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  1. Fractional Order Bagley-TorvikMechanics(2) Jiaguo Liu School of Math. and Statics., Shandong University, Mesa Lab, UC Merced, Ca, 9/26/2013. Thursday 09:00-11:15, KL217

  2. Fractional Calculus Approach to the Analysis of Viscoelastically Damped Structures

  3. 2.1 Introduction The successful determination of the response of a visco-elastically damped structure to prescribed loading time histories hinges on the successful solution of two problems: • Describing the viscoelastic material’s mechanical properties in a mathematically rational manner • Solving the resulting equations of motion for the structure.

  4. Yes. Fractionalderivative models can do it. Fractionalderivative stress-strain constitutive relations for viscoelastic solids • not only describe the mechanical properties of some materials, • but lead to closed-form solutions of the finite element equations of motion for viscoelastically damped structures.

  5. Popular approaches at that time: • Complex constant as material modulus (restricted to sinusoidal motion of material, or else leads to serious mathematical problems) • Numerical methods in the transform domain (cumbersome, substantial effort required to calculate numerically the transform inversion integral) • Standard linear viscoelastic model (cumbersome, effort to obtain the eigenvalues and eigenvectors of the equations of motion)

  6. We will show thatTheformulation of fractional equations of motion also produces higher order matrix equations to solve. Comparing to the standard linear viscoelastic model the fractional derivative model requires only five empirical parameters.

  7. 2.2 The fractional viscoelastic model The general form of fractional derivative viscoelastic model is (2.1) Experimental observations indicate that many viscoelastic materials can be modeled by a viscoelastic model with five parameters, and . (2.2) . .

  8. By using the relation =, taking the Fourier transform of (2.2), yields (2.3) then (2.4) .

  9. Fig. 1 Mechanical properties of Corning 10.

  10. Figure 1 demonstrates the excellent agreement between the model and the mechanical properties at 550 ℃ of a Corning glass doped with oxides of aluminum, sodium, and cobalt ( 7.5% , 3% , 1% ). The parameters are which are determined by a least squares fit of this model to the frequency-dependent mechanical properties of material.

  11. , The form of Laplace transform reads (2.5) where (2.6) The modulus is used to construct the stiffness matrices of the viscoelastic elements in the damped structure. In the following derivations, it is assumed that a single viscoelastic material undergoing uniform, uniaxial stress (strain) is used to damp the structure. .

  12. 2.3 The construction and solution of the equation of motion The cornerstone of any finite element approach is the construction of the stiffness matrices for each of the finite elements in the structure. The stiffness matrix for each elastic finite element can be constructed by • assumed displacement methods-----to be used. • assumed stress methods-----( not consistent with the assumption that stress be dependent on the local strain history).

  13. The stiffness matrix of a viscoelastic finite element is constructed using the elastic-viscoelastic correspondence principle, and is seperated into two matrices, the terms proportional to the Lamé constant the terms proportional to the Laméconstant . (2.7) The development is limited to the consideration of uniform, uniaxial shear strain, then the stiffness matrix takes the form (2.8) =. =.

  14. =. Substituting (2.6) into (2.8) produces (2.9) The resulting equations of motion for the total structure in the transform domain are (2.10) [M]{X(s)}+[K(s)]{X(s)}={F(s)}.

  15. ={f(t)}, The equations of motion having non-proportional viscous damping are (2.11) which can be posed in the form (2.12) The efficacy of this approach is that an orthogonal transformation can be found that diagonalizes the two real, square, symmetric matrices in (2.12), while obtaining an orthogonal transformation for (2.11) is usually not possible. +=.

  16. Multiplying the equation of motion by (1+b), the denominator of , results (2.13) Let is the smallest common denominator of the fractions and . The above equation becomes (2.14) Clearly, some of the matrices will be zero, and nonzero come from and through in (2.13). Note that . • [M]+ +]]{X(s)} • =(1+b){F(s)}. =(1+b){F(s)}

  17. To pose the equations of motion in terms of two real, square symmetric matrices, for which we can obtain an orthogonal transformation and decouple the equations of motion, the equations of motion become (2.15) (2.16) }=}.

  18. (2.17) (2.18)

  19. (2.19) Notice: • The two large matrices are real, square, and symmetric; • The lowest sets of partitioned matrix equations are the equations of motion (2.14), all of the upper sets of matrix equations are satisfied identically.

  20. The expanded form of the equations of motion differs from Eq. (2.12) in that the equations are posed in the transform domain, but the fractional powers of in the submatrices of the column matrices correspond to derivatives of fractional order. The major difference between Eq. (2.15), the expanded equations of motion, and Eq. (2.12) is that, the latter are posed in terms of derivatives of integer order.

  21. It is clear that the order of the expanded equations for structures of engineering interest could be very large and the size of the matrices prohibitive to manipulate even on the computer. However, this does not diminish the value of the expanded equations of motion for the orthogonal transformation decoupling the expanded equations of motion can be constructed and the decoupling procedure accomplished without directly using the expanded equations of motion.

