a Ranjbaran - Department of Civil Engineering, Shiraz University, Shiraz, Iran
m Ranjbaran - Department of Chemical Engineering, Shiraz University, Shiraz, Iran

In damage detection of civil, mechanical, aerospace, nuclear, bio-mechanic, and offshore engineering dynamic characteristics of beam-spring structures (BSS) play an important rule. The BSS is a model for beam segments plus springs, cracked beams and human or robot arms.Recently a new and innovative method for the free axial vibration of cracked bars is proposed. The method is extended for the free vibration of BSS. The model is considered as a sum of beam segments and springs. The governing equation (GE) for the free vibration of beam segments are combined with the compatibility conditions at spring positions. By introducing a new and innovative conjugate beam through defining a new variable, as function of lateral displacement, a single ordinary differential equation, golden equation, is obtained. The solution for the golden governing equation (GGE) is the same as that for an intact beam and so great simplicity and generality is obtained. Using the GGE both closed form and numerical solutions are obtained. The solutions are very much accurate and simpler than the conventional methods in the literature. Through applying the work to specific example and comparison of the results with others the accuracy, efficiency and robustness of the work is verified.

Beam-spring system, dynamic characteristics, crack, equivalent mass method,stiffness reduction method, golden equation, golden derivative