Transverse free vibration analysis of a tapered Timoshenko beam on visco-Pasternak foundations using the interpolating matrix method
|
摘要
The characteristics of transverse free vibration of a tapered Timoshenko beam under an axially conservative compression resting on visco-Pasternak foundations are investigated by the interpolating matrix method. The research is executed in view of a three-parameter foundation which includes the effects of the Winkler coefficient, Pasternak coefficient and damping coefficient of the elastic medium. The governing equations of free vibration of a non-prismatic Timoshenko beam under an axially conservative force resting on visco-Pasternak foundations are transformed into ordinary differential equations with variable coefficients in light of the bending rotation angle and transverse displacement. All the natural frequencies orders together with the corresponding mode shapes of the beam are calculated at the same time, and a good convergence and accuracy of the proposed method is verified through two numerical examples. The influences of foundation mechanical characteristics together with rotary inertia and shear deformation on natural frequencies of the beam with different taper ratios are analyzed. A comprehensive parametric numerical study is carried out emphasizing the primary parameters that describe the dynamic property of the beam.
The characteristics of transverse free vibration of a tapered Timoshenko beam under an axially conservative compression resting on visco-Pasternak foundations are investigated by the interpolating matrix method. The research is executed in view of a three-parameter foundation which includes the effects of the Winkler coefficient, Pasternak coefficient and damping coefficient of the elastic medium. The governing equations of free vibration of a non-prismatic Timoshenko beam under an axially conservative force resting on visco-Pasternak foundations are transformed into ordinary differential equations with variable coefficients in light of the bending rotation angle and transverse displacement. All the natural frequencies orders together with the corresponding mode shapes of the beam are calculated at the same time, and a good convergence and accuracy of the proposed method is verified through two numerical examples. The influences of foundation mechanical characteristics together with rotary inertia and shear deformation on natural frequencies of the beam with different taper ratios are analyzed. A comprehensive parametric numerical study is carried out emphasizing the primary parameters that describe the dynamic property of the beam.
The characteristics of transverse free vibration of a tapered Timoshenko beam under an axially conservative compression resting on visco-Pasternak foundations are investigated by the interpolating matrix method. The research is executed in view of a three-parameter foundation which includes the effects of the Winkler coefficient, Pasternak coefficient and damping coefficient of the elastic medium. The governing equations of free vibration of a non-prismatic Timoshenko beam under an axially conservative force resting on visco-Pasternak foundations are transformed into ordinary differential equations with variable coefficients in light of the bending rotation angle and transverse displacement. All the natural frequencies orders together with the corresponding mode shapes of the beam are calculated at the same time, and a good convergence and accuracy of the proposed method is verified through two numerical examples. The influences of foundation mechanical characteristics together with rotary inertia and shear deformation on natural frequencies of the beam with different taper ratios are analyzed. A comprehensive parametric numerical study is carried out emphasizing the primary parameters that describe the dynamic property of the beam.
引文
Auciello NM and De Rosa MA(2004),“Two Approaches to the Dynamic Analysis of Foundation Beams Subjected to Subtangential Forces,”Computers and Structures82(6):519-524.
?alim FF(2009),“Dynamic Analysis of Beams on Viscoelastic Foundation,”European Journal o Mechanics A/Solids,28(3):469-476.
CaliòI and Greco A(2013),“Free Vibrations of Timoshenko Beam-columns on Pasternak Foundations,”Journal of Vibration and Control,19(5):686-696.
Chen Changnew(2000),“Vibration of Prismatic Beam on an Elastic Foundation by the Differential Quadrature Element Method,”Computers and Structures,77(1)1-9.
Chen Changnew(2001),“DQEM Vibration Analysis of Non-prismatic Shear Deformable Beams Resting on Elastic Foundations,”American Society of Mechanica Engineers,430:11-20.
Chen Liqun,Peng Li,Zhang Aqiang and Ding Hu(2015)“Transverse Vibration of Viscoelastic Timoshenko Beam-columns,”Journal of Vibration and Control23(10):1572-1584.
Clementi F,Demeio L,Mazzilli CEN and Lenci S(2015),“Nonlinear Vibrations of Non-Uniform Beams by the MTS Asymptotic Expansion Method,”Continuum Mechanics and Thermodynamics,27(4-5):703-717.
De Rosa MA(1995),“Free Vibration of Timoshenko Beams on Two-Parameter Elastic Foundation,”Computers and Structures,57:151-156.
Ding Hu,Chen Liqun and Yang Shaopu(2012)“Convergence of Galerkin Truncation for Dynamic Response of Finite Beams on Nonlinear Foundations under a Moving Load,”Journal of Sound and Vibration331(10):2426-2442.
Engel RS(1991),“Dynamic Stability of an Axially Loaded Beam on an Elastic Foundation with Damping,”Journal of Sound and Vibration,146(3):463-477.
