Free vibration analysis of sandwich beam carrying sprung masses

International Journal of Mechanical Sciences 52 (2010) 1620–1633

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Free vibration analysis of sandwich beam carrying sprung masses S.M.R. Khalili a,b,n, A.R. Damanpack a, N. Nemati a, K. Malekzadeh a a

Centre of Excellence for Research in Advanced Materials and Structures, Faculty of Mechanical Engineering, K.N. Toosi University of Technology, No. 19, Pardis Street, Molasadra Avenue, Vanak Sqare, Tehran, Iran Faculty of Engineering, Kingston University, London, UK

b

a r t i c l e in f o

a b s t r a c t

Article history: Received 1 November 2009 Received in revised form 5 June 2010 Accepted 21 August 2010 Available online 27 August 2010

In this paper, free vibration of three-layered symmetric sandwich beam carrying sprung masses is investigated using the dynamic stiffness method and the finite element formulation. First the governing partial differential equations of motion for one element are derived using Hamilton’s principle. Closed form analytical solution of these equations is determined. Applying the effect of sprung masses by replacing each sprung mass with an effective spring on the boundary condition of the element, the element dynamic stiffness matrix is developed. These matrices are assembled and the boundary conditions of the beam are applied, so that the dynamic stiffness matrix of the beam is derived. Natural frequencies and mode shapes are computed by the use of numerical techniques and the well known Wittrick–Williams algorithm. Free vibration analysis using the finite element method is carried out by increasing one degree of freedom for each sprung mass. Finally, some numerical examples are discussed using the dynamic stiffness method and the finite element formulation. After verification of the present model, the effect of various parameters such as mass and stiffness of the sprung mass is studied on the natural frequencies. & 2010 Elsevier Ltd. All rights reserved.

Keywords: Free vibration Sandwich beam Sprung mass Dynamic stiffness method

1. Introduction In our everyday situation, many systems may be modeled as a uniform beam or plate carrying various concentrated elements. There is a number of works dealing with the problem of free and forced transverse vibration of such a constrained beam or plate. The type of concentrated elements includes elastically mounted point masses, rigidly attached point masses, translational springs and/or rotational springs. The effects of vibration absorbers on the vehicle suspensions or the rotating machineries and the dynamic behavior of components of long span bridges or tall buildings under the influence of serve wind loads due to excitations are the important information that the designers hope to obtain. Since beams carrying one- or two-degrees-of-freedom spring–mass systems are good examples to provide this information, many researchers have devoted themselves to the studies of this area [1–7]. Some of the key references that are directly relevant to this work are briefly reviewed below. Wu and Chou [1,2] presented the analytical–numericalcombined method (ANCM) and the finite element method to calculate the natural frequencies and mode shapes of a uniform Bernoulli beam carrying any number of masses and sprung

n Corresponding author at: Centre of Excellence for Research in Advanced Materials and Structures, Faculty of Mechanical Engineering, K.N. Toosi University of Technology, No. 19, Pardis Street, Molasadra Avenue, Vanak Sqare, Tehran, Iran, Tel.: + 98 2188677272; fax: + 98 2188677273. E-mail address: [email protected] (S.M.R. Khalili).

0020-7403/$ - see front matter & 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijmecsci.2010.08.003

masses. In ANCM, the effect of each sprung mass is applied with an effective spring. Free vibration analysis using the finite element method is carried out by (i) increasing one degree of freedom for each sprung mass and (ii) replacing each sprung mass by an effective spring. Cha [3] obtained the natural frequencies of a uniform Bernoulli beam carrying any number of masses and sprung masses by solving an eigenvalue problem. Rossit and Laura [4] obtained an exact solution for the cantilever beam carrying one sprung mass using the Euler–Bernoulli theory of the beam vibrations. Wu and Hsu [5] analyzed the free vibration of simply supported beams carrying multiple point masses and spring–mass systems with the consideration of mass of each of the helical spring. Lin and Tsai [6] presented the numerical-assembly-method (NAM) to calculate the natural frequencies and mode shapes of a uniform multi-span Bernoulli beam carrying any number of sprung masses. Qiao et al. [7] presented the exact method for the analysis of free vibration of non-uniform multi-step Euler–Bernoulli beams carrying an arbitrary number of single-degree-of-freedom and twodegree-of-freedom spring–mass systems. Recently, the application of sandwich beams as members with low weight and high strength and stiffness has widespread in the engineering and the industrial fields. Hence, the free vibration analysis of these beams has been carried out by many scientists [8–16]. Di Taranto [8] and Mead and Marcus [9] are some of the earlier investigators who studied the free vibration of sandwich beams using the classical theory. Mead [10] made assessing and

