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Exact solutions for free longitudinal vibration of bars with non-uniform Cross-sections

100%

Li Q. S.

2000

tom Vol. 5, no 3

521-541

Using appropriate transformations, the differential equations of free longitudinal vibration of bars with variably distributed mass and axial stiffness are reduced to Bessel?s equations or ordinary differential equations with constant coefficients by selecting suitable expressions, such as power functions and exponential functions, for the distributions of mass and axial stiffness. Exact analytical solutions to determine the longitudinal natural frequencies and mode shapes for a one step non-uniform bar are derived and used to obtain the general solution and the frequency equation of a multi-step non-uniform bar with various boundary conditions, including non-classical cases. This approach which combines the transfer matrix method and closed form solutions of one step bars leads to a single frequency equation for any number of steps.

Modal analysis of vibrating structures impregnated with crack

84%

Das H. C. , Parhi D. R.

2008

tom Vol. 13, no 3

639-652

In the current investigation a dynamic analysis of the cracked cantilever beam is carried out. Local flexibility is being introduced due to the presence of a crack in the structural member that affects its dynamic response. For finding out the deviation in mode shapes and natural frequencies of the cracked cantilever beam the local stiffness matrices are taken into account. Theoretical expressions have been developed to calculate the natural frequencies and mode shapes of the cracked cantilever beam using local stiffness matrices. The strain energy release rate has been used for calculating the local stiffnesses of the beam. Suitable boundary conditions are taken into account at the crack location. Comparisons made between the numerical results and the corresponding experimental results show very good agreement and authenticate the theory developed.