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Exact closed-form solutions for the natural frequencies and stability of elastically connected multiple beam system using Timoshenko and high-order shear deformation theory
Journal of Sound and Vibration 332 (2013) 563–576
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Exact closed-form solutions for the natural frequencies and stability of elastically connected multiple beam system using Timoshenko and high-order shear deformation theory Vladimir Stojanovic´, Predrag Kozic´ n, Goran Janevski University of Niˇs, Department of Mechanical Engineering, A. Medvedeva 14, 18000 Niˇs, Serbia
a r t i c l e i n f o
abstract
Article history: Received 21 February 2012 Received in revised form 29 August 2012 Accepted 5 September 2012 Handling editor: S. Ilanko Available online 12 October 2012
A general procedure for the determination of the natural frequencies and buckling load for a set of beam system under compressive axial loading is investigated using Timoshenko and high-order shear deformation theory. It is assumed that the set beams of the system are simply supported and continuously joined by a Winkler elastic layer. The model of beam includes the effects of axial loading, shear deformation and rotary inertia. In the special case of identical beams, explicit expressions for the natural frequencies and the critical buckling load are determined using a trigonometric method. The influences of the compressive axial loading and the number of beams in the system on the natural frequencies and the critical buckling load are discussed. These results are of considerable practical interest and have wide application in engineering practice of frameworks. & 2012 Elsevier Ltd. All rights reserved.
1. Introduction Vibration and buckling problems of beams or beam–columns on elastic foundations occupy an important place in many fields of structural and foundation engineering. This problem is very often encountered in aeronautical, mechanical, and civil engineering applications. Its solution demands the modeling of (a) the mechanical behavior of the beam, (b) the mechanical behavior of the soil and (c) the form of interaction between the beam and the soil. As far as the beam is concerned, most engineering analyses based on the classical Bernoulli–Euler beam theory, in which straight lines or planes normal to the neutral beam axis remain straight and normal after deformation. This theory thus neglects the effect of transverse shear deformations, a condition that holds only in the case of slender beams. To confront this problem, the wellknown Timoshenko-beam model can be used, in which the effect of transverse shear deflections is considered. A critical scrutiny of such studies, conducted for mechanical systems with a single beam, is presented in the review article by Chandra Duta and Roy [1]. Zhang et al. [2] derive explicit expressions for the natural frequencies and the associated amplitude ratios of the two beams, and the analytical solutions of the critical buckling load are obtained. The influences of the compressive axial loading on the responses of the double-beam system are discussed. It is shown that the critical buckling load of the system is related to the axial compression ratio of the two beams and the Winkler elastic layer, and the properties of free transverse vibration of the system greatly depend on axial compressions. Stojanovic´ et al. [3] study the influence of rotary inertia and shear on the free vibration and buckling of a double-beam system under axial loading.
n
Corresponding author. Tel.: þ381 18 500 634; fax: þ 381 18 588 244. E-mail addresses: [email protected] (V. Stojanovic´), [email protected] (P. Kozic´).
0022-460X/$ - see front matter & 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.jsv.2012.09.005
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Stojanovic´ and Kozic´ [4] found analytical solutions for forced vibration of a double beam system under the rotary inertia and shear effects. It is assumed that the system under consideration is composed of two parallel and homogeneous simply supported beams continuously joined by a Winkler elastic layer. Both beams have the same length. It is also supposed that the buckling can only occur in the plane where the double-beam system lies. The explicit expressions are derived for natural frequencies and associated amplitude ratio of the two beams, and the analytical solution of the critical buckling is obtained. The influence of the characteristics of the Winkler elastic layer on natural frequencies, and the critical buckling force is determined. Li et al. [5] establish an exact dynamic stiffness matrix established for an elastically connected threebeam system, which is composed of three parallel beams of uniform properties with uniformly distributed-connecting springs among them. The formulation includes the effects of shear deformation and rotary inertia of the beams. The dynamic stiffness matrix is derived by rigorous use of the analytical solutions of the governing differential equations of motion of the three-beam system in free vibration. The use of the dynamic stiffness matrix to study the three vibration characteristics of the three-beam system is demonstrated by applying the Muller root search algorithm. Kelly and Srinivas [6] investigate the problem concerning the free vibrations of a set of n axially loaded stretched Bernoulli–Euler beams connected by elastic layers. A normal-mode solution is applied to the governing partial differential equations to derive a set of coupled ordinary differential equations which are used to determine the natural frequencies and mode shapes. It is shown that the set of differential equations can be written in a self-adjoint form with an appropriate inner product. An exact solution for the general case is obtained, but numerical procedures must be used to determine the natural frequencies and mode shapes. The numerical procedure is difficult to apply, especially in determining higher frequencies. For the special case of identical beams, an exact expression for the natural frequencies is obtained in terms of the natural frequencies of a corresponding set of unstretched beams and the eigenvalues of the coupling matrix. The Adomian modified decomposition method (AMDM) is employed by Mao [7] in order to investigate the free vibrations of N elastically connected parallel Euler–Bernoulli beams, which are continuously joined by a Winkler-type elastic layer. The proposed AMDM method is used to analyze the vibration of beam system consisting of an arbitrary number of beams. By using boundary conditions, the natural frequencies and corresponding mode shapes are easily obtained simultaneously. Ariaei et al. [8] investigate the dynamic response of an elastically connected multiple beam system based on Timoshenko-beam theory. The identical prismatic beams are assumed to be parallel and connected by a finite number of springs. Assuming n parallel Timoshenko-beams, the motion of the system is described by a coupled set of 2n partial differential equations. The method involves a change of variables and modal analysis to decouple and to solve the governing differential equations, respectively. Miranda and Taghavi [9] present an approximate procedure to estimate floor acceleration demands in multistory buildings with the use of only a small number of parameters. Floor acceleration demands are computed using approximations of the first three modes of vibration of the building based on those of a continuum model consisting of a cantilever flexural beam connected laterally to a cantilever shear beam and models with uniform stiffness along the height. In this paper, the effect of compressive axial load on the properties of free natural frequencies and buckling load of a set Timoshenko and Reddy-Bickford beam system is investigated. It is assumed that the system under consideration is composed of a set of parallel, slender prismatic, and homogeneous simply supported beams continuously joined by a Winkler elastic layer. The beams have the same length l, and it is also supposed that the buckling can only occur in the plane where the system beams lie. The natural frequencies of the system with a different number of beams are determined using the numerical solution of the frequency equation. Also, using the trigonometric method, the explicit analytical expressions for the natural frequencies and the critical buckling load are determined. Numerical solution of the frequency equation determines the natural system frequencies and they are compared to the values obtained through the trigonometric method. The influence of the number of beams on the lowest natural frequency of the system is also determined.
2. Formulation of the differential equations and structural model using Timoshenko theory It can be seen from the literature that the dynamic analysis of the elastically parallel-beam system is concentrated primarily on the case of a double-beam system of two parallel simply supported beams continuously joined by a Winkler elastic layer. Very few research papers can be found to deal with the problem related to the elastically connected threebeam system. Those studies of this region are limited to the particular cases of identical beams with some prescribed boundary conditions. It is also shown that in most of the references, the simple Bernoulli–Euler beam theory has been used in deriving the necessary equation. The basic differential equations of motion for the analysis will be deduced by considering the Timoshenko-beam of length l, Fig. 1a, subjected to axial compressive force F, and to a distributed lateral loads of intensity q1 and q2 which vary with the distance z along the beam. This will be applied on the basis of the following assumptions: (a) the behavior of the beam material is linear elastic; (b) the cross-section is rigid and constant throughout the length of the beam and has one plane of symmetry; (c) shear deformations of the cross-section of the beam are taken into account while elastic axial deformations are ignored; (d) the equations are derived bearing in mind the geometric axial deformations; (e) the axial forces F acting on the ends of the beam are not changed with time. An element of length dz between two cross-sections taken normal to the deflected axis to the beam is shown in Fig. 1b. Since the slope of the beam is small, the normal forces acting on the sides of the element can be taken to be equal to the
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Fig. 1. The coordinate system and notation for Pinned–Pinned beam. (a) Timoshenko-beam subjected to an axial compressive force F and to a distributed lateral loads of intensity q1 and q2 ; (b) Deflected differential layered-beam element of length dz.
Fig. 2. A set of m elastically connected Timoshenko-beams on a Winkler type foundation.
axial compressive force F. The shearing force F T is related to the following relationship @w c , F T ¼ kGA @z
(1)
where w ¼ wðz,tÞ is the displacement of a cross-section in y direction, @w=@z is the global rotation of the cross-section, c is the bending rotation, G is the shear modulus, A is the area of the beam cross section, and k is the shear factor. Analogously, the relationship between bending moments M and bending angles c ¼ cðz,tÞ is given by M ¼ EIx
@c , @z
(2)
where E is the Young’s modulus and Ix is the second moment of the area of the cross-section. Finally, forces and moments of inertia are given by f I ¼ A
@2 w , @t 2
J I ¼ Ix
@2 , @t 2
(3)
respectively, where r is the mass density. The forces acting on a differential layered-beam element are shown in Fig. 1b. The dynamic-force equilibrium conditions of these forces are given by the following equations ! @2 w @2 w @c @2 w rA 2 kGA (4a) þ F 2 q1 ðzÞ þ q2 ðzÞ ¼ 0, 2 @z @z @z @t
rI
@2 c @2 c @w EI kGA c ¼ 0: x @z @z2 @t 2
(4b)
This paper presents a general theory for the determination of the natural frequencies and the critical buckling load of a set of m of elastically connected, identically axially loaded, Timoshenko-beams connected by elastic layers to a Winkler type foundation as illustrated in Fig. 2.
