Beam Vibration Theory Research Articles

Radial Error Feedback Control for Bar Turning in CNC Turning Centers1

In-process measurement and control of CNC machines can improve machining accuracy while increasing productivity and eliminating waste in a cost effective way. This paper describes an in-process measurement and control system called radial error feedback control (REFC) for bar turning in CNC turning centers. REFC was designed to compensate for the radial error due to the time-varying bar deflection caused by the cutting force, which moves along the bar at a constant velocity. In a dry cutting environment, a noncontact laser sensor was used to directly measure the radial error of the workpiece while it is being cut. To compensate for the radial error, a gain-scheduled Kalman filter for prediction together with PI control law was designed, based on a time-varying model of the deflection of the bar. Analytical modelling (using classical beam vibration theory) was employed to develop a simple model for the bar bending. For non-uniform bars finite element modelling can be used. Experimental results show that the dimensional and geometric accuracy of the workpiece is substantially improved by this radial error feedback control technique.

VIBRATION OF A BEAM WITH A BREATHING CRACK

A continuous cracked beam vibration theory is used for the prediction of changes in transverse vibration of a simply supported beam with a breathing crack. The equation of motion and the boundary conditions of the cracked beam considered as a one-dimensional continuum were used. The eigenfrequency changes due to a breathing edge-crack are shown to depend on the bi-linear character of the system. The associated linear problems are solved over their respective domains of definition and the solutions are matched through the initial conditions. The changes in vibration frequencies for a fatigue-breathing crack are smaller than the ones caused by open cracks. The method has been tested for the evaluation of the lowest natural frequency of lateral vibration for beams with a single-edge breathing crack. Experimental results from aluminium beams with fatigue cracks are used for comparison with the analytical results.

Elastic modulus measurement of multilayer metallic thin films

Two- and three-layer composite models were developed using a beam vibration theory, and the models were applied for measuring Young's moduli of thin metallic films. The Cr, Ni, and Co-coated Si wafer composites (two-layer composite) and (Cr/Ti/Si) composites (three-layer composite) were produced by radio-frequency (rf) magnetron sputtering and used to test the developed models. Young's moduli of (Cr) films obtained by the three-layer composite model agree well with those of (Cr) films obtained by the two-layer composite model, considering (Ti/Si) as the one layer and (Cr) as the other layer. This suggests that moduli of multilayer films may be obtained by using a two-layer composite model repeatedly.

A CONTINUOUS CRACKED BEAM VIBRATION THEORY

A continuous cracked beam vibration theory is developed for the lateral vibration of cracked Euler–Bernoulli beams with single-edge or double-edge open cracks. The Hu–Washizu–Barr variational formulation was used to develop the differential equation and the boundary conditions of the cracked beam as a one-dimensional continuum. The displacement field about the crack was used to modify the stress and displacement field throughout the bar. The crack was modelled as a continuous flexibility using the displacement field in the vicinity of the crack, found with fracture mechanics methods. The results of two independent evaluations of the lowest natural frequency of lateral vibrations for beams with a single-edge crack are presented: the continuous cracked beam vibration theory developed here, and a lumped cracked beam vibration analysis. Experimental results from aluminum beams with fatigue cracks are very close to the values predicted. A steel beam with a double-edge crack was also investigated with the above mentioned methods, and results compared well with existing experimental data.

Elastic modulus measurement of thin film using a dynamic method

A two layer composite model was developed using a beam vibration theory and the model was applied for measuring the Young’s modulus of thin films. The Ti coated Si wafer composites were produced by radio frequency magnetron sputtering and used to test the developed model. The measured film modulus using a dynamic method was checked with that using the static method utilizing the pure bending of a cantilever composite beam. The film modulus values measured in both methods were in good agreement. The film modulus values for the specimens with different film thicknesses were also in good agreement when they were measured by the same method.

Another second order beam vibration theory: Explicit bending warping flexibility and restraint

A new symmetric beam bending vibration theory which explicitly includes bending warping is presented. The governing differential equations are of sixth order and the boundary conditions allow enforcement of total fixity of the end of a beam. The motivation for the study was the desire to determine if consideration of bending warping restraint in bending theory is as important as is torsion warping restraint in torsion theory. Natural frequency predictions are given and compared with other theories for constant cross-section, cantilevered “beams” of two types: short, deep rectangular beams; and, short, thin, shallow, circular cylindrical shells. The accuracy of the new theory is comparable with Murty's, Levinson's, Bickford's and Rehfield's theories for beams with rectangular cross-sections. It is also comparable with Cowper's theory for general symmetric cross-sections.

On a “paradox” in beam vibration theory

On a “paradox” in beam vibration theory