This paper presents a comprehensive investigation of the free vibrations of stepped straight and curved beams with different shapes and different materials. The beams are assumed to be Euler-Bernoulli type, and Finite Element Displacement Method (FEDM) is used as a computational approach. In-plane and out-of-plane vibration analyses are handled with stepped straight and curved beams at different end conditions. Material pairs of the stepped curved beam are considered as (i) steel-steel, (ii) steel-aluminum, (iii) steel-brass and (iv) steel-araldite. Results are given in tabular form and compared with those in literature and computations obtained by Ansys. The effects of beam shape and different material type on the vibration characteristics are also investigated.

Dynamics of stepped beams have been commonly investigated by many researchers due to their importance of industrial applications in many engineering areas such as the automotive world or nautical world. Since the stepped beams are used in many structures and engineering applications, their vibration characteristics have been of great interest to researchers. One one hand, there have been many studies on the straight stepped beams. Some of them are presented as follows. Kisa and Gurel1 present a novel numerical technique to analyze the free vibration of uniform and stepped cracked beams with a circular cross-section. Mao and Pietrzko2 investigate free vibration of a stepped Euler-Bernolli beam consisting of two uniform sections by using the Adomian Decomposition Method (ADM). Mao3 states that ADM offers an accurate and effective method of free vibration analysis of multiple-stepped beams with arbitrary boundary conditions. Suddoung et al.4 study on free vibration response of stepped beams made from functionally graded materials. It is indicated that the differential transformation method is useful for solving the governing differential equations of such beams. Lee5 applies a Chebyshev-tau method based on Euler-Bernoulli and Timoshenko beam theories to the free vibration analysis of stepped beams. Lin and Ng6 introduce a novel numerical method to the prediction of vibration modes of general stepped beams with arbitrary steps and general elastic supports. Özyiğit et al.7 work on the out-of-plane vibrations of curved uniform and curved tapered beams.

On the other hand, there are rare studies for stepped curved beams in literature. One of them is presented by Noori et al.8 where they investigate damped, and undamped transient response of in-plane and out-of-plane loaded stepped curved rods with circular cross-sections. The novelty of this study is to address the vibration characteristics of a curved beam with different materials regarding stepped wise condition. In-plane and out-of-plane free vibrations of stepped straight and curved beams are investigated. The vibration analysis consists of two main parts: (i) A stepped straight beam is taken into account concerning different boundary conditions. The results are compared with the literature. (ii) A stepped curved beam is taken into consideration regarding different boundary conditions and different materials.

The in-plane elastic and kinetic energy equations of straight (s) and curved (c) beam can be expressed as follows (Fig. 1) where E, I and A are the modulus of elasticity, mass moment of inertia and cross-section area of beams, respectively. ρ is the density of the material.

Us=12ExAεs2+Iκs2dx
(1)
Ts=12ρAxU̇s2+V̇s2dx
(2)
Uc=12EsAεc2+Iκc2ds
(3)
Tc=12ρsAU̇s2+V̇s2+Iβ̇c2ds
(4)

Strain, cross-sectional rotation, and curvature change of straight and curved beams are

εs=usdxβs=vsxκs=βsx=2vsx2
(5)
εc=ucs+vcRβc=vcsucRκc=βcs=2vcs21Rucs
(6)

The out-of-plane elastic and kinetic energy equations of curved (c) beam can be expressed as follows;7 

Ucout=12EISκcout2ds+12GJSφc2ds
(7)
Tcout=12ρAsω̇c2ds+12ρIsΨ̇c2ds+12ρJsΦ̇c2ds
(8)

where G is the modulus of shear and J is the polar moment of inertia, respectively. The term (  ) denotes differentiation with respect to time t. Out-of-plane curvature change, torsion, and slope terms are;

κcout=ΦcR2wcs2,φc=Φcs+1Rwcs,Ψc=wcs
(9)

where Φc is the torsional displacement of the curved element.