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Each beam is made of some material with Young’s modulus Ei, and mass density ri and has a cross-section with a uniform cross-section of area Ai and moment of inertia Ii ¼ Ix , and beams are subjected to the same compressive axial loading F. The first beam is connected to the ground by a Winkler foundation of the stiffness per length K0, and other beams are also connected by a continuous linear elastic layer of Winkler type of stiffness per length Ki. The transverse displacement of the beams is wi ¼ wi ðz,tÞ, i ¼ 1,2,3 m and ci ¼ ci ðz,tÞ,i ¼ 1,2,3 m are bending rotations. If we apply the abovementioned procedure to a differential element of each beam, the following set of coupled differential equations will be obtained: ! @2 w @2 w1 @c1 @2 w1 r1 A1 2 1 kG1 A1 þK 0 w1 K 1 ðw2 w1 Þ ¼ 0, (5a) þF 2 @z @z @z2 @t @2 c1 @2 c1 @w1 E I kG A c ¼ 0, 1 1 1 1 1 @z @z2 @t 2
(5b)
! @2 wi @2 wi @ci @2 wi kG A þ K i1 ðwi wi1 ÞK i ðwi þ 1 wi Þ ¼ 0, þF i i 2 2 @z @z @z2 @t
(6a)
r1 I1 ri Ai
ri Ii rm Am
@2 ci @2 ci @wi E I kG A c ¼ 0, i i i i i @z @z2 @t 2
i ¼ 2,3,. . .,m1
(6b)
! @2 wm @2 wm @m @2 wm kG A þ K m1 ðwm wm1 Þ ¼ 0, þF m m 2 2 @z @z @z2 @t
(7a)
@2 cm @2 cm @wm cm ¼ 0: Em Im kGm Am 2 2 @z @z @t
(7b)
rm Im
3. Governing equations and solution of the problem The equations of motion, i.e., Eqs. (5a–7b), which are coupled together, are reduced by standard procedure, eliminating
ci , i ¼ 1,2,3,. . .,m, to the following system of differential equations, E1 I1 1
4 F @ w1 r1 I1 @2 w1 K 1 r1 I1 @2 w2 E1 I1 @2 w1 E1 I1 @2 w2 þ r A þ ðK þ K Þ þ FðK þK Þ þK 1 1 0 1 0 1 1 kA1 G1 @z4 kA1 G1 @t 2 kA1 G1 @t 2 kA1 G1 @z2 kA1 G1 @z2 4 r2 I1 @4 w1 E1 F @ w1 r1 I1 1þ þ 1 þ xðK 0 þ K 1 Þw1 K 1 w2 ¼ 0 (8a) 2 2 kG1 kA1 G1 @z @t kG1 @t 4
Ei Ii 1
4 F @ wi K i1 ri Ii @2 wi1 r I @2 wi K i ri Ii @2 wi þ 1 K i1 Ei Ii @2 wi1 þ ri Ai þ ðK i1 þ K i Þ i i þ 4 2 kAi Gi @z kAi Gi kAi Gi @t 2 kAi Gi @t 2 kAi Gi @z2 @t 2 4 2 4 2 r Ii @ wi þ 1 E I @ wi K i Ei Ii @ wi þ 1 E F @ wi þ FðK i1 þ K i Þ i i þ ri Ii 1þ i þ i 2K i1 wi1 kAi Gi @z2 kAi Gi @z2 kGi kAi Gi @z2 @t 2 kGi @t 4 þ ðK i1 K i Þwi K i wi þ 1 ¼ 0,i ¼ 2,3,. . .,m1
Em Im 1
@4 wm K m1 rm Im @2 wm1 K m1 rm Im @2 wm K m1 Em Im @2 wm1 þ r A þ þ m m kAm Gm kAm Gm kAm Gm kAm Gm @z4 @z2 @t 2 @t 2 2 4 4 2 K m1 Em Im @ wm Em F @ wm r Im @ wm þ F rm Im 1 þ þ m K m1 wm1 þ K m1 wm ¼ 0: kAm Gm kGm kAm Gm @z2 @t 2 kGm @t 4 @z2 F
(8b)
(8c)
The initial conditions in general form and boundary conditions for simply supported beams of the same length l are assumed as follows wi ðz,0Þ ¼ wi0 ðzÞ,
_ i ðz,0Þ ¼ vi0 ðzÞ, w
ci ðz,0Þ ¼ ci0 ðzÞ, c_ i ðz,0Þ ¼ oi0 ðzÞ,
wi ð0,tÞ ¼ wi ðl,tÞ ¼ w00i ð0,tÞ ¼ w00i ðl,tÞ ¼ 0,
i ¼ 1,2,3,. . .,m:
(9) (10)
Assuming time harmonic motion and using separation of variables, the solutions of Eqs. (8a–c) with the governing boundary conditions (10) can be written in the form wi ðz,tÞ ¼
1 X n¼1
X n ðzÞT in ðtÞ,
i ¼ 1,2,3,. . .,m
(11)
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where T in ðtÞ,i ¼ 1,2,3. . . is the unknown time function, and Xn(z) is the known mode shape function for simply supported single beam, which is defined as kn ¼ np=l,
X n ðxÞ ¼ sinðkn zÞ,
n ¼ 1,2,3,. . .:
(12)
Introducing the general solutions (11) into Eqs. (8a–c) one gets the system of ordinary differential equations # ( " ! 4 2 2 1 X 1 d T 1n C 2b1 1 d T 1n H1 d T 2n 2 2 n þ 1 þC 1þ þ ðH þ H F Z Þ 2 k 1 r1 0 1n n 2 4 2 4 2 2 dt C s1 C r1 C s1 dt 2 C s1 n ¼ 1 C s1 dt " !# ! ) C2 C2 4 2 2 þ C 2b1 kn þ ðHn0 þ H1 F Z1n Þ 1þ 2 b12 kn T 1n H1 1 þ 2 b12 kn T 2n X n ¼ 0, C s1 C r1 C s1 C r1 # " ! 2 2 4 2 1 d T in Hni1 d T ði1Þn C 2bi 1 d T in Hi d T ði þ 1Þn 2 2 n þ 1 þ C 1 þ þ ðH þH F Z Þ k i ri n i1 in dt 2 dt 2 dt 2 C 2si C 2si C 4ri C 2si C 2si dt 4 C 2si ! " !# 2 2 C C 2 4 2 Hnði1Þ 1 þ 2 bi 2 kn T ði1Þn þ C 2bi kn þðHni1 þHi F Zin Þ 1 þ 2 bi 2 kn T in C si C ri C si C ri ! ) C 2bi 2 Hi 1 þ 2 2 kn T ði þ 1Þn X n ¼ 0, i ¼ 2,3,. . .,m1 C si C ri
(13a)
(
1 X n¼1
(13b)
# " ! 2 4 2 1 d T mn Hnm1 d T ðm1Þn C 2bm 1 d T mn 2 2 n þ 1 þ C 1 þ þ ðH F Þ k mn rm m1 n 2 4 2 2 2 4 2 dt dt 2 C sm C sm C rm C sm n ¼ 1 C sm dt ! " !# ) C2 C2 2 4 2 Hnm1 1 þ 2 bm2 kn T ðm1Þn þ C 2bm kn þ ðHnm1 F Zmn Þ 1 þ 2 bm2 kn T mn X n ¼ 0, C sm C rm C sm C rm (
1 X
(13c)
where K i1 , ri Ai
Hni1 ¼
Ki
Hi ¼
ri Ai
2
Zin ¼
,
kn
ri Ai
,
i ¼ 1,2,3,. . .,m:
The coefficients sffiffiffiffiffi Ii C ri ¼ , Ai
sffiffiffiffiffiffiffiffiffiffiffiffi Gi Ai k , C si ¼ ri I i
sffiffiffiffiffiffiffiffiffi Ei Ii , C bi ¼ ri Ai
i ¼ 1,2,3,. . .,m:
related to bending stiffness, shear stiffness and rotational effects, respectively, are now introduced. The shear beam model, the Rayleigh beam model and the simple Euler beam model can be obtained from the Timoshenko-beam model by setting C ri to zero (that is, ignoring the rotational effect), C si to infinity (ignoring the shear effect) and setting both C ri to zero and C si to infinity, respectively. The solutions of Eqs. (13a–c) can be assumed to have the following forms pffiffiffiffiffiffiffi T in ¼ Ain ejon t , i ¼ 1,2,3,. . .,m, j ¼ 1, (14) where on denotes the natural frequency of the system. Substituting Eq. (14) into Eqs. (13a–c) results in the following system of homogeneous algebraic equations for the unknown constants A1n ,A2n ,A3n ,. . .Amn , (
o4n
C 2s1
" 1 þ C 2r1
1þ
C 2b1
! 2 kn þ
C 2s1 C 4r1
#
1
n
C 2s1
ðH0 þ H1 F Z1n Þ o "
þ H1 " Hni1 "
o2n C 2si
1þ
C 2bi C 2si C 2ri
4
"
o2n C 2sm
1þ
C 2bm C 2sm C 2rm
C 2s1
!# 2
kn
þ C 2bi kn þ ðHni1 þ Hi F Zin Þ 1þ
Hnm1
o2n
o4n
C 2bi C 2si C 2ri
!#) 2
kn
!# 2
C 2si
" 4
Z
"
o4n
C 2sm
o2n C 2si
2
kn
!#)
C 2b1
2 kn C 2s1 C 2r1
A1n
A2n ¼ 0,
C 2bi C 2si C 2ri
"
C 2bm C 2sm C 2rm
1
2
C 2si C 4ri
1 þC 2rm 1þ
(15a)
!
C 2bi
1þ
þ C 2bm kn þ ðHnm1 F Zmn Þ 1 þ
1þ
!#
1þ C 2ri 1 þ
Ain þHi (
Aðm1Þn þ
C 2b1 2 2 C s1 C r1
4 C 2b1 kn þðHn0 þ H1 F 1n Þ
"
( Aði1Þn þ
kn
1þ
" 2 nþ
kn þ !# 2
kn
Aði þ 1Þn ¼ 0, !
C 2bm
2
C 2sm C 4rm !#) 2
kn
C 2si
# ðHni1 þ Hi F Zin Þ o2n
kn þ
1 C 2sm
Amn ¼ 0:
i ¼ 2,3,. . .,m1
(15b)
# ðHnm1 F Zmn Þ o2n (15c)
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3.1. Identical beams When all beams are identical Ai ¼ A, Ei ¼ E, Gi ¼ G, Ii ¼ I, K i1 ¼ K, K m ¼ 0, ri ¼ r, i ¼ 1,2,3,. . .,m, then substituting these value in Eqs. (15a–c) and rewriting in matrix form, one obtains the following homogeneous set of m algebraic equations 9 9 8 8 0 > > > A1n > > > > 2 3> > > > > > 0 > > > > A2n > un 0 0 0 0 0 0 0 Sn > > > > > > > > > > > > > > 6 u 7> > > > > > > > S u 0 0 0 0 0 0 A 0 n n n 3n 6 7> > > > > > > > 6 7> > > > > > > > 6 7> > > > > > > > > > > 6 7> > > > > > > > > 6 0 7 > > > > u 0 0 0 0 A 0 0 S 0 n n > > 6 7> ði1Þn > = < = < 6 7 6 0 7 S u 0 0 0 A 0 0 u 0 (16) ¼ n n n in 6 7> > > > > > > >A 6 0 > > > Sn 0 0 0 7 0 0 0 un 0 > > > > 6 7> ði þ 1Þn > > > > > > > 6 7> > > > > > > > 6 7 > > > > > > > > 6 7> > > > > > > > > 6 0 7 > > > Sn un 5> A 0 0 0 0 0 un > > > > 0 4 ðm2Þn > > > > > > > > > > > > > > > > > > > 0 0 0 0 0 0 0 un Sn un > A 0 ðm1Þn > > > > > > > > ; ; : : Amn 0 where 2
K i1 Ki k ¼ H, Hi ¼ ¼ H, Zin ¼ n ¼ Zn , i ¼ 1,2,3,. . .,m, ri Ai ri Ai ri Ai sffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffi Ei Ii Gi Ai k Ii C bi ¼ ¼ C s , C ri ¼ ¼ C b , C si ¼ ¼ C r , i ¼ 1,2,3,. . .,m, ri Ai ri Ii Ai ! " !# " # o4 C2 1 C2 2 2 4 Sn ¼ 2n 1 þC 2r 1 þ 2 b 4 kn þ 2 ð2HF Zn Þ o2n þ C 2b kn þ ð2HF Zn Þ 1 þ 2 b 2 kn , Cs Cs Cr Cs Cr Cs " !# 2 o2 C 2 un ¼ H 2n þ 1 þ 2 b 2 kn : Cs Cs Cr Hni1 ¼
Eq. (16) represents both the free-vibration and stability eigenvalue problem of a set of m identically elastically connected Timoshenko-beams with generalized end conditions (10). By making the determinant of the mxm matrix equal to zero yields an ordinary polynomial of the 4mth order in terms of the undamped natural frequencies on : Even the analytical determination of natural frequencies for a mechanical system comprising three same Timoshenko-beams presents a certain problem. For a system with more than three same Timoshenko-beams, natural frequencies can be determined only numerically for specific physical system parameters. For a homogenous system of algebraic Eq. (16), analytical expressions for natural frequencies of the system can be determined using the trigonometric method [10, pp. 157–166]. According to [10], let us assume that the value of unknown constants Ain ,i ¼ 1,2,3. . .m changes oscillatory Ain ¼ N sinðijÞ,
