Input shaping is an effective means for suppressing motion-induced residual vibration of lightly damped structures. Here, to demonstrate the ideas of various input shaping schemes for continuous structures, the model system of a cantilever beam, whose base is to be displaced by a prescribed distance, is considered. The cantilever-beam motion is modeled by the damped Bernoulli-Euler beam equation, and is then decomposed into normal vibration modes. For the particular system set up here, the modal equations of motion are linear and uncoupled, and consequently are integrated analytically. It is then shown that, by completing the cantilever base movement in a series of properly calculated steps (i.e., by shaping the input command of the dynamical system), so as to annihilate the dominant vibration modes through destructive interference, the overall induced vibration of the cantilever can be significantly suppressed. In particular, the "zero-vibration" (ZV) and "zero-vibration-and-derivative" (ZVD) input shaping schemes previously proposed for discrete systems are adapted and applied to the continuous beam here. The theoretical results are also supported by experiments.
Various input-shaping schemes such as the "zero-vibration" (ZV), "zero-vibration-andderivative" (ZVD), "negative ZV" (NZV), and "negative ZVD" (NZVD) schemes have previously been proposed to suppress motion-induced residual vibration of lightly damped structures. In such schemes, the input command of the dynamical system in question is properly administered (i.e., shaped), so that the dominant induced vibration modes are annihilated through destructive interference. Here we are concerned with the effects of system payload on the vibration reduction capabilities of the aforementioned input-shaping schemes, especially when they are applied to continuous systems. By use of the simple structure of a linearly elastic rod as a specific example, it is demonstrated that both the minimum achievable residual vibration amplitude and the tolerance of detuning parameter errors of the inputshaping schemes are sensitive to the amount of payload on the system. It is therefore imperative to take the factor of system payload into account in the design of practical input shapers.
Absorption and photoconductivity in GaAs and InP have been studied as functions of light intensity I up to 20 MW/cm2 at 1.06 μ. The two-photon-absorption coefficient is determined to be ≈ 0.3I cm−1 for n-GaAs and ≈ 0.2I cm−1 for n-InP. These values are much larger than that predicted by the existing theories. For the n-InP samples, the absorption by the two-photon-excited free carriers is found to be important, giving an absorption coefficient of ≈ 0.15I2 cm−1. From the photoconductivity measurements, the concentration and the absorption cross section of holes are determined.
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