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Typical free responses of underdamped, critically damped and overdamped SDOF systems with ω n t 4.8 rad/s
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A curve-fitting technique using a minimum error criterion was presented for identifying higher damping ratios near the critical damping ratio in single-degree-of-freedom systems. Free vibration decay tests were performed on a linear spring–mass–damper system with constant initial conditions using a modified commercial vibration apparatus. A vibrati...
Contexts in source publication
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... magnitude of viscous damping forces was altered and controlled using an electromagnetic (or eddy-current) damper installed in the apparatus. 2 Identification of lower damping ratios by logarithmic decrement Figure 1 shows the typical free responses of the underdamped, critically damped, and overdamped SDOF systems (Appendix: Eqs. (23-25)). ...
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... magnitudes of the peak displacements were determined by substitutingt p into Eq. (23). After some manipulations, we obtain the ratio of any two successive amplitudes with the same or different sign (Fig. 1) ...
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... in Eq. (12) is useful for the lower values of the damping ratio ζ for which the difference between the two successive amplitudes can be so small that they cannot be accurately differentiable by a measuring instrument. In such cases, the expression in Eq. (6) may not accurately predict the values of the logarithmic decrement δ. As shown in Fig. 1, the critically damped and overdamped free vibration responses do not have two successive peak amplitudes; hence, ζ values cannot be identified using Eqs. (8) and (9). The logarithmic decrement method is not applicable for estimating higher damping ratios, as suggested by Karnopp and Fisher [9]. Figure 2 gives two images of a modified ...
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... induced by eddy currents [11] in the conductive copper plate as a Lorentz force. The magnitude of the viscous damping forces was controlled by varying the damping currents ranging from I 0-2.8 A flowing in the magnetic coils with a control box (not shown). The undamped natural frequency ω n of the SDOF system was determined by ω n 2π where T d (Fig. 1) was measured from the oscilloscope records of the free vibration decay curves. Substituting δ from Eq. (12) into Eq. (8) yields ζ , and substituting ζ and T d into Eq. (13) yields ω n (theoretical ω n k eq /m √ 46.4/2.032 4.779 rad/s of the SDOF system or the vibration apparatus). When 0.3 < ζ < 0.7, two successive amplitudes with ...
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... (13) yields ω n (theoretical ω n k eq /m √ 46.4/2.032 4.779 rad/s of the SDOF system or the vibration apparatus). When 0.3 < ζ < 0.7, two successive amplitudes with different signs are observed in the mass displacement history. Then ζ is determined from Eq. (9) using Eq. (7). When almost no oscillations occur in the case of ζ > 0.8, as shown in Fig. 1, the logarithmic decrement method cannot be applied; accordingly, an alternative technique is needed to identify higher damping ratios. Figure 3 shows a typical oscilloscope record from the free vibration decay tests for I 0 A using the modified vibration apparatus. The output voltage on the vertical axis corresponds to the mass ...
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Citations
... where ξ is the damping ratio, t is time, ω(K) and ω D ðKÞ are the natural undamped and damped frequencies of the system, 71 respectively, and are defined as functions of K as follows An important consideration related to the mitigation of earthquake effects is the use of hydraulic jacks to control the undesirable vibration effects that arise from this kind of excitation. For instance, consider the seismic event known as the "El Centro Earthquake," which occurred in North America in 1940. ...
The aim of this work was to assess the capability of a hydraulic bottle jack to control vibrations and the effects of earthquakes. The first stage of the present investigation focused on determining the axial stiffness of the apparatus, in order to relate its behavior to that of a linear elastic element. To do this, uniaxial compression tests were carried out using a manually controlled servo system. In the second stage, a conceptual application based on the field of mechanical vibrations was considered. The experimental results confirmed the hypothesis of linear behavior and showed that the stiffness coefficient had distinct values of magnitude that depended on the cylinder height. Simulations were conducted in which the frequencies and the peak displacements of an idealized SDOF system subjected to earthquake excitation were adjusted based on the experimental stiffness. It was possible to conclude that the equipment analyzed here is an effective tool for controlling vibration and reducing displacement during a seismic event, and can be calibrated according to the specific conditions of operation. This is an important aspect of safety and functionality for mechanical and structural systems.