i ¼ 1,2,3,. . .,m:
(17)
Let us observe that the first and the last equation of system (16) have a structure different to the structure of the other equations. Substituting assumed solutions (17) into the ith equation of system (16) for i ¼ 2,3. . .m1 yields un N sinði1Þj þ Sn N sinðijÞun N sinði þ 1Þj ¼ 0,
i ¼ 2,3,. . .,m1:
(18)
After some algebra, the following is obtained ðSn 2un cos jÞN sinðijÞ ¼ 0,
i ¼ 2,3,. . .,m1:
(19)
Eq. (19) has to have Na0, and sinðijÞa0, because then Ain a0, for i ¼ 2,3. . .m1: From there Sn 2un cos j ¼ 0 ) Sn ¼ 2un cos j:
(20)
Eq. (20) is a frequency equation, which is a fourth order polynomial in on . The unknown j can be determined based on the condition that the assumed solutions (17) satisfy the first and the last equation of system (16). When A1n ¼ N sin j and A2n ¼ N sin 2j are substituted into the first equation of system (16), the following is obtained Sn N sin jun N sin 2j ¼ 0 ) ðSn 2un cos jÞNsin j 0:
(21)
Eq. (21) is identical since it is determined that Sn 2un cos j ¼ 0, where Na0, and sin ja0: When the assumed solutions for Aðm1Þn and Amn are substituted into the mth equation of system (16), the following is obtained un N sin½ðm1jÞ þ ðSn un ÞN sinðmjÞ ¼ 0:
(22)
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After some algebra, the following is obtained 2m þ 1 Nun sinðj=2Þcos j ¼ 0: 2
(23)
Solutions of Eq. (23) are cos
2mþ 1 2s1 j ¼ 0 ) js ¼ p, s ¼ 1,2,3,. . .,m, 2 2mþ 1
since Na0, un a0 and sinðj=2Þa0: By substituting solutions (24) into (20), this frequency equation follows " # ! " !# o4n,s C 2b 1 C 2b 2 2 2 2 4 2 1 þ C r 1 þ 2 4 kn þ 2 ð2HF Zn Þ on,s þ C b kn þ ð2HF Zn Þ 1 þ 2 2 kn C 2s Cs Cr Cs Cr Cs " !# o2 C2 2 2s1 þ 1þ 2 b 2 kn cos p : ¼ 2H n,s 2 2m þ 1 Cs Cs Cr After some algebra, biquadratic algebraic equation is obtained 2s1 2s1 o4n,s C 2s ðRn þ C 2r k2n Þ þ 4Hsin2 p F Zn o2n,s þ C 2s C 2b k4n þ4HRn sin2 p F Zn Rn ¼ 0, 2ð2m þ 1Þ 2ð2m þ1Þ
(24)
(25)
(26)
s ¼ 1,2,3,. . .,m: where Rn ¼ 1 þ
C 2b C 2s C 2r
2
kn :
Then from frequency Eq. (26), we obtain
i 1nh 2 2 2s1 C s ðRn þ C 2r kn Þ þ 4H sin2 2ð2m þ 1Þ p F Zn 2 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi9 2 = 2s1 2s1 2 p F Zn 4C 2s C 2b k4n þ4HRn sin2 p F Zn Rn , C 2s ðRn þ C 2r kn Þ þ 4H sin2 ; 2ð2m þ 1Þ 2ð2m þ 1Þ ½o2n,s l ¼
(27)
s ¼ 1,2,3,. . .,m:
i 1nh 2 2 2s1 C s ðRn þ C 2r kn Þ þ 4H sin2 2ð2m þ 1Þ p F Zn 2 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi9 2 = 2s1 2s1 2 þ C 2s ðRn þ C 2r kn Þ þ 4H sin2 p F Zn 4C 2s C 2b k4n þ 4HRn sin2 p F Zn Rn , ; 2ð2m þ1Þ 2ð2m þ 1Þ ½o2n,s h ¼
(28)
s ¼ 1,2,3,. . .,m: where ½on,s l is the lower natural frequency of the system, and ½on,s h is the higher natural frequency [11, pp. 403]. When the natural frequency of the system vanishes under the axial loading, the system begins to buckle. Introduction of on,s ¼ 0 into Eq. (26) gives 2s1 4 C 2b kn þ 4HRn sin2 p F Zn Rn ¼ 0, s ¼ 1,2,3,. . .,m: (29) 2ð2mþ 1Þ It follows from Eq. (29) that the value of the buckling stress corresponding to vibration mode n of m identically elastically connected Timoshenko-beams can be obtained by 4 C 2b kn H 2s1 F bkr ¼ min þ 4 sin2 p : (30) 2ð2m þ 1Þ Zn n ¼ 1,2,3 Rn Zn s ¼ 1,2,3 m
For K ¼0 from Eq. (30) we obtain Pn ¼
EIp2 n2 2
2
l ð1 þ ðEI=GAkÞðp2 n2 =l ÞÞ
:
(31)
V. Stojanovic´ et al. / Journal of Sound and Vibration 332 (2013) 563–576
570
P n is the critical buckling load corresponding to the buckled mode number n of the Timoshenko-beam. By setting n ¼ 1 Eq. (31) is reduced to EIp2
P¼
2
ð1 þðEI=GAkÞðp2 =l ÞÞl
2
:
and coincides with Eq. (2–58) on page 134 of Timoshenko and Gere [12]. 4. Formulation of equations and structural model using Reddy–Bickford beam theory Introducing the higher order shear deformation theory by Reddy–Bickford [13], equations of motion for the system presented on Fig. 2 are ! @2 w1 @3 c1 @4 w1 @c1 @2 w1 @2 w1 2 ^ þ F1 þ r1 A1 2 aF zz 3 þ a Hzz 4 Azy þK 0 w1 K 1 ðw2 w1 Þ ¼ 0, (32a) @z @z @z @z2 @z2 @t 3 @2 c1 @2 c1 ^ zz @ w1 Azy c þ @w1 ¼ 0, þD a F zz 1 @z @z2 @z3 @t 2 ! @2 w @3 c @4 w @ci @2 wi @2 wi þ ri Ai 21 aF^ zz 3 i þ a2 Hzz 4 i Azy þ K i1 ðwi wi1 ÞK i ðwi þ 1 wi Þ ¼ 0, þ Fi @z @z @z @z2 @z2 @t
r1 I1
3 @2 ci @2 ci ^ zz @ wi Azy c þ @wi ¼ 0, i ¼ 2,3,. . .,m1 F þ D a zz i @z @z2 @z3 @t 2 ! @2 wm @3 cm @4 wm @cm @2 wm @2 wm 2 ^ þ rm Am 2 aF zz 3 þ a Hzz 4 Azy þ K m1 ðwm wm1 Þ ¼ 0, þ Fm @z @z @z @z2 @z2 @t
ri Ii
rm Im
3 @2 cm @2 cm ^ zz @ wm Azy c þ @wm ¼ 0, F þ D a zz m @z @z2 @z3 @t 2
(32b)
(33a)
(33b)
(34a)
(34b)
where ^ zy , Azy ¼ A^ zy bD
a¼
4 2
3h
^ zz aF^ zz , Dzz ¼ D
^ zz , F zz ¼ F^ zz aH
b ¼ 3a, D^ zz ¼ Dzz aF zz , F^ zz ¼ F zz aHzz , A^ zy ¼ Azy bDzy , D^ zy ¼ Dzy bF zy
,
ðAzz ,Dzz ,F zz ,Hzz Þ ¼
Z
ð1,y2 ,y4 ,y6 ÞEz dA,
ðAzy ,Dzy ,F zy Þ ¼
Z
A
ð1,y2 ,y4 ÞGzy dA:
A
After some algebra, the equations of motion can be reduced, eliminating ci , i ¼ 1,2,3, m, to the following system of differential equations ! ! 2 a2 F^ zz a2 Dzz Hzz @6 w1 a2 r1 I1 Hzz @6 w1 Dzz F 1 @4 w1 2 ^ F þ D þ 2 a þ a H zz zz zz 6 2 4 @z @z4 @z @t Azy Azy Azy Azy ! " # r A1 Dzz r1 I1 F 1 @4 w1 K 1 Dzz @2 w2 Dzz ðK 0 þ K 1 Þ @2 w1 þ r1 I1 1 þ þ F1 2 2 2 @z2 @z @t Azy Azy Azy @z Azy " # r2 A1 I1 @4 w1 r1 I1 K 1 @2 w2 r I1 ðK 0 þ K 1 Þ @2 w1 1 þ þ r1 A1 1 þðK 0 þ K 1 Þw1 K 1 w2 ¼ 0 (35a) 4 2 @t @t @t 2 Azy Azy Azy !
2
a2 F^ zz Azy
a2 Dzz Hzz @6 wi @z2
Azy
a2 ri Ii Hzz @6 wi Azy
@z4 @t
þ Dzz
Azy "
! þ2aF^ zz þ a2 Hzz
@4 wi @z4
# @ wi K i Dzz @ wi þ 1 Dzz ðK i1 þ K i Þ @2 wi þ þ F i @z2 @z2 @z2 @t 2 Azy Azy Azy Azy " # Dzz K i1 @2 wi1 r2i Ai Ii @4 wi ri Ii K i @2 wi þ 1 r I ðK þ K i Þ @2 wi ri Ii K i1 @2 wi1 þ þ þ ri Ai i i i1 þ 2 4 2 @z @t @t 2 @t 2 Azy Azy @t Azy Azy Azy þ ri Ii
ri Ai Dzz
ri Ii F i
!
2
Dzz F i
4
2
2K i1 wi1 þ ðK i1 K i Þwi K i wi þ 1 ¼ 0, !
2
a2 F^ zz Azy
a2 Dzz Hzz @6 wm Azy
@z6
a2 rm Im Hzz @6 wm Azy
@z4 @t 2
þ Dzz
i ¼ 2,3, m1 Dzz F m Azy
(35b) !
þ 2aF^ zz þ a2 Hzz
@4 wm @z4
V. Stojanovic´ et al. / Journal of Sound and Vibration 332 (2013) 563–576
þ rm Im
2 m Am Im
r
Azy
rm Am Dzz Azy
rm Im F m
!
Azy
4
2
571
@ wm rm Im K m1 @ wm1 þ @t 4 @t 2 Azy
! @4 wm Dzz K m1 @2 wm Dzz K m1 @2 wm1 þ F þ m @z2 @z2 @z2 @t 2 Azy Azy ! r Im K m1 @2 wm þ rm Am m K m1 wm1 þ K m1 wm ¼ 0: @t 2 Azy
(35c)
Following the same procedure as in case for the system of identical Timoshenko-beams, assuming time harmonic motion and using separation of variables for the same boundary conditions, new homogeneous set of m algebraic equations is 9 8 9 8 > A1n > > 2 RB 3> > 0 > > > > > > > RB > > > > > > > > un 0 0 0 0 0 0 0 Sn > > > 0 > > > > A2n > 6 7 > > > > > > > > RB RB 6 uRB 7 > > > > S u 0 0 0 0 0 0 A > > > > 0 3n n n n 6 7> > > > > > > > 6 7> > > > > > > > 7> 6 > > > > > > > > > > > 6 7 > > > > RB RB > > > > 6 0 7 > > un 0 0 0 0 0 0 Sn Aði1Þn > 0 > > > > 6 7> = < = < 6 7 RB RB RB 6 0 7 A 0 0 u S u 0 0 0 0 (36) ¼ in n n n 6 7> > > > > > > 6 7> RB > > > > RB A 0 > > > > ði þ 1Þn 6 0 7 Sn 0 0 0 0 0 0 un > > > > > > > 6 7> > > >> > > > > 6 > > > 7 > > > > 6 7> > > > > > 6 > > 0 > > > > Aðm2Þn > RB RB RB 7 > > > > 6 0 7 S u 0 0 0 0 0 u > > > n n n > > > > 4 5> > > > > > > > RB > > > > RB RB > 0 A > > ðm1Þn > 0 0 0 0 0 0 0 un Sn un > > > > > ; : > > > > : Amn ; 0 where,
(
SRB n ¼
þ
)
r½Ið2KFk2n þ a2 k4n Hzz Þ þ Azy ðIk2n AÞAk2n Dzz ~ 2 r2 AI ~ 4 on þ on Azy
2 Dzz ð2Kkn þ
Azy
4 6 2 2 2 6 kn Hzz Fkn Þ 2 kn F^ zz þAzy ð2KFkn þ
a
a
a2 k4n Hzz þk4n Dzz þ2ak4n F^ zz Þ
Azy
,
!
uRB n ¼
2 rIK ~ 2 k D o n þK 1þ n zz :
Azy
Azy
Trigonometric solutions of the system are shown for Timoshenko case and by substituting solutions (24) into RB SRB n 2un cos j ¼ 0
RB SRB n ¼ 2un cos j:
)
(37)
new, for high shear order deformable multi beam equation follows ( ) 2 4 2 2 Ið2KFkn þ a2 kn Hzz Þ þ Azy ðIkn AÞAkn Dzz r2 AI ~ 4 on þ o~ 2n Azy Azy þ
2 6 4 6 2 2 4 4 4 Dzz ð2Kkn þ a2 kn Hzz Fkn Þa2 kn F^ zz þAzy ð2KFkn þ a2 kn Hzz þkn Dzz þ2akn F^ zz Þ
Azy
"
2 rIK ~ 2 k D ¼2 o n þK 1þ n zz
Azy
Azy
!#
2s1 p , cos 2m þ 1
(38)
Then from Eq. (38), we obtain frequency equation pffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2A3 o~ n,s ¼ A2 þ 2B2 f ðs,mÞ ½A2 2B2 f ðs,mÞ2 4A3 ½A1 2B1 f ðs,mÞ, 2A3 where
(39)
2s1 f ðs,mÞ ¼ cos p , 2m þ1 2
A1 ¼
6
4
6 2
2
4
4
4
Dzz ð2Kkn þ a2 kn Hzz Fkn Þa2 kn F^ zz þAzy ð2KFkn þ a2 kn Hzz þkn Dzz þ2akn F^ zz Þ Azy A2 ¼ A3 ¼
r½Ið2KFk2n þ a2 k4n Hzz Þ þ Azy ðIk2n AÞAk2n Dzz
r2 AI Azy
2
,B1 ¼ K 1 þ
Azy !
