This paper presents an analytical solution for the forced vibration of beams carrying a number of two degrees-of-freedom (DOF) spring–damper–mass (SDM) systems. The beam is divided into a series of distinct sub-beams at the spring connection points and the point of force action. The 2DOF SDM systems are replaced with a set of effective springs with complex stiffness, and the compatibility of the placement and the force at the common interface of two adjacent sub-beams is systematically organized. Then, the boundary conditions are enforced, and the governing matrix equation is formulated. Next, the closed-form expression for the frequency response function (FRF) was determined analytically. The presented method can simultaneously consider arbitrary boundary conditions and any number of 2DOF SDM systems. Furthermore, regardless of the number of subsystems, none of the associated matrices is larger than 4 × 4, which provides a significant computational advantage. To validate the accuracy and reliability of the proposed method, some results are compared with the corresponding results obtained using the conventional finite element method (FEM); good agreement is observed between the results of the two approaches. Finally, the effects of the system parameters on the vibration transmission in the beam and subsystem vibration reduction are studied.
Issue Section:
Research Papers
References
1.
Wu
, J. S.
, and Chou
, H. M.
, 1998
, “Free Vibration Analysis of a Cantilever Beam Carrying Any Number of Elastically Mounted Point Masses With the Analytical-and-Numerical-Combined Method
,” J. Sound Vib.
, 213
(2
), pp. 317
–332
.2.
Wu
, J. S.
, and Chou
, H. M.
, 1999
, “A New Approach for Determining the Natural Frequencies and Mode Shapes of a Uniform Beam Carrying Any Number of Sprung Masses
,” J. Sound Vib.
, 220
(3
), pp. 451
–468
.3.
Lin
, H. Y.
, and Tsai
, Y. C.
, 2007
, “Free Vibration Analysis of a Uniform Multi-Span Beam Carrying Multiple Spring–Mass Systems
,” J. Sound Vib.
, 302
(3
), pp. 442
–456
.4.
Wu
, J. S.
, and Hsu
, T. F.
, 2007
, “Free Vibration Analyses of Simply Supported Beams Carrying Multiple Point Masses and Spring-Mass Systems With Mass of Each Helical Spring Considered
,” Int. J. Mech. Sci.
, 49
(7
), pp. 834
–852
.5.
Banerjee
, J. R.
, 2012
, “Free Vibration of Beams Carrying Spring-Mass Systems-A Dynamic Stiffness Approach
,” Comput. Struct.
, 104–105
, pp. 21
–26
.6.
Wu
, J. S.
, and Chang
, B. H.
, 2013
, “Free Vibration of Axial-Loaded Multi-Step Timoshenko Beam Carrying Arbitrary Concentrated Elements Using Continuous-Mass Transfer Matrix Method
,” Eur. J. Mech. A: Solids
, 38
, pp. 20
–37
.7.
Torabi
, K.
, Jazi
, A. J.
, and Zafari
, E.
, 2014
, “Exact Closed Form Solution for the Analysis of the Transverse Vibration Modes of a Timoshenko Beam With Multiple Concentrated Masses
,” Appl. Math. Comput.
, 238
, pp. 342
–357
.8.
Jen
, M. U.
, and Magrab
, E. B.
, 1993
, “Natural Frequencies and Mode Shapes of Beams Carrying a Two Degree-of-Freedom Spring-Mass System
,” ASME J. Vib. Acoust.
, 115
(2
), pp. 202
–209
.9.
Wu
, J. J.
, 2002
, “Alternative Approach for Free Vibration of Beams Carrying a Number of Two-Degree of Freedom Spring-Mass Systems
,” J. Struct. Eng.
, 128
(12
), pp. 1604
–1616
.10.
Wu
, J. J.
, 2006
, “Use of Equivalent Mass Method for Free Vibration Analyses of a Beam Carrying Multiple Two-dof-Spring–Mass Systems With Inertia Effect of the Helical Springs Considered
,” Int. J. Numer. Methods Eng.
, 65
(5
), pp. 653
–678
.11.
Wu
, J. J.
, 2004
, “Free Vibration Analysis of Beams Carrying a Number of 2DOF Spring-Damper-Mass Systems
,” Finite Elem. Anal. Des.
, 40
(4
), pp. 363
–381
.12.
Wu
, J. J.
, 2005
, “Use of Equivalent-Damper Method for Free Vibration Analysis of a Beam Carrying Multiple Two Degree-of-Freedom Spring–Damper–Mass Systems
,” J. Sound Vib.
, 281
(1
), pp. 275
–293
.13.
Chen
, D. W.
, 2006
, “The Exact Solution for Free Vibration of Uniform Beams Carrying Multiple 2DOF Spring–Mass Systems
,” J. Sound Vib.
