• Structural Engineering and Mechanics
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• v.40 no.5
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• pp.689-703
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• 2011

A new perturbation method is introduced to study the undamped free vibration of a non-prismatic Timoshenko beam for its natural frequencies and vibration modes. For simplicity, the natural modes of vibration of its corresponding prismatic Euler-Bernoulli beam with the same length and boundary conditions are used as Ritz base functions with necessary modifications to account for shear strain in the Timoshenko beam. The new method can transform two coupled partial differential equations governing the transverse vibration of the non-prismatic Timoshenko beam into a set of nonlinear algebraic equations. It significantly simplifies the solution process and is applicable to non-prismatic beams with various boundary conditions. Three examples indicated that the new method is more accurate than the previous perturbation methods. It successfully takes into account the effect of shear deformation of Timoshenko beams particularly at the free end of cantilever structures.

• Transactions of the Korean Society for Noise and Vibration Engineering
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• v.15 no.11 s.104
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• pp.1268-1275
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• 2005

Vibration in Engine module could be reduced by introducing a balance shaft module which has one or more unbalanced rotors. Since the unbalanced rotor is installed in an opposite direction of the free force or unbalance moment by engine component, the unexpected vibration could be decreased kinematically. The essential equation of the unbalanced rotor was Presented for two cases, 3 in-line and 4 in-line cylinder engine type, And the efficiency of the balance shaft is investigated by the vehicle testing that is focused on measuring the reduced vibration level when adapting a balancing module. With the signal processing of measured signals, some important issues on design the balancing shaft could be derived and the overall design process is explained in the final part including the peripheral component, i.e. housing and bush.

• Journal of Welding and Joining
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• v.30 no.2
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• pp.65-69
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• 2012

Environmental and health concerns over the lead have led to investigation of the alternative Pb-free solders to replace commonly used Pb-Sn solders in microelectronic packaging application. The leading candidates for lead-free solder alloys are presently the near eutectic Sn-Ag-Cu alloys. Therefore, extensive studies on reliability related with the composition have been reported. However, the insufficient drop property of the near eutectic Sn-Ag-Cu alloys has demanded solder compositions of low Ag content. In addition, the solder interconnections in automobile applications like a smart box require significantly improved vibration resistance. Therefore, this study investigated the effect of alloying elements (Ag, Bi, In) on the vibration fatigue strength. The vibration fatigue was conducted in 10~1000Hz frequency and 20Grms. The interface of the as-soldered cross section close to the Cu pad indicated the intermetallic compound ($Cu_6Sn_5$) regardless of solder composition. The type and thickness of IMC was not significantly changed after the vibration test. It indicates that no thermal activities occurred significantly during vibration. Furthermore, as a function of alloying composition, the vibration crack path was investigated with a focus on the IMCs. Vibration crack was initiated from the fillet surface of the heel for QFP parts and from the plating layer of chip parts. Regardless of the solder composition, the crack during a vibration test was propagated as same as that during a thermal fatigue test.

• Journal of the Computational Structural Engineering Institute of Korea
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• v.33 no.1
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• pp.9-16
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• 2020

The objective of this study is to propose closed-form solutions to the free vibration response of single-degree-of-freedom (SDOF) systems, as part of fundamental research on dynamic systems with Coulomb friction. The motion of a dynamic system with Coulomb friction is described by a nonlinear differential equation, and, due to the variation in the sign of friction force term with the direction of motion, it is difficult to obtain the closed-form solution. To solve this problem, the nonlinear differential equation is directly computed by numerical integration, or an approximated solution is indirectly obtained using a linear differential equation wherein the damping effect due to Coulomb friction is replaced by an equivalent viscous damping term. However, these conventional methods do not provide a closed-form solution from a mathematical point of view. In this regard, closed-form solutions to the free vibration response of SDOF systems with Coulomb friction are derived herein by considering that the sign of the friction force term is reversed in each half-cycle of motion and by expanding it to the entire time history using the power series function. In addition, for a given initial condition, both the number of free vibration half-cycles and the response at the instant when free vibration motion stops are predicted under the condition that the motion of free vibration is stopped when the amplitude of the friction force is higher than that of the restoring force due to stiffness.

• Proceedings of the Korean Geotechical Society Conference
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• 2005.03a
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• pp.1148-1158
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• 2005

This work reports results of our study on the dynamic responses of the buried pipelines both along the axial and the transverse directions under various boundary end conditions. We have considered three cases, i.e., the free ends, the fixed ends, and the fixed-free ends. We have studied the seismic responses of the buried pipelines with the various boundary end conditions both along the axial and the transverse direction. We have considered three cases, i.e., the free ends, the fixed ends, and the fixed-free ends for the axial direction, and three more cases including the guided ends, the simply supported ends, and the supported-guided ends for the transverse direction. The buried pipelines are modeled as beams on elastic foundation while the seismic waves as a ground displacement in the form of a sinusoidal wave. The natural frequency and its mode, and the effect of parameters have been interpreted in terms of free vibration. The natural frequency varies most significantly by the soil stiffness and the length of the buried pipelines in the case of free vibration, which increases with increasing soil stiffness and decreases with increasing length of the buried pipeline. Such a behavior appears most prominently along the axial rather than the transverse direction of the buried pipelines. The resulting frequencies and the mode shapes obtained from the free vibration for the various boundary end conditions of the pipelines have been utilized to derive the mathematical formulae for the displacements and the strains along the axial direction, and the displacements and the bending strains along the transverse direction in case of the forced vibration. The negligibly small difference of 6.2% between our result and that of Ogawa et. al. (2001) for the axial strain with a one second period confirms the accuracy of our approach in this study.