S.M.R. Khalili et al. / International Journal of Mechanical Sciences 52 (2010) 1620–1633

Nomenclature

M(x,t)  M0

a2

Ml

e

Ac Af Aj b b2f b2c Bj c D Ef Fe 9 F0t > > > F0b = Flt > > > ; Flb Gc hc hf i i If j J J0 Jm 9 k0t > > > k0b = klt > > > ; klb 9 > keff 0t > > > eff > = k0b > > keff lt > > > > eff > klb ;

non-dimensional variable element constant vector vertical area of the core vertical area of each face jth constant index width of the element non-dimensional variable non-dimensional variable jth constant index on-dimensional variable differential operator Young’s modulus of each face element force vector

effective forces

shear’s modulus of the core thickness of the core thickness of each face p ffiffiffiffiffiffiffi 1 counter for natural numbers second moment of area of each face counter for natural numbers number of natural frequencies number of natural frequencies when DSM ¼ 0 number of natural frequencies of one element

stiffness of sprung masses

effective stiffness

KDSM e

element dynamic stiffness matrix

KFEM e l L m mc mf 9 m0t > > > m0b =

element stiffness matrix length of an element length of the beam counter for natural numbers mass per unit length of the core mass per unit length of each face

mass of sprung mass

element mass matrix

comparison of different models that are used to investigate the free vibration of sandwich beams. A simple model assumes that the top and the bottom faces of a sandwich beam deform according to the Euler–Bernoulli beam theory, whereas the core deforms only in shear. This model has been used by many researchers. With the advent of digital computer, finite element based solutions became available [11,12]. Sakiyama et al. [13] derived the characteristic equation for finding natural frequencies

bending moment bending moment of the nodes axial force axial force of the nodes

r non-dimensional variable s2n o non-dimensional variable s KDSM number of negative elements on the diagonal of the upper triangular matrix obtained from KDSM without row interchanges t time T kinetic energy element displacement vector e overall displacement vector u(x,t) axial displacement  u0 axial displacement of the nodes ul U V(x,t)  V0 Vl w(x,t)  w0 wl x X y 9 y0t > > > y0b =

Y Z

aj gxy d

overall dynamic stiffness matrix

MFEM e

Pl

ylt > > > ; ylb

KDSM

mlt > > > ; mlb

P(x,t)  P0

1621

ex Z lj y(x,t) y0  yl

rc rf sx txy o o F

strain energy shear force shear force of the nodes volume of element vertical displacement vertical displacement of the nodes member axis axial direction member axis

displacements of sprung masses

vertical direction width direction non-dimensional variable shear strain first variation operator normal strain non-dimensional variable jth roots of characteristic equation rotation displacement rotation displacement of the nodes density of the core density of each face normal stress shear stress natural frequency natural frequency of sprung mass displacement function

and modes in the sandwich beam by introducing a discrete-type Green function. It is a discrete solution to the differential equation governing the flexural behavior of a sandwich beam. In the recent years, scientists have investigated free vibrations of sandwich beams using the dynamic stiffness method [14–16]. Ability to obtain better results with the minimum element is advantage of this method. For example, Banerjee [14] and Howson and Zare [15] analyzed free vibration of symmetric sandwich beams using

1622

S.M.R. Khalili et al. / International Journal of Mechanical Sciences 52 (2010) 1620–1633

first the sandwich beam is subdivided into (n+1) elements and then governing equations of motion for an element are determined. When deriving the governing equations of motion of a sandwich beam element, the following general assumptions are made:

