1. Nonlinear Structural Dynamics: The Fundamentals Tutorial Douglas Adams Distinguished Professor and Chair Civil and Environmental Engineering Professor of Mechanical Engineering Vanderbilt University International Modal Analysis Conference, Orlando, FL February 3, 2014

2. 2 Nonlinearity,friendorfoe? It is fair to say that nonlinearity is viewed as a detriment by most designers and experimentalists perhaps for good reason. 3. 3 Exhaust systems have a number of flexible elements that facilitate resonant response but only for increasing engine speed how is this possible? Exhaust system subject to powertrain excitation Decreasing Increasing f(t) Nonlinearityfoe 4. 4 Nonlinearityfriend If you hit a tuning fork twice as hard it will ring twice as loud but still at the same frequency. That's a linear response. If you hit a person twice as hard they're unlikely just to shout twice as loud. That property lets you learn more about the person than the tuning fork. Neil Gershenfeld, MIT Credit: Wikipedia 5. 5 Anexample: Findingdamageincompositesstructures Suppose you want to find a small defect in a large structure. How would you go about finding it? Bond, R., Underwood, S., Cummins, J., and Adams, D. E., Structural Health Monitoring-Based Methodologies for Managing Uncertainty in Aircraft Structural Life Assessment, Structural Health Monitoring, submitted for review. 6. 6 Youcouldtryvisuallyinspectingthematerial Is this sandwich panel damaged? Polypropylene honeycomb core sandwiched between two fiberglass face sheets. Core cracking Disbond 7. 7 Youcouldalsotrylinearvibrationmeasurements =c/f But local changes can only be detected using high frequency vibrations (100 kHz 1 in wavelength). The problem with high frequency vibration is that it is heavily damped (n=-2fn) preventing the effects of damage from being sensed in large structures. c=2740 m/s 8. 8 Nonlinearity,friendorfoe? Is there a different [nonlinear] way? Perform a mental exercise: push down on the material where there is damage and then pull back up again. Eric Dittman, Douglas E. Adams mage mage 5 4 0 eing F ig. 10 Example of motion in a disbonded panel that leads to bilinear stiffness Bending stiffness of face sheet Bending stiffness of face sheet + Stiffness of core 9. Damagemechanismsarenonlinearinnature 9 10. 10 Anexperimenttostudythesemechanisms Sandwich panel on vacuum surface to emulate single degree of freedom system.4 Eric F ig. 1 Testing Apparatus of Disbonded Aluminum Panels This form of the equation once again allows for easy Table 1 Primary Resonances Damage Size in mm Prima Undamaged 1161 25.0 467.5 63.5 359.5 101 296.0 Table 2 Testing Envelope of t Damage in mm Testing Fre 25.0 467.50 233.75 155.80 63.5 359.50 179.75 119.80 101 296.0 148.0 Dittman, E., and Adams, D. E., Identification of Cubic Nonlinearity in Disbonded Aluminum Honeycomb Panels using Single Degree-of-Freedom Models, Nonlinear Dynamics, submitted for review . M K? 11. 11 Sourcesofnonlinearbehaviorincompositematerialdamage The source of quadratic stiffness nonlinearity is obvious, but are their any cubic nonlinearities?Eric Dittman, Douglas E. Adams Values by Damage 3 ) R2 0.9828 0.9975 0.8672 Values by Damage s3 / m3 ) R2 0.8645 0.9884 0.9930 the stiffness being F ig. 10 Example of motion in a disbonded panel that leads to bilinear stiffness Table 8 Quadratic Stiffness and R-squared Values by Dam- cubic nonlinearity. 