Lumped parameter models representing impedance functions at the interface of a rod on a viscoelastic medium

Abstract In this study, a lumped parameter model that properly simulates the impedance characteristics at the extremity of a uniform, isotropic, homogeneous rod on a viscoelastic medium is proposed. The lumped parameter model consists of springs, dashpots, and so-called “gyro-mass elements”. The gyro-mass elements generate a reaction force proportional to the relative acceleration of the nodes between which they are placed. This model consists of units arranged in series, each unit consisting of a spring, a damper, and a gyro-mass element arranged in parallel. A formula is proposed for determining the properties of the elements in the units calculated from a closed-form solution based on a modal expansion. The impedance function simulated by the proposed model shows good agreement with the rigorous impedance function derived from the differential equation of motion of the rod. The results obtained by employing this model in some example applications show that the accuracy of the model is appreciably high when compared with conventional finite element models. A great advantage of this model is that a significant reduction of the number of degrees of freedom can be achieved for solving recent vibration problems with high-frequency excitations, such as ultrasonic vibrations.

[1]  J. W. Tedesco,et al.  Structural Dynamics: Theory and Applications , 1998 .

[2]  Isaac Elishakoff,et al.  Constructing the axial stiffness of longitudinally vibrating rod from fundamental mode shape , 2001 .

[3]  Milos Novak,et al.  Dynamic Stiffness and Damping of Piles , 1974 .

[4]  J. F. Howell,et al.  TORSIONAL VIBRATION ON PILE FOUNDATIONS , 1977 .

[5]  Lars Vabbersgaard Andersen,et al.  Assessment of lumped-parameter models for rigid footings , 2010 .

[6]  Nicos Makris,et al.  DYNAMIC PILE-SOIL-PILE INTERACTION. PART I: ANALYSIS OF AXIAL VIBRATION , 1991 .

[7]  M. Gürgöze,et al.  Continuous and discrete models for longitudinally vibrating elastic rods viscously damped in-span , 2002 .

[8]  Milos Novak.  Vertical Vibration of Floating Piles , 1977 .

[9]  J. Enrique Luco,et al.  Discrete models for vertical vibrations of surface and embedded foundations , 1990 .

[10]  W. Thomson.  Theory of vibration with applications , 1965 .

[11]  Wen-Hwa Wu,et al.  Systematic lumped‐parameter models for foundations based on polynomial‐fraction approximation , 2002 .

[12]  N. A. Hizal,et al.  LUMPED PARAMETER REPRESENTATION OF A LONGITUDINALLY VIBRATING ELASTIC ROD VISCOUSLY DAMPED IN-SPAN , 1998 .

[13]  Toyoaki Nogami,et al.  Dynamic Soil Reactions for Plane Strain Case , 1978 .

[14]  Dikai Liu,et al.  Exact solutions for longitudinal vibration of rods coupled by translational springs , 2000 .

[15]  Milos Novak,et al.  Soil‐pile interaction in vertical vibration , 1976 .

[16]  Masato Saitoh,et al.  Simple Model of Frequency-Dependent Impedance Functions in Soil-Structure Interaction Using Frequency-Independent Elements , 2007 .

[17]  H. Tajimi,et al.  Dynamic analysis of structure embedded in elastic stratum , 1969 .

[18]  S. Inceoglu,et al.  On the vibrations of an axially vibrating elastic rod with distributed mass added in-span , 2000 .

[19]  John P. Wolf,et al.  Spring‐dashpot‐mass models for foundation vibrations , 1997 .