dynamic stiffness method. In more recent research done by Khalili et al. [16], using same approach to analyze the free vibration of sandwich beams, the effect of core density was also considered in deriving the equation of motions. Frostig and Baruch [17] presented different model for sandwich beams analysis. They were analyzed the free vibrations of sandwich beams with flexible core based on the high-order theory. Sokolinsky et al. [18] used the same dynamic analysis formulation as Ref. [17] to examine the influence of boundary conditions on the free vibrations of sandwich beams. Sokolinsky et al. [19] predicated the natural frequencies and corresponding vibration modes of a cantilever sandwich beam with a soft polymer core using the higher-order theory by the two-dimensional finite element analysis, and the experimental measurements. Bekuit et al. [20] presented a quasitwo-dimensional finite element formulation for the static and dynamic analysis of sandwich beams. The through-the-thickness variation of each displacement field in each layer was expanded in polynomials and the spanwise variation was interpolated by the use of Lagrange cubic shape functions. Specific studies on sandwich beam carrying sprung masses have not been carried out yet. Hence, in the present paper, the free vibrations of sandwich beams carrying sprung masses are carried out using the dynamic stiffness method and the finite element formulation. First the governing partial differential equations of motion for one element are derived using Hamilton’s principle. For the harmonic motion, these equations are combined into one ordinary differential equation, which applies to both axial and bending deformations. Closed form analytical solution of this equation is determined by MATLAB 7.1 software in the parametric form. By replacing each sprung mass with an effective spring on the boundary condition of the element, the element dynamic stiffness matrix is developed. The dynamic stiffness matrix of the beam is derived, after the element dynamic stiffness matrices are assembled and the boundary conditions of the beam are applied. Natural frequencies and mode shapes are computed by the use of calculations techniques and the well known Wittrick–Williams algorithm [21–24]. The presented model is able to study the effect of various boundary conditions on the beam. Free vibration analysis using the finite element method is carried out by increasing one degree of freedom for each sprung mass. The effect of various parameters such as mass and stiffness of the sprung mass on the natural frequencies is studied.

(1) All deformations and strains are very small, so that the theory of linear elasticity applies. (2) The faces and the core of the sandwich beam are made of isotropic and homogeneous materials and the variation of strain within them is linear. (3) Transverse normal strains are negligible in the faces and the core. (4) There is no slippage or delamination between the layers. (5) The faces do not deform in shear and the core carries only shear. The elastic modulus of the core is usually much lower than that of the skins. (6) The transverse flexural inertia of the faces is predominant so that the rotary inertia of the faces may be neglected. Fig. 2 shows the cross sectional displacements for a symmetric sandwich beam element with length l and width b. As shown in Fig. 2, the thickness of each face is equal to hf and the thickness of the core is hc. In the coordinate system shown, the X-axis is the direction of the beam axis, the Y-axis is the bending or thickness direction (positive upward) and the Z-axis is in the beam width direction (perpendicular to and out of plane of the paper). All deformations of the beam are considered in the XY plane. The top and the bottom faces are deformed according to the Euler–Bernoulli beam theory, whereas the core is deformed only

w′ Y

Top face (Layer 1)

ut

hf hc

Core (Layer 2)

X 2. Theory

y1t

ub

hf

θ = w′

Fig. 1 shows a symmetric sandwich beam carrying sprung masses on n point through the length. For free vibrations analysis,

m1t

Bottom face (Layer 3)

Z

2.1. Derivation of governing partial equations of motion

Fig. 2. Coordinate system and symbols for a symmetric sandwich beam.

yit

mit

mnt

ynt

Y k1t x1

kit x2

knt

xi

xn

X

Z θ

k2b

y2b m2b

kib

yib mib

Fig. 1. Symmetric sandwich beam carrying sprung masses on n point through the length.

S.M.R. Khalili et al. / International Journal of Mechanical Sciences 52 (2010) 1620–1633

1623

in shear, so that the normal stresses in the core and in the axial direction are zero. With respect to the displacement field, each layer is considered to have the same transverse displacement w(x,t), whereas the mid-plane displacements of the top and the bottom faces in the axial direction are ut(x,t) and ub(x,t), respectively. It can be shown [10] from the symmetry of the motion that ut(x,t)¼ ub(x,t). Therefore,

The kinetic energy of this sandwich beam element is (Z Z 1 1 2 2 _ _ 2 þðuy _ _ 2 þ 2w _ 2 d T¼ rðX_ þ Y_ Þ d ¼ rf ½ðuy wuÞ wuÞ 2 2 t

ut ¼ ub ¼ uðx,tÞ

where rf is the density for each face sheet and rc is the density for the core. Also, by calculating the integrals of Eq. (7), the kinetic energy of this sandwich beam element is derived as Z o 1 ln m _ 2 dx _ 2 þ c ð2u_ þhf wuÞ T¼ ð8Þ 2mf u_ 2 þ ð2mf þ mc Þw 2 0 12

