Damping Performance of Dynamic VibrationAbsorber in Nonlinear Simple Beam with 1:3Internal ResonanceYi-Ren WangDepartment of Aerospace Engineering, Tamkang University, NewTaipei City, Tamsui Dist., 25137, Taiwan.
Hsueh-Ghi LuRD Center-Environmental Reliability QA Div., Enterprise Products Testing Department, Pegatron Corporation,Taipei City 11261, Taiwan.
(Received 20 October 2014; accepted 17 November 2016)
This study investigated the damping effects of a dynamic vibration absorber (DVA) attached to a hinged-hingednonlinear Euler-Bernoulli beam. The model constructed in this study was used to simulate suspended nonlinearelastic beam systems or vibrating elastic beam systems on a nonlinear Winkler-type foundation. This makesthe modeling in this study applicable to suspension bridges, railway tracks, and even carbon nanotubes. Thehinged-hinged beam in this study includes nonlinear stretching effects, which is why we adopted the methodof multiple scales (MOMS) to analyze the frequency responses of fixed points in various modes. The use ofamplitudes and vibration modes made it possible to examine the internal resonance. Our results indicate thatparticular elastic foundations or suspension systems can cause 1:3 internal resonance in a beam. The use of 3Dmaximum amplitude contour plots (3DMACPs) enabled us to obtain a comprehensive understanding of variousDVA parameters, including mass, spring coefficient, and the location of DVA on the beam, and thereby combinethem for optimal effect. Our results were verified using numerical calculations.
Engineering from the scale of small mechanical componentsto that of aircraft wings and bridges must take into accountthe effects of vibration, which can cause fatigue, loosening ofparts, and failure. Vibration can be linear or nonlinear, withthe latter involving internal resonance that is complex and dif-ficult to analyze. As described by Nayfeh and Mook,1 in theevent that the natural vibration frequencies of various degreesof freedom (DOFs) are multiples of one another, excitationat high frequencies can increase amplitudes at low frequen-cies. This study sought to account for this unique nonlinearphenomenon. Nayfeh and Balachandran2 defined several sta-ble conditions in nonlinear systems and presented a numberof methods that could be used to determine system stability,and are therefore highly valuable as a reference in the anal-ysis of system stability. Nayfeh and Pai3 investigated vibra-tions in nonlinear Euler-Bernoulli beams. According to New-tons laws, Eulers angle transformation, and the Karman-typestrain-displacement relationship in order to derive 3D and 2Dequations associated with nonlinear beams. These theoreti-cal models have greatly benefited related research. Nayfehand Mook1 also proposed a number of methods with whichto solve nonlinear systems, including the Poincare method, theLindstedt method, the average method, and MOMS, which ishighly conducive to the analysis of damped vibration systems.Ji and Zu4 studied the rotating shaft system of a Timoshenkobeam using MOMS to analyze the natural frequency responsesof nonlinear systems. Nayfeh and Nayfeh5 employed MOMSto identify nonlinear modes and nonlinear frequencies, where-
upon they applied the Galerkin method to the analysis of dy-namic responses in a nonlinear beam. Analysis of vibrationsin an elastic beam involves many factors. Van Horssen andBoertjens6 considered a suspension bridge under the influenceof nonlinear aerodynamic effects using the theoretical modelof a Euler-Bernoulli beam with linear springs to simulate vi-brations in the bridge. They discovered that energy can shiftbetween the first and third modes and that this is typical of non-linear internal resonance. Their study prompted a great dealof research based on theoretical models. Theories similar tothis, in which a beam is placed on an elastic foundation, havebeen widely applied to civil, mechanical, and aerospace engi-neering, by researchers such as Mundrey.7 Fu et al.8 reportednonlinear vibrations in embedded carbon nanotubes (CNTs),based on research involving the use of nonlinear 2D Euler-Bernoulli beams to simulate vibrations in CNTs on elastic ma-trices. Shen9 used the model of a nonlinear 2D Euler-Bernoullibeam to analyze vibrations in a post-buckling beam placed on adouble-layered elastic foundation, thereby demonstrating thatthe stiffness of elastic foundations has a significant influenceon the vibration behavior of nonlinear beams.
