Browsing by Subject "EULER-BERNOULLI BEAM"

Now showing 1 - 4 of 4
  • No Thumbnail Available
    Item
    Three-to-one internal resonance in multiple stepped beam systems
    (SHANGHAI UNIV) Tekin, A; Özkaya, E; Bagdatli, SM
    In this study, the vibrations of multiple stepped beams with cubic nonlinearities are considered. A three-to-one internal resonance case is investigated for the system. A general approximate solution to the problem is found using the method of multiple scales (a perturbation technique). The modulation equations of the amplitudes and the phases are derived for two modes. These equations are utilized to determine steady state solutions and their stabilities. It is assumed that the external forcing frequency is close to the lower frequency. For the numeric part of the study, the three-to-one ratio in natural frequencies is investigated. These values are observed to be between the first and second natural frequencies in the cases of the clamped-clamped and clamped-pinned supports, and between the second and third natural frequencies in the case of the pinned-pinned support. Finally, a numeric algorithm is used to solve the three-to-one internal resonance. The first mode is externally excited for the clamped-clamped and clamped-pinned supports, and the second mode is externally excited for the pinned-pinned support. Then, the amplitudes of the first and second modes are investigated when the first mode is externally excited. The amplitudes of the second and third modes are investigated when the second mode is externally excited. The force-response, damping-response, and frequency-response curves are plotted for the internal resonance modes of vibrations. The stability analysis is carried out for these plots.
  • No Thumbnail Available
    Item
    Investigation of stepped microbeam vibration motions according to modified couple stress theory
    (WALTER DE GRUYTER GMBH) Polat, SC; Bagdatli, SM
    In this study, linear and nonlinear vibration movements of simply supported stepped microbeams placed in a magnetic field have been analyzed using Modified Couple Stress Theory. By taking into account the step radius ratio, step-change position, and material size parameter, the equations of motion of the stepped-microbeam are obtained using the Hamilton principle. The resulting equations of motion are nondimensionalized to eliminate dependence on material type and geometric structure. The approximate solution of the dimensionless equations of motion is calculated using the method of multiple scales, one of the perturbation methods. The solution stages of the study are divided into two separate parts as linear and nonlinear problems. Firstly, the linear issue of the stepped microbeam is addressed. The natural frequencies of the system are derived by solving the linear problem. Linear and nonlinear effects of step radius ratio, step-change position, and microbeam coefficient are investigated and frequency-amplitude graphs are presented. The resonance state where forcing frequency is equal to natural frequency is examined and stability analysis has been made.
  • No Thumbnail Available
    Item
    Non linear vibrations of stepped beam system under different boundary conditions
    (TECHNO-PRESS) Özkaya, E; Tekin, A
    In this study, the nonlinear vibrations of stepped beams having different boundary conditions were investigated. The equations of motions were obtained using Hamilton's principle and made non dimensional. The stretching effect induced non-linear terms to the equations. Forcing and damping terms were also included in the equations. The dimensionless equations were solved for six different set of boundary conditions. A perturbation method was applied to the equations of motions. The first terms of the perturbation series lead to the linear problem. Natural frequencies for the linear problem were calculated exactly for, different boundary conditions. Second order non-linear terms of the perturbation series behave as corrections to the linear problem. Amplitude and phase modulation equations were obtained. Non-linear free and forced vibrations were investigated in detail. The effects of the position and magnitude of the step, as well as effects of different boundary conditions on the vibrations, were determined.
  • No Thumbnail Available
    Item
    Non linear vibrations of stepped beam systems using artificial neural networks
    (TECHNO-PRESS) Bagdatli, SM; Özkaya, E; Özyigit, HA; Tekin, A
    In this study, the nonlinear vibrations of stepped beams having different boundary conditions were investigated. The equations of motions were obtained by using Hamilton's principle and made non dimensional. The stretching effect induced non-linear terms to the equations. Natural frequencies are calculated for different boundary conditions, stepped ratios and stepped locations by Newton-Raphson Method. The corresponding nonlinear correction coefficients are also calculated for the fundamental mode. At the second part, an alternative method is produced for the analysis. The calculated natural frequencies and nonlinear corrections are used for training an artificial neural network (ANN) program which has a inulti-layer, feed-forward, back-propagation algorithm. The results of the algorithm produce errors less than 2.5% for linear case and 10.12% for nonlinear case. The errors are much lower for most cases except clamped-clamped end condition. By employing the ANN algorithm, the natural frequencies and nonlinear corrections are easily calculated by little errors, and the computational time is drastically reduced compared with the conventional numerical techniques.