The Analysis of beam vibrations in micro machines

The vibrations of magnetically actuated microbeams under nonideal boundary conditions have beenstudied. Non ideal boundary conditions can be defined as the quantities like displacements and/ormoments are not exactly the assumed ideal values.The microbeams that are parts of micromachines typically can be modelled as normal beams. Invibration analysis of normal beams, the boundary conditions can be accepted as ideal. However, inthe systems at micro scale, accepting the boundary conditions as ideal can cause higher deviationsthan normal systems.In our work, nonideal boundary conditions are defined as very small transverse displacements to beallowed at the boundaries of microbeam and assumed helical springs that have small stiffnesscoefficient at both ends.Microbeams can be actuated by different methods at different systems. In our analysis, the microbeamis assumed to be actuated magnetically. The fundamental natural frequency of a magneticallyactuated microbeam is very sensitive to the axial strain. External loads such as pressure, temperature,force and acceleration apply axial strains on a microbeam, leading to shifts in its natural frequencies.Hence, our model contains an axial force. Also the stretching effects are considered due to immovableboundaries.Under all of these conditions, equation of motion of the microbeam is obtained by using Hamilton? sprinciple firstly. Then, the equation of motion and nonideal boundary conditions are converted tonondimensional form. So the problem is independent from its geometry. To find a uniform expansionto solution of the problem the method of multiple time scales that is a perturbation method is applieddirectly to the nonlinear equation of motion and to the nonideal boundary conditions.Two fundamental cases are considered in our analysis:Magnetic forcing frequency is to be near to one of the system? s natural frequencies (primaryresonances) and magnetic forcing frequency is to be near to the system? s subharmonic andsuperharmonic natural frequencies. (secondary resonances)The amplitude and phase modulation equations are obtained for each cases. By that equations,frequency-response graphics are plotted. It? s been seen clearly at this graphics that the effect ofnonideal boundary conditions make significant deviations according to the case of ideal cases inresponses.