3 Establishment of a SDOF nonlinear model In prior work by Dittman and Adams [6], a disbonded aluminum honeycomb panel exhibited a strong nonlin- ear responsethat indicates a quadratic stiffness nonlin- earity was present. The response data also contained evidence of a cubic nonlinearity, although this nonlin- earity was not identied by the previous SDOF models that were proposed. Two new SDOF models are pro- posed to explain the behaviors of the system: u + 2 u + u2 + 2 N u3 + 2 0u = F cos(t) (1) u + 2 u + u2 + 2 N u3 + 2 0u = F cos(t) (2) 12. 12 Sourcesofnonlinearbehaviorincompositematerialdamage When the disbonded face sheet is free to vibrate, what are the effects of the strains in the adhesive?Eric Dittman, Douglas E. Adams F ig. 11 Simplied cross-section of a honeycomb panel show- ing how the epoxy llet binds the facesheet to the core cubic nonlinearity. 3 Establishment of a SDOF nonlinear model In prior work by Dittman and Adams [6], a disbonded aluminum honeycomb panel exhibited a strong nonlin- ear responsethat indicates a quadratic stiffness nonlin- earity was present. The response data also contained evidence of a cubic nonlinearity, although this nonlin- earity was not identied by the previous SDOF models that were proposed. Two new SDOF models are pro- posed to explain the behaviors of the system: u + 2 u + u2 + 2 N u3 + 2 0u = F cos(t) (1) u + 2 u + u2 + 2 N u3 + 2 0u = F cos(t) (2) 13. 13 Polymers P =s A, s = Ee If P- then A- this causes true stress s to increase, which means e and thus d and this creates a "hardening" spring Nonlinearstiffness 14. 14 Acloserlookatlinearspringswiththephaseplane We can use phase-plane diagrams as a means of studying free and forced vibrations. Displacement Velocity Rolling disk on an incline: M + ICM a2 x + Kx = 0 x2 1/ K + x2 1/ Meff = Constant Phase plane shape: Stiffness is constant in linear system 15. 15 Acloserlookatnonlinearspringswiththephaseplane We generate the phase plane by conserving the sum of the kinetic energy T(v) and potential energy V(x): 1 2 mv2 + V(x) = C OR v = 2C - 2V(x) M + ICM a2 x + Kx +mx3 = 0 16. 16 Phase plane diagrams reveal how stiffness (and damping) characteristics change with amplitude. - Which phase plane diagram contains nonlinearity? -1.5 -1 -0.5 0 0.5 1 1.5 -1.5 -1 -0.5 0 0.5 1 1.5 position velocity -1.5 -1 -0.5 0 0.5 1 1.5 -1.5 -1 -0.5 0 0.5 1 1.5 position velocity M + ICM a2 x + K +mx2 ( )x = 0 Whatistheeffectofnonlinearstiffnessonthefreeresponse? 17. 17 Plug x(t)=Acos(t) into Duffings equation to find out! -w2 AM cos wt( )+ AK cos wt( )= -mK Acos wt( ) 3 K -w2 M Acos wt( )= -mKA3 3 4 cos wt( )+ 1 4 cos 3wt( ) Mx+ Kx = -mKx3 If the stiffness [and damping] can change with amplitude, then what happens to the modal properties of the system? Use harmonic balance Howdoesthistranslateintochangesinnaturalfrequency? 18. 18 9 9.5 10 10.5 11 11.5 12 12.5 13 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 Frequency, [rad/s] Amplitude,A[m] w K M 1+ 3m 4 A2 Backbone curve Free oscillation frequency: Shootingatamovingtargetmodulationofnaturalfrequency 19. 19 in Hz -20 -15 -10 -5 0 5 10 15 20 0 Figure 2.1. Amplitude of the response of a cubic stiffness EOM excited at the primary resonance vs. detuning parameter. Coefficient values are 0 = 12, = 1, F = 5, and = 1. in Hz AmplitudeofResponseatPrimaryResonance -20 -15 -10 -5 0 5 10 15 20 0 0.05 0.1 0.15 0.2 0.25 Figure 2.2. Amplitude of the response of a cubic stiffness EOM If the n changes as a function of the amplitude of response, how does the frequency response change? Howdoesthistranslateintochangesinforcedresponse? Tracks the backbone 20. 