By following the finite element procedure, the stiffness and inertia matrices are obtained for straight and curved beam elements for in-plane and out-of-plane vibrations.

Matrix equation for the free vibrations of beam starts with an equation form

KV+Md2Vdt2=0
(10)

where {V} denotes global displacement vector, [K] and [M] are global stiffness and inertia matrices, respectively.7 The solution of Eq. (7) is assumed as

V=V¯ejωnt
(11)

where j=1, ωn is natural frequency and {V¯} is displacement amplitude vector of all nodes. Then, one obtains the eigenvalue equation giving the natural frequencies for in-plane and out-of-plane vibrations

Kωn2M=0
(12)

In this part, natural frequencies are obtained for the clamped side(s) of clamped-clamped (C-C) and clamped-free (C-F) boundary conditions corresponding to the following equation:

us=vs=vsx=0
(13)

Vibration behavior of stepped straight beam is analyzed by considering natural frequencies. Results are compared with those in literature as presented in Tables I–III. One can say that there is a good agreement between results of the present study and those in the literature.5,6

Then, a single-stepped steel beam with a square cross-section is regarded as shown in Figure 2. The beam is 1 m in length, and it is stepped at mid-point. (a=20 mm, E=2*1011 Pa, and ρ=7800 kg/m3). The thin part of the beam is connected centrally to the right side of the thick part. The boundary condition of the beam is assumed to be clamped-free (namely C-F, clamped end at left and the free end at the right side). Results are given in Table IV because there is a smaller cross-section at the right side of the beam. It is noted that (i) fundamental frequencies increase with decreasing cross-section area, (ii) the other frequencies, however, decrease with decreasing cross-section area.

In this part, in-plane natural frequencies are obtained for the clamped side(s) of C-C and C-F boundary conditions corresponding to the following equation:

uc=vc=vcs=0
(14)

The similar analysis is performed for a stepped curved steel beam. The stepped curved beam is shown in Figure 3 with 900 arc angle (α). Cross-sectional and material properties are assumed to be the same as the previous part for the straight stepped beam. The quarter circle beam is 1 m in length, and stepped at mid-point. The cross-section of the beam is still square (i.e., a=20 mm at left part with decreasing ratio to the right part) and the connection between two cross-sections is central. The results of in-plane natural frequencies are obtained under C-C and C-F boundary conditions as given in Tables V and VI respectively.

Table V says that in-plane natural frequencies of all modes decrease due to the thinner right side part of the stepped beam. However, results of the beam under C-F boundary condition in Table VI indicate that there is an increase for the first mode up to a1=0.5a.

In this part, computational analysis is carried to half circle steel beams which are symmetrically stepped at two different arc angle types. As shown in Figures 4 and 5, arc angles of parts are considered 45-90-45 (A) and 60-60-60 (B) in degrees, respectively. While the cross-section at the bottom is still 20×20 mm2 at both sides, the cross-section of the mid-part of the curved beam is less than 20×20 mm2. The length of the central axis of half circled beam is 1 m. The mid-part is connected to left and right sides centrally.

On the one hand, results in Table VII say that the fundamental frequencies decrease moderately at first and then increase due to the thinner mid-part of the beam (A). However, other frequencies decrease consistently. On the other hand, results in Table VIII indicate that a consistent decrease is observed at modes 1, 3, 4 and 5. However, mode 2 shows an increase at the beginning and later a decrease due to the thinner mid-part of the beam.

1. Beam with different materials

In this part, different materials are considered for mid-part of the beams (A) and (B) (see Figures 4 and 5). Brass and aluminum are chosen as a material type of the mid-part (E=1011 Pa, ρ=8000 kg/m3 for brass and E=0.7*1011 Pa, ρ=2700 kg/m3 for aluminum). Results are presented in Tables IX and X for the steel-brass-steel beam (A) and the steel-brass-steel beam (B), respectively.