kn Dzz Azy
,B2 ¼
rIK Azy
,
:s ¼ 1,2,3 m:
,
V. Stojanovic´ et al. / Journal of Sound and Vibration 332 (2013) 563–576
572
When the natural frequency of the system vanishes under the axial loading, the system begins to buckle. Introduction of on,s ¼ 0 into Eq. (38) gives 2 3 h i 2 2 4 2 6 2 6^ 2 4 ^ 1 6Dzz ð2Kkn þ a kn Hzz Þa kn F zz þ Azy 2K þ a kn Hzz þ kn ð2aF zz þ Dzz Þ 7 RB min 2Kf ðs,mÞ5: (40) F kr ¼ 4 2 m ¼ 1,2,3 k2 A þ k D zy zz n n s ¼ 1,2,3,. . .,m
5. Numerical results The values for the parameters of the system which are used in the numerical calculations are given in [2] E ¼ 1 1010 N m2 , 5
K 0 ¼ 2 10 2
A ¼ 5 10
2
m ,
Nm
G ¼ 0:417 1010 N m2 , 2
,
3
k ¼ 5=6,
3
r ¼ 2 10 kg m , l ¼ 10 m,
I ¼ 4 10
4
4
m ,
1 b¼ 8
rffiffiffi 5 m, 3
2 h¼ 5
rffiffiffiffiffi 3 m: 5
(41)
For system parameters (41), and through numerical solution of the appropriate characteristic equation for system Eq. (16), the natural lower and higher modal frequencies for the system with m ¼3,5,7 and 9 beams are determined and shown in Table 1 for Timoshenko system. Table 1 also shows these frequencies determined using the trigonometric method through Eqs. (27, 28). Table 2 shows differences between frequencies using Timoshenko and high order shear deformation theory. It can be observed from Table 1 that the differences between natural frequencies determined through numerical solution of the characteristic equation and using analytical expressions obtained through the trigonometric method are very small. Another observation is that the value of the smallest natural frequency decreases with the increase with the number of beams m in the system. In the following, let us illustrate the effect of compressive axial loading at the lowest natural frequency of the system with one, three and five equal beams, m ¼1,3,5. In order to investigate the influence of compressive axial loading at the lowest natural frequency for stiffness modulus K ¼1,5K0,K0,0,5K0, the necessary analytical expressions are determined when s¼1 n ¼1 and m ¼1,3,5 are substituted in that order in Eq. (27). After the substitution, the following is obtained: i 1 nh 2 p 2 C s ðR1 þC 2r k1 Þ þ4Hsin2 F Z1 ½o21,1 l ¼ 2 6 Table 1 Timoshenko low and high natural frequencies [Hz] for mode n¼ 1 and for system with number of beam m ¼3, 5, 7, 9 and F¼ 0. Beam numberm
Lower modal frequency
Higher modal frequency
Calculated using Eq. (28)
Calculated using Eq. (17)
Calculated using Eq. (29)
Calculated using Eq. (17)
m¼3
½o1,1 l ¼ 28:0046632875 ½o1,2 l ¼ 59:1263182936 ½o1,3 l ¼ 82:9295278423
28:0046632873 59:1263182935 82:9295278423
½o1,1 h ¼ 14,759:81289 ½o1,2 h ¼ 14,759:81296 ½o1,3 h ¼ 14,759:81305
14,759:8129651 14,759:8129651 14,759:8129651
m¼5
½o1,1 l ¼ 23:4595097487 ½o1,2 l ¼ 42:0466472348 ½o1,3 l ¼ 61:7777785430 ½o1,4 l ¼ 77:7538496710 ½o1,5 l ¼ 88:0208916175
23:4595097486 42:0466472347 61:7777785429 77:7538496709 88:0208916176
½o1,1 h ¼ 14,759:81288 ½o1,2 h ¼ 14,759:81291 ½o1,3 h ¼ 14,759:81296 ½o1,4 h ¼ 14,759:81302 ½o1,5 h ¼ 14,759:81307
14,759:8129722 14,759:8129722 14,759:8129722 14,759:8129722 14,759:8129722
m¼7
½o1,1 l ¼ 21:8125810709 ½o1,2 l ¼ 33:9378813493 ½o1,3 l ¼ 48:8556749524 ½o1,4 l ¼ 62:9882585212 ½o1,5 l ¼ 74:9693492803 ½o1,6 l ¼ 84:0221781219 ½o1,7 l ¼ 89:6471858944
21:8125810709 33:9378813494 48:8556749523 62:9882585213 74:9693492802 84:0221781219 89:6471858945
½o1,1 h ¼ 14,759:81287 ½o1,2 h ¼ 14,759:81289 ½o1,3 h ¼ 14,759:81292 ½o1,4 h ¼ 14,759:81297 ½o1,5 h ¼ 14,759:81301 ½o1,6 h ¼ 14,759:81305 ½o1,7 h ¼ 14,759:81308
14,759:8129739 14,759:8129739 14,759:8129739 14,759:8129739 14,759:8129739 14,759:8129739 14,759:8129739
m¼9
½o1,1 l ¼ 21:0466074643 ½o1,2 l ¼ 29:4987136192 ½o1,3 l ¼ 40:9671530516 ½o1,4 l ¼ 52:7236801181 ½o1,5 l ¼ 63:6808309227 ½o1,6 l ¼ 73:2562413513 ½o1,7 l ¼ 81:0640517350 ½o1,8 l ¼ 86:8296550265 ½o1,9 l ¼ 90:3637299233
21:0466074644 29:4987136191 40:9671530516 52:7236801181 63:6808309228 73:2562413513 81:0640517351 86:8296550263 90:3637299234
½o1,1 h ¼ 14,759:81287 ½o1,2 h ¼ 14,759:81288 ½o1,3 h ¼ 14,759:81291 ½o1,4 h ¼ 14,759:81294 ½o1,5 h ¼ 14,759:81297 ½o1,6 h ¼ 14,759:81300 ½o1,7 h ¼ 14,759:81304 ½o1,8 h ¼ 14,759:81306 ½o1,9 h ¼ 14,759:81308
14,759:8128375 14,759:8128375 14,759:8128375 14,759:8128375 14,759:8128375 14,759:8128375 14,759:8128375 14,759:8128375 14,759:8128375
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573
Table 2 Timoshenko and Reddy Bickford natural frequencies [Hz] for mode n ¼ 1 and for system with number of beam m ¼ 3,5,7,9 and F ¼ 0. Beam number m
Timoshenko Calculated using Eq. (28)
Reddy–Bickford Calculated using Eq. (39)
m¼3
½o1,1 l ¼ 28:0046632875 ½o1,2 l ¼ 59:1263182936 ½o1,3 l ¼ 82:9295278423
½o1,1 l ¼ 28:0340499486 ½o1,2 l ¼ 59:1763262258 ½o1,3 l ¼ 82:9972672789
m¼5
½o1,1 l ¼ 23:4595097487 ½o1,2 l ¼ 42:0466472348 ½o1,3 l ¼ 61:7777785430 ½o1,4 l ¼ 77:7538496710 ½o1,5 l ¼ 88:0208916175
½o1,1 l ¼ 23:4867433497 ½o1,2 l ¼ 42:0846292831 ½o1,3 l ¼ 61:8297234917 ½o1,4 l ¼ 77:8176815808 ½o1,5 l ¼ 88:0924938871
m¼7
½o1,1 l ¼ 21:8125810709 ½o1,2 l ¼ 33:9378813493 ½o1,3 l ¼ 48:8556749524 ½o1,4 l ¼ 62:9882585212 ½o1,5 l ¼ 74:9693492803 ½o1,6 l ¼ 84:0221781219 ½o1,7 l ¼ 89:6471858944
½o1,1 l ¼ 21:839180854 ½o1,2 l ¼ 33:9706521525 ½o1,3 l ¼ 48:8983327711 ½o1,4 l ¼ 63:0410919649 ½o1,5 l ¼ 75:0310883912 ½o1,6 l ¼ 84:0907450889 ½o1,7 l ¼ 89:7200254756
m¼9
½o1,1 l ¼ 21:0466074643 ½o1,2 l ¼ 29:4987136192 ½o1,3 l ¼ 40:9671530516 ½o1,4 l ¼ 52:7236801181 ½o1,5 l ¼ 63:6808309227 ½o1,6 l ¼ 73:2562413513 ½o1,7 l ¼ 81:0640517350 ½o1,8 l ¼ 86:8296550265 ½o1,9 l ¼ 90:3637299233
½o1,1 l ¼ 21:0729477466 ½o1,2 l ¼ 29:528903779 ½o1,3 l ¼ 41:0044146338 ½o1,4 l ¼ 52:7690725794 ½o1,5 l ¼ 63:7341738108 ½o1,6 l ¼ 73:3166966242 ½o1,7 l ¼ 81:1303803448 ½o1,8 l ¼ 86:900351992 ½o1,9 l ¼ 90:4371151478
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi) h i2 h i p p 2 4 C 2s ðR1 þC 2r k1 Þ þ4Hsin2 F Z1 4C 2s C 2b k1 þ 4HR1 sin2 F Z1 R1 , 6 6
i 1nh 2 p 2 C s ðR1 þC 2r k1 Þ þ4Hsin2 F Z1 2 14 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi) h i2 h i p p 2 4 F Z1 4C 2s C 2b k1 þ 4HR1 sin2 F Z1 R1 , C 2s ðR1 þ C 2r k1 Þ þ 4Hsin2 14 14
(42)
½o21,1 l ¼
i 1 nh 2 p 2 C s ðR1 þC 2r k1 Þ þ4Hsin2 F 1 2 22 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi) h i h i 2 p p 2 4 F Z1 4C 2s C 2b k1 þ 4HR1 sin2 F Z1 R1 , C 2s ðR1 þ C 2r k1 Þ þ 4Hsin2 22 22
(43)
½o21,1 l ¼
(44)
with
l ¼ F=F bkr : It can be observed from Fig. 3 that with the increase in compressive axial loading the lower natural frequency ½o21,1 I decreases. It can also be found that the decrease in the stiffness modulus K decreases the lower natural frequency. This decrease is the smallest in the system with a single beam m¼1. With the rising number of beams, the influence of the stiffness modulus K on the lower natural frequency decreases. It can be concluded from Fig. 4 that with the increasing number of beams m, the lower natural frequency decreases. When s¼1 and n ¼1 are substituted in Eq. (30), one obtains analytical expressions for the critical buckling force for systems with different numbers of beams m¼1,3,5, respectively, 4
F bkr ¼
C 2b k1 H p þ 4 sin2 , R1 Z1 Z1 6
F bkr ¼
C 2b k1 H p , þ 4 sin2 R1 Z1 Z1 14
(45)
4
(46)
574
V. Stojanovic´ et al. / Journal of Sound and Vibration 332 (2013) 563–576
Fig. 3. Effect of the stiffness modulus K at the lowest natural frequency for system with. (a) m ¼ 1 beam, Eq. (33); (b) m ¼ 3 beams, Eq. (34); (c) m ¼ 5 beams, Eq. (35).
Fig. 4. Effect of the number of elastically connected axially loaded beams at the lowest natural frequency for K ¼ K 0 .
4
F bkr ¼
C 2b k1 H p , þ4 sin2 R1 Z1 Z1 22
(47)
In Fig. 5, static stability regions for the first vibration mode n ¼1 are represented for a system with one, three and five Timoshenko-beams supported on a Winkler elastic layer. It can be seen that the static stability region is largest in the case when one –Timoshenko-beam is supported on a Winkler elastic layer, Eq. (45). In that case, the critical buckling load F bkr is also the largest. For the system with three Timoshenko-beams supported on a Winkler elastic layer, the static stability region and critical buckling load are reduced, Eq. (46). This trend of the reduction of the static stability region and critical buckling load continues for the system with five Timoshenko-beams, Eq. (47). We can conclude that in the studied
V. Stojanovic´ et al. / Journal of Sound and Vibration 332 (2013) 563–576
575
Fig. 5. Effect of the number of elastically connected axially loaded beams on the critical buckling load F bkr .
oscillatory system the region of stability is reduced with the increasing number of beams, since each added beam introduces additional instability to the system, that is, influences the reduction of the critical buckling force. 6. Conclusions The problem of free transverse vibration and buckling of an elastically connected simply supported set of beams under compressive axial load are studied using Timoshenko and high order shear deformation theory. Governing partial differential equations are formulated based on the following influences: constant axial forces at the ends of the same beams (second order theory), the Winkler elastic foundation, and elastic layers between the beams. A normal-mode solution is applied to the governing partial differential equations to derive a set of coupled ordinary differential equations which are used to determine the natural frequencies and the critical buckling load. The analytical solution for the natural frequencies and the critical buckling load for the special case of identical beams is derived using the trigonometric method. The effects of compressive axial load and number beams at the lowest natural frequency are investigated. The explicit expressions for natural frequencies and critical buckling load for a system with three identical Timoshenko-beams can be determined with a lot of difficulty. For systems with a greater number of beams, these values can be determined only numerically. However, using the trigonometric method, it is possible to approximately determine the explicit expressions for natural frequencies and critical buckling load for a system with a random number of beams m. With the increase in the number m of Timoshenko-beams in a system, the lower and higher natural frequencies decrease. It is found that the increase in compressive axial load affects the decrease in the natural frequencies of the system. Still, the increase in stiffness modulus K of the Winkler elastic layer causes the increase in the natural frequencies of the system. It is presented that there are no significant differences between natural frequencies between Timoshenko and Reddy–Bickford used theory. Layers between beams and effects of high modes don’t make big differences between frequencies for the same number of beams and in the same mode considered. It is observed that the critical buckling load is influenced by stiffness modulus K and the number m of beams. With the increase in stiffness modulus K critical buckling load also increases, but this value decreases rapidly with the increase in the number of beams m in the system. The most stable is the system with a single beam.
Acknowledgments This research was supported by the research grant of Serbian Ministry of Science and Environment Protection under the number of ON 174011. The authors are grateful to the referees for the useful remarks which helped to improve this paper. References [1] S. Chandra Dutta, R. Roy, A critical review on idealization and modeling for interaction among soil–foundation–structure system, Computers and Structures 80 (2002) 1579–1594. [2] Y.Q. Zhang, Y. Lu, S.L. Wang, X. Liu, Vibration and buckling of a double-beam system under compressive axial loading, Journal of Sound and Vibration 318 (2008) 341–352. [3] V. Stojanovic´, P. Kozic´, R. Pavlovic´, G. Janevski, Effect of rotary inertia and shear on vibration and buckling of a double beam system under compressive axial loading, Archive of Applied Mechanics 81 (12) (2011) 1993–2005.
