Parametric instability of twisted Timoshenko beams with localized damage

• Increasing twist angle always reduces the first instability region.
• The location and extent of damage greatly affect parametric instability behaviors.
• Increasing static load has a destabilizing effect on the parametric stability.
• The presence of extent of damage has a destabilizing effect on the stability.
• Effects of damage location on instability zones vary depending on restraint types.

The parametric instability of twisted Timoshenko beams with localized damage under an axial pulsating force is studied. The bending vibration equations of motion in the twisted frame are established using the Hamilton׳s principle. The partial differential equations of motion are then discretized into a set of second-order ordinary differential equations with periodic coefficients of Mathieu–Hill type by using a finite element approach. An eigenvalue problem is formulated to determine the instability regions of the twisted Timoshenko beams with localized damage based on Bolotin׳s method. The effects of the twist angle, the damage location, the extent of the damage, the static component of axial force and boundary conditions on the instability regions of the damaged twisted beam are investigated. Two parameters associated with the parametric instability are used to assess the destabilizing effect of damage location and extent of damage on the parametric resonance of twisted beams with localized damage.