20 Frequencyresponsedistortion Nonlinear systems respond at frequencies other than the excitation frequency, this results in distortion. Storer and Tomlinson, Mech. Systems and Signal Proc., 1993 ooooooo oooo ooooo oo oo o HHKH H HHKH H CjMK H 53,,,, 0,,, 3,, term(bilinear)tynonlineariquadraticnoistherebecause0, 1 1 5 1 2 5 4 1 3 13 2 21 tj o o eFyyKyCyM 2 1 tjnn oooon tj oooo tj ooo tj oo n o ooo eFH eFHeFHeFH tytytytyty ,,, ,,, )()()()()( 33 3 22 21 321 21. 21 Frequency [Hz] H1()H2(,)H3(,,) We can calculate and plot so-called higher order FRFs, for example, for the quarter car model. H2 -Peaks at half the linear resonance frequencies, indicating quadratic nonlinearity H3 -Peaks at one third the linear resonance frequencies, indicating cubic nonlinearity. n n n X nY H 1 2 Higher-orderfrequencyresponsefunctions 22. 22 Arenonlinearvibrationmeasurementssensitivetodamage? When the panel vibrates at two amplitudes, does the modal stiffness change or is it constant? According to the modal stiffness changes leading to a change in frequency response. w K M 1+ 3m 4 A2 23. Damage introduces nonlinear properties u F 3D vibrometer used to collect frequency response data from panel Nonlinear behavior in multi-amplitude frequency response data used to locate damage 200 300 400 500 600 700 800 0 0.005 0.01 0.015 0.02 0.025 Frequency(Hz) Magnitude(mm/s/N) Damage indices are created based on the magnitude and location of the nonlinear behavior 23 Usingamplitude-dependentnonlinearitytodetectdamage 24. 24 Testonvibratingfiberglasssandwichpanel Actuator Location (Front Side) Disbond Damage (Front Side) Measurement Area (Front Side) Fiberglass Panel Measurement Area (Back Side) Underwood, S., Plumlee, M., Adams, D., E., and Koester, D., Structural Damage Detection in a Sandwich Honeycomb Composite Rotor Blade Material Using Three-Dimensional Laser Velocity Measurements, 2009, Proceedings of the Annual Forum of the American Helicopter Society. 25. 25 Nonlinearityduetolocalstiffnesschangewithamplitude Difference between frequency response function measurement at high and low amplitudes. 5 10 15 20 1 4 7 10 -2 -1 0 1 2 x 10 -3 Grid X Grid Y Amplitude(mm/s/N) 26. 26 Takingacloserlookatacommonsourceofnonlinearity ML2 q +MgLsinq = 0 sinMgL MgLsin qo +Dq( )= MgL Dq - Dq3 3! + Dq5 5! - @ MgLDq qo 8 Eric Dittman, Douglas E. Adams F ig. 10 Example of motion in a disbonded panel that leads to bilinear stiffness Table 8 Quadratic Stiffness and R-squared Values by Dam- age Size Damage (mm) Quadratic Stiffness (N/ m2 ) R2 25 1,480*106 0.9510 63.5 460*106 0.9992 101 89.9*106 0.9125 ing nonlinearity. While previous work was able to sug- gest the existence of a quadratic stiffness nonlinearity and a cubic nonlinearity in a disbonded aluminum hon- eycomb panel, the data and analysis results in this pa- per have conrmed the quadratic stiffness nonlinearity as well as determined that the cubic nonlinearity is a stiffness nonlinearity. The quadratic stiffness nonlinear- ity can also be described as a bi-linear stiffness. This arises from the change in stiffness seen on the facesheet as it rises off the honeycomb core and then descends back to the core and presses into it, as seen in Figure 10. This action creates two distinct stiffness regimes, which can also be modeled as a quadratic stiffness. This behavior is most noticeable when the system is excited at one half the primary frequency. When the system is excited at the primary frequency, the response ampli- F ig. 11 Simplied cross-section of a honeycomb panel show- ing how the epoxy llet binds the facesheet to the core t is not as good as the damage size increases. There is also a scaling factor that has not been accounted for in this determination of , which is . From a comparison of the values of , is around 0.5. The physical behavior behind of the cubic stiffness is not yet isolated. Since it is a cubic phenomenon, the physical mechanism resists the displacement of the sys- tem equally against the positive and negative displace- ment. With the data t strengthening as the damage grows larger, it is thought that the epoxy may play a role. As the damage grows, more of the epoxy llets that bind the facesheet to the core, seen in Figure 11 become exposed to the vibratory input. It is thought that these exposed llets provide the basis of the cubic stiffness. 