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[4] V. Stojanovic´, P. Kozic´, Forced transverse vibration of Rayleigh and Timoshenko-double-beam system with effect of compressive axial load, International Journal of Mechanical Sciences 60 (2012) 59–71. [5] J. Li, Y. Chen, H. Hua, Exact dynamic stiffness matrix of a Timoshenko-three-beam system, International Journal of Mechanical Sciences 50 (2008) 1023–1034. [6] S.G. Kelly, S. Srinivas, Free vibrations of elastically connected stretched beams, Journal of Sound and Vibration 326 (2009) 883–893. [7] Q. Mao, Free vibration analysis of elastically connected multiple-beams by using the Adomian modified decomposition method, Journal of Sound and Vibration http://dx.doi.org/10.1016/j.jsv.2012.01.028. [8] A. Ariaei, S. Ziaei-Rad, M. Ghayour, Transverse vibration of a multiple-Timoshenko-beam system with intermediate elastic connections due to a moving load, Archive of Applied Mechanics 81 (3) (2011) 263–281. [9] E. Miranda, S. Taghavi, Approximate floor acceleration demands in multistory buildings. I: Formulation, Journal of Structural Engineering& ASCE 131 (2) (2005) 203–211. [10] D. Raˇskovic´, Teorija Oscilacija, (Theory of Oscillations), (Naucˇna knjiga, 1965, in Serbian). /http://www.2shared.com/document/qLnDdzVq/ Teorija_oscilacija__DRaskovic_.htmlS. [11] S.G. Kelly, Advanced Vibration Analysis, CRC Press, Taylor & Francis Group, Boca Raton, FL, 2007. [12] P.S. Timoshenko, M.J. Gere, Theory of Elastic Stability, second ed. International student edition, McGraw-Hill, 1964. [13] C.M. Wang, J.N. Reddy, K.H. Lee, Shear Deformable Beams and Plates, Relationships with Classical Solutions, ELSEVIER, 2000.
Contents lists available at SciVerse ScienceDirect
Journal of Sound and Vibration journal homepage: www.elsevier.com/locate/jsvi
Exact closed-form solutions for the natural frequencies and stability of elastically connected multiple beam system using Timoshenko and high-order shear deformation theory Vladimir Stojanovic´, Predrag Kozic´ n, Goran Janevski University of Niˇs, Department of Mechanical Engineering, A. Medvedeva 14, 18000 Niˇs, Serbia
a r t i c l e i n f o
abstract
Article history: Received 21 February 2012 Received in revised form 29 August 2012 Accepted 5 September 2012 Handling editor: S. Ilanko Available online 12 October 2012
A general procedure for the determination of the natural frequencies and buckling load for a set of beam system under compressive axial loading is investigated using Timoshenko and high-order shear deformation theory. It is assumed that the set beams of the system are simply supported and continuously joined by a Winkler elastic layer. The model of beam includes the effects of axial loading, shear deformation and rotary inertia. In the special case of identical beams, explicit expressions for the natural frequencies and the critical buckling load are determined using a trigonometric method. The influences of the compressive axial loading and the number of beams in the system on the natural frequencies and the critical buckling load are discussed. These results are of considerable practical interest and have wide application in engineering practice of frameworks. & 2012 Elsevier Ltd. All rights reserved.
1. Introduction Vibration and buckling problems of beams or beam–columns on elastic foundations occupy an important place in many fields of structural and foundation engineering. This problem is very often encountered in aeronautical, mechanical, and civil engineering applications. Its solution demands the modeling of (a) the mechanical behavior of the beam, (b) the mechanical behavior of the soil and (c) the form of interaction between the beam and the soil. As far as the beam is concerned, most engineering analyses based on the classical Bernoulli–Euler beam theory, in which straight lines or planes normal to the neutral beam axis remain straight and normal after deformation. This theory thus neglects the effect of transverse shear deformations, a condition that holds only in the case of slender beams. To confront this problem, the wellknown Timoshenko-beam model can be used, in which the effect of transverse shear deflections is considered. A critical scrutiny of such studies, conducted for mechanical systems with a single beam, is presented in the review article by Chandra Duta and Roy [1]. Zhang et al. [2] derive explicit expressions for the natural frequencies and the associated amplitude ratios of the two beams, and the analytical solutions of the critical buckling load are obtained. The influences of the compressive axial loading on the responses of the double-beam system are discussed. It is shown that the critical buckling load of the system is related to the axial compression ratio of the two beams and the Winkler elastic layer, and the properties of free transverse vibration of the system greatly depend on axial compressions. Stojanovic´ et al. [3] study the influence of rotary inertia and shear on the free vibration and buckling of a double-beam system under axial loading.
n
Corresponding author. Tel.: þ381 18 500 634; fax: þ 381 18 588 244. E-mail addresses: [email protected] (V. Stojanovic´), [email protected] (P. Kozic´).
0022-460X/$ - see front matter & 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.jsv.2012.09.005
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Stojanovic´ and Kozic´ [4] found analytical solutions for forced vibration of a double beam system under the rotary inertia and shear effects. It is assumed that the system under consideration is composed of two parallel and homogeneous simply supported beams continuously joined by a Winkler elastic layer. Both beams have the same length. It is also supposed that the buckling can only occur in the plane where the double-beam system lies. The explicit expressions are derived for natural frequencies and associated amplitude ratio of the two beams, and the analytical solution of the critical buckling is obtained. The influence of the characteristics of the Winkler elastic layer on natural frequencies, and the critical buckling force is determined. Li et al. [5] establish an exact dynamic stiffness matrix established for an elastically connected threebeam system, which is composed of three parallel beams of uniform properties with uniformly distributed-connecting springs among them. The formulation includes the effects of shear deformation and rotary inertia of the beams. The dynamic stiffness matrix is derived by rigorous use of the analytical solutions of the governing differential equations of motion of the three-beam system in free vibration. The use of the dynamic stiffness matrix to study the three vibration characteristics of the three-beam system is demonstrated by applying the Muller root search algorithm. Kelly and Srinivas [6] investigate the problem concerning the free vibrations of a set of n axially loaded stretched Bernoulli–Euler beams connected by elastic layers. A normal-mode solution is applied to the governing partial differential equations to derive a set of coupled ordinary differential equations which are used to determine the natural frequencies and mode shapes. It is shown that the set of differential equations can be written in a self-adjoint form with an appropriate inner product. An exact solution for the general case is obtained, but numerical procedures must be used to determine the natural frequencies and mode shapes. The numerical procedure is difficult to apply, especially in determining higher frequencies. For the special case of identical beams, an exact expression for the natural frequencies is obtained in terms of the natural frequencies of a corresponding set of unstretched beams and the eigenvalues of the coupling matrix. The Adomian modified decomposition method (AMDM) is employed by Mao [7] in order to investigate the free vibrations of N elastically connected parallel Euler–Bernoulli beams, which are continuously joined by a Winkler-type elastic layer. The proposed AMDM method is used to analyze the vibration of beam system consisting of an arbitrary number of beams. By using boundary conditions, the natural frequencies and corresponding mode shapes are easily obtained simultaneously. Ariaei et al. [8] investigate the dynamic response of an elastically connected multiple beam system based on Timoshenko-beam theory. The identical prismatic beams are assumed to be parallel and connected by a finite number of springs. Assuming n parallel Timoshenko-beams, the motion of the system is described by a coupled set of 2n partial differential equations. The method involves a change of variables and modal analysis to decouple and to solve the governing differential equations, respectively. Miranda and Taghavi [9] present an approximate procedure to estimate floor acceleration demands in multistory buildings with the use of only a small number of parameters. Floor acceleration demands are computed using approximations of the first three modes of vibration of the building based on those of a continuum model consisting of a cantilever flexural beam connected laterally to a cantilever shear beam and models with uniform stiffness along the height. In this paper, the effect of compressive axial load on the properties of free natural frequencies and buckling load of a set Timoshenko and Reddy-Bickford beam system is investigated. It is assumed that the system under consideration is composed of a set of parallel, slender prismatic, and homogeneous simply supported beams continuously joined by a Winkler elastic layer. The beams have the same length l, and it is also supposed that the buckling can only occur in the plane where the system beams lie. The natural frequencies of the system with a different number of beams are determined using the numerical solution of the frequency equation. Also, using the trigonometric method, the explicit analytical expressions for the natural frequencies and the critical buckling load are determined. Numerical solution of the frequency equation determines the natural system frequencies and they are compared to the values obtained through the trigonometric method. The influence of the number of beams on the lowest natural frequency of the system is also determined.
2. Formulation of the differential equations and structural model using Timoshenko theory It can be seen from the literature that the dynamic analysis of the elastically parallel-beam system is concentrated primarily on the case of a double-beam system of two parallel simply supported beams continuously joined by a Winkler elastic layer. Very few research papers can be found to deal with the problem related to the elastically connected threebeam system. Those studies of this region are limited to the particular cases of identical beams with some prescribed boundary conditions. It is also shown that in most of the references, the simple Bernoulli–Euler beam theory has been used in deriving the necessary equation. The basic differential equations of motion for the analysis will be deduced by considering the Timoshenko-beam of length l, Fig. 1a, subjected to axial compressive force F, and to a distributed lateral loads of intensity q1 and q2 which vary with the distance z along the beam. This will be applied on the basis of the following assumptions: (a) the behavior of the beam material is linear elastic; (b) the cross-section is rigid and constant throughout the length of the beam and has one plane of symmetry; (c) shear deformations of the cross-section of the beam are taken into account while elastic axial deformations are ignored; (d) the equations are derived bearing in mind the geometric axial deformations; (e) the axial forces F acting on the ends of the beam are not changed with time. An element of length dz between two cross-sections taken normal to the deflected axis to the beam is shown in Fig. 1b. Since the slope of the beam is small, the normal forces acting on the sides of the element can be taken to be equal to the
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Fig. 1. The coordinate system and notation for Pinned–Pinned beam. (a) Timoshenko-beam subjected to an axial compressive force F and to a distributed lateral loads of intensity q1 and q2 ; (b) Deflected differential layered-beam element of length dz.
Fig. 2. A set of m elastically connected Timoshenko-beams on a Winkler type foundation.
axial compressive force F. The shearing force F T is related to the following relationship @w c , F T ¼ kGA @z
(1)
where w ¼ wðz,tÞ is the displacement of a cross-section in y direction, @w=@z is the global rotation of the cross-section, c is the bending rotation, G is the shear modulus, A is the area of the beam cross section, and k is the shear factor. Analogously, the relationship between bending moments M and bending angles c ¼ cðz,tÞ is given by M ¼ EIx
@c , @z
(2)
where E is the Young’s modulus and Ix is the second moment of the area of the cross-section. Finally, forces and moments of inertia are given by f I ¼ A
@2 w , @t 2
J I ¼ Ix
@2 , @t 2
(3)
respectively, where r is the mass density. The forces acting on a differential layered-beam element are shown in Fig. 1b. The dynamic-force equilibrium conditions of these forces are given by the following equations ! @2 w @2 w @c @2 w rA 2 kGA (4a) þ F 2 q1 ðzÞ þ q2 ðzÞ ¼ 0, 2 @z @z @z @t
rI
@2 c @2 c @w EI kGA c ¼ 0: x @z @z2 @t 2
(4b)
This paper presents a general theory for the determination of the natural frequencies and the critical buckling load of a set of m of elastically connected, identically axially loaded, Timoshenko-beams connected by elastic layers to a Winkler type foundation as illustrated in Fig. 2.