Wang and Chen10 developed a damping approach to vibra-tion reduction without the need to change the damper, butmerely enable its relocation by having a mass-spring-dampervibration absorber hung from the optimal damping location ona rotating mechanism (a CD-ROM drive or the coupling sys-tem of rotary wings and swashplates). Wang and Chang11 stud-ied nonlinear vibrations in a rigid plate using cubic springsto simulate supports at the four corners with two single-mass dampers hung from beneath the plate. The locations
International Journal of Acoustics and Vibration, Vol. 22, No. 2, 2017 (pp. 167185) https://doi.org/10.20855/ijav.2017.22.2462 167
Y.-R. Wang, et al.: DAMPING PERFORMANCE OF DYNAMIC VIBRATION ABSORBER IN NONLINEAR SIMPLE BEAM WITH 1:3 INTERNAL. . .
of the dampers were adjusted to achieve optimal damping ef-fects. Dynamic vibration absorber (DVA) is a practical passivedamping device for vibrating systems. For a linear vibratingbeam, DVA can be pre-tuned to the modal frequency of thevibrating structure or the disturbance frequency to damp outthe beam vibration. Wang and Chang12 discussed a hinged-free linear Euler-Bernoulli beam placed on a nonlinear elas-tic foundation. They found that placing a DVA of appropriatemass could prevent internal resonance and suppress vibrationsin the beam. Nambu et al.13 studied the smart self-sensingactive DVA attached on a flexible beams supported string. Asmall cantilevered piezoelectric transducer was used to real-ize this active DVA. They showed improvement control ro-bustness upon passive DVAs. Tso et al.14 proposed a noveldesign method of hybrid dynamic vibration absorber (HVA) tosuppress broadband vibration of the whole structure instead ofjust one point of the structure. Samani and Pellicano15 identi-fied the optimal locations of a DVA applied to a simple beamsubjected to regularly spaced and concentrated moving loads.The nonlinear stiffness of a nonlinear DVA possesses com-plex dynamic properties such as quasi-periodic, chaotic, andsub-harmonic responses. They discovered that the dampingeffects of a nonlinear non-symmetric dissipation DVA are su-perior to those of a linear DVA. In contrast, Wang and Lin16
used internal resonance contour plots (IRCPs) and flutter speedcontour plots (FSCPs) to analyze nonlinear dynamic stability.Their results demonstrated that changing the location of thedamper can prevent internal resonance and inhibit vibrationsin the main body. Wang and Kuo17 examined the vibrationsof a hinged-free nonlinear beam placed on a nonlinear elasticfoundation. The 1:3 internal resonance in the first and secondmodes of the beam was found. An optimal location of an at-tached DVA was proposed to prevent internal resonance andsuppress vibrations. Wang and Tu18 investigated the damp-ing effects of a tuned mass damper (TMD) on a fixed-free 3Dnonlinear beam resting on a nonlinear elastic foundation withexisting internal resonances. They proposed that the locationsof the maximum amplitudes between node points in the modeshapes and near the free end of the beam display the best damp-ing effect for the fixed-free 3D beam. In the works of Wangand Kuo17 and Wang and Tu,18 the nonlinear geometry andnonlinear inertia were included in these two nonlinear beams.The conditions to trig the internal resonance were also inves-tigated in their researches. The various beam boundary condi-tions were examined for better understanding the DVA damp-ing effects on beam vibrations. Wang and Liang19 investigatedthe damping effects of vibration absorbers with a lumped masson a hinged-hinged beam. This kind of vibration absorber isable to mitigate vibrations in mechanical or civil engineeringstructures on an elastic foundation; however, they are not ap-plicable to all suspension systems. It is this shortcoming thatwe sought to address in this research.
Further consideration of resonance analysis of structures us-ing the analytical and numerical strategies can be found in thework of Ansari et al.20 They derived a geometrically nonlinearbeam model to simulate the nanobeam vibration under ther-mal environment. The material properties of nanobeams varythrough the thickness direction with functionally graded (FG)distribution. The method of multiple scales was applied to de-
termine the frequency response of the temperature and materialproperties. A 2-DOF nonlinear system (the main body and thecontroller (absorber)) with quadratic and cubic nonlinearitiesunder external and parametric bounded excitations was con-sidered in the work of Sayed and Kamel.21 The method ofmultiple scales was applied to determine the amplitudes andphases in the existence of internal resonance. The stabilityof the controlled system was studied. Ghayesh et al.22 in-vestigated the axially moving beam with a three-to-one inter-nal resonance. The coupled longitudinal and transverse dis-placements of the beam were studied numerically. The fre-quency response of the nonlinear forced system was solvedby the pseudo-arc length continuation method and direct timeintegration. Ghayesh and Amabili23 provided a detailed dis-cussion of an axially moving Timoshenko beam with an inter-nal resonance. They used the Galerkins scheme to discretizethe nonlinear partial differential equations into a set of non-linear ordinary differential equations. The pseudo-arc lengthcontinuation method was also employed to find the freq