7 Summary A single degree-of-freedom model has been identied that dictates the motion of a disbonded aluminum hon- eycomb panel when excited at the primary resonance and thesuperharmonic frequencies. Using vibration data coefficients for the equation of motion were calculated using a least-squares method. The high R-squared val- ueson thecoefficientsgivestrong condencein both the model and the coefficients. The disbond exhibits both quadratic and cubic stiffness. These damage modes are most easily seen when the panel is excited at one-half and one-third the primary resonance, respectively. The quadratic stiffness arises from the two different stiff- 27. 27 Nonlinear systems detune their own resonance as they respond with different amplitudes. Pendulum resonates and slowly falls out of resonance Effectsofgeometricnonlinearityonfreeresponse 28. 28 he following form: u + 2 u + u3 + 2 0u = F cos(t) hod of Multiple Time Scales introduces Tn asthenth timescale. the other times scales are orders of so that T1 = t, etc. It is ponse of u could be rewritten as u = u0 + u1 and that the diff ritten asoperatorsof theform d dt = D0 + D1, where Dn is thede sion with respect to Tn. By keeping only order 0 and 1 terms, E tten as: 0 : D2 u + 2 u = 0 sintroducesTn asthenth timescale. Starting orders of so that T1 = t, etc. It is assumed en as u = u0 + u1 and that the differentials orm d dt = D0 + D1, whereDn isthederivative keeping only order 0 and 1 terms, Equation (2.5) 1u0 2D0u0 u3 0 + F cos(t). (2.6) u + 2 u + u3 + 2 0u = TheMethod of Multiple Time Scales introduces with T0 = t, the other times scales are orders of s that the response of u could be rewritten as u = u could berewritten asoperatorsof theform d dt = D0 of theexpression with respect to Tn. By keeping onl (2.4) is rewritten as: 0 : D2 0u0 + 2 0u0 = 0 1 : D2 0u1 + 2 0u1 = 2D0D1u0 2D0u Solving Equation (2.5) for u0 yields: u + u3 + 2 0u = F cos(t) (2.4) e Scales introduces Tn asthenth timescale. Starting es are orders of so that T1 = t, etc. It is assumed rewritten as u = u0 + u1 and that the differentials of theform d dt = D0 + D1, whereDn isthederivative Tn. By keeping only order 0 and 1 terms, Equation 0 (2.5) 2D0D1u0 2D0u0 u3 0 + F cos(t). (2.6) Method of MultipleTime Scales introduces Tn as the nth time = t, the other times scales are orders of so that T1 = t, etc e response of u could be rewritten as u = u0 + u1 and that t erewritten asoperatorsof theform d dt = D0 + D1, where Dn i xpression with respect to Tn. By keeping only order 0 and 1 t rewritten as: 0 : D2 0u0 + 2 0u0 = 0 1 : D2 0u1 + 2 0u1 = 2D0D1u0 2D0u0 u3 0 + F cos( Equation (2.5) for u0 yields: n t, the other times scales are orders of so that T1 = t, etc. It is assu esponse of u could be rewritten as u = u0 + u1 and that the differe ewritten as operators of theform d dt = D0 + D1, where Dn is thederiv ression with respect to Tn. By keeping only order 0 and 1 terms, Equ written as: 0 : D2 0u0 + 2 0u0 = 0 1 : D2 0u1 + 2 0u1 = 2D0D1u0 2D0u0 u3 0 + F cos(t). quation (2.5) for u0 yields: u0 = A(T1)e0T0 + A(T1)e 0T0 is the