The beam displacement field is derived using the compatibility conditions for deformations: 8 hf hf > > ry r Top face uywu > > > 2 2 > > < y hc hc ry r ð2u þ hf wuÞ ð2aÞ X : Core hc 2 2 > > > > > hf hf > > ry r : Bottom face uywu 2 2 ð2bÞ

Zðfor all layerÞ : 0

ð2cÞ

 y2 _2 d þ w h2c

_ 2 rc ð2u_ þ hf wuÞ c

ð1Þ

Yðfor all layerÞ : w



Z þ

t

) ð7Þ

c

where ðmf Þ ¼ rf Af ,

ðmc Þ ¼ rc Ac

ð9Þ

By applying Hamilton’s principle, the governing partial equations of motion and the boundary conditions of sandwich beam element are derived as follows: Z t2 d ðTUÞ dt ¼ 0 ð10Þ t1

According to the presented assumptions, the strain energy of sandwich beam is given by (Z ) Z Z 1 U¼ sx ex d t þ txy gxy d c þ sx ex d b 2 c b (Z t ) Z Z 1 2 2 ð3Þ ¼ Ef ex d t þ Gc gxy d c þ Ef e2x d b 2 t c b

In the above equation, d is the first variation operator and t1 and t2 are defined as the time intervals. Substituting Eqs. (4) and (8) in Eq. (10) yields Ef Af u00 

hc

2

u

Gc Ac ðhf þhc Þ hc

2

wu

ðmf þ

Ef If w0000 

Gc Ac ðhf þ hc Þ2 2hc

2

w00 

Gc Ac ðhf þ hc Þ hc

2

uu

2

þ ðmf þ

mc hf 00 mc hf mc € €  € ¼0 Þw w uu 2 24 12

P ¼ 2Ef Af uu dA ¼ bdy

Z At,b

ð1,y,y2 Þ dAt,b ;

ð5Þ V ¼

Z

dAc

ðAc Þ ¼

ð6Þ

ð13Þ

Gc Ac ðhf þhc Þ  hc

2

mc hf € ð2u€ þ hf wuÞ 2u þ ðhf þhc Þwu þ2Ef If w000  12 ð14Þ

Ac

V0

P0 M0

ylt

y0t

m0t

mlt

wl

u0

0

k0b

Vl

klt

k0t

w0

ð12Þ

The axial force P(x,t), the shearing force V(x,t) and the bending moment M(x,t) for the sandwich beam element can be defined as follows:

where d ¼ dAdx,

mc hf mc € € ¼0 Þu wu 6 12 ð11Þ

where Ef is Young’s modulus of each face sheet and Gc is the shear modulus of the core. By substituting the strain values and calculating the integrals of Eq. (3), the strain energy for a beam element is derived as "  2 Z ( hf þ hc 1 l 4 2 00 2 U¼ 2Ef Af uu þ 2Ef If w þGc Ac 2 u2 þ wu2 2 0 hc hc #)   4 hf þ hc þ uwu dx ð4Þ hc hc

ðAf ,0,If Þ ¼

2Gc Ac

ul

Pl

l

klb

ylb

y0b m0b

mlb l Fig. 3. Symmetric sandwich beam element carrying sprung masses.

Ml

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S.M.R. Khalili et al. / International Journal of Mechanical Sciences 52 (2010) 1620–1633

M ¼ 2Ef If w00

and

ð15Þ

D ¼ d=dZ Then, the boundary conditions are considered as 8 or Pð0,tÞ ¼ 0 > < duð0,tÞ ¼ 0 x ¼ 0 : dwð0,tÞ ¼ 0 or Vð0,tÞ ¼ 0 > : dwuð0,tÞ ¼ 0 or Mð0,tÞ ¼ 0 8 > < duðl,tÞ ¼ 0 x ¼ l : dwðl,tÞ ¼ 0 > : dwuðl,tÞ ¼ 0

or

Pðl,tÞ ¼ 0

or or

Vðl,tÞ ¼ 0 Mðl,tÞ ¼ 0

ð22Þ

By comparing Eqs. (11) and (12) with Eqs. (19) and (20), it can be obtained that the number of variables are reduced in Eqs. (19) and (20). Eqs. (19) and (20) are the homogenous differential equations with constant coefficients. With the help of computational software MATLAB 7.1, Eqs. (19) and (20) are combined to form a sixth order differential equation as follows:

