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On the determination of natural frequencies of a cantilever beam in free bending vibration: a rigid multibody approach

Über die Bestimmung der natürlichen Frequenz von Konsolen bei freien Schwingungen: rigides Mehrkörpersystem

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Abstract

A new approximate method for the determination of natural frequencies of a cantilever beam in free bending vibration by a rigid multibody system is proposed. Uniform Euler-Bernoulli cantilever beams with and without a lumped mass at the tips are considered. The modelling method consists of two steps. In the first step, the cantilever beam is replaced by lumped masses interconnected by massless flexible beams. In the second step, the massless flexible beams are replaced by massless rigid beams connected through revolute and prismatic joints with corresponding springs in them. Elastic properties of the massless flexible beams are modelled by the springs introduced. The method proposed is compared with similar ones in the literature.

Zusammenfassung

Es wird eine neue Methode angezeigt, für die approximative Bestimmung der Kreisfrequenz der elastischen, transversal schwingenden Konsole. Diese Methode stützt sich auf die Theorie des Mehrkörpersystems. Es wird eine homogene Konsole betrachtet, mit und ohne konzentrierte Masse auf ihrer freien Ende. Diese Methode besteht aus zwei Schritten. In erstem Schritt wird die Traverse durch ein System von konzentrierten Massen, die gegenseitig mit elastischen, leichten Kolben verbunden sind, ersetzt. Im zweiten Schritt werden die elastischen Kolben durch leichte, steife Kolben ersetzt, die gegenseitig mit rotierenden und prismenförmigen Gelenken mit entsprechenden Federn verbunden sind. Durch diese Federn werden die elastischen Merkmale der leichten, flexiblen Kolben modelliert. Diese Methode wurde auch mit verwandten Methoden in der Literatur verglichen.

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Abbreviations

x, y, z :

Axes of the inertial coordinate frame

F :

Force [N]

M :

Torque of a couple [Nm]

u :

Deflection [m]

L :

Length [m]

E :

Young’s modulus of elasticity [N/m2]

I z :

Axial moment of inertia for the principal axis z [m4]

i, j, k :

Unit vectors of axes x, y, and z, respectively

u, u P :

Displacement vectors of points B and P, respectively

p i :

Local vector of the ith massless beam

K B , K P :

Stiffness matrices corresponding to points B and P

q i :

Displacement in the ith joint

c i :

Stiffness of the spring in the ith joint

m :

Mass of the beam [kg]

m i :

Mass of the particle M i [kg]

m T :

Lumped mass at the tip of the cantilever beam

n :

Number of equal parts

e i :

Unit vector of the axis of the ith joint

T :

Kinetic energy [J]

\(\mathbf{V}_{M_{\alpha}}\) :

Velocity of the particle M α

\(\mathbf{r}_{M_{\alpha}}\) :

Position vector of the particle M α

q :

Vector of generalized coordinates

q 0 :

Vector of zero values of generalized coordinates

\(\dot{\mathbf{q}}\), \(\ddot{\mathbf{q}}\) :

First and second time derivatives of the vector q

\(\mathbf{\overline{q}}_{1}\) :

Vector of odd generalized coordinates

\(\overline{\mathbf{q}}_{2}\) :

Vector of even generalized coordinates

\(\ddot{\mathbf{\overline{q}}_{1}}\), \(\ddot{\mathbf{\overline{q}}_{2}}\) :

Second time derivatives of the vectors \(\mathbf{\overline{q}}_{1}\) and \(\overline{\mathbf{q}}_{2}\)

M :

Mass matrix

m ij :

Components of the mass matrix M

K, \(\overline{\mathbf{K}}\), K ⋆ :

Stiffness matrices of the system

0 :

Zero matrix

I :

Identity matrix

\(\overline{\mathbf{M}}_{ij}\) :

Block matrices of the matrix \(\overline{\mathbf{M}}\)

\(\overline{\mathbf{K}}_{ij}\) :

Block matrices of the matrix \(\overline{\mathbf{K}}\)

\(\mathbf{K}_{ij}^{\star}\) :

Block matrices of the matrix K ⋆

\(\mathbf{T}_{\varLambda}, \mathbf{T}_{\varPi}, \tilde{\mathbf{B}}, \tilde{\tilde{\mathbf{B}}}\) :

Matrices

v :

Eigenvector

w i :

The ith root of the transcendental equation

θ :

Slope

Π :

Potential energy of the cantilever beam

Π c :

Potential energy of the system of springs in the joints

\(\overline{\chi}_{i}\), χ i :

Identifiers of the joint type

ω i :

The ith natural frequency [rad/s]

\(\overline{\omega}_{i}\) :

The ith non-dimensional natural frequency

T :

Matrix transposition operation

−1:

Inverse matrix operation

q 0 :

Zero values of generalized coordinates

References

  1. Schiehlen WO, Rauh J (1986) Modeling of flexible multibeam systems by rigid-elastic superelements. Rev Bras Cienc Mec 8(2):151–163