complex conjugate of A. uning parameter, , is introduced here. , with a value o understand how theresponseamplitudeand phasing ch y changes in the neighborhood of a known value. Upon etuning parameter = 0 + into Equation (2.6), thef ed: D2 0u1 + 2 0u1 = ( 20A 20A 3A2 A + 1 2 FeT A3 e30T0 + c.c ng = T1 , the equations are written with the detuning parameter, y as follows: a = a + F 20 sin() (2.1 a = a 3 80 a3 + F 20 cos(). (2.1 se equations are examined in the steady state to determine the response tation force. Instead of solving for a in terms of , the equation is solved f ms of a, the response amplitude: = 3 80 a2 1 a F 2 40 2a2. (2.1 sponse with respect to the detuning parameter is shown in Figures 2.1 a Now,letsaddaforceMethodofmultipletimescales Credit: E. Dittman, 2013 29. 29 Failureofthesuperpositionprinciple written as: 0 : D2 0u0 + 2 0u0 = 0 1 : D2 0u1 + 2 0u1 = 2D0D1u0 2D0u0 u3 0 + F cos(t). quation (2.5) for u0 yields: u0 = A(T1)e0T0 + A(T1)e 0T0 s the complex conjugate of A. ning parameter, , is introduced here. , with a value in Hz, allow o understand how theresponseamplitudeand phasing changeasthefo changes in the neighborhood of a known value. Upon substitution 30. 30 values are 0 = 12, = 1, F = 5, and = 1. in Hz AmplitudeofResponseatPrimaryResonance -20 -15 -10 -5 0 5 10 15 20 0 0.05 0.1 0.15 0.2 0.25 Figure 2.2. Amplitude of the response of a cubic stiffness EOM equations are examined in the steady state to determine the response to ion force. Instead of solving for a in terms of , the equation is solved for s of a, the response amplitude: = 3 80 a2 1 a F 2 40 2a2. (2.14) onse with respect to the detuning parameter is shown in Figures 2.1 and sponse exhibits a backbone curve, which indicates that the maximum am- the response moves away from the primary resonance with increasing force Shiftingresonance,multipleamplitudesofresponse 31. 31 A linear system is the sum of its parts: If y = f x( ) then f ax1 + bx2( ) = f ax1( )+ f by2( ) = af x1( )+bf x2( ) Linearsuperpositionprinciple 32. 32 In the linear equation of motion for a pendulum we expect to see one steady state response regardless of the initial conditions. Not-So-Simple Pendulum (cont.) 3.7 The steady state response of this linearized pendulum with damping at the excitation frequency is UNIQUE it does not depend on what the initial conditions are. Also note there is only one frequency in the response after the transient decays. 0 1000 2000 3000 4000 5000 6000 7000 -1 -0.5 0 0.5 1 TIME[s] THETA[rad] 0 1000 2000 3000 4000 5000 6000 7000 -1 -0.5 0 0.5 1 TIME[s] THETA[rad] t q t q Same amplitude for different ICs ML2 q +MgLq =t(t) attraction for the alternative responses. This type of primary resonance because it occurs near the linea 0 1000 -1 -0.5 0 0.5 1 THETA[rad] 0 1000 -1 -0.5 0 0.5 1 THETA[rad] q q 0.8 0.9 1 1.1 1.2 1.3 10 -2 10 -1 10 0 10 1 FREQUENCY[rad/s] MAGNITUDE[rad] 0.8 0.9 1 1.1 1.2 1.3 10 -2 10 -1 10 0 10 1 FREQUENCY[rad/s] MAGNITUDE[rad] w oq oq Linear Nonlinear Nonlinear Vibration D. E. Adams, 7/21/04 Whatweexpecttoseein[linear]pendulum 33. 33 In the actual equation of motion for a nonlinear pendulum we see two steady state responses when we are just beneath the linear resonance. ML2 q +MgLsinq =t(t) Not-So-Simple Pendulum (cont.) steady state response of this nonlinear system is NOT UNIQUE it s depend on what the initial conditions are; there are domains of action for the alternative responses. This type of response is called a mary resonance because it occurs near the linear resonance. 