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Structural Engineering and Mechanics
  Volume 16, Number 2, August 2003 , pages 153-176

The exact solutions for the natural frequencies and mode shapes of non-uniform beams carrying
Der-Wei Chen

    From the equation of motion of a \"bare\" non-uniform beam (without any concentrated elements), an eigenfunction in term of four unknown integration constants can be obtained. When the last eigenfunction is substituted into the three compatible equations, one force-equilibrium equation, one governing equation for each attaching point of the concentrated element, and the boundary equations for the two ends of the beam, a matrix equation of the form [B]{C} = {0} is obtained. The solution of (where
Key Words
    non-uniform beam; natural frequencies; mode shapes; bare beam; constrained beam; eigenfunction.
Department of Naval Architecture and Marine Engineering, Chung Cheng Institute of Technology,
National Defense University, Yuansulin, Dashi, Taoyuan, Taiwan 335, Republic of China

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