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Each beam is made of some material with Young’s modulus Ei, and mass density ri and has a cross-section with a uniform cross-section of area Ai and moment of inertia Ii ¼ Ix , and beams are subjected to the same compressive axial loading F. The first beam is connected to the ground by a Winkler foundation of the stiffness per length K0, and other beams are also connected by a continuous linear elastic layer of Winkler type of stiffness per length Ki. The transverse displacement of the beams is wi ¼ wi ðz,tÞ, i ¼ 1,2,3 m and ci ¼ ci ðz,tÞ,i ¼ 1,2,3 m are bending rotations. If we apply the abovementioned procedure to a differential element of each beam, the following set of coupled differential equations will be obtained: ! @2 w @2 w1 @c1 @2 w1 r1 A1 2 1 kG1 A1 þK 0 w1 K 1 ðw2 w1 Þ ¼ 0, (5a) þF 2 @z @z @z2 @t @2 c1 @2 c1 @w1 E I kG A c ¼ 0, 1 1 1 1 1 @z @z2 @t 2
(5b)
! @2 wi @2 wi @ci @2 wi kG A þ K i1 ðwi wi1 ÞK i ðwi þ 1 wi Þ ¼ 0, þF i i 2 2 @z @z @z2 @t
(6a)
r1 I1 ri Ai
ri Ii rm Am
@2 ci @2 ci @wi E I kG A c ¼ 0, i i i i i @z @z2 @t 2
i ¼ 2,3,. . .,m1
(6b)
! @2 wm @2 wm @m @2 wm kG A þ K m1 ðwm wm1 Þ ¼ 0, þF m m 2 2 @z @z @z2 @t
(7a)
@2 cm @2 cm @wm cm ¼ 0: Em Im kGm Am 2 2 @z @z @t
(7b)
rm Im
3. Governing equations and solution of the problem The equations of motion, i.e., Eqs. (5a–7b), which are coupled together, are reduced by standard procedure, eliminating
ci , i ¼ 1,2,3,. . .,m, to the following system of differential equations, E1 I1 1
4 F @ w1 r1 I1 @2 w1 K 1 r1 I1 @2 w2 E1 I1 @2 w1 E1 I1 @2 w2 þ r A þ ðK þ K Þ þ FðK þK Þ þK 1 1 0 1 0 1 1 kA1 G1 @z4 kA1 G1 @t 2 kA1 G1 @t 2 kA1 G1 @z2 kA1 G1 @z2 4 r2 I1 @4 w1 E1 F @ w1 r1 I1 1þ þ 1 þ xðK 0 þ K 1 Þw1 K 1 w2 ¼ 0 (8a) 2 2 kG1 kA1 G1 @z @t kG1 @t 4
Ei Ii 1
4 F @ wi K i1 ri Ii @2 wi1 r I @2 wi K i ri Ii @2 wi þ 1 K i1 Ei Ii @2 wi1 þ ri Ai þ ðK i1 þ K i Þ i i þ 4 2 kAi Gi @z kAi Gi kAi Gi @t 2 kAi Gi @t 2 kAi Gi @z2 @t 2 4 2 4 2 r Ii @ wi þ 1 E I @ wi K i Ei Ii @ wi þ 1 E F @ wi þ FðK i1 þ K i Þ i i þ ri Ii 1þ i þ i 2K i1 wi1 kAi Gi @z2 kAi Gi @z2 kGi kAi Gi @z2 @t 2 kGi @t 4 þ ðK i1 K i Þwi K i wi þ 1 ¼ 0,i ¼ 2,3,. . .,m1
Em Im 1
@4 wm K m1 rm Im @2 wm1 K m1 rm Im @2 wm K m1 Em Im @2 wm1 þ r A þ þ m m kAm Gm kAm Gm kAm Gm kAm Gm @z4 @z2 @t 2 @t 2 2 4 4 2 K m1 Em Im @ wm Em F @ wm r Im @ wm þ F rm Im 1 þ þ m K m1 wm1 þ K m1 wm ¼ 0: kAm Gm kGm kAm Gm @z2 @t 2 kGm @t 4 @z2 F
(8b)
(8c)
The initial conditions in general form and boundary conditions for simply supported beams of the same length l are assumed as follows wi ðz,0Þ ¼ wi0 ðzÞ,
_ i ðz,0Þ ¼ vi0 ðzÞ, w
ci ðz,0Þ ¼ ci0 ðzÞ, c_ i ðz,0Þ ¼ oi0 ðzÞ,
wi ð0,tÞ ¼ wi ðl,tÞ ¼ w00i ð0,tÞ ¼ w00i ðl,tÞ ¼ 0,
i ¼ 1,2,3,. . .,m:
(9) (10)
Assuming time harmonic motion and using separation of variables, the solutions of Eqs. (8a–c) with the governing boundary conditions (10) can be written in the form wi ðz,tÞ ¼
1 X n¼1
X n ðzÞT in ðtÞ,
i ¼ 1,2,3,. . .,m
(11)
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where T in ðtÞ,i ¼ 1,2,3. . . is the unknown time function, and Xn(z) is the known mode shape function for simply supported single beam, which is defined as kn ¼ np=l,
X n ðxÞ ¼ sinðkn zÞ,
n ¼ 1,2,3,. . .:
(12)
Introducing the general solutions (11) into Eqs. (8a–c) one gets the system of ordinary differential equations # ( " ! 4 2 2 1 X 1 d T 1n C 2b1 1 d T 1n H1 d T 2n 2 2 n þ 1 þC 1þ þ ðH þ H F Z Þ 2 k 1 r1 0 1n n 2 4 2 4 2 2 dt C s1 C r1 C s1 dt 2 C s1 n ¼ 1 C s1 dt " !# ! ) C2 C2 4 2 2 þ C 2b1 kn þ ðHn0 þ H1 F Z1n Þ 1þ 2 b12 kn T 1n H1 1 þ 2 b12 kn T 2n X n ¼ 0, C s1 C r1 C s1 C r1 # " ! 2 2 4 2 1 d T in Hni1 d T ði1Þn C 2bi 1 d T in Hi d T ði þ 1Þn 2 2 n þ 1 þ C 1 þ þ ðH þH F Z Þ k i ri n i1 in dt 2 dt 2 dt 2 C 2si C 2si C 4ri C 2si C 2si dt 4 C 2si ! " !# 2 2 C C 2 4 2 Hnði1Þ 1 þ 2 bi 2 kn T ði1Þn þ C 2bi kn þðHni1 þHi F Zin Þ 1 þ 2 bi 2 kn T in C si C ri C si C ri ! ) C 2bi 2 Hi 1 þ 2 2 kn T ði þ 1Þn X n ¼ 0, i ¼ 2,3,. . .,m1 C si C ri
(13a)
(
1 X n¼1
(13b)
# " ! 2 4 2 1 d T mn Hnm1 d T ðm1Þn C 2bm 1 d T mn 2 2 n þ 1 þ C 1 þ þ ðH F Þ k mn rm m1 n 2 4 2 2 2 4 2 dt dt 2 C sm C sm C rm C sm n ¼ 1 C sm dt ! " !# ) C2 C2 2 4 2 Hnm1 1 þ 2 bm2 kn T ðm1Þn þ C 2bm kn þ ðHnm1 F Zmn Þ 1 þ 2 bm2 kn T mn X n ¼ 0, C sm C rm C sm C rm (
1 X
(13c)
where K i1 , ri Ai
Hni1 ¼
Ki
Hi ¼
ri Ai
2
Zin ¼
,
kn
ri Ai
,
i ¼ 1,2,3,. . .,m:
The coefficients sffiffiffiffiffi Ii C ri ¼ , Ai
sffiffiffiffiffiffiffiffiffiffiffiffi Gi Ai k , C si ¼ ri I i
sffiffiffiffiffiffiffiffiffi Ei Ii , C bi ¼ ri Ai
i ¼ 1,2,3,. . .,m:
related to bending stiffness, shear stiffness and rotational effects, respectively, are now introduced. The shear beam model, the Rayleigh beam model and the simple Euler beam model can be obtained from the Timoshenko-beam model by setting C ri to zero (that is, ignoring the rotational effect), C si to infinity (ignoring the shear effect) and setting both C ri to zero and C si to infinity, respectively. The solutions of Eqs. (13a–c) can be assumed to have the following forms pffiffiffiffiffiffiffi T in ¼ Ain ejon t , i ¼ 1,2,3,. . .,m, j ¼ 1, (14) where on denotes the natural frequency of the system. Substituting Eq. (14) into Eqs. (13a–c) results in the following system of homogeneous algebraic equations for the unknown constants A1n ,A2n ,A3n ,. . .Amn , (
o4n
C 2s1
" 1 þ C 2r1
1þ
C 2b1
! 2 kn þ
C 2s1 C 4r1
#
1
n
C 2s1
ðH0 þ H1 F Z1n Þ o "
þ H1 " Hni1 "
o2n C 2si
1þ
C 2bi C 2si C 2ri
4
"
o2n C 2sm
1þ
C 2bm C 2sm C 2rm
C 2s1
!# 2
kn
þ C 2bi kn þ ðHni1 þ Hi F Zin Þ 1þ
Hnm1
o2n
o4n
C 2bi C 2si C 2ri
!#) 2
kn
!# 2
C 2si
" 4
Z
"
o4n
C 2sm
o2n C 2si
2
kn
!#)
C 2b1
2 kn C 2s1 C 2r1
A1n
A2n ¼ 0,
C 2bi C 2si C 2ri
"
C 2bm C 2sm C 2rm
1
2
C 2si C 4ri
1 þC 2rm 1þ
(15a)
!
C 2bi
1þ
þ C 2bm kn þ ðHnm1 F Zmn Þ 1 þ
1þ
!#
1þ C 2ri 1 þ
Ain þHi (
Aðm1Þn þ
C 2b1 2 2 C s1 C r1
4 C 2b1 kn þðHn0 þ H1 F 1n Þ
"
( Aði1Þn þ
kn
1þ
" 2 nþ
kn þ !# 2
kn
Aði þ 1Þn ¼ 0, !
C 2bm
2
C 2sm C 4rm !#) 2
kn
C 2si
# ðHni1 þ Hi F Zin Þ o2n
kn þ
1 C 2sm
Amn ¼ 0:
i ¼ 2,3,. . .,m1
(15b)
# ðHnm1 F Zmn Þ o2n (15c)
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3.1. Identical beams When all beams are identical Ai ¼ A, Ei ¼ E, Gi ¼ G, Ii ¼ I, K i1 ¼ K, K m ¼ 0, ri ¼ r, i ¼ 1,2,3,. . .,m, then substituting these value in Eqs. (15a–c) and rewriting in matrix form, one obtains the following homogeneous set of m algebraic equations 9 9 8 8 0 > > > A1n > > > > 2 3> > > > > > 0 > > > > A2n > un 0 0 0 0 0 0 0 Sn > > > > > > > > > > > > > > 6 u 7> > > > > > > > S u 0 0 0 0 0 0 A 0 n n n 3n 6 7> > > > > > > > 6 7> > > > > > > > 6 7> > > > > > > > > > > 6 7> > > > > > > > > 6 0 7 > > > > u 0 0 0 0 A 0 0 S 0 n n > > 6 7> ði1Þn > = < = < 6 7 6 0 7 S u 0 0 0 A 0 0 u 0 (16) ¼ n n n in 6 7> > > > > > > >A 6 0 > > > Sn 0 0 0 7 0 0 0 un 0 > > > > 6 7> ði þ 1Þn > > > > > > > 6 7> > > > > > > > 6 7 > > > > > > > > 6 7> > > > > > > > > 6 0 7 > > > Sn un 5> A 0 0 0 0 0 un > > > > 0 4 ðm2Þn > > > > > > > > > > > > > > > > > > > 0 0 0 0 0 0 0 un Sn un > A 0 ðm1Þn > > > > > > > > ; ; : : Amn 0 where 2
K i1 Ki k ¼ H, Hi ¼ ¼ H, Zin ¼ n ¼ Zn , i ¼ 1,2,3,. . .,m, ri Ai ri Ai ri Ai sffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffi Ei Ii Gi Ai k Ii C bi ¼ ¼ C s , C ri ¼ ¼ C b , C si ¼ ¼ C r , i ¼ 1,2,3,. . .,m, ri Ai ri Ii Ai ! " !# " # o4 C2 1 C2 2 2 4 Sn ¼ 2n 1 þC 2r 1 þ 2 b 4 kn þ 2 ð2HF Zn Þ o2n þ C 2b kn þ ð2HF Zn Þ 1 þ 2 b 2 kn , Cs Cs Cr Cs Cr Cs " !# 2 o2 C 2 un ¼ H 2n þ 1 þ 2 b 2 kn : Cs Cs Cr Hni1 ¼
Eq. (16) represents both the free-vibration and stability eigenvalue problem of a set of m identically elastically connected Timoshenko-beams with generalized end conditions (10). By making the determinant of the mxm matrix equal to zero yields an ordinary polynomial of the 4mth order in terms of the undamped natural frequencies on : Even the analytical determination of natural frequencies for a mechanical system comprising three same Timoshenko-beams presents a certain problem. For a system with more than three same Timoshenko-beams, natural frequencies can be determined only numerically for specific physical system parameters. For a homogenous system of algebraic Eq. (16), analytical expressions for natural frequencies of the system can be determined using the trigonometric method [10, pp. 157–166]. According to [10], let us assume that the value of unknown constants Ain ,i ¼ 1,2,3. . .m changes oscillatory Ain ¼ N sinðijÞ,
