In this article, a solution to the problem of finding the shape of piezoelectric modal sensors for a non-uniform beam with continuously exponential varying width under general boundary conditions is proposed using the Adomian modified decomposition method (AMDM). A general expression for designing the shape of a piezoelectric modal sensor is presented, in which the output signal of the designed sensor is proportional to the response of the target mode. Other modes are filtered out. The modal sensor shape is expressed as a linear function of the second spatial derivative of the structural mode shape function and the beam width function. Based on the AMDM and employing some simple mathematical operations, the closed-form series solution of the second spatial derivative of the mode shapes can be determined. Then, the shapes of the designed modal sensors are obtained. Finally, several numerical examples are given to demonstrate the feasibility of the proposed modal sensors with various boundary conditions.
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