch k For xa/ 0.2 x2 fdh c x a2 a y ft Damping coefficient is dependent on the square of the amplitude Summary • Historical perspective • Nonlinear stiffness • Symmetry • Asymmetry • Nonlinear damping .