H. Beyer et S. Kempfle, DEFINITION OF PHYSICALLY CONSISTENT DAMPING LAWS WITH FRACTIONAL DERIVATIVES, Zeitschrift fur angewandte Mathematik und Mechanik, 75(8), 1995, pp. 623-635
The generalized damping equation E: (D-2 + aD(q) + b) x(t) = f(t); q i
s an element of (0, 2) is treated It is shown that for q not equal 1 a
nd x, f is an element of L(C)(2)(R) there are arbitrarily many proper
definitions of E corresponding to the choice of branches of (i omega)(
q) in the definition of the characteristic functions p(omega) = (i ome
ga)(2) + a(i omega)(q) + b. The only restriction is that p(omega) is m
easurable. General conditions and results concerning uniqueness and ca
usality of the solutions of E are developed Physically reasonable ones
are: E has unique solutions if p(omega) is continuous and has no real
zeros. If, furthermore, p is restricted to the principal branch, the
solutions then become causal if and only if a, b > 0. For demonstratio
n purposes a general analytic solution of the causal impulse response
is given and discussed.