Kargarnovin MH and Younesian D(2004),“Dynamics of Timoshenko Beams on Pasternak Foundation under Moving Load,”Mechanics Research Communications31(6):713-723.
Lee JW and Lee JY(2016),“Free Vibration Analysis using the Transfer-matrix Method on a Tapered Beam,”Computers and Structures,164:75-82.
Lenci S,Clementi F,and Mazzilli CEN(2013),“Simple Formulas for the Natural Frequencies of Non-Uniform Cables and Beams,”International Journal of Mechanica Sciences,77:155-163.
Leung AYT,Zhou WE,Lim CW,Yuen KK and Lee U(2001),“Dynamic Stiffness for Piecewise NonUniform Timoshenko Column by Power Series-part IConservative Axial Force,”International Journal for Numerical Methods in Engineering,51(5):505-529.
Magrab EB(2007),“Natural Frequencies and Mode Shapes of Timoshenko Beams with Attachments,”Journal of Vibration and Control,13(7):905-934.
Mahmoud AA,Awadalla R and Nassar MM(2011)“Free vibration of Non-Uniform Column Using DQM,”Mechanics Research Communications,38(6):443-448.
Malekzadeh P,Karami G and Farid M(2003),“DQEMfor Free Vibration Analysis of Timoshenko Beams on Elastic Foundations,”Computational Mechanics,31(3-4):219-228.
Mohammadimehr M,Monajemi AA and Moradi M(2015),“Vibration Analysis of Viscoelastic Tapered Micro-rod Based on Strain Gradient Theory Resting on Visco-Pasternak Foundation using DQM,”Journal o Mechanical Science and Technology,29(6)2297-2305.
Morfidis K(2010),“Vibration of Timoshenko Beams on Three-parameter Elastic Foundation,”Computers and Structures,88(5-6):294-08.
Niu Zhongrong(1993),“Interpolating Matrix Method for Multi-point BVPs and Its Error Analysis,”Chinese Journal of Computational Physics,10(3)336-344.
Niu Zhongrong,Ge Dali,Cheng Changzhen and Hu Zongjun(2008),“Determining Stress Singularity Exponents of Plane V-Notches in Bonded Biomaterial Journal of University of Science and Technology o China,”38(3):314-319.
Peng Li,Ding Hu and Chen Liqun(2014),“Transverse Free Vibration of a Beam Resting on a Three-parameter Viscoelastic Foundation,”Journal of Vibration and Shock,33(1):101-105.(in Chinese)
Pradhan M and Dash PR(2016),“Stability of an Asymmetric Tapered Sandwich Beam Resting on a Variable Pasternak Foundation Subjected to a Pulsating Axial Load with Thermal Gradient,”Composite Structures,140:816-834.
Pradhan M,Dash PR and Pradhan PK(2016),”Static and Dynamic Stability Analysis of an Asymmetric Sandwich Beam Resting on a Variable Pasternak Foundation Subjected to Thermal Gradient,”Meccanica,51(3):725-739.
Reza A,Ahmad S and Shabnam JS(2010),“Application of Differential Transform in Free Vibration Analysis of Timoshenko Beams Resting on Two-parameter Elastic Foundation,”The Arabian Journal for Science and Engineering,35(2B):12-132.
Ruta P(2006),“Application of Chebyshev Polynomials to the Solution of the Non-prismatic Timoshenko Beam Vibration Problem,”Journal of Sound and Vibration296:243-263.
Tang AY,Wu JX,Li X,and Lee KY(2014),“Exac Frequency Equations of Free Vibration of Exponentially Non-uniform Functionally Graded Timoshenko Beams,”International Journal Mechanical Sciences,89:1-11.
Valsangkar AJ and Pradhanang R(1988),“Vibrations of Beam-columns on Two-parameter Elastic Foundations,”Earthquake Engineering and Structural Dynamics,16217-225.
Yokoyama T(1996),“Vibrations Analysis of Timoshenko Beam-columns on Two-parameter Elastic Foundations,”Computers and Structures,61(6):995-1007.
?alim FF(2009),“Dynamic Analysis of Beams on Viscoelastic Foundation,”European Journal o Mechanics A/Solids,28(3):469-476.
CaliòI and Greco A(2013),“Free Vibrations of Timoshenko Beam-columns on Pasternak Foundations,”Journal of Vibration and Control,19(5):686-696.
Chen Changnew(2000),“Vibration of Prismatic Beam on an Elastic Foundation by the Differential Quadrature Element Method,”Computers and Structures,77(1)1-9.
Chen Changnew(2001),“DQEM Vibration Analysis of Non-prismatic Shear Deformable Beams Resting on Elastic Foundations,”American Society of Mechanica Engineers,430:11-20.
Chen Liqun,Peng Li,Zhang Aqiang and Ding Hu(2015)“Transverse Vibration of Viscoelastic Timoshenko Beam-columns,”Journal of Vibration and Control23(10):1572-1584.