i ¼ 1,2,3,. . .,m:
(17)
Let us observe that the first and the last equation of system (16) have a structure different to the structure of the other equations. Substituting assumed solutions (17) into the ith equation of system (16) for i ¼ 2,3. . .m1 yields un N sinði1Þj þ Sn N sinðijÞun N sinði þ 1Þj ¼ 0,
i ¼ 2,3,. . .,m1:
(18)
After some algebra, the following is obtained ðSn 2un cos jÞN sinðijÞ ¼ 0,
i ¼ 2,3,. . .,m1:
(19)
Eq. (19) has to have Na0, and sinðijÞa0, because then Ain a0, for i ¼ 2,3. . .m1: From there Sn 2un cos j ¼ 0 ) Sn ¼ 2un cos j:
(20)
Eq. (20) is a frequency equation, which is a fourth order polynomial in on . The unknown j can be determined based on the condition that the assumed solutions (17) satisfy the first and the last equation of system (16). When A1n ¼ N sin j and A2n ¼ N sin 2j are substituted into the first equation of system (16), the following is obtained Sn N sin jun N sin 2j ¼ 0 ) ðSn 2un cos jÞNsin j 0:
(21)
Eq. (21) is identical since it is determined that Sn 2un cos j ¼ 0, where Na0, and sin ja0: When the assumed solutions for Aðm1Þn and Amn are substituted into the mth equation of system (16), the following is obtained un N sin½ðm1jÞ þ ðSn un ÞN sinðmjÞ ¼ 0:
(22)
V. Stojanovic´ et al. / Journal of Sound and Vibration 332 (2013) 563–576
569
After some algebra, the following is obtained 2m þ 1 Nun sinðj=2Þcos j ¼ 0: 2
(23)
Solutions of Eq. (23) are cos
2mþ 1 2s1 j ¼ 0 ) js ¼ p, s ¼ 1,2,3,. . .,m, 2 2mþ 1
since Na0, un a0 and sinðj=2Þa0: By substituting solutions (24) into (20), this frequency equation follows " # ! " !# o4n,s C 2b 1 C 2b 2 2 2 2 4 2 1 þ C r 1 þ 2 4 kn þ 2 ð2HF Zn Þ on,s þ C b kn þ ð2HF Zn Þ 1 þ 2 2 kn C 2s Cs Cr Cs Cr Cs " !# o2 C2 2 2s1 þ 1þ 2 b 2 kn cos p : ¼ 2H n,s 2 2m þ 1 Cs Cs Cr After some algebra, biquadratic algebraic equation is obtained 2s1 2s1 o4n,s C 2s ðRn þ C 2r k2n Þ þ 4Hsin2 p F Zn o2n,s þ C 2s C 2b k4n þ4HRn sin2 p F Zn Rn ¼ 0, 2ð2m þ 1Þ 2ð2m þ1Þ
(24)
(25)
(26)
s ¼ 1,2,3,. . .,m: where Rn ¼ 1 þ
C 2b C 2s C 2r
2
kn :
Then from frequency Eq. (26), we obtain
i 1nh 2 2 2s1 C s ðRn þ C 2r kn Þ þ 4H sin2 2ð2m þ 1Þ p F Zn 2 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi9 2 = 2s1 2s1 2 p F Zn 4C 2s C 2b k4n þ4HRn sin2 p F Zn Rn , C 2s ðRn þ C 2r kn Þ þ 4H sin2 ; 2ð2m þ 1Þ 2ð2m þ 1Þ ½o2n,s l ¼
(27)
s ¼ 1,2,3,. . .,m:
i 1nh 2 2 2s1 C s ðRn þ C 2r kn Þ þ 4H sin2 2ð2m þ 1Þ p F Zn 2 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi9 2 = 2s1 2s1 2 þ C 2s ðRn þ C 2r kn Þ þ 4H sin2 p F Zn 4C 2s C 2b k4n þ 4HRn sin2 p F Zn Rn , ; 2ð2m þ1Þ 2ð2m þ 1Þ ½o2n,s h ¼
(28)
s ¼ 1,2,3,. . .,m: where ½on,s l is the lower natural frequency of the system, and ½on,s h is the higher natural frequency [11, pp. 403]. When the natural frequency of the system vanishes under the axial loading, the system begins to buckle. Introduction of on,s ¼ 0 into Eq. (26) gives 2s1 4 C 2b kn þ 4HRn sin2 p F Zn Rn ¼ 0, s ¼ 1,2,3,. . .,m: (29) 2ð2mþ 1Þ It follows from Eq. (29) that the value of the buckling stress corresponding to vibration mode n of m identically elastically connected Timoshenko-beams can be obtained by 4 C 2b kn H 2s1 F bkr ¼ min þ 4 sin2 p : (30) 2ð2m þ 1Þ Zn n ¼ 1,2,3 Rn Zn s ¼ 1,2,3 m
For K ¼0 from Eq. (30) we obtain Pn ¼
EIp2 n2 2
2
l ð1 þ ðEI=GAkÞðp2 n2 =l ÞÞ
:
(31)
V. Stojanovic´ et al. / Journal of Sound and Vibration 332 (2013) 563–576
570
P n is the critical buckling load corresponding to the buckled mode number n of the Timoshenko-beam. By setting n ¼ 1 Eq. (31) is reduced to EIp2
P¼
2
ð1 þðEI=GAkÞðp2 =l ÞÞl
2
:
and coincides with Eq. (2–58) on page 134 of Timoshenko and Gere [12]. 4. Formulation of equations and structural model using Reddy–Bickford beam theory Introducing the higher order shear deformation theory by Reddy–Bickford [13], equations of motion for the system presented on Fig. 2 are ! @2 w1 @3 c1 @4 w1 @c1 @2 w1 @2 w1 2 ^ þ F1 þ r1 A1 2 aF zz 3 þ a Hzz 4 Azy þK 0 w1 K 1 ðw2 w1 Þ ¼ 0, (32a) @z @z @z @z2 @z2 @t 3 @2 c1 @2 c1 ^ zz @ w1 Azy c þ @w1 ¼ 0, þD a F zz 1 @z @z2 @z3 @t 2 ! @2 w @3 c @4 w @ci @2 wi @2 wi þ ri Ai 21 aF^ zz 3 i þ a2 Hzz 4 i Azy þ K i1 ðwi wi1 ÞK i ðwi þ 1 wi Þ ¼ 0, þ Fi @z @z @z @z2 @z2 @t
r1 I1
3 @2 ci @2 ci ^ zz @ wi Azy c þ @wi ¼ 0, i ¼ 2,3,. . .,m1 F þ D a zz i @z @z2 @z3 @t 2 ! @2 wm @3 cm @4 wm @cm @2 wm @2 wm 2 ^ þ rm Am 2 aF zz 3 þ a Hzz 4 Azy þ K m1 ðwm wm1 Þ ¼ 0, þ Fm @z @z @z @z2 @z2 @t
ri Ii
rm Im
3 @2 cm @2 cm ^ zz @ wm Azy c þ @wm ¼ 0, F þ D a zz m @z @z2 @z3 @t 2
(32b)
(33a)
(33b)
(34a)
(34b)
where ^ zy , Azy ¼ A^ zy bD
a¼
4 2
3h
^ zz aF^ zz , Dzz ¼ D
^ zz , F zz ¼ F^ zz aH
b ¼ 3a, D^ zz ¼ Dzz aF zz , F^ zz ¼ F zz aHzz , A^ zy ¼ Azy bDzy , D^ zy ¼ Dzy bF zy
,
ðAzz ,Dzz ,F zz ,Hzz Þ ¼
Z
ð1,y2 ,y4 ,y6 ÞEz dA,
ðAzy ,Dzy ,F zy Þ ¼
Z
A
ð1,y2 ,y4 ÞGzy dA:
A
After some algebra, the equations of motion can be reduced, eliminating ci , i ¼ 1,2,3, m, to the following system of differential equations ! ! 2 a2 F^ zz a2 Dzz Hzz @6 w1 a2 r1 I1 Hzz @6 w1 Dzz F 1 @4 w1 2 ^ F þ D þ 2 a þ a H zz zz zz 6 2 4 @z @z4 @z @t Azy Azy Azy Azy ! " # r A1 Dzz r1 I1 F 1 @4 w1 K 1 Dzz @2 w2 Dzz ðK 0 þ K 1 Þ @2 w1 þ r1 I1 1 þ þ F1 2 2 2 @z2 @z @t Azy Azy Azy @z Azy " # r2 A1 I1 @4 w1 r1 I1 K 1 @2 w2 r I1 ðK 0 þ K 1 Þ @2 w1 1 þ þ r1 A1 1 þðK 0 þ K 1 Þw1 K 1 w2 ¼ 0 (35a) 4 2 @t @t @t 2 Azy Azy Azy !
2
a2 F^ zz Azy
a2 Dzz Hzz @6 wi @z2
Azy
a2 ri Ii Hzz @6 wi Azy
@z4 @t
þ Dzz
Azy "
! þ2aF^ zz þ a2 Hzz
@4 wi @z4
# @ wi K i Dzz @ wi þ 1 Dzz ðK i1 þ K i Þ @2 wi þ þ F i @z2 @z2 @z2 @t 2 Azy Azy Azy Azy " # Dzz K i1 @2 wi1 r2i Ai Ii @4 wi ri Ii K i @2 wi þ 1 r I ðK þ K i Þ @2 wi ri Ii K i1 @2 wi1 þ þ þ ri Ai i i i1 þ 2 4 2 @z @t @t 2 @t 2 Azy Azy @t Azy Azy Azy þ ri Ii
ri Ai Dzz
ri Ii F i
!
2
Dzz F i
4
2
2K i1 wi1 þ ðK i1 K i Þwi K i wi þ 1 ¼ 0, !
2
a2 F^ zz Azy
a2 Dzz Hzz @6 wm Azy
@z6
a2 rm Im Hzz @6 wm Azy
@z4 @t 2
þ Dzz
i ¼ 2,3, m1 Dzz F m Azy
(35b) !
þ 2aF^ zz þ a2 Hzz
@4 wm @z4
V. Stojanovic´ et al. / Journal of Sound and Vibration 332 (2013) 563–576
þ rm Im
2 m Am Im
r
Azy
rm Am Dzz Azy
rm Im F m
!
Azy
4
2
571
@ wm rm Im K m1 @ wm1 þ @t 4 @t 2 Azy
! @4 wm Dzz K m1 @2 wm Dzz K m1 @2 wm1 þ F þ m @z2 @z2 @z2 @t 2 Azy Azy ! r Im K m1 @2 wm þ rm Am m K m1 wm1 þ K m1 wm ¼ 0: @t 2 Azy
(35c)
Following the same procedure as in case for the system of identical Timoshenko-beams, assuming time harmonic motion and using separation of variables for the same boundary conditions, new homogeneous set of m algebraic equations is 9 8 9 8 > A1n > > 2 RB 3> > 0 > > > > > > > RB > > > > > > > > un 0 0 0 0 0 0 0 Sn > > > 0 > > > > A2n > 6 7 > > > > > > > > RB RB 6 uRB 7 > > > > S u 0 0 0 0 0 0 A > > > > 0 3n n n n 6 7> > > > > > > > 6 7> > > > > > > > 7> 6 > > > > > > > > > > > 6 7 > > > > RB RB > > > > 6 0 7 > > un 0 0 0 0 0 0 Sn Aði1Þn > 0 > > > > 6 7> = < = < 6 7 RB RB RB 6 0 7 A 0 0 u S u 0 0 0 0 (36) ¼ in n n n 6 7> > > > > > > 6 7> RB > > > > RB A 0 > > > > ði þ 1Þn 6 0 7 Sn 0 0 0 0 0 0 un > > > > > > > 6 7> > > >> > > > > 6 > > > 7 > > > > 6 7> > > > > > 6 > > 0 > > > > Aðm2Þn > RB RB RB 7 > > > > 6 0 7 S u 0 0 0 0 0 u > > > n n n > > > > 4 5> > > > > > > > RB > > > > RB RB > 0 A > > ðm1Þn > 0 0 0 0 0 0 0 un Sn un > > > > > ; : > > > > : Amn ; 0 where,
(
SRB n ¼
þ
)
r½Ið2KFk2n þ a2 k4n Hzz Þ þ Azy ðIk2n AÞAk2n Dzz ~ 2 r2 AI ~ 4 on þ on Azy
2 Dzz ð2Kkn þ
Azy
4 6 2 2 2 6 kn Hzz Fkn Þ 2 kn F^ zz þAzy ð2KFkn þ
a
a
a2 k4n Hzz þk4n Dzz þ2ak4n F^ zz Þ
Azy
,
!
uRB n ¼
2 rIK ~ 2 k D o n þK 1þ n zz :
Azy
Azy
Trigonometric solutions of the system are shown for Timoshenko case and by substituting solutions (24) into RB SRB n 2un cos j ¼ 0
RB SRB n ¼ 2un cos j:
)
(37)
new, for high shear order deformable multi beam equation follows ( ) 2 4 2 2 Ið2KFkn þ a2 kn Hzz Þ þ Azy ðIkn AÞAkn Dzz r2 AI ~ 4 on þ o~ 2n Azy Azy þ
2 6 4 6 2 2 4 4 4 Dzz ð2Kkn þ a2 kn Hzz Fkn Þa2 kn F^ zz þAzy ð2KFkn þ a2 kn Hzz þkn Dzz þ2akn F^ zz Þ
Azy
"
2 rIK ~ 2 k D ¼2 o n þK 1þ n zz
Azy
Azy
!#
2s1 p , cos 2m þ 1
(38)
Then from Eq. (38), we obtain frequency equation pffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2A3 o~ n,s ¼ A2 þ 2B2 f ðs,mÞ ½A2 2B2 f ðs,mÞ2 4A3 ½A1 2B1 f ðs,mÞ, 2A3 where
(39)
2s1 f ðs,mÞ ¼ cos p , 2m þ1 2
A1 ¼
6
4
6 2
2
4
4
4
Dzz ð2Kkn þ a2 kn Hzz Fkn Þa2 kn F^ zz þAzy ð2KFkn þ a2 kn Hzz þkn Dzz þ2akn F^ zz Þ Azy A2 ¼ A3 ¼
r½Ið2KFk2n þ a2 k4n Hzz Þ þ Azy ðIk2n AÞAk2n Dzz
r2 AI Azy
2
,B1 ¼ K 1 þ
Azy !