0 1000 2000 3000 4000 5000 6000 7000 -1 -0.5 0 0.5 1 TIME[s] THETA[rad] 0 1000 2000 3000 4000 5000 6000 7000 -1 -0.5 0 0.5 1 TIME[s] THETA[rad] t q t q 1 1.1 1.2 1.3 FREQUENCY[rad/s] 1 1.1 1.2 1.3 FREQUENCY[rad/s] w Linear Nonlinear 0 1000 -1 -0.5 0 THET 0 1000 -1 -0.5 0 0.5 1 THETA[rad] q 0.8 0.9 1 1.1 1.2 1.3 10 -2 10 -1 FREQUENCY[rad/s] MAGNIT 0.8 0.9 1 1.1 1.2 1.3 10 -2 10 -1 10 0 10 1 FREQUENCY[rad/s] MAGNITUDE[rad] w oq Linear Nonlinear Nonlinear VibrationD. E. Adams, 7/21/04 Whatweactuallyseeina[nonlinear]pendulum 34. 34 Thin aircraft panels can experience sonic fatigue, which is exacerbated by large deflections why? Credit: S.M. Spottswood, AFRL y K Due to bending only Stiffness y With stretching of neutral axis Stiffness Whymightthistypeofsteadystatesolutionposeaproblem? 35. 35 Response at mid point Response Excitation Credit: Increasing amplitude 36. 36 In a nonlinear system, we even see new resonances because of the interaction of the two components of the response (free and forced). ML2 q +MgLsinq =t(t) 6000 6020 6040 6060 6080 6100 -0.1 -0.05 0 0.05 0.1 TIME [s] THETA[rad] 6000 6020 6040 6060 6080 6100 -1 -0.5 0 0.5 1 TIME [s] THETA[rad] t t 0 1 2 3 4 5 6 10 -2 10 -1 10 0 10 1 10 2 10 3 Frequency [rad/s] MAGH()[rad/N-m] A o Superharmonic resonance Nonlinearresonancetheresmoresuper-harmonics 37. 37 In a nonlinear system, we even see new resonances because of the interaction of the two components of the response (free and forced). ML2 q +MgLsinq =t(t) 2400 2410 2420 2430 2440 2450 -1 -0.5 0 0.5 1 TIME [s] THETA[rad] 2400 2410 2420 2430 2440 2450 -0.02 -0.01 0 0.01 0.02 TIME [s] THETA[rad] t t 0 1 2 3 4 5 6 10 -2 10 -1 10 0 10 1 10 2 10 3 Frequency [rad/s] MAGH()[rad/N-m] A o Subharmonic resonance Nonlinearresonancesub-harmonicresonance 38. 38 Carbon Reinforced Fiberglass Youngs Modulus : 4.75~4.95 GPa Failure strength : 29.6~57.5GPa Howcanthislackofsuperpositionbebeneficial? 39. 39 Windenergytestbed Chen Fleeter Kim, S., Adams, D. E., Sohn, H., Rodriguez Rivera, G., Vitek, J., Carr, S., and Grama, A., Validation of Vibro- Acoustic Modulation of wind turbine blades for structural health monitoring using operational vibration as a pumping signal, 2013, Proc of the 9th Intl. Workshop on Structural Health Monitoring, Palo Alto, CA . 40. 40 Smartrotor 24 circuit slipring (12 chns) Installed in turbine nacelle Attached with a flexible coupling DC Triaxial Accelerometers and Macro Fiber Composite Actuators Anemometer Optical Tachometer Shaft and Bearing Assembly Slip Ring 41. Usinglackofsuperpositionfordetectionofcracking Sensor Actuator crack Structural vibration due to wind turbine operation Probing signal from actuator Sidebands from crack Pumping signal from shear & bending loads 41 42. Quadratic damping Linear damping 42 Most damping mechanisms are nonlinear but we describe them using equivalent viscous damping. Fluid drag damping: 0 KyyyyM Effective damping Letsnotforgetaboutdampingnonlinearities 43. 43 Oscillations take place in the vocal folds due to air flow allowing humans to vocalize how? Human vocal system Close-up of fundamental voice production system Credit: Lucero, TEMA Tend. Mat. Apl. Comput., 2005 Animportantclassofnonlineardampers 44. 44 Oscillations took place in the Tacoma Narrows Bridge in 1940 due to a similar phenomena. Koughan, J., The University of Texas at Austin, 1996 Astructuralexamplethemostinfamousoscillation 45. 45 Oscillations take place in reed vales in reciprocating compressors for similar reasons. Lets look closer. Piston in cylinder with reed valve to manifold Suction valve Cylinder wall Suction manifold Discharge manifold Discharge valve Control volume Piston Amechanicalexampleinvolvingnonlineardamping Park, Bilal and Adams, ASME JVA, 2008 46. 