8 Resonant Analysis of Systems Equipped with Nonlinear Displacement-Dependent (NDD) Dampers 71 By adding a small term to the mathematical description (" in this paper), a perturbation method can be used to find an approximate solution to the governing differential Eq. (8.11). The parameter " is directly proportional to¯and, accordingly, to the fluid viscosity. It is also dependent to . Hence, increasing the viscosity or increasing , causes increasing ", and strength of nonlinearity in Eq. (8.11), successively. The shape parameters, i.e. n and s, effect on ˇ as ˇ D 2 d:n 1 s and successively on the damping coefficient c as given by Eq. (8.4). Accordingly, the dimensionless form of the governing equation of the vibration system utilizing the NDD damper with an external periodic excitation is affected by these shape parameters (see Eq. (8.11)). It should be noted that the values of the shape parameters n and s do not have any effect on " the one parameter which describes the strength of nonlinearity of the governing equation. Since the main focus of this paper is to analyze forced-resonant vibration of a system equipped with NDD damper, the shape parameters have been selected as a fixed set for a general application. However, the couple of the values of these parameters can be optimized according to the desire goal and intended particular application. For the forced-resonant analysis of the mass-spring-NDD damper system, the excitation frequency is considered as: D!n Cı" (8.12) where, ı is the detuning parameter to present the deviation of the external force’s frequency from the natural frequency of the system. Also, the external force amplitude coefficient, i.e. K, can be expressed as K D"k without any loss of the generality of the mathematical model. Therefore, the term related to the periodic-resonant external force in Eq. (8.11) can be expressed as following equation: Kcos b t D"kcos b t C "ı !nb t (8.13) Substituting Eq. (8.13) in Eq. (8.11) leads to: d2u d b t 2 Cu D " 1Cˇ1u2 Cˇ2u4 Cˇ3u6 Cˇ4u8 du d b t C "kcosh b t 1C"ı !n i (8.14) where, ˇi D ˛iC1 ˛1 are given in Appendix. In the following section, the procedure of employing MSM as a perturbation technique is illustrated to solve Eq. (8.14). 8.4 Forced-Resonant Vibration Analysis of the Mass-Spring-NDD Damper Using MSM This method is based on the idea of representing multiple independent variables, which are all functions of the time variable, and express all other time dependent functions including the response, as functions of the represented variables [54, 59–61]. For this aim, the independent variables are introduced as: Tn D" n b t for n D0;1;2;3 (8.15) Thus, the term related to the periodic-resonant external force in Eq. (8.14) can be determined using the terms T0 and T1 as follows "kcos b t 1C "ı !n D"kcos T0 C ı !n T1 (8.16) Assumingn D0and1, the solution of Eq. (8.14) can be expressed as u Du0 .T0; T1/ C"u1 .T0; T1/ CO " 2 (8.17)
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