Moreover, results are presented in Table XI and Table XII for the steel-aluminum-steel beams (A) and (B) respectively.

As it is seen in Tables VII–X, one can conclude that the replacement of mid-part with brass causes a decrease at all frequencies for beam A and B. However, when noting Tables VII, VIII, XI, and XII, one can observe that replacing the steel part (mid-part) with aluminum causes an increase for the beam (A), but a decrease for the beam (B) at fundamental frequencies. On the other hand, for the upper modes, lower or higher values are obtained as natural frequencies for different cross-sections of mid-parts.

The work in Sec. V is reconsidered concerning out-of-plane vibration of a half circle stepped beam under clamped side(s) of C-C and C-F boundary conditions such that

Φc=wc=wcs=0
(15)

Results of the out-of-plane natural frequencies for steel beams (A) and (B) are presented in Tables XIII and XIV (G=0.84*1011 Pa).

Tables XIII and XIV indicate that (i) the fundamental natural frequencies increase because of the thinner mid-part of the beam, (ii) other frequencies of the modes shows a decrease trend.

At last, three different materials are taken into account for mid-part of the beams (A) and (B) namely Brass, Aluminum and Araldite (G=0.39*1011 Pa for Brass, G=0.26*1011Pa for Aluminum, E=979*106 Pa, ρ=1000 kg/m3, G=623*106 Pa for Araldite). Results of the out-of-plane vibration are presented in Table XV and Table XVI for the steel-brass-steel beam (A) and (B) respectively.

The out-of-plane analysis states that (i) the steel-brass-steel beam shows similar frequency behavior with the steel beam, (ii) when comparing Tables XIII and XIV with Tables XV and XVI respectively, one can see that the decrease of natural frequencies is due to the brass effect in mid-part of the curved beam.

Results of the out-of-plane natural frequencies for the steel-aluminum-steel beams A and B are presented in Tables XVII and XVIII.

On one hand, Tables XVII and XVIII present that (i) for the first mode, the effect of material (aluminum) on frequency shows mostly an increasing tendency due to the smaller sizes of the mid-part of the beam, (ii) conversely, for the other modes, aluminum effect on frequency indicates fully a decreasing tendency. On the other hand, when comparing Tables XVII and XVIII with Tables XIII and XIV respectively, mode three possesses lower frequency values for the beam (A) and modes second and fifth possess lower frequency values for the beam (B).

Results of the out-of-plane natural frequency for the steel-araldite-steel beams A and B are presented in Tables XIX and XX, respectively. One can note that (i) making mid-part thinner reduces the frequencies at almost all modes, (ii) frequency gap between sequential modes becomes smaller than that of other bimaterial beams.

In order to obtain high potential functionality of a beam under the dynamic condition, bimaterial approach is one of the solutions to improve damping, energy absorption, and structural stability characteristics. Consequently, investigation of vibration characteristics of a curved beam with different materials regarding stepped wise condition is a key point to understand how to design a beam to resist or enhance in-plane and out-of-plane free vibrational behavior. As engineering materials, one may say that brass and araldite are useful materials for these purposes. While the former material may be chosen for improvement of in-plane free vibrational behavior, the latter may be selected for that of out-of-plane free vibrational behavior. One may also note that there is a critical point for araldite usage. That is, because modes of steel-araldite-steel beam have close frequency values, there may occur catching resonance.

Despite there are many studies about curved beams and stepped beams separately, this study provides a practical solution for vibrations of beams which are stepped and curved together. In-plane and out-of-plane free vibrations of half circle stepped beams are investigated because they are rare in literature. Stepped parts of the beams are also considered concerning different engineering materials, e.g., steel, brass, aluminum, and araldite. Both geometrical and material effects on natural frequencies are determined and interpreted. Results conclude that this finite element solution proposed here is suitable to investigate the vibrations of stepped and curved bimaterial beams.

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