    Google Scholar 

  2. Rauh J, Schiehlen WO (1987) Various approaches for the modelling of flexible robot arms. In: Elishahoff I, Irretier H (eds) Refined dynamical theories of beams, plates and shells and their applications. Springer, Berlin, pp 420–429

    Google Scholar 

  3. Molenaarx DP, Sj D (1998) Modeling the structural dynamics of the lagerwey LW-50/750m wind turbine. Wind Eng 22(6):253–264

    Google Scholar 

  4. Zhao X, Maisser P, Wu J (2007) A new multibody modelling methodology for wind turbine structures using a cardanic joint beam element. Renew Energy 32:532–546

    Article  Google Scholar 

  5. Wang Y, Huston RL (1994) A lumped parameter method in the nonlinear analysis of flexible multibody systems. Comput Struct 50(3):421–432

    Article  MATH  Google Scholar 

  6. Plosa J, Wojciech S (2000) Dynamics of systems with changing configuration and with flexible beam-like links. Mech Mach Theory 35:1515–1534

    Article  MathSciNet  Google Scholar 

  7. Routh EJ (1905) Treatise on the dynamics of a system of rigid bodies, elementary part. Dover Publication Inc, New York

    Google Scholar 

  8. Dimentberg FM (1965) The screw calculus and its applications in mechanics. Nauka, Moscow (in Russian)

    Google Scholar 

  9. Meirovitch L (2001) Fundamentals of vibrations. McGraw-Hill, New York

    Google Scholar 

  10. Chernousko FL, Bolotnik NN, Gradetsky VG (1989) Manipulation robots: dynamics, control, optimization. Nauka, Moscow (in Russian)

    Google Scholar 

  11. Nikravesh PE (1988) Computer aided analysis of mechanical systems. Prentice Hall, Englewood Cliffs

    Google Scholar 

  12. Attia HA, El-Mistikawy TMA (2001) A matrix formulation for the dynamic analysis of planar mechanisms using point coordinates and velocity transformation. Eur J Mech A, Solids 20(6):981–995

    Article  MATH  Google Scholar 

  13. Chaudhary H, Saha SK (2008) Balancing of shaking forces and shaking moments for planar mechanisms using the equimomental systems. Mech Mach Theory 43(3):310–334

    Article  MATH  Google Scholar 

  14. Ding X, Selig JM, Zhan Q (2002) Study on dynamic modeling of compliant manipulators. In: 4th world congress on intelligent control and automation, Shanghai, P.R. China, pp 1347–1353

    Google Scholar 

  15. Ding X, Selig JM (2004) Lumped parameter dynamic modelling for the flexible manipulator. In: 5th world congress on intelligent control and automation, Hangzhou, P.R. China, pp 280–284

    Google Scholar 

  16. Przemineiecki JS (1968) Theory of matrix structural analysis. McGraw-Hill, New York

    Google Scholar 

  17. Čović V, Lazarević M (2009) Mechanics of robots. Faculty of Mechanical Engineering, University of Belgrade (in Serbian)

  18. Thomas M, Tesar D (1982) Dynamic modeling of serial manipulator arms. ASME J Dyn Syst Meas Control 104:218–228

    Article  Google Scholar 

  19. Lurie AI (2002) Analytical mechanics. Springer, New York

    MATH  Google Scholar 

  20. Belonozhko PA, Zhechev MM, Tarasov SV (1986) Mathematical modeling of the dynamics of a system of two bodies coupled by an elastic multiple-element chain. Int Appl Mech 22(7):683–688

    MATH  Google Scholar 

  21. Zhechev MM (2007) Equations of motion for singular systems of massed and massless bodies. J Multibody Dyn 221:591–597

    Google Scholar 

  22. Laura PAA, Pombo JL, Susemihl EA (1974) A note on the vibrations of a clamped-free beam with a mass at the free end. J Sound Vib 37(2):161–168

    Article  Google Scholar 

  23. Wu JS, Lin TL (1990) Free vibration analysis of a uniform cantilever beam with point masses by an analytical and numerical combined method. J Sound Vib 136(2):201–213

    Article  MathSciNet  MATH  Google Scholar 

  24. Sehgal S, Kumar H (2012) Structural dynamic analysis of cantilever beam structure. Int J Res Eng Appl Sci 2(2):1110–1114

    Google Scholar 

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Acknowledgements

This research was supported under grants no. TR35006 and no. ON174016 by the Ministry of Education, Science and Technological Development of the Republic of Serbia. This support is gratefully acknowledged.

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Authors and Affiliations

  1. Faculty of Mechanical and Civil Engineering in Kraljevo, University of Kragujevac, Dositejeva 19, 36000, Kraljevo, Serbia

    Slaviša Šalinić & Aleksandar Nikolić

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  1. Slaviša Šalinić
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  2. Aleksandar Nikolić
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Correspondence to Slaviša Šalinić.

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Šalinić, S., Nikolić, A. On the determination of natural frequencies of a cantilever beam in free bending vibration: a rigid multibody approach. Forsch Ingenieurwes 77, 95–104 (2013). https://doi.org/10.1007/s10010-013-0168-0

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