46 Steady state response of the reed valve. Acloserlookatthisnonlineardampingmechanism 47. 47 A simple nonlinear damping law, in the Van der Pohl oscillator, can be used to replicate this behavior. -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 -500 -400 -300 -200 -100 0 100 200 THETA [deg] a*(theta. 2 -b)[N-m-s] Nonlinear Damping Coefficient + POS - NEG Nonlineardampingrestoringforce 48. 48 Equation of motion for Van der Pohl oscillator: Energy change per cycle (for a periodic motion) = 0: q +c q2 -b( )q +wn 2 q = 0 q c q2 -b( )q0 2p wn dt = 0 Assume q = Acoswnt cA2 wn 2 A2 cos2 wnt - b( )0 2p wn sin2 wntdt = 0 49. 49 cA2 wn 2 A2 1 2 + cos2wnt 2 1 2 - sin2wnt 2 - b 2 + bsin2wnt 2 0 2p wn dt = 0 cA2 wn 2 A2 4 - b 2 0 2p wn dt = 0 pcwn A2 A2 2 - b = 0 A = 0,A = 2b So we conclude that oscillations take place with an amplitude of: And these oscillations are stable if: c > 0 A = 2b 50. 50 Reference textbooks Nayfeh and Mook, Nonlinear Oscillations, Wiley, 1979. Davis, Introduction to Nonlinear Differential and Integral Equations, Dover, 1998. There are a few big-picture things to remember when approaching analytical work in nonlinear systems: - Nearly all methods build upon the response of the corresponding linear system, i.e., nonlinearity = 0 - All methods require only trigonometry and solution of linear differential equations (sometimes integration too) - Must define the order of magnitude of nonlinear terms relative to linear terms (damping, forces) - Must solve for possible responses and then check the stability of those responses (route to steady state) Quickasideonnonlinearanalyticalmethods 51. LADSS What You Need to KnowNLs, July 19-20, 2010 52. 52 Nonlinearities are common in: - Some materials Plasticity, hysteresis - Flexible structures Geometry - Structures with joints and fasteners Friction and geometry - Structures with preload with buckling Multiple equilibria - Structures that interact with fluids Boundary condition - Every structure (over large operating range) aM F E Material properties Physics (forcing) Kinematics (motions) Sourcesofnonlinearity 53. 53 Nonlinearity is more difficult to understand - We use linear modes to understand linear systems; we can use normal forms to understand most nonlinearities. - Algebra, trig, and linear differential equations are needed. Nonlinearity is only important 5% of the time - Squeak & rattle and brake squeal only occur for certain operating conditions, but lead to serious warranty problems. - Small changes in analytical or experimental initial conditions can lead to major qualitative changes in dynamics. Nonlinearity can be treated like noise in our data - Chaos was believed to be random noise until 1961 but has since been shown to be a nonlinear phenomenon. - When systems are nonlinear, noise will never be Gaussian in nature and we need to be extra careful about estimation. 2 1 1 cos cos2 2 2 t t Nonlinearmyths(Jackson,1989) 54. 54 Nonlinearity is only important for large amplitudes - The effects of Coulomb (dry) friction due to rubbing are most apparent for small amplitudes of response. - Nonlinear resonance phenomena are often quite nonlinear for small response amplitudes (jump bifurcations). - A common practice in experimental dynamics is to add dither signals to the excitation to reduce the effects of nonlinearities. Nonlinear analysis is pointless because no closed- form solutions are available - Qualitative solutions are available and are often more useful for design purposes than closed-form solutions. - Approximate methods to first-order accuracy are readily available and accessible. Refer to Jackson (1989) for other myths in nonlinear dynamics. Nonlinearmyths(Jackson,1989) 55. 