On the dynamics of a Duffing oscillator with an exponential - PowerPoint PPT Presentation

On the dynamics of a Duffing oscillator with an exponential non-viscous damping model D J Wagg and S Adhikari Department of Aerospace Engineering, University of Bristol, Bristol, U.K. Email: S.Adhikari@bristol.ac.uk Non-viscous Duffing oscillator – p.1/15 Gran Canaria, Spain, September 2006

Contents of talk Motivation Review of linear systems The governing non-linear equation Computing solutions The effect of non-viscous damping Conclusions Non-viscous Duffing oscillator – p.2/15 Gran Canaria, Spain, September 2006

Linear non-viscous system The equation of motion: t � c µe − µ ( t − τ ) ˙ m ¨ u ( t ) + u ( τ ) d τ + k u ( t ) = f ( t ) (1) 0 Non-viscous Duffing oscillator – p.3/15 Gran Canaria, Spain, September 2006

Frequency domain representation d ( s ) u ( s ) = p ( s ) (2) where � � ω n d ( s ) = s 2 + s 2 ζω n + ω 2 (3) n sβ + ω n p ( s ) is the equivalent forcing function and � k c β = ω n ω n = m, ζ = , and µ . (4) √ 2 k m ω n : undamped natural frequency, ζ : viscous damping factor and β : non-viscous damping factor. Non-viscous Duffing oscillator – p.4/15 Gran Canaria, Spain, September 2006

Conditions for oscillatory motion 0.25 A β c =1/(3(3) 1/2 ) 0.2 Non−viscous damping factor: β C 2 D > 0 0.15 ζ U B 0.1 C 1 ζ c =1 (viscous) D < 0 ζ c =4/(3(3) 1/2 ) 0.05 ζ L 0 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 Viscous damping factor: ζ Critical values of ζ and β for oscillatory (periodic) motion. Non-viscous Duffing oscillator – p.5/15 Gran Canaria, Spain, September 2006

Frequency response function 12 5 4.5 β = 1 β = 1 10 4 β = 0.75 3.5 β = 0.75 8 β = 0.5 3 β = 0.5 |G (i ω )| |G (i ω )| 6 β = 0.25 2.5 2 β = 0 (viscous) β = 0.25 β = 0 (viscous) 4 1.5 1 2 0.5 0 0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Normalized frequency: ω / ω n Normalized frequency: ω / ω n 2.5 1.4 1.2 β = 1 2 β = 1 β = 0.75 1 β = 0.75 β = 0.5 1.5 β = 0.5 0.8 β = 0 (viscous) |G (i ω )| |G (i ω )| β = 0.25 0.6 1 β = 0.25 0.4 β = 0 (viscous) 0.5 0.2 0 0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Normalized frequency: ω / ω n Normalized frequency: ω / ω n (a) ζ = 0 . 1 (b) ζ = 0 . 25 (c) ζ = 0 . 5 (d) ζ = 1 . 0 Non-viscous Duffing oscillator – p.6/15 Gran Canaria, Spain, September 2006

Partial summary of new results A non-viscously damped oscillator will have oscillatory 4 1 motion if ζ < 3 or β > 3 . √ √ 3 3 1 If β < 3 , the oscillatory motion is possible if and only if √ 3 ζ / ∈ [ ζ L , ζ U ] . ζ L and ζ U are the lower and upper critical damping factors. If β > 1 / 4 , the natural frequency of a non-viscously damped oscillator will be more than that of an equivalent undamped oscillator. The amplitude of the frequency response function of a non-viscously damped oscillator can reach a maximum � √ √ � value if ζ < 1 5 − 1 or β > 1 3 3 − 4 . 2 2 Non-viscous Duffing oscillator – p.7/15 Gran Canaria, Spain, September 2006

Some References 1. Adhikari, S., AIAA Journal , Vol. 39, No. 5, May 2001, pp. 978–980. 2. Adhikari, S., AIAA Journal , Vol. 39, No. 8, August 2001, pp. 1624–1630. 3. Adhikari, S., ASCE Journal of Engineering Mechanics , Vol. 128, No. 3, March 2002, pp. 328–339. 4. Wagner, N. and Adhikari, S., AIAA Journal , Vol. 41, No. 5, 2003, pp. 951–956. 5. Adhikari, S. and Wagner, N., Transactions of ASME, Journal of Applied Mechanics , Vol. 70, No. 6, December 2003, pp. 885–893. 6. Adhikari, S. and Wagner, N., Computer and Structures , Vol. 82, No. 29-30, November 2004, pp. 2453–2461. 7. Adhikari, S., Proceedings of the Royal Society of London, Series - A , Vol. 461, No. 2059, July 2005, pp. 2269–2288. 8. Lei, Y., Friswell, M. I., and Adhikari, S., International Journal of Solids and Structures , Vol. 43, No. 11-12, 2006, pp. 3381–3400. 9. Adhikari, S., Transactions of ASME, Journal of Applied Mechanics , 2006, accepted. 10. Adhikari, S., Lei, Y., and Friswell, M. I., Transactions of ASME, Journal of Applied Mechanics , 2006, accepted. Non-viscous Duffing oscillator – p.8/15 Gran Canaria, Spain, September 2006