kn Dzz Azy
,B2 ¼
rIK Azy
,
:s ¼ 1,2,3 m:
,
V. Stojanovic´ et al. / Journal of Sound and Vibration 332 (2013) 563–576
572
When the natural frequency of the system vanishes under the axial loading, the system begins to buckle. Introduction of on,s ¼ 0 into Eq. (38) gives 2 3 h i 2 2 4 2 6 2 6^ 2 4 ^ 1 6Dzz ð2Kkn þ a kn Hzz Þa kn F zz þ Azy 2K þ a kn Hzz þ kn ð2aF zz þ Dzz Þ 7 RB min 2Kf ðs,mÞ5: (40) F kr ¼ 4 2 m ¼ 1,2,3 k2 A þ k D zy zz n n s ¼ 1,2,3,. . .,m
5. Numerical results The values for the parameters of the system which are used in the numerical calculations are given in [2] E ¼ 1 1010 N m2 , 5
K 0 ¼ 2 10 2
A ¼ 5 10
2
m ,
Nm
G ¼ 0:417 1010 N m2 , 2
,
3
k ¼ 5=6,
3
r ¼ 2 10 kg m , l ¼ 10 m,
I ¼ 4 10
4
4
m ,
1 b¼ 8
rffiffiffi 5 m, 3
2 h¼ 5
rffiffiffiffiffi 3 m: 5
(41)
For system parameters (41), and through numerical solution of the appropriate characteristic equation for system Eq. (16), the natural lower and higher modal frequencies for the system with m ¼3,5,7 and 9 beams are determined and shown in Table 1 for Timoshenko system. Table 1 also shows these frequencies determined using the trigonometric method through Eqs. (27, 28). Table 2 shows differences between frequencies using Timoshenko and high order shear deformation theory. It can be observed from Table 1 that the differences between natural frequencies determined through numerical solution of the characteristic equation and using analytical expressions obtained through the trigonometric method are very small. Another observation is that the value of the smallest natural frequency decreases with the increase with the number of beams m in the system. In the following, let us illustrate the effect of compressive axial loading at the lowest natural frequency of the system with one, three and five equal beams, m ¼1,3,5. In order to investigate the influence of compressive axial loading at the lowest natural frequency for stiffness modulus K ¼1,5K0,K0,0,5K0, the necessary analytical expressions are determined when s¼1 n ¼1 and m ¼1,3,5 are substituted in that order in Eq. (27). After the substitution, the following is obtained: i 1 nh 2 p 2 C s ðR1 þC 2r k1 Þ þ4Hsin2 F Z1 ½o21,1 l ¼ 2 6 Table 1 Timoshenko low and high natural frequencies [Hz] for mode n¼ 1 and for system with number of beam m ¼3, 5, 7, 9 and F¼ 0. Beam numberm
Lower modal frequency
Higher modal frequency
Calculated using Eq. (28)
Calculated using Eq. (17)
Calculated using Eq. (29)
Calculated using Eq. (17)
m¼3
½o1,1 l ¼ 28:0046632875 ½o1,2 l ¼ 59:1263182936 ½o1,3 l ¼ 82:9295278423
28:0046632873 59:1263182935 82:9295278423
½o1,1 h ¼ 14,759:81289 ½o1,2 h ¼ 14,759:81296 ½o1,3 h ¼ 14,759:81305
14,759:8129651 14,759:8129651 14,759:8129651
m¼5
½o1,1 l ¼ 23:4595097487 ½o1,2 l ¼ 42:0466472348 ½o1,3 l ¼ 61:7777785430 ½o1,4 l ¼ 77:7538496710 ½o1,5 l ¼ 88:0208916175
23:4595097486 42:0466472347 61:7777785429 77:7538496709 88:0208916176
½o1,1 h ¼ 14,759:81288 ½o1,2 h ¼ 14,759:81291 ½o1,3 h ¼ 14,759:81296 ½o1,4 h ¼ 14,759:81302 ½o1,5 h ¼ 14,759:81307
14,759:8129722 14,759:8129722 14,759:8129722 14,759:8129722 14,759:8129722
m¼7
½o1,1 l ¼ 21:8125810709 ½o1,2 l ¼ 33:9378813493 ½o1,3 l ¼ 48:8556749524 ½o1,4 l ¼ 62:9882585212 ½o1,5 l ¼ 74:9693492803 ½o1,6 l ¼ 84:0221781219 ½o1,7 l ¼ 89:6471858944
21:8125810709 33:9378813494 48:8556749523 62:9882585213 74:9693492802 84:0221781219 89:6471858945
½o1,1 h ¼ 14,759:81287 ½o1,2 h ¼ 14,759:81289 ½o1,3 h ¼ 14,759:81292 ½o1,4 h ¼ 14,759:81297 ½o1,5 h ¼ 14,759:81301 ½o1,6 h ¼ 14,759:81305 ½o1,7 h ¼ 14,759:81308
14,759:8129739 14,759:8129739 14,759:8129739 14,759:8129739 14,759:8129739 14,759:8129739 14,759:8129739
m¼9
½o1,1 l ¼ 21:0466074643 ½o1,2 l ¼ 29:4987136192 ½o1,3 l ¼ 40:9671530516 ½o1,4 l ¼ 52:7236801181 ½o1,5 l ¼ 63:6808309227 ½o1,6 l ¼ 73:2562413513 ½o1,7 l ¼ 81:0640517350 ½o1,8 l ¼ 86:8296550265 ½o1,9 l ¼ 90:3637299233
21:0466074644 29:4987136191 40:9671530516 52:7236801181 63:6808309228 73:2562413513 81:0640517351 86:8296550263 90:3637299234
½o1,1 h ¼ 14,759:81287 ½o1,2 h ¼ 14,759:81288 ½o1,3 h ¼ 14,759:81291 ½o1,4 h ¼ 14,759:81294 ½o1,5 h ¼ 14,759:81297 ½o1,6 h ¼ 14,759:81300 ½o1,7 h ¼ 14,759:81304 ½o1,8 h ¼ 14,759:81306 ½o1,9 h ¼ 14,759:81308
14,759:8128375 14,759:8128375 14,759:8128375 14,759:8128375 14,759:8128375 14,759:8128375 14,759:8128375 14,759:8128375 14,759:8128375
V. Stojanovic´ et al. / Journal of Sound and Vibration 332 (2013) 563–576
573
Table 2 Timoshenko and Reddy Bickford natural frequencies [Hz] for mode n ¼ 1 and for system with number of beam m ¼ 3,5,7,9 and F ¼ 0. Beam number m
Timoshenko Calculated using Eq. (28)
Reddy–Bickford Calculated using Eq. (39)
m¼3
½o1,1 l ¼ 28:0046632875 ½o1,2 l ¼ 59:1263182936 ½o1,3 l ¼ 82:9295278423
½o1,1 l ¼ 28:0340499486 ½o1,2 l ¼ 59:1763262258 ½o1,3 l ¼ 82:9972672789
m¼5
½o1,1 l ¼ 23:4595097487 ½o1,2 l ¼ 42:0466472348 ½o1,3 l ¼ 61:7777785430 ½o1,4 l ¼ 77:7538496710 ½o1,5 l ¼ 88:0208916175
½o1,1 l ¼ 23:4867433497 ½o1,2 l ¼ 42:0846292831 ½o1,3 l ¼ 61:8297234917 ½o1,4 l ¼ 77:8176815808 ½o1,5 l ¼ 88:0924938871
m¼7
½o1,1 l ¼ 21:8125810709 ½o1,2 l ¼ 33:9378813493 ½o1,3 l ¼ 48:8556749524 ½o1,4 l ¼ 62:9882585212 ½o1,5 l ¼ 74:9693492803 ½o1,6 l ¼ 84:0221781219 ½o1,7 l ¼ 89:6471858944
½o1,1 l ¼ 21:839180854 ½o1,2 l ¼ 33:9706521525 ½o1,3 l ¼ 48:8983327711 ½o1,4 l ¼ 63:0410919649 ½o1,5 l ¼ 75:0310883912 ½o1,6 l ¼ 84:0907450889 ½o1,7 l ¼ 89:7200254756
m¼9
½o1,1 l ¼ 21:0466074643 ½o1,2 l ¼ 29:4987136192 ½o1,3 l ¼ 40:9671530516 ½o1,4 l ¼ 52:7236801181 ½o1,5 l ¼ 63:6808309227 ½o1,6 l ¼ 73:2562413513 ½o1,7 l ¼ 81:0640517350 ½o1,8 l ¼ 86:8296550265 ½o1,9 l ¼ 90:3637299233
½o1,1 l ¼ 21:0729477466 ½o1,2 l ¼ 29:528903779 ½o1,3 l ¼ 41:0044146338 ½o1,4 l ¼ 52:7690725794 ½o1,5 l ¼ 63:7341738108 ½o1,6 l ¼ 73:3166966242 ½o1,7 l ¼ 81:1303803448 ½o1,8 l ¼ 86:900351992 ½o1,9 l ¼ 90:4371151478
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi) h i2 h i p p 2 4 C 2s ðR1 þC 2r k1 Þ þ4Hsin2 F Z1 4C 2s C 2b k1 þ 4HR1 sin2 F Z1 R1 , 6 6
i 1nh 2 p 2 C s ðR1 þC 2r k1 Þ þ4Hsin2 F Z1 2 14 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi) h i2 h i p p 2 4 F Z1 4C 2s C 2b k1 þ 4HR1 sin2 F Z1 R1 , C 2s ðR1 þ C 2r k1 Þ þ 4Hsin2 14 14
(42)
½o21,1 l ¼
i 1 nh 2 p 2 C s ðR1 þC 2r k1 Þ þ4Hsin2 F 1 2 22 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi) h i h i 2 p p 2 4 F Z1 4C 2s C 2b k1 þ 4HR1 sin2 F Z1 R1 , C 2s ðR1 þ C 2r k1 Þ þ 4Hsin2 22 22
(43)
½o21,1 l ¼
(44)
with
l ¼ F=F bkr : It can be observed from Fig. 3 that with the increase in compressive axial loading the lower natural frequency ½o21,1 I decreases. It can also be found that the decrease in the stiffness modulus K decreases the lower natural frequency. This decrease is the smallest in the system with a single beam m¼1. With the rising number of beams, the influence of the stiffness modulus K on the lower natural frequency decreases. It can be concluded from Fig. 4 that with the increasing number of beams m, the lower natural frequency decreases. When s¼1 and n ¼1 are substituted in Eq. (30), one obtains analytical expressions for the critical buckling force for systems with different numbers of beams m¼1,3,5, respectively, 4
F bkr ¼
C 2b k1 H p þ 4 sin2 , R1 Z1 Z1 6
F bkr ¼
C 2b k1 H p , þ 4 sin2 R1 Z1 Z1 14
(45)
4
(46)
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Fig. 3. Effect of the stiffness modulus K at the lowest natural frequency for system with. (a) m ¼ 1 beam, Eq. (33); (b) m ¼ 3 beams, Eq. (34); (c) m ¼ 5 beams, Eq. (35).
Fig. 4. Effect of the number of elastically connected axially loaded beams at the lowest natural frequency for K ¼ K 0 .
4
F bkr ¼
C 2b k1 H p , þ4 sin2 R1 Z1 Z1 22
(47)
In Fig. 5, static stability regions for the first vibration mode n ¼1 are represented for a system with one, three and five Timoshenko-beams supported on a Winkler elastic layer. It can be seen that the static stability region is largest in the case when one –Timoshenko-beam is supported on a Winkler elastic layer, Eq. (45). In that case, the critical buckling load F bkr is also the largest. For the system with three Timoshenko-beams supported on a Winkler elastic layer, the static stability region and critical buckling load are reduced, Eq. (46). This trend of the reduction of the static stability region and critical buckling load continues for the system with five Timoshenko-beams, Eq. (47). We can conclude that in the studied
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575
Fig. 5. Effect of the number of elastically connected axially loaded beams on the critical buckling load F bkr .
oscillatory system the region of stability is reduced with the increasing number of beams, since each added beam introduces additional instability to the system, that is, influences the reduction of the critical buckling force. 6. Conclusions The problem of free transverse vibration and buckling of an elastically connected simply supported set of beams under compressive axial load are studied using Timoshenko and high order shear deformation theory. Governing partial differential equations are formulated based on the following influences: constant axial forces at the ends of the same beams (second order theory), the Winkler elastic foundation, and elastic layers between the beams. A normal-mode solution is applied to the governing partial differential equations to derive a set of coupled ordinary differential equations which are used to determine the natural frequencies and the critical buckling load. The analytical solution for the natural frequencies and the critical buckling load for the special case of identical beams is derived using the trigonometric method. The effects of compressive axial load and number beams at the lowest natural frequency are investigated. The explicit expressions for natural frequencies and critical buckling load for a system with three identical Timoshenko-beams can be determined with a lot of difficulty. For systems with a greater number of beams, these values can be determined only numerically. However, using the trigonometric method, it is possible to approximately determine the explicit expressions for natural frequencies and critical buckling load for a system with a random number of beams m. With the increase in the number m of Timoshenko-beams in a system, the lower and higher natural frequencies decrease. It is found that the increase in compressive axial load affects the decrease in the natural frequencies of the system. Still, the increase in stiffness modulus K of the Winkler elastic layer causes the increase in the natural frequencies of the system. It is presented that there are no significant differences between natural frequencies between Timoshenko and Reddy–Bickford used theory. Layers between beams and effects of high modes don’t make big differences between frequencies for the same number of beams and in the same mode considered. It is observed that the critical buckling load is influenced by stiffness modulus K and the number m of beams. With the increase in stiffness modulus K critical buckling load also increases, but this value decreases rapidly with the increase in the number of beams m in the system. The most stable is the system with a single beam.
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