Natural Frequencies, Sensitivity and Mode Shape Details of an Euler–bernoulli Beam With One-Step Change in Cross-Section and With Ends on Classical Supports

Free vibration analysis of multiple-stepped beams by the differential quadrature element method

“…Free vibration analysis of beams with continuous and discontinuous variation in cross-section has drawn attentions of researchers for many years. Particularly the free vibration problem of multiple-stepped beams has been extensively investigated theoretically, numerically, and experimentally [3][4][5][6][7][8].…”

Three-to-one internal resonance in multiple stepped beam systems

“…Studies on stepped beam systems are usually linear.Özkaya and Tekin [4] investigated the non-linear vibrations of a clamped-clamped EulerBernoulli beam with n steps on the arbitrary points and researched the contributions of the non-linear terms on natural frequencies. Naguleswaran [5] obtained motion equations of three different Euler-Bernoulli stepped beams with all states of boundary conditions and computed three natural frequencies using the motion equation. In his other study, Naguleswaran [6] considered three different types of stepped beams and investigated the vibration of a beam with up to three step changes.…”

Reference Value Selection in a Perturbation Theory Applied to Nonuniform Beams

“…Physically it is expected that the solution 1 will be continuous and have continuous derivatives. Additional continuity conditions can be determined mathematically by integrating the ODE in (10). The continuity conditions at an arbitrary point are listed in (13a), (13b), (13c), and (13d).…”

“…In [9] it was demonstrated that when the area and moment of inertia of a beam structure are of a specific polynomial form, the equation of motion for the transverse bending could be transformed to that of a homogeneous beam for which an analytical solution is easily obtainable. With regard to a system with a step discontinuity in the mass per unit length and bending stiffness, [10,11] demonstrated that an analytical solution can be found for the transverse vibrations by applying continuity conditions at the location of the step. Other times, the solution to a nonuniform, Euler-Bernoulli beam structure can be expressed in terms of special functions such as Bessel functions and Chebyshev polynomials [12][13][14].…”

“…Additionally, the moment and shear in the system are investigated and compared for the cases with and without a first-order correction to the mode shapes. For the case of a stepped beam, an analytical solution can be found [10,11], which serves as an exact solution to which the perturbation theory results can be compared. Determining a solution for a system with piecewise continuous spatial properties, for which in some sections an analytical solution cannot be found, is the goal of the current work.…”