ð16aÞ

ðc6 D6 þ c4 D4 þ c2 D2 þc0 Þc ¼ 0

ð16bÞ

ð23Þ

where c ¼u or w and 2.2. Solving the governing equations of motion for free vibration

c6 ¼ 12r 2 c4 ¼ 4r 2 ½3b2f þ2b2c 24a2 ½3c2 þ r 2 

In Fig. 3, a sandwich beam element carrying sprung masses at both end nodes has been shown. k0t, m0t, y0t ; klt, mlt and ylt are the spring constants, the masses and displacements of the sprung masses attached at both end nodes of the top face and k0b, m0b, y0b; klb, mlb, ylb are the similar parameters at the end nodes of the bottom face of sandwich beam element, respectively. P0, V0, M0 ; Pl, Vl, Ml are the external axial force, the external shear force and the external bending moment at both end nodes of sandwich beam element, too. Assume harmonic variation for u,w,y0t ,y0b ,ylt ,ylb , then uðx,tÞ ¼ uðxÞeiot ;

c2 ¼ 6b2c ½3 þ 2a2 ðcrÞ2 6b2f ½6 þ 12a2 c2 b2c r 2  c0 ¼ 36a2 ½2b2f þ b2c 12b2f ½3b2f þ2b2c 3b4c The solution of Eq. (24) can be found to be in the form of

c ¼ elZ

6

Z ¼ x=l

ð26Þ

uðZÞ ¼

6 X

Aj elj Z

ð27Þ

j¼1

ð18Þ wðZÞ ¼

6 X

Bj elj Z

ð28Þ

j¼1

ð6D2 12a2 þ 6b2f þ b2c Þu þ ð12a2 c þb2c rÞDw ¼ 0

ð19Þ By substituting Eqs. (27) and (28) in Eq. (19), it can be shown that the constants Aj and Bj are related as follows:

ð12a2 c þ b2c rÞDu þ ½2r 2 D4 ð12a2 c2 b2c r 2 ÞD2 3ð2b2f þ b2c Þw ¼ 0 ð20Þ

Bj ¼ aj Aj

where

ð29Þ

where

Ac Gc l2 Ef Af hc

2

The characteristic equation is a sixth order polynomial equation, so that it has six roots. Thus the solutions for u and wcan be written as

and substituting Eq. (17) into Eqs. (11) and (12), gives

a ¼

4

c6 l c4 l c2 l c0 ¼ 0

ð17Þ

where u,w,y0t ,y0b ,ylt ,ylb are the amplitude functions of u, w, y0t, y0b, ylt, ylb, respectively, and o is the natural frequency of the beam. In order to make simplification and a brief description about the partial differential equations of motion, the non-dimensional variable Z is introduced as follows:

2

ð25Þ

Substituting Eq. (25) in Eq. (24), yields the characteristic equation as follows:

wðx,tÞ ¼ wðxÞeiot

ðy0t ,y0b ,ylt ,ylb Þ ¼ ðy0t ,y0b ,ylt ,ylb Þeiot

ð24Þ

2

;

b2f

mf o2 l2 ¼ ; Ef Af

b2c

mc o2 l2 ¼ ; Ef Af

hf þ hc c¼ ; 2l

hf r¼ 2l

2

aj ¼

ð21Þ

V0

P0

w0

M0

6lj 12a2 þ6b2f þb2c

ð30Þ

ð12a2 cb2c rÞlj

F0t

Flt

u0

wl

θ0

Vl

ul

Ml

θl F0b

Flb l

Fig. 4. Free body diagram of the symmetric sandwich beam element carrying sprung masses.

Pl

S.M.R. Khalili et al. / International Journal of Mechanical Sciences 52 (2010) 1620–1633

Referring to Eqs. (36) and (37) and Fig. 5, it can be seen that, each sprung mass replaced by an effective spring with spring

