Three mass system excited by an actuator acting on a relative motion and b absolute motion. Taking the Laplace transform of Eq. It can be shown that the numerator of Eq. For example, for the three mass system Fig. This corresponds to the system with an additional constraint at the degree of freedom the damper acts on.
For the three mass system of Fig. Again, this corresponds to a system that is equivalent to the structure for which the actuator is replaced by a rigid link. So, an equivalent expression as in Eq. By combination of Eqs. Eigenvalues of a structure with a single viscous damper In this section, the possibilities of obtaining the eigenvalues of a structure including a viscous damper are discussed. First, the classical approach is repeated, where solutions are found by solving a quadratic eigenvalue problem.
Next it is shown how the eigenvalues can be derived by solving a root locus problem, which is computationally less time consuming. Finally, simple approximate solutions are proposed, which are very useful in the design of dampers.
The free response of a structure including a viscous damper Fig. Complex eigenvalues by solving root locus problem Taking the Laplace transform of Eq. Typical examples of such root locus plots are shown in Fig. It is seen that the poles and zeros alternate along the imaginary axis, which is characteristic for an undamped collocated system [1].
It should be clear that in contrast to the resonance frequencies, the values of the anti-resonance frequencies do depend on the location of the damper in the structure and there is always exactly one anti-resonance between two consecutive resonances. For zero damping constant, the closed-loop poles coincide with the undamped resonance frequencies of the structure. Typical root locus plots of an undamped discrete structure degrees of freedom n43 with attached viscous damper for varying damping constant c.
Only the upper half of the s-plane is shown, the diagram is symmetrical with respect to the real axis. In case the mode is critically damped at this point, this mode shall typically remain critically damped by further increasing c, as can be seen for example in Fig. For all the other cases shown in Fig. The closed loop poles then coincide with the anti-resonance frequencies of the structure and the damper acts like a support. Because a viscous damper can only dissipate energy, the branches of the root-locus diagram are all contained in the left half-plane.
The form of the root locus diagram of Fig. Here, the anti-resonance frequencies differ only slightly from the resonance frequencies and moderate values of modal damping are achieved for the optimal damping constant. As will be seen in the numerical example of a cantilever beam with attached translational viscous damper, in some cases it is possible to achieve critical damping for a mode, as can be seen for example in Fig.
Hereby the branch of the second mode goes to the third zero. It is clearly seen in these examples that the amount of achievable damping is small for poles that have a zero in the near neighborhood and gets larger when they are more separated.
When a zero coincides with a pole, the mode is uncontrollable for the corresponding damper location. This will be demonstrated by numerical examples in Section 4.
Approximating formulas for the optimal damping In this section, approximating formulas for the optimal damping constant and maximum modal damping ratio are derived. These are very useful in the design of dampers. From the knowledge of the values of the poles and zeros, it is not trivial to predict whether a pole will be critically damped, or attracted to one of the zeros.
However, it can be stated that it is most probable that a pole will be attracted by the nearest zero. Certainly when the distance is small compared to the other poles and zeros, this probability is large. From this reduced root locus problem, approximate values for the maximum modal damping and the optimal damping constant can be derived. Substitution of Eq. This root locus plot can be derived from the root locus plots of Fig.
For each eigenmode, there is an optimal value for c that results in maximum modal damping. By substitution of Eq. The importance of inducing such frequency shifts has been demonstrated before for active devices controlled by the IFF algorithm [1], for a taut cable with attached viscous damper [4,9] and for a general discrete structure including viscous dampers [3]. A comparison with the results obtained in this paper is made in Section 5. Such explicit approximations are derived in this section for the two cases considered in the previous section, in a similar way as in Ref.
Krenk derived previously explicit approximations in the same form for the special case of a taut cable [4] and a beam with rotational viscous dampers at the end [5].
An explicit equation for the damping ratio can be deduced from Eq. This results in equations of exactly the same form as Eq. The quadratic eigenvalue problem Eq. The stiffness and mass matrices Kb and Mb are given in Appendix A.
The number of beam elements is chosen 50 for the following numerical examples, resulting in a total of degrees of freedom.
Two cases are considered for this numerical example: a beam with a damper acting on the translational degrees of freedom n and on the rotational degrees of freedom y. Cantilever beam with translational viscous damper The exact solution of the modal damping ratio is compared to the approximate solutions in Fig.
Also plotted in Fig. The optimum of these curves corresponds to the approximating formulas for the maximum modal damping Eq. The approximations are very good for all cases, except for the case of Fig. This can be explained by the fact that the closest zero is relatively far away from the pole for this case, which seems to be critically damped for values of c larger than a certain critical value.
Modal damping ratio for a cantilever beam with translational damper. Exact, —; explicit approximation Eq. The maximum modal damping ratios obtained by the approximating formulas for case 1 Eq.
