The behavior of higher order mode shape derivatives in damaged, beam-like structures

This work considers the effect that damage has upon the higher order derivatives of mode shapes of structures having primarily beam-like vibration. Via numerical investigations, the sensitivity of various damage related parameters in inducing changes in these higher order modal derivatives is determined, leading to a more complete understanding of what factors make the most contribution to significant changes in these derivatives. It is concluded that higher order mode shape derivatives (e.g., modal curvature, third derivative, and fourth derivative) are better indicators of damage than the mode shapes. Three distinct types of response for the damage-induced higher order derivative discontinuities are identified as three key parameters (the mass loss, stiffness loss, and damage radius scale) vary. From this, formal approximations are obtained for the expected forms of the higher order derivative discontinuities based upon the underlying behavior predicted by a simple relation among these three parameters. These approximations are checked with numerical simulations, and an excellent level of agreement is observed under appropriate conditions. Finally, the potential of these higher order derivative changes for indicating the presence and location of damage in a global setting is examined.