The rotation of the beam can be written as follows:

constant keff . ð0,lÞðt,bÞ

6 dw 1 dw 1X ¼ yðZÞ ¼ ¼ A a l elj Z dx l dZ l j¼1 j j j

ð31Þ

2.3. Dynamic stiffness formulation for sandwich beam carrying sprung masses

Also, the inner moment and forces can be written as

pðZÞ ¼

6 2Ef Af X Aj lj elj Z l j¼1

VðZÞ ¼

6 2Ef Af X Aj mj elj Z ; l j¼1

mj ¼ 2a2 c

1625

Referring to Fig. 3, the displacements of both end nodes of the beam element are 8 6 X > > > Aj > u0 ¼ uð0Þ ¼ > > > j¼1 > > > > > 6 X < Aj aj Z ¼ 0 : w0 ¼ wð0Þ ¼ ð39aÞ > > j¼1 > > > > 6 > X > > > y ¼ yð0Þ ¼ l1 Aj aj lj 0 > > :

ð32Þ

  b2c r b2 r 2 r2 3 þ ð2a2 c2  c Þlj  lj aj 6 6 3

ð33Þ

j¼1

MðZÞ ¼

6 2Ef If X 2 Aj aj lj elj Z l2 j ¼ 1

8 6 X > > > ul ¼ uð1Þ ¼ Aj elj > > > > j ¼ 1 > > > > > 6 X < Aj aj elj Z ¼ 1 : wl ¼ wð1Þ ¼ > > j¼1 > > > > 6 > X > > y ¼ yð1Þ ¼ l1 > Aj aj lj elj l > > : j¼1

ð34Þ

Referring to Fig. 4, the effect of each sprung mass of both end nodes of the beam element have been replaced by the forces (F0t, F0b, Flt, Flb). Assume harmonic variation for F0t , F0b , Flt , Flb , then,  F0ðt,bÞ ¼ m0ðt,bÞ y€ 0ðt,bÞ ¼ k0ðt,bÞ y0ðt,bÞ ðtÞwð0,tÞ , h i m0ðt,bÞ o2 y0ðt,bÞ eiot ¼ k0ðt,bÞ y0ðt,bÞ wð0Þ eiot ) y0ðt,bÞ ¼

k0ðt,bÞ wð0Þ k0ðt,bÞ m0ðt,bÞ o2

If the element displacement vector e and the element constant vector e are defined in the following way, ð35Þ

u0

w0

y 0 ul

wl

yl

e

¼ A1

A2

A3

A5

A6

A4

ð40Þ

T

ð41Þ

ð36Þ Eqs. (39) can be written in the matrix form as follows:

keff wð1Þ lðt,bÞ

ð37Þ

e

where ¼ keff ð0,lÞðt,bÞ

oT

¼

and similar to the above: F lðt,bÞ ¼

n

e

hence, F 0ðt,bÞ ¼ keff wð0Þ 0ðt,bÞ

ð39bÞ

¼S

ð42Þ

e

where mð0,lÞðt,bÞ kð0,lÞðt,bÞ o2 kð0,lÞðt,bÞ mð0,lÞðt,bÞ o2

V0

P0 M0

S1j ¼ 1,

ð38Þ

S4j

eff

S2j ¼ aj , S5j ¼ aj

elj ,

S3j ¼ l1 aj lj S6j ¼ l1 aj lj elj ,

klt

wl

u0

θ0

ul

Ml

θl eff

eff

k0b

j ¼ 1,2,:::,6

Vl

eff

k0t

w0

¼ elj ,

klb l

Fig. 5. Symmetric sandwich beam element (the sprung masses are replaced by effective springs).

Pl

ð43Þ

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S.M.R. Khalili et al. / International Journal of Mechanical Sciences 52 (2010) 1620–1633

Also, the external forces and moment at both end nodes of the beam element are 8 6 X > > > P0 ¼ Pð0Þ ¼ 2Ef Af Aj lj > > > l > j¼1 > > > > > 6 < 2Ef Af X Aj mj ð44aÞ Z ¼ 0 : V0 þF0t þ F0b ¼ Vð0Þ ¼ l > > j¼1 > > > > 6 > 2Ef If X > 2 > > M0 ¼ Mð0Þ ¼ Aj aj lj > 2 > l : j¼1 8 6 > 2Ef Af X > > Pl ¼ Pð1Þ ¼ Aj lj elj > > > l > j¼1 > > > > > 6 < 2Ef Af X Aj mj elj Z ¼ 1 : Vl þ Flt þ Flb ¼ Vð1Þ ¼ l > > j¼1 > > > > 6 > 2Ef If X > 2 > > Ml ¼ Mð1Þ ¼ 2 Aj aj lj elj > > l :