Over the larger part of the beam, good approximations are found, except for the regions where the modes are critically damped. It should be noted here, that the root locus diagram for a cantilever beam with a translational damper has the form depicted by Fig. The exact optimal values of the damping constant are compared to the approximations for case 1 and case 2 by, respectively, Eq. For damper locations where a mode can be critically damped, the minimal value of c for which critical damping is achieved is plotted here.
As for the prediction of the optimal damping constant, it is seen that when the closest zero is located within a relative distance of compared to the distance of the next or previous pole, the difference between the approximate damping constant and the exact value is very small. The largest differences are visible at locations where the maximum modal damping approaches zero, thus the least interesting regions. And even here, the maximal difference is smaller than 2 dB, which coincides with a factor 1.
Because of the typical robust character of the modal damping versus the damping constant at the optimal point, as illustrated for example in Fig.
Note that if no restriction would be made on the allowed relative distance of the considered pole and zero, the prediction of the optimal damping constant would be good over the entire range of the beam length, even in regions where critical damping is achieved. The error on the approximate value of the damping constant for the case the formulas of the closest zero are used, is smaller than 2 dB over the entire beam length.
Cantilever beam with rotational viscous damper The same calculations are repeated here for a cantilever beam with attached absolute rotational viscous damper. In contrast to the previous example, the subsequent zeros are always closer here than the previous zeros, as shown in Fig. The root locus plots all have the form of Fig. As depicted in Fig. This can also be concluded for the prediction of the optimal damping constant Fig. The largest difference in the region at the end of the beam for mode two and three is smaller than 3 dB, which coincides with a factor 1.
Again in these regions the maximum modal damping approaches zero. Maximum modal damping for beam with translational damper. Optimal damping constant for beam with translational damper. Comparison with previous research The approximate solutions for the complex eigenvalues and the formulas for the optimal damping are very similar to results obtained in the work of Preumont [1] and Main and Krenk [3]. In this paper however, an alternate approach is used, which leads to new insights.
A comparison is made in this section. On the other hand, it should be clear that the method represented in this paper is restricted to a single damper, while the works of Preumont [1] and Main and Krenk [3] propose solutions where multiple devices are used. Maximum modal damping for beam with rotational damper. Optimal damping constant for beam with rotational damper.
Whereas these are obviously the most remarkable differences, the discussion below reveals some other important differences. Preumont [1] investigated the effect of active devices controlled with the integral force feedback control algorithm, which is equivalent to viscous damping.
By using a diagonalization technique, Preumont found equations with exactly the same form as Eq. The equations have exactly the same form, only kk;k is replaced in Ref.
Very important is that the approximations obtained in Ref. The usage of the symbols ok and Ok is reversed here in comparison with Ref. Exact, —; approximation by Eq. According to the authors it is not possible to use the same diagonalization technique as in Ref.
In other words, at higher frequencies dashpots are more of a viscoelastic damper than a viscous damper. Viscoelastic attributes of a realistic damper can be described by a combination of springs and ideal viscous dampers. Kelvin-Voight model A spring and a viscous damper in parallel is commonly used to characterize dashpots at a single frequency.
The damping coefficient and stiffness used in the Kelvin-Voight model are identified, experimentally, at various frequencies. The viscoelastic dashpot model is extended to all frequencies by fitting a generalized three-parameter also known as generalized Maxwell viscoelastic model to the experimentally evaluated damping coefficient and stiffness at various frequencies. Figure 3 shows the experimentally evaluated damping and stiffness coefficients of a dashpot at various frequencies the blue marks as well as a five-term generalized three-parameter viscoelastic model fitted to that the experimentally evaluated data.
With the low VTC of around 0. Nevertheless, there is some temperature dependency on the rheological properties of silicone fluid. Figure 4 shows the dependency of silicone fluid viscosity on temperature over the temperature range of deg C. Viscosity-Temperature Coefficient VTC is used to characterize the variation of viscosity of a fluid with temperature.
Thus, the lower the VTC. Viscous dampers may be used as a stand-alone damping unit to dampen a single or multiple resonances of underdamped structures such as piping systems, buildings to reduce interstory drift , and floor systems or in conjunction with spring elements in vibration isolation applications and realization of tuned mass dampers. The blue traces in Figure 5 depict the magnitude and phase of frequency response function of an underdamped structure.
They clearly show two resonant modes at 6 and 12 Hz. The red traces on the same figure present the same frequency response function after two dashpots were installed to the structure. Comparison of the blue and red traces in Figure 5 shows the effectiveness of the viscous dampers in dampening both resonant modes. Figure 5 The experimentally measured frequency response functions of a structure without blue traces and with red traces viscous damping. Email us : info deicon. Viscous Dampers Viscous Dampers Viscous damping has been widely used as the energy dissipation mechanism of choice in abating resonant vibration in structures.
Following the design of viscous dampers, they are prototyped and their damping effectiveness verified, experimentally.
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