• Smart Structures and Systems
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• v.21 no.4
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• pp.493-519
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• 2018

This article presents a finite element (FE) model to assess the free vibration and static response of a functionally graded skew magneto-electro-elastic (FGSMEE) plate. Through the thickness material grading of FGSMEE plate is achieved using power law distribution. The coupled constitutive equations along with the total potential energy approach are used to develop the FE model of FGSMEE plate. The transformation matrix is utilized in bringing out the element matrix corresponding to the global axis to a local axis along the skew edges to specify proper boundary conditions. The effect of skew angle on the natural frequency of an FGSMEE plate is analysed. Further, the study includes the evaluation of the static behavior of FGSMEE plate for various skew angles. The influence of skew angle on the primary quantities such as displacements, electric potential, and magnetic potential, and secondary quantities such as stresses, electric displacement and magnetic induction is studied in detail. In addition, the effect of power-law gradient, thickness ratio, boundary conditions and aspect ratio on the free vibration and static response characteristics of FGSMEE plate has been investigated.

• Structural Engineering and Mechanics
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• v.62 no.1
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• pp.65-77
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• 2017

The free vibration analysis of fluid conveying Timoshenko pipeline with different boundary conditions using Differential Transform Method (DTM) and Adomian Decomposition Method (ADM) has not been investigated by any of the studies in open literature so far. Natural frequencies, modes and critical fluid velocity of the pipelines on different supports are analyzed based on Timoshenko model by using DTM and ADM in this study. At first, the governing differential equations of motion of fluid conveying Timoshenko pipeline in free vibration are derived. Parameter for the nondimensionalized multiplication factor for the fluid velocity is incorporated into the equations of motion in order to investigate its effects on the natural frequencies. For solution, the terms are found directly from the analytical solution of the differential equation that describes the deformations of the cross-section according to Timoshenko beam theory. After the analytical solution, the efficient and easy mathematical techniques called DTM and ADM are used to solve the governing differential equations of the motion, respectively. The calculated natural frequencies of fluid conveying Timoshenko pipelines with various combinations of boundary conditions using DTM and ADM are tabulated in several tables and figures and are compared with the results of Analytical Method (ANM) where a very good agreement is observed. Finally, the critical fluid velocities are calculated for different boundary conditions and the first five mode shapes are presented in graphs.

• Steel and Composite Structures
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• v.29 no.3
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• pp.363-377
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• 2018

An efficient and free of shear locking finite element model is developed and employed to study free vibration of tapered bidirectional functionally graded material (BFGM) beams. The beam material is assumed to be formed from four distinct constituent materials whose volume fraction continuously varies along the longitudinal and thickness directions by power-law functions. The finite element formulation based on the first-order shear deformation theory is derived by using hierarchical functions to interpolate the displacement field. In order to improve efficiency and accuracy of the formulation, the shear strain is constrained to constant and the exact variation of the cross-sectional profile is employed to compute the element stiffness and mass matrices. A comprehensive parametric study is carried out to highlight the influence of the material distribution, the taper and aspect ratios as well as the boundary conditions on the vibration characteristics. Numerical investigation reveals that the proposed model is efficient, and it is capable to evaluate the natural frequencies of BFGM beams by using a small number of the elements. It is also shown that the effect of the taper ratio on the fundamental frequency of the BFGM beams is significantly influenced by the boundary conditions. The present results are of benefit to optimum design of tapered FGM beam structures.

• Transactions of the Korean Society of Mechanical Engineers A
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• v.30 no.1 s.244
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• pp.18-25
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• 2006

The purpose of this research work is to demonstrate a successful application of hybrid-mixed formulation and nodeless degrees of freedom in developing a very accurate in-plane curved beam element for free vibration analysis. To resolve the numerical difficulties due to the spurious constraints, the present element, based on the Hellinger-Reissner variational principle and considering the effect of shear deformation, employed consistent stress parameters corresponding to cubic displacement polynomials with additional nodeless degrees. The stress parameters were eliminated by the stationary condition, and the nodeless degrees were condensed by Guyan Reduction. Several numerical examples indicated that the property of the mass matrix as well as that of the stiffness matrix have a great effect on the numerical performance. The element with consistent mass matrix produced best results on convergence and accuracy in the numerical analysis of Eigenvalue problems. Also, the higher-order mixed curved beam element showed a superior numerical behavior for the free vibration analyses.

• Steel and Composite Structures
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• v.24 no.1
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• pp.113-128
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• 2017

This paper highlights a study undertaken on the free vibration of a precast steel-concrete composite slab panel for track support. The steel-concrete composite slab track is an evolvement from the slab track, a form of ballastless track which is becoming increasingly attractive to asset owners as they seek to reduce lifecycle costs and deal with increasing rail traffic speeds. The slender nature of the slab panel due to its reduced depth of construction makes it susceptible to vibration problems. The aim of the study is driven by the need to address the limited research available to date on the dynamic behaviour of steel-concrete composite slab panels for track support. Free vibration analysis of the track slab has been carried out using ABAQUS. Both eigenfrequencies and eigenmodes have been extracted using the Lanczos method. The fundamental natural frequencies of the slab panel have been identified together with corresponding mode shapes. To investigate the sensitivity of the natural frequencies and mode shapes, parametric studies have been established, considering concrete strength and mass and steel's modulus of elasticity. This study is the world first to observe crossover phenomena that result in the inversion of the natural orders without interaction. It also reveals that replacement of the steel with aluminium or carbon fibre sheeting can only marginally reduce the natural frequencies of the slab panel.