## Definition of physically consistent damping laws with fractional derivatives.(English)Zbl 0865.70014

Summary: The generalized damping equation $$E:(D^2+aD^q+b)x(t)=f(t)$$; $$q\in (0,2)$$ is treated. It is shown that for $$q\neq 1$$ and $$x,f\in \mathbb{L}_C^2(\mathbb{R})$$ 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 $$p(\omega)=(i\omega)^2+a(i\omega)^q+b$$. The only restriction is that $$p(\omega)$$ 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 $$p(\omega)$$ 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:

 70J99 Linear vibration theory 70J40 Parametric resonances in linear vibration theory 47A60 Functional calculus for linear operators 26A33 Fractional derivatives and integrals
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### References:

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