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In this paper, the stability of two weakly coupled elastic beams connected vertically by a spring is investigated via the frequency domain method and the multiplier technique. When the two beams have partially local damping, the operator   A   is obtained via variable conversion, and it generating a semigroup is proved, then we obtain that the semigroup is exponentially stable by reduction to absurdity.

Stability of Two Weakly Coupled Elastic Beams with Partially Local Damping