  22. The homogeneous solutions for the original equations of motion (2.13) satisfy the equation (2.20) The eigenvalues and the eigenvectors for this equation can be found using matrix iteration techniques similar to those commonly used for finding mode shapes and resonant frequencies for undamped structures. • [M]+ • +]]{}=0.

  23. Given the expanded equations of motion can be decoupled using constructions based on homogeneous solutions to (2.13), the general form of the solution to expanded equations of motion, (2.21) Here is the order of the matrices and in the expanded equations of motion. is a modal constant, defined as (2.22) where is the th eigenvector of the expanded equations. {X(s)}={}. • =,

  24. The existence of the inverse transform The existence of the inverse transform of the structural response, is namely the existence of the time history of the motion of the modes of the finite element model, particularly, in the structural response for impulsive loading. The applied forces are the delta function

  25. {(s)}={}. The transform of the impulse response is (2.23) The inverse transform of {(s)} exists and is real, continuous, and causal when 1. {(s)} is analytical for ; 2. {(s)} is real for real and positive; 3. {(s)} is order , where , for large in the right half plane.

  26. Using contour integration and the residue theorem to determine the form of the impulse response produces (2.24) The response of the structure has two parts, one part being a sum of decaying sinusoids and the other an integral that decreases with increasing time. The integral does not decrease exponentially, and is asymptotic to for large , where Im +{}.

  27. 2.4 Example Consider elastic rod shown in Fig. 2. The tension member is fixed at both ends, but supported at two interior points by shear dampers, consisting of pads of a material described by the five-parameter fractional derivative model. Only longitudinal motion resulting from forces applied at the nodal points is considered.

  28. The finite element representation of this system in Laplace domain is (2.25) The mass and stiffness matrices are developed from rod elements, and damping matrix is developed from the nodal force relationship that results when the shear pads are described by the five parameter model, (2.26)

  29. Introduce a dimensionless scaled time defined as (2.27) and is taken as the dimensionless Laplace transform parameter corresponding to the dimensionless time. Then (2.28) (2.29) (2.30)

  30. Numerical parameters: (2.31) Scale factor:

  31. (2.32) Clearing the Laplace parameter

  32. (2.33)

  33. The characteristic value problem corresponding to the homogeneous equations of motion is where corresponds to square roots of the Laplace parameter . (2.34)

  34. In expanded equation of motion for this example,

  35. The eigenvector associated with the eigenvalues identifying resonant behavior, ( the first, second, fifth, and sixth) also occur in conjugate pairs. The first and second eigenvectors combine to produce the first mode shape and indicate real, in-phase motion of the nodes. Similarly, the fifth and sixth eigenvectors combine to produce the second mode shape and indicates real, out-of-phase motion. The remaining six eigenvectors combine with the conjugate temporal terms to show that the nonoscillatory portion of the motion of the structure is also real.

  36. 2.5 Conclusion The equations of motion for viscoelastically damped structures can be constructed and solved in a reasonably straightforward manner when the mechanical behavior of the viscoelastic material is portrayed by the fractional calculus model. Attractiveness: • consistent with the physical principles • accurately describe the mechanical properties over wide ranges of frequency • Small number of empirical parameters ( facilitate on lease square and solving equations of motion)

  37. The fractional calculus approach is an encompassing method for the analysis of damped structures: • Begin with the molecular theory of polymer solid • Generate accurate viscoelastic mathematical models • Well-posed equation of motion • Closed-form solutions for structures of engineering interest

  38. Appendix: Calculation of the Response of Impulsive Loading Form the following equation (2.23) we calculate the response function of impulsive loading by using the residue theorem and contour integral. ds. (A.1) The integral contour is shown in Fig. 3. {(s)}={},

  39. Positive imaginary axis S Plane Positive real axis Fig. 3Integration contour for inverse Transform

  40. Segments 3-5 of the contour are required, because the branch cut and branch points of the function are taken to be along the negative real axis and at the origin of the plane, respectively. By the residue theorem, (A.2) where are the residues of the poles of enclosed by the contour.

  41. When the length of segment is extended indefinitely in the positive and negative imaginary directions , (A.3) and (A.2) becomes (A.2)

  42. To maintain the continuity of the closed contour, the radii of segments and are increased indefinitely and segments and are extended indefinitely in the negative real direction. It can be shown that, • when the radii of contours and are increased indefinitely, the resulting value of integrals along these two segments , is zero; • when the radius of contour is decreased indefinitely, the integral along segment goes to zero; • the integrals along contours and are not zero.

  43. Rather, let , (A.3) then . (A.4) Similarly, . (A.5)

  44. Notice that and are complex conjugates of each other, we have . (A.6) The residues (A.7)

  45. Thus we have from (A.2), Im +{}. (2.24)