55 Reference textbook Worden and Tomlinson, Nonlinearity in Structural Dynamics, Taylor & Francis, 2001 There are a few big-picture things to remember when approaching experimental work in nonlinear systems: - Record time histories, not spectra - Do not average the data upfront, be careful of noise - Sample quite a bit faster than you think you need to - Remember that everything from the stiffness to the steady- state response can depend on amplitude and initial conditions - Harmonic (or slow swept sine) testing is generally better than random testing due to distortion - Must sufficiently observe (spatially) nonlinear elements. Experimentalguidelinesinnonlinearsystems 56. 56 Acceleration response of diaphragm with NL stiffness Aliasing of 3rd order harmonic Must ensure that sampling frequency is high enough to observe both the excitation frequency and the distortion, i.e., nonlinear harmonics, to avoid bias. Samplefastenough 57. 57 Donotoverlooktheobvious(removetimefromyourplots) Restoring Force Technique (or Force State Mapping) Masri and Caughey, Journal of Applied Mechanics, 1979. Haroon, Adams, and Luk, ASME JVA, 2005. M2 x2 = F x1 - x2 ,x1 - x2( ) x2 x1 xb 58. 58 3 1 3 cos cos 3 cos 4 4 o o ot t t tt oo 2cos 2 1 2 1 cos2 Ordinary and partial coherence measurements are a means of identifying sources of correlated noise. Bendat, Nonlinear Systems Techniques, Wiley, 1998. Donotoverlooktheobvious(usespectralcoherence) 59. 59 Timedatahasalottooffer,e.g.,freeresponsesteprelaxation Hilbert transforms can be used to identify modulation due to nonlinearity as a function of free response amplitude. Feldman, Mechanical Systems and Signal Proc., 1994 1 h y y t y t d t H 2 2 ( ) ( ) ( ) ( ) ( ) arctan ( ) h h A t y t y t y t t y t 03 KyKyyCyM 60. 60 Acknowledgethenon-stationarynatureofnonlinearsystems Time-frequency maps (spectrograms, wavelets, etc.) are also used to visualize non-stationary signals from the free and forced response. deytwtY j sp , x2 x1 xb 61. 61 Frequencyresponseconceptsarestilluseful When nonlinear feedback is considered, a new set of FRF-like expressions can be derived that account for nonlinearity in the feedback loop. Adams and Allemang, Journal of Sound and Vibration, 1999. Spottswood and Allemang, Journal of Sound and Vibration, 2006. X (w){ }= HL (w) m1 (w) HL (w) Bn1{ } mNn (w) HL (w) BnNn { } F(w){ } -Xn1 (w) -XnNn (w) Nonlinear generating functions Nonlinear components NL 0 1 2 3 4 5 6 10 -2 10 -1 10 0 10 1 10 2 10 3 Frequency [rad/s] MAGH()[rad/N-m] - Capable of MIMO NL modeling without need for large number of excitations but must sufficiently observe spatial response 62. 62 Butbecarefulabouthowyoutreatnoiseindata Which technique you use to estimate linear and nonlinear parameters determines the types of noise that can be accommodated. + kn ( ) 2 X F A w( ) + Reverse path Method (Bendat) - Noise on force NIFO Method (Adams) - Noise on response + kn ( ) 2 F X H w( )- Haroon, M. and Adams, D., E., A Modified H2 Algorithm for Improved Frequency Response Function and Nonlinear Parameter Estimation, 2008, JSV. 63. 63 There are many other noteworthy topics and articles for your consideration, including but not limited to: Staszweski and Chance, Wavelets, IMAC, 1997. Vinh and Liu, Modal analysis, IMAC, 1989. Wright and Hammond, FRF estimators, IMAC, 1991. Bedrossan and Rice, Volterra series, IEEE, 1971. Strogatz, Nonlinear dynamics, Addison-Wesley, 1994. And many many more Someotherresourcestoconsider