Constructing the axial stiffness of longitudinally vibrating rod from fundamental mode shape

It is shown that inhomogeneous rods of uniform cross-section with the same material density variation and the same length, but with different boundary conditions and different elastic modulus variations along the axes of the rods have coincident fundamental natural frequency. The rods have different mode shapes that stem from the static displacements of the associated unif orm rods, under uniformly distributed loads. These interesting properties a ppear in combination with other unanticipated results; namely, the natural frequency expression for any polynomial variation of the material density c an be expressed in a unified manner, and depends solely on two coefficients : one coefficient describing material density and the other associated with elastic modulus variation. (C) 2001 Elsevier Science Ltd. All rights reser ved.