Fractional Order Bagley- Torvik Mechanics (1)

1. Fractional Order Bagley-TorvikMechanics(1) Jiaguo Liu School of Math. and Statics., Shandong University, Mesa Lab, UC Merced, Ca liujiaguo@sdu.edu.cn, jliu68@ucmerced.edu 9/24/2013. Tuesday 09:00-11:15, KL217

2. This is an introduction to Bagley- Torvik's work on applications of fractional calculus to viscoelastic mechanics, which shows that the fractional calculus is a potentially powerful description of the viscoelastic phenomenon, and promoted greatly the development of fractional calculus and its applications.

3. Ronald L. Bagley, Ph.D. Professor Department of Mechanical Engineering The University of Texas at San Antonio Educational Background: Ph.D., Air Force Institute of Technology, Ohio. Areas of Research Interest: • Material Characterization • Engineering Mathematics Peter J. Torvik Current Consultant at Self (1996-) Past Professor at Air Force Institute of Technology Education Wright State University University of Minnesota- Twin Cities

4. Outline • Fractional viscoelastic models • Generalized derivative model for an elastomer damper • Fractional calculus approach to viscoelastically damped structures • Theoretical basis for the application of fractional calculus to viscoelasticity • Appearance of the fractional derivative in the behavior of real materials • Thermodynamic constrains on fractional models • Power law and fractional viscoelastic models • Thermorheologically complex material

5. Bagley-Torvik’s work list: • A generalized derivative model for an elastomer damper, Shock Vibr. Bull, 49(2), 135-143, 1979. • A theoretical basis for the application of fractional calculus to viscoelasticity, Journal of rheology, 27(3), 201-210. 1983. • Fractional calculus-A different approach to the analysis of viscoelastically damped structures, AIAA journal, 21(5), 741-748, 1983. • On the appearance of the fractional derivative in the behavior of real materials, Transaction of the ASME, 51: 294-298, 1984. • Fractional calculus in the transient analysis of viscoelastically damped structures, AIAA Journal, 23(6): 918-925, 1984. • On the fractional calculus model of viacoelastic behavior, Journal of rheology, 30(1): 133-155, 1986. • Power law and fractional calculus model of viscoelasticity, AIAA Journal, 28(10): 1412-1417, 1988. • The thermorheologically complex material, Int. J. Engng. Sci., 29(7): 797-806, 1991. • Fractional order calculus model of the generalized theodorsen function, J. Aircraft, 30(6): 1003-1005, 1992.

6. A generalized derivative model for an elastomer damper

7. 1.1 Preliminary knowledge • SDOF Oscillator • Viscoelasticity • Fourier Transform and Laplace Transform • Inverse Laplace Transform, Contour Integral

8. 1.1.1 SDOF Oscillator Fig. 1 Schematic of SDOF mass-spring-dashpot system. (b) Dashpot force is and opposes the direction of positive velocity. (1.1)

9. 1.1.2 Viscoelastic behaviors • Relaxation • Creep • Hysteresis

10. Relaxation

11. Creep

12. w t Hysteresis s e Elastic response eand s are synchronous! How about viscous response??

13. w t Viscous response s e dalay/2

14. Stress or strain p/2 p 2p 3p/2 0 wt Comparing d viscoelastic Ideal elastic 0  /2 Ideal viscous

15. Viscoelastic Models Fig. 2b Spring Fig. 2a FractionalSpring-pot (1.2) Fig. 2c Dashpot

16. Fig.3a Maxwell Fig. 3b Fractional Maxwell Model (1.3)

17. Fig.4a Kelvin-Voigt Fig. 4b Fractional Kelvin-Voigt (1.4)

18. Fig. 5a Zener Fig. 5b Fractional Zener (1.5)

19. Most general fractional viscoelastic model (1.6)

20. 1.2.1 Introduction For a given temperature, a typical elastomer (Fig. 6) exhibits: • At low frequencies, relatively low and frequency independent stiffness, and has relatively small damping that increases with frequency; • At high frequencies, has relatively high stiffness that is frequency independent and relatively small damping that decreases as frequency increases • At intermediate frequencies, has stiffness that increases with frequency and relatively high damping

21. Fig. 6 Typical elastomer properties

22. Complex modulus method The equation of motion in the complex frequency domain is (1.7) (1.8) which is also the equation for a SDOF oscillator with “structural damping”.

23. Complex modulus method’s advantages: • can yield acceptable results in some cases • can predict steady-state, sinusoidal responses that are in many cases in good agreement with experimental observations • some transient responses can be closely approximated

24. Complex modulus method’s inadequacies: • Non-causal response to impulsive loading • Its expression in the time domain is (1.9) which mixes time and frequency. Hence, any solution obtained with integral transforms using frequency as a parameter has no precise mathematical meaning.