ð44bÞ

j¼1

Eqs. (44) can be written in the matrix form as 9 8 P0 > > > > > > > > > V0 þF0t þ F0b > > > > > > > > > = < M0 ¼ Q DSM e Pl > > > > > > > > > > > > V þ Flt þFlb > > > > l > > ; : M

ð45Þ

l

where 2

Q1j ¼ 2Ef Af l1 lj ,

Q2j ¼ 2Ef Af l1 mj ,

Q3j ¼ 2Ef If l2 aj lj

Q4j ¼ 2Ef Af l1 lj elj ,

Q5j ¼ 2Ef Af l1 mj elj ,

Q6j ¼ 2Ef If l2 aj lj elj

2

ð46Þ By substituting Eq. (42) in Eq. (45), it yields 9 8 P0 > > > > > > > > > > V þF þ F > > 0 0t 0b > > > > > > = < M0 ¼ K DSM e Pl > > > > > > > > > > > Vl þ Flt þFlb > > > > > > > ; : Ml

After the element dynamic stiffness matrices were assembled and the boundary conditions of the beam were applied, the dynamic stiffness matrix of the beam KDSM is derived. Natural frequencies and mode shapes are computed by the use of calculation techniques and the Wittrick–Williams algorithm. 2.4. Finite element formulation for a sandwich beam carrying sprung masses The results obtained by finite element formulation are the most appropriate for validation of the results obtained by dynamic stiffness approach. The shape functions C0 continuity and C1 continuity [25] are applied to analyze the sandwich beam shown in Fig. 3 as follows: u ¼ B1 þ B2 x w ¼ B3 þ B4 x þB5 x2 þ B6 x3

ð52Þ

The constants (B1,...,B6) may be determined at the two ends of the sandwich beam in terms of u, w and y. For applying the effect of sprung masses in the strain and kinetic energy (4) and (8), it is necessary to increase one degree of freedom for each sprung mass: "  2 Z ( hf þ hc 1 l 4 2 00 2 U¼ 2Ef Af uu þ 2Ef If w þ Gc Ac 2 u2 þ wu2 2 0 hc hc #)   4 hf þ hc uwu dx þ hc hc n o 1 k0t ðw0 y0t Þ2 þ k0b ðw0 y0b Þ2 þ klt ðwl ylt Þ2 þklb ðwl ylb Þ2 þ 2 ð53Þ Z o 1 ln m _ 2 dx _ 2 þ c ð2u_ þ hf wuÞ 2mf u_ 2 þ ð2mf þ mc Þw T¼ 2 0 12  1

m0t y20t þ m0b y20b þ mlt y2lt þ mlb y2lb ð54Þ þ 2 Substituting the shape functions in the strain and kinetic energy (53) and (54), then applying Hamilton’s principle to the discretized system by assumption of harmonic motion, gives FEM

fe ð47Þ

FEM

FEM e

K ¼ Q S1

ð48Þ

If the element force vector f e is defined as follows:

T DSM f e ¼ P0 V0 M0 Pl Vl Ml

ð49Þ

FEM e

ð55Þ

and fe

where

¼ ðKFEM o2 MFEM Þ e e

¼ P0 ¼

n

u0

V0 w0

M0

y0

0

0

y0t

Pl y0b

Vl ul

Ml wl

0

yl

0



ylt

ð56Þ ylb

o

ð57Þ

and the element mass The element stiffness matrix KFEM e matrix MFEM are derived which are symmetric and 10  10. The e members of the element stiffness matrix and the element mass matrix are shown in the appendix.

and substituting Eqs. (36) and (37) into Eq. (47), it gives DSM

fe

¼ KDSM e

DSM e

2.5. Free vibration analysis of simply supported sandwich beam ð50Þ

the dynamic stiffness matrix KDSM which is symmetric and real is e derived as follows: 2 3 K12 K13 K14 K15 K16 K11 6 7 eff eff K22 k0t k0b K23 K24 K25 K26 7 6 6 7 6 K33 K34 K35 K36 7 6 7 KDSM ¼6 ð51Þ 7 e K44 K45 K46 7 6 6 7 eff eff 6 7 K55 klt klb K56 5 4 Symmetric K66

In this section, the exact solution for sandwich beam element of Fig. 3 is presented that is employed later for the application of the Wittrick–Williams algorithm. By assuming the harmonic variation for u and w of a simply supported sandwich beam element, then mp x eiot um ðx,tÞ ¼ Am cos ml p x eiot , wm ðx,tÞ ¼ Bm sin l

m ¼ 1,2,3,. . .