Equation of motion The governing equation is � ˆ τ =ˆ m d 2 x t τ ) d x τ + α 1 kx + α 2 kx 3 = A cos(Ωˆ µ e − µ (ˆ t − ˆ t 2 + c τ d ˆ t ) , d ˆ d ˆ τ =0 ˆ x : the displacement of mass m ; k : linear spring stiffness α 1 , α 2 : strength of linear and nonlinear spring stiffness c : viscous damping coefficient The non-viscous damping effect is represented by the parameter µ via the convolution integral. µ → ∞ implies viscous damping, i.e., classical Duffing oscillator Non-viscous Duffing oscillator – p.9/15 Gran Canaria, Spain, September 2006

The nondimensional equation The nondimensional governing equation is � t e − 1 β ( t − τ ) xdτ + α 1 x + α 2 x 3 = x 0 cos( ωt ) , x + 2 ζ ¨ ˙ β 0 We now define the integral term as � t e − 1 β ( t − τ ) y = xdτ ˙ β 0 Then by using the Leibniz rule for differentiation of an integral we can write y = 1 x − 1 ˙ β ˙ β y Non-viscous Duffing oscillator – p.10/15 Gran Canaria, Spain, September 2006

The first-order form We can then write a set of three first order ordinary differential equations x 1 = x 2 , ˙ x 2 = − 2 ζy − α 1 x 1 − α 2 x 3 ˙ 1 + x 0 cos( ωt ) , = 1 β x 2 − 1 y ˙ β y, Note that if we multiply through the last line by β , then as β → 0 , y → x 2 and the viscous damping case is obtained. Non-viscous Duffing oscillator – p.11/15 Gran Canaria, Spain, September 2006

Computing solutions 4th order Runge-Kutta integration algorithm Start at the lowest ω value Compute transient periods (typically 100–200) Max displacement recorded for 20–50 steady state periods Increase ω and repeat At max ω , reverse increment and the process continued to ω min Non-viscous Duffing oscillator – p.12/15 Gran Canaria, Spain, September 2006

Weak coupling: α 1 = 1 . 0 and α 2 = 0 . 05 12 12 maximum displacement per forcing period maximum displacement per forcing period 10 (a) beta=0 10 (b) beta=0.1 8 8 6 6 4 4 2 2 0 0 0.5 1 1.5 2 2.5 3 0.5 1 1.5 2 2.5 3 nondimensional frequency nondimensional frequency 16 16 maximum displacement per forcing period maximum displacement per forcing period 14 14 12 12 10 (c) beta=0.5 10 (d) beta=1.0 8 8 6 6 4 4 2 2 0 0 0.5 1 1.5 2 2.5 3 0.5 1 1.5 2 2.5 3 nondimensional frequency nondimensional frequency Non-viscous Duffing oscillator – p.13/15 Gran Canaria, Spain, September 2006

Strong coupling: α 1 = 0 and α 2 = 1 5 5 4 (a) 4 (b) 3 3 2 2 1 1 0 0 -1 -1 -2 -2 -3 -3 -4 -4 -5 -5 -4 -3 -2 -1 0 1 2 3 4 -3 -2 -1 0 1 2 3 8 6 (c) 6 4 (d) 4 2 2 0 0 -2 -2 -4 -4 -6 -8 -6 -4 -3 -2 -1 0 1 2 3 4 -4 -3 -2 -1 0 1 2 3 4 (a) Viscous case: β = 0 (b) β = 0 . 1 (c) β = 0 . 3 (d) β = 0 . 8 Non-viscous Duffing oscillator – p.14/15 Gran Canaria, Spain, September 2006

Conclusions Qualitative changes in dynamics have been observed Non-viscous Duffing oscillator – p.15/15 Gran Canaria, Spain, September 2006

Conclusions Qualitative changes in dynamics have been observed Many new features cannot be predicted (or institutively guessed) by ’simple extension’ of the classical results known for viscously damped systems Non-viscous Duffing oscillator – p.15/15 Gran Canaria, Spain, September 2006

Conclusions Qualitative changes in dynamics have been observed Many new features cannot be predicted (or institutively guessed) by ’simple extension’ of the classical results known for viscously damped systems More new dynamical features are yet to be discovered in the future ... this is far from over! Non-viscous Duffing oscillator – p.15/15 Gran Canaria, Spain, September 2006