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Definition of physically consistent damping laws with fractional derivatives. (English) Zbl 0865.70014
Summary: The generalized damping equation E:(D 2 +aD q +b)x(t)=f(t); q(0,2) is treated. It is shown that for q1 and x,f𝕃 C 2 () there are arbitrarily many proper definitions of E corresponding to the choice of branches of (iω) q in the definition of the characteristic functions p(ω)=(iω) 2 +a(iω) q +b. The only restriction is that p(ω) is measurable. General conditions and results concerning uniqueness and causality of the solutions of E are developed. Physically reasonable ones are: E has unique solutions if p(ω) is continuous and has no real zeros. If, furthermore, p is restricted to the principal branch, the solutions become causal if and only if a,b>0. For demonstration purposes, a general analytic solution of the causal impulse response is given and discussed.
MSC:
70J99Linear vibration theory
70J40Parametric resonances (linear vibrations)
47A60Functional calculus of operators
26A33Fractional derivatives and integrals (real functions)