Clementi F,Demeio L,Mazzilli CEN and Lenci S(2015),“Nonlinear Vibrations of Non-Uniform Beams by the MTS Asymptotic Expansion Method,”Continuum Mechanics and Thermodynamics,27(4-5):703-717.
De Rosa MA(1995),“Free Vibration of Timoshenko Beams on Two-Parameter Elastic Foundation,”Computers and Structures,57:151-156.
Ding Hu,Chen Liqun and Yang Shaopu(2012)“Convergence of Galerkin Truncation for Dynamic Response of Finite Beams on Nonlinear Foundations under a Moving Load,”Journal of Sound and Vibration331(10):2426-2442.
Engel RS(1991),“Dynamic Stability of an Axially Loaded Beam on an Elastic Foundation with Damping,”Journal of Sound and Vibration,146(3):463-477.
Kargarnovin MH and Younesian D(2004),“Dynamics of Timoshenko Beams on Pasternak Foundation under Moving Load,”Mechanics Research Communications31(6):713-723.
Lee JW and Lee JY(2016),“Free Vibration Analysis using the Transfer-matrix Method on a Tapered Beam,”Computers and Structures,164:75-82.
Lenci S,Clementi F,and Mazzilli CEN(2013),“Simple Formulas for the Natural Frequencies of Non-Uniform Cables and Beams,”International Journal of Mechanica Sciences,77:155-163.
Leung AYT,Zhou WE,Lim CW,Yuen KK and Lee U(2001),“Dynamic Stiffness for Piecewise NonUniform Timoshenko Column by Power Series-part IConservative Axial Force,”International Journal for Numerical Methods in Engineering,51(5):505-529.
Magrab EB(2007),“Natural Frequencies and Mode Shapes of Timoshenko Beams with Attachments,”Journal of Vibration and Control,13(7):905-934.
Mahmoud AA,Awadalla R and Nassar MM(2011)“Free vibration of Non-Uniform Column Using DQM,”Mechanics Research Communications,38(6):443-448.
Malekzadeh P,Karami G and Farid M(2003),“DQEMfor Free Vibration Analysis of Timoshenko Beams on Elastic Foundations,”Computational Mechanics,31(3-4):219-228.
Mohammadimehr M,Monajemi AA and Moradi M(2015),“Vibration Analysis of Viscoelastic Tapered Micro-rod Based on Strain Gradient Theory Resting on Visco-Pasternak Foundation using DQM,”Journal o Mechanical Science and Technology,29(6)2297-2305.
Morfidis K(2010),“Vibration of Timoshenko Beams on Three-parameter Elastic Foundation,”Computers and Structures,88(5-6):294-08.
Niu Zhongrong(1993),“Interpolating Matrix Method for Multi-point BVPs and Its Error Analysis,”Chinese Journal of Computational Physics,10(3)336-344.
Niu Zhongrong,Ge Dali,Cheng Changzhen and Hu Zongjun(2008),“Determining Stress Singularity Exponents of Plane V-Notches in Bonded Biomaterial Journal of University of Science and Technology o China,”38(3):314-319.
Peng Li,Ding Hu and Chen Liqun(2014),“Transverse Free Vibration of a Beam Resting on a Three-parameter Viscoelastic Foundation,”Journal of Vibration and Shock,33(1):101-105.(in Chinese)
Pradhan M and Dash PR(2016),“Stability of an Asymmetric Tapered Sandwich Beam Resting on a Variable Pasternak Foundation Subjected to a Pulsating Axial Load with Thermal Gradient,”Composite Structures,140:816-834.
Pradhan M,Dash PR and Pradhan PK(2016),”Static and Dynamic Stability Analysis of an Asymmetric Sandwich Beam Resting on a Variable Pasternak Foundation Subjected to Thermal Gradient,”Meccanica,51(3):725-739.
Reza A,Ahmad S and Shabnam JS(2010),“Application of Differential Transform in Free Vibration Analysis of Timoshenko Beams Resting on Two-parameter Elastic Foundation,”The Arabian Journal for Science and Engineering,35(2B):12-132.
Ruta P(2006),“Application of Chebyshev Polynomials to the Solution of the Non-prismatic Timoshenko Beam Vibration Problem,”Journal of Sound and Vibration296:243-263.
Tang AY,Wu JX,Li X,and Lee KY(2014),“Exac Frequency Equations of Free Vibration of Exponentially Non-uniform Functionally Graded Timoshenko Beams,”International Journal Mechanical Sciences,89:1-11.
Valsangkar AJ and Pradhanang R(1988),“Vibrations of Beam-columns on Two-parameter Elastic Foundations,”Earthquake Engineering and Structural Dynamics,16217-225.
Yokoyama T(1996),“Vibrations Analysis of Timoshenko Beam-columns on Two-parameter Elastic Foundations,”Computers and Structures,61(6):995-1007.