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Initial undamped resonant frequency of slender structures considering nonlinear geometric effects: the case of a 60.8 m-high mobile phone mast


In this study, an analytical approach based on the Rayleigh method is adopted to calculate the first resonant frequency of a 60.80 m-high mobile phone mast system, by considering the geometric stiffness, functions of the concentrated forces, and self-weight of the structure. Such a technique is applicable to continuous systems with infinite degrees of freedom. However, it is important to bear in mind that actual structures are more complex than simple systems, such as beams and columns, because the properties of actual structures vary with their length. For comparison, a finite element method (FEM)-based computer simulation is performed. First, the axial forces on each segment of the structure are compared. Then, under geometric nonlinearity, the vibration frequency of the fundamental mode is calculated analytically. Finally, the structural stiffness is evaluated. The results of the analytical approach are found to differ slightly from those of the FEM-based model.

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Fig. 1
Fig. 2
Fig. 3


D :

External diameter, cm

E :

Elastic modulus of the material, N/m2

e :

Axial displacement

d :

Elementary, infinitesimal, internal diameter (cm)

F :

Force, N

f :

Frequency, Hz

g :

Gravitational acceleration, m/s2

I :

Moment of the section, m4

K :

Stiffness, N/m

L :

Length, cm

M :

Total mass generalized total mass (kg)

N :

Normal force, N

m :

Mass, kg

\(\bar{m}\) :

Distributed mass, kg/m

q :


t :


u :

Axial displacement

v :

Transversal displacement

x :

Independent variable

δ :

Virtual work (J)

ϕ :


ω :

Frequency (rd/s)

π :


ρ :

Density (kg/m3)

0 :

Relative to elastic, lumped

i, I :

Relative to internal

g :

Relative to geometric

t :

Relative to time

1,2,3 :

Denote the first, second, and third respectively


Relative to spatial derivate


Relative to derivate in relation to time


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Correspondence to Alexandre de M. Wahrhaftig.

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Technical Editor: Kátia Lucchesi Cavalca Dedini.

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M. Wahrhaftig, A.d., Brasil, R.M.L.R.F. Initial undamped resonant frequency of slender structures considering nonlinear geometric effects: the case of a 60.8 m-high mobile phone mast. J Braz. Soc. Mech. Sci. Eng. 39, 725–735 (2017).

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