25. Bagley and Torvik’s intention: to obtain a mathematical model for elastomeric materials with frequency dependent stiffness and damping properties that is free of mathematical contradictions.

26. 1.2.2 Generalized derivatives as constitutive relations A general constitutive relation for one dimensional deformation suggested by Caputo: (1.10) Here, is the stress, is the strain and and are parameters describing the material. When , it models materials properties that are strongly frequency dependent, which is not characteristic of the elastomer of interest.

27. Question 1: How to derive the order of model ?

28. The constitutive relation corresponding to a much weaker frequency dependence is adopted, (1.11) or (1.12) which is referred to as the RT model. (Rubbery region into the Transition region)

29. The next tasks is to demonstrate • The RT model produces a hysteresis loop; i.e., when , becomes sinusoidal? • The stress is physically realizable. Is the stress function a bounded and continuous?

30. The hysteresis of the RT model Let (1.13) then (1.14) By the triangle formula and Fourier cosine and sine transformations of , we have

31. (1.15) Thus, (1.16) Then, the existence of a hysteresis loop is established since the superposition of any number of out of phase sine waves can be expressed as a single sine wave.

32. The Laplace Transform of Eq. (2.8) produces Theorem Let be [a Laplace transform and ] any function of the complex variable that if analytic and order for all () over a half plane , where ; also let be real when . Then for all real , the [inverse transform is ] is a real valued function, …… Furthermore, is a continuous function of exponential order and when . L{}=+ (1.17)

33. Then is a continuousand causal. Notice that is sinusoidal for large, we conclude that is a boundedfor all time.

34. The total energy dissipated by a unit volume of material undergoing a homogeneous strain given by (1.13) is (1.18) The maximum energy stored at any point in the cycle is (1.19) where is the portion of the stress (1.16) which is in phase with the strain. U

35. The loss factor, (1.20) Question 2: How about the loss factor of fractional springpot?

36. 1.2.3 The RT model and damping The elastomer damper is shown in Fig. 7. (1.21) Fig. 7 A damper employing an elastomer undergoing pure shear strain (t)=x(t)+{x(t)}.

37. m+{x(t)}+x(t)=(t), Then the equation of motion for system shown in Fig.8 is (1.22) where (1.23) +K. Fig. 8 A SDOF oscillator using an elastomer damper

38. =1/2 = Particular case: Let , the impulsive loading response (1.24) =c+ +dr .

39. dr = . Although not readily apparent, is a continuous, real functionwhere for . (1.25) (1.26) .

40. d The displacement time history of the elastomer damped oscillator for sinusoidal loading applied at (1.27) (1.28) (1.29) d Re +Re.

41. Why can the RT model lead to correct predictions? • The response is causal • A sinusoidal input leads to a sinusoidal response • The stiffness (modulus) and damping are frequency dependent

42. 1.2.4 The RT model for 3M-467 Let (1.30) The values of the parameters, and were determined by choosing initial values match asymptotes and slopes and then making iterative changes to parameters until an acceptable fit was obtained. .

43. For a sinusoid strain history , (1.31) where . The loss factor (1.32) The predicted frequency dependent modulus, and frequency dependent damping, are compared to the initial data in Fig.4. The agreement is seen to be acceptable, both qualitatively and quantitatively. ,

44. Fig. 9 Material properties of at ℃ (USAF Materials Lab Test –Dec 77) with generalized derivative model superimposed.

45. 1.2.5. Conclusion • The generalized derivative is capable of modeling the frequency dependent stiffness and damping properties in the rubbery region and into the transition region of 3M-467. • The impulsive response of the generalized derivative model of elastomer damper is causal, superior to “structural damping” models. • The essential mathematical feature is that, it is a time domain model which properly predicts a frequency dependence.

46. Appendix: Residue theorem, contour integral, and inverse Laplace transform Cauchy’s Residue Theorem Let be a domain containing a simple loop and the points inside . Suppose that is meromorphic on with finitely many isolated poles at inside . Then . (A.1)

47. m+{x(t)}+x(t)=(t), Now we calculate the response function to an impulse loading, . The Laplace transform of Eq. (1.22), (1.22) for impulsive loading is (A.2) where . (A.3)

48. Then . (A.4) The inverse Laplace transform, ds, (A.5) is evaluated by using the residue theorem and integration contour given by Fig. 10.

49. Positive imaginary axis S Plane Positive real axis Fig. 10 Integration contour to evaluate the inverse Transform of the impulsive response

50. (A.6) where are the residues of the poles of enclosed by the closed contour, and . (A.7) = (A.8) Then . (A.9)