Two-dimensional stick-slip motion of an oscillator subjected to dry friction is investigated in this paper. The equations of motion of the non-smooth system are discretized in the time domain by means of the implicit Bozzak-Newmark scheme. The system state equations in a time step are written in the incremental displacements to model the frictional constraints in accordance with Coulomb's law. With the help of a coordinate transformation and introduction of paired non-negative and complementary variables, the non-smooth vibration problem is reduced to a mathematical programming problem for which a numerical solution can be obtained. Numerical results for a single body oscillator under a harmonic excitation are obtained using the proposed method and compared with those in the literature; excellent agreement is achieved. The proposed method is then applied to a general two-dimensional oscillator with stiffness and viscous coupling in addition to the frictional coupling. Experiments are conducted for free vibration of a single body vibration system subjected to two-dimensional dry friction. Good agreement between the measurements and numerical results obtained using the proposed scheme is observed.
A precision testing method based on a seven-sensor configuration is presented in this paper to quantify thermal errors (drifting, elongation, tilting) of the mandrel cross-section, for an example horizontal machining center with linear optical grating scale. Three tri-axial displacement sensors with 120° spread angles and 36 thermal couples are mounted on judiciously chosen locations to record the temperatures and thermal expansions for various operating conditions. Based on the measurements covering a wide range of spindle locations, environment temperatures, coolant temperature, and spindle rotational speeds, we found that (i) the maximum thermal drifts of the mandrel are 11.3 µm in the x direction with a compensation rate of 62%, and 165.3 µm in the y direction with a compensation rate of 93%, (ii) the maximum thermal tilt of the mandrel is 0.005°, and (iii) the thermal elongation of the mandrel in the z direction, which could not be compensated by the linear optical grating scale, is 51.9 µm. From a correlation study, the thermal elongation of the mandrel is most closely correlated to the temperatures recorded for the thermal couple mounted at the front surface of the spindle bearing with a correlation coefficient of 0.83.
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