Skip to main content
SpringerLink
Log in

Calculation of natural frequencies of beam structures including concentrated mass effects by the dynamic stiffness matrix method

  • Scientific Reports
  • Published:
Materials and Structures Aims and scope Submit manuscript

Abstract

In this paper, the dynamic stiffness matrix method for a 2-node and 6-DOF (Degree Of Freedom) per node beam element is presented along with a numerical method to include the effect of concentrated masses. We treat as examples the case of free vibrations of beam structures with and without the concentrated masses effect. By means of a parametric study, we assess the quantitative effect of concentrated masses to the natural frequencies of the structure.

Résumé

Dans cet article, nous présentons la méthode des matrices de rigidité dynamique pour un élément poutre à 2 nœuds 6 DDL (Degrés De Libertés) par nœud ainsi qu'une méthode numérique pour modéliser des masses concentrées. Nous traitons le cas des vibrations libres de structures à poutres avec et sans prise en compte des masses ponctuelles. La variation des fréquences propres en fonction de l'accroissement quantitatif des masses concentrées a fait l'objet d'une étude paramétrique.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Abbreviations

[]:

matrix

{}:

vector

〈〉:

Dirac operator

A m2 :

cross sectional area of beam

DOF:

Degree Of Freedom

E N/m2 :

elastic modulus of beam

fi Hz:

eigenfrequency number i

G N/m2 :

shear modulus

i N/m2 :

imaginary unit i2=−1

IG m4 :

polar inertia moment

IY m4 :

inertia moment of the section A around 0y

IZ m4 :

inertia moment of the section A around 0z

J m4 :

torsional inertia

[Kd] m4 :

dynamic stiffness matrix

L m:

length of beam member

m kg:

mass of beam member

Mci kg:

concentrated mass at the node i

[Mc] kg:

overall concentrated mass matrix

n kg:

total number of degree of freedom

t s:

time

u(x,t) m:

dynamic axial displacement

U(x) m:

axial displacement

x,y,z m:

principle axes

v m:

Poisson's ratio

ϱ kg/m3 :

density

ω rad/s:

angular velocity

References

  1. Batoz, J. L. and Dhatt, G., ‘Structures modelization by finite element method—Beams and plates’, (only available in French), Vol. 2, (Hermès, Paris, 1990).

    Google Scholar 

  2. Banerjee, J. R. and Williams, F. W., ‘An exact dynamic stiffness matrix for coupled extensional-torsional vibration of structural members’,Computers & Structures 50 (2) (1994) 161–166.

    Article  MATH  Google Scholar 

  3. Henchi, K., ‘Dynamic analysis of bridges by finite element method under solicitations of mobile vehicles’, (only available in French), Ph. D. thesis UTC-MNM, France (1995).

    Google Scholar 

  4. Hoorpah, W., Henchi K. and Dhatt, G., ‘Exact calculation of vibration frequencies of beam structures by dynamic stiffness matrix method. Application to mixed bridges’, (only available in French),Construction Métallique 4 (1994) 19–41.

    Google Scholar 

  5. Banerjee, J. R., ‘Coupled bending-torsional dynamic stiffness matrix for beam elements’,Int. J. Num. Methods in Engng 28 (1989) 1283–1298.

    Article  MATH  Google Scholar 

  6. Eisenberger, M., ‘Vibration frequencies for beams on variable one-and-two-parameter elastic foundations’,J. Sound and Vibration 176 (5) (1994) 577–584.

    Article  MATH  Google Scholar 

  7. Lundblad, H. M., ‘Forced harmonic vibration of rotating beam systems in space analyses by use of exact finite elements’,Int. J. Num. Methods in Engng. 32 (1991)571–594.

    Article  Google Scholar 

  8. Srinivas, S., Joga, C. V. and Rao, A. K., ‘An exact analysis for vibration of simply-supported homogeneous and laminated thick rectangular plates’,J. Sound and Vibration 12 (2) (1970) 187–199.

    Article  MATH  Google Scholar 

  9. Wittrick, W. H. and Williams, F. W., ‘A general algorithm for computing natural frequencies of elastic structures’,Quart. J. Mechanics and Applied Math. 24 (3) (1971) 263–284.

    MATH  MathSciNet  Google Scholar 

  10. Gürgöze, M., ‘A note on the vibrations of restrained beams and rods with point masses’,J. Sound and Vibration 96 (4) (1984) 461–468.

    Article  MATH  Google Scholar 

  11. Wu, J. S. and Lin, T. L, ‘Free vibration analysis of a uniform cantilever beam with point masses by an analytical and numerical combined method’,J. Sound and Vibration 13 (2) (1990) 201–213.

    Article  MathSciNet  Google Scholar 

  12. Corn, S., Piranda, J. and Bouhaddi, N., ‘Simplification of finite element method models of the behaviour of beam structures by equivalent beam element’, (only available in French) in 3e Colloque National en Calcul des Structures, Giens, France, May 1997, 259–264.

Download references

Author information

Authors and Affiliations

  1. LMSC, IUP-GCI Université de Cergy-Pontoise, 5, Mail Gay Lussac, Neuville-Sur-Oise, 95031, Cergy-Pontoise Cedex, France

    R. Aït-Djaoud, I. Djeran-Maigre & R. Cabrillac

Authors
  1. R. Aït-Djaoud
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. I. Djeran-Maigre
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. R. Cabrillac
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Editorial Note Prof. Richard Cabrillac is a RILEM Senior Member.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Aït-Djaoud, R., Djeran-Maigre, I. & Cabrillac, R. Calculation of natural frequencies of beam structures including concentrated mass effects by the dynamic stiffness matrix method. Mat. Struct. 34, 71–75 (2001). https://doi.org/10.1007/BF02481554

Download citation

  • Received:

  • Accepted:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02481554

Keywords

Search

Navigation