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An equivalent line-beam vibration model for frequency analysis of the V-and Z-shaped electrothermal microactuators

This paper presents an equivalent analytical model for vibration analysis of the V-and Z-shaped electrothermal microactuators. Considering the geometric analogy between the line-shaped and V/Z-shaped beam, the analytical model for vibration of the line-shaped beam that characterize the V-and Z-shaped beams is established first. The line-shaped beam provides completely the same boundary conditions and geometric dimensions (beam span, width, thickness, and shuttle dimensions) as the V-and Z-shaped beams, except for the inclined angle and the middle step beam, namely the stiffness. By writing the frequency of the line-shaped beam as a product of its stiffness and the term determined by the boundary conditions, the first-order natural frequency of the V-and Z-shaped beams can be derived by substituting their stiffness into the frequency expression. As the movement of the shuttle (i.e., the stiffness) does not contribute higher order mode shapes, corresponding higher-order natural frequencies of the V-and Z-shaped beams can be predicted directly with the line-shaped beam model. Verification of the vibration model for the line-shaped beam as well as the equivalent approach to predicting the natural frequencies of the V-and Z-shaped beams was conducted by comparing the results with finite-element simulation data using ANSYS software. Results show that, apart from some inaccuracies in predicting the higher order frequencies for the Z-shaped beam, this equivalent approach is accurate for the first-order frequency of the V-and Z-shaped beam as well as the higher order frequencies for the V-shaped beam. With this simple equivalent model, it is possible to investigate the frequency characteristics of the two typical beams thus paving the way for control and improving the design of the V-and Z-shaped electrothermal microactuators.