Transverse Vibration Analysis of a Self-Excited Beam Subjected to Delayed Distributed and a Singular Load Using Differential Transformation Method

PurposeIn this article, the transverse vibration motion of a self-excited beam which is subjected to a distributed and a singular load is analyzed using the differential transformation method (DTM).MethodsThe Euler-Bernoulli beam model is employed. The beam is modeled to represent the cutting tool holder motion in machining. A delayed distributed load and a vibration velocity-dependent singular load are considered as forcing. Analysis is performed for different time delays, widths of distributed load, and beam lengths in the time domain. The Laplace transform method is deployed for the stability analysis. Multi-step DTM is applied for the mathematical solution. Matlab (R) ddesd solutions is used for mathematical comparison.ResultsWhen the width of the distributed load increases and then the vibration amplitude increases. An increase in the beam length causes the amplitude to increase. The vibration amplitude increases as the delay time decreases. However, the reduction of some delay values reduces the amplitude.ConclusionThe equation that expresses the variation of the distributed load width with respect to the delay time is compatible results in accordance with the experimental studies in the literature. If beam length increases, the stability region of the width of the distributed load will decrease. The effect of beam length for stability can be adjusted by changing the time delay.