, 295
(1
), pp. 342
–361
.14.
Cha
, P. D.
, 2007
, “Free Vibration of a Uniform Beam With Multiple Elastically Mounted 2DOF Systems
,” J. Sound Vib.
, 307
(1
), pp. 386
–392
.15.
Chang
, T. P.
, and Chang
, C. Y.
, 1998
, “Vibration Analysis of Beams With a Two Degree-of-Freedom Spring-Mass System
,” Int. J. Solids Struct.
, 35
(5
), pp. 383
–401
.16.
Bambill
, D. V.
, and Rossit
, C. A.
, 2002
, “Forced Vibrations of a Beam Elastically Restrained Against Rotation and Carrying a Spring–Mass System
,” Ocean Eng.
, 29
(6
), pp. 605
–626
.17.
Muscolino
, G.
, Benfratello
, S.
, and Sidoti
, A.
, 2002
, “Dynamics Analysis of Distributed Parameter System Subjected to a Moving Oscillator With Random Mass, Velocity and Acceleration
,” Probab. Eng. Mech.
, 17
(1
), pp. 63
–72
.18.
Mohebpour
, S. R.
, Fiouz
, A. R.
, and Ahmadzadeh
, A. A.
, 2011
, “Dynamic Investigation of Laminated Composite Beams With Shear and Rotary Inertia Effect Subjected to the Moving Oscillators Using FEM
,” Compos. Struct.
, 93
(3
), pp. 1118
–1126
.19.
Nguyen
, K. V.
, 2013
, “Comparison Studies of Open and Breathing Crack Detections of a Beam-Like Bridge Subjected to a Moving Vehicle
,” Eng. Struct.
, 51
, pp. 306
–314
.20.
Lu
, Z. R.
, and Liu
, J. K.
, 2013
, “Parameters Identification for a Coupled Bridge-Vehicle System With Spring-Mass Attachments
,” Appl. Math. Comput.
, 219
(17
), pp. 9174
–9186
.21.
Zhu
, D. Y.
, Zhang
, Y. H.
, and Ouyang
, H.
, 2015
, “A Linear Complementarity Method for Dynamic Analysis of Bridges Under Moving Vehicles Considering Separation and Surface Roughness
,” Comput. Struct.
, 154
, pp. 135
–144
.22.
Manikanahally
, D. N.
, and Crocker
, M. J.
, 1991
, “Vibration Absorbers of Hysteretically Damped Mass-Load Beams
,” ASME J. Vib. Acoust.
, 113
(1
), pp. 116
–122
.23.
Wu
, J. J.
, 2006
, “Study on the Inertia Effect of Helical Spring of the Absorber on Suppressing the Dynamic Responses of a Beam Subjected to a Moving Load
,” J. Sound Vib.
, 297
(3
), pp. 981
–999
.24.
Wong
, W. O.
, Tang
, S. L.
, Cheung
, Y. L.
, and Cheng
, L.
, 2007
, “Design of a Dynamic Vibration Absorber for Vibration Isolation of Beams Under Point or Distributed Loading
,” J. Sound Vib.
, 301
(3
), pp. 898
–908
.25.
Febbo
, M.
, and Vera
, S. A.
, 2008
, “Optimization of a Two Degree of Freedom System Acting as a Dynamic Vibration Absorber
,” ASME J. Vib. Acoust.
, 130
(1
), p. 011013
.26.
Tursun
, M.
, and Eşkinat
, E.
, 2014
, “H2 Optimization of Damped-Vibration Absorbers for Suppressing Vibrations in Beams With Constrained Minimization
,” ASME J. Vib. Acoust.
, 136
(2
), p. 021012
.27.
Pai
, P. F.
, Peng
, H.
, and Jiang
, S.
, 2014
, “Acoustic Metamaterial Beams Based on Multi-Frequency Vibration Absorbers
,” Int. J. Mech. Sci.
, 79
, pp. 195
–205
.28.
Wu
, J. S.
, and Chen
, D. W.
, 2000
, “Dynamic Analysis of a Uniform Cantilever Beam Carrying a Number of Elastically Mounted Point Masses With Dampers
,” J. Sound Vib.
, 229
(3
), pp. 549
–578
.29.
Zhang
, Z.
, Huang
, X.
, Zhang
, Z.
, and Hua
, H.
, 2014
, “On the Transverse Vibration of Timoshenko Double-Beam Systems Coupled With Various Discontinuities
,” Int. J. Mech. Sci.
, 89
, pp. 222
–241
.30.
Bathe
, K. J.
, 1982
, Finite Element Procedure in Engineering Analysis
, Prentice-Hall
, Englewood Cliffs, NJ
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