ð58Þ

S.M.R. Khalili et al. / International Journal of Mechanical Sciences 52 (2010) 1620–1633

and io t

ðy0t ,y0b ,ylt ,ylb Þ ¼ ðy0t ,y0b ,ylt ,ylb Þe

ð59Þ

where Am, Bm, y0t, y0b, ylt, ylb are constants and o is natural frequency of the beam. Substituting Eqs. (58) and (59) into Eqs. (11) and (12), and rewriting them in the matrix form, gives the followings: " # 6m2 p2 þ 12a2 12a2 cmp 12a2 cmp 2r 2 m4 p4 þ 12a2 c2 m2 p2 2 31 2 2 2 6b þb b rm p c c 6 f 7C o2 4 2 2 2 2 5A bc rmp bc r 2 m2 p2 þ 6bf þ 3bc ( )   Am 0  ¼ , m ¼ 1,2,3,::: Bm 0

ð60Þ

Ac Gc l2

Ef Af hc hf þhc , c¼ 2l

2

2

,

bf ¼ r¼

mf l2 , Ef Af

2

bc ¼

mc l2 Ef Af

k0t B6 B6 0 B6 B6 0 @4 0

hf 2l

ð66Þ

where J is the number of structure natural frequencies, and J0 is the number of structure natural frequencies when all the elements have fixed supports or and also, s{KDSM} is the sign function of the dynamic stiffness matrix. s{KDSM} is equal to the number of negative elements on the leading diagonal of the upper triangular matrix obtained from KDSM without row interchanges. However, s{KDSM} is simply assigned by the use of Gauss elimination method [22], but assigning J0 is too difficult.

ð61Þ

0 k0b

0 0

0 0

klt 0

22 3 m0t 0 66 7 0 7 66 0 2 7o 66 66 0 0 7 44 5 0 klb

0 m0b

0 0

0 0

mlt 0

331 0 77C 0 77C 77C 7C 0 7 55A mlb

8 9 8 9 y0t > > 0 > > > > > > > > > < y0b > = > <0> = ¼ y > > > 0 lt > > > > > > > > > > :y > ; :0; lb

ð62Þ Eqs. (60) and (62) can be rewritten in the following form ð o2

Þ ¼0

1

In the above equation, Jm is the number of natural frequencies of a component element when this element has fixed supports and the summation extends over all the elements. In some cases, the direct definition of Jm is possible [23]. Instead, some results are obtained using Eq. (66), where it is possible to obtain the Jm [24]. For example, when an element is simply supported (Sections 2–5), Jm can easily compute the exact natural frequencies. In this case, if the stiffness matrix of this element is shown with KDSM e_SS , then, according to Eq. (66), JSS ¼ Jm þsfKDSM e_SS g

ð63Þ

The squared natural frequencies are calculated from the eigenvalues of ð

J ¼ J0 þ sfKDSM g

For the definition of J0, the following equation can be applied: X Jm ð67Þ J0 ¼

and 02

involves computing the determinant of the dynamic stiffness matrix and noting when it changes sign through zero, some roots may be missed. This problem is removed completely by applying the well-known Wittrick–Williams algorithm [21]. This algorithm is defined as

4. Determination of J0

where a2 ¼

1627

ð68Þ

where JSS is the number of natural frequencies of this element when it is simply supported. So, Jm ¼ JSS sfKDSM e_SS g

ð69Þ

Þ: After the determination of Jm, J0 can be obtained by Eq. (67), n o X ð70Þ JSS s KDSM J0 ¼ e_SS

3. Wittrick–Williams algorithm After the stiffness matrix KDSM is derived from assembling the element stiffness matrices KDSM , an expression in the form of e Eq. (64) should be solved to assign the natural frequencies: KDSM ¼ 0

ð64Þ

is the vector of amplitudes of the element nodal where displacements and KDSM is a square matrix and a function of natural frequencies o. In most cases, the determinant of the dynamic stiffness matrix 9KDSM9 is used to assign the natural frequencies and it should be equal to zero: 9KDSM 9 ¼ 0

where this summation extends to all elements.

5. Behavior of sprung mass attached to the beam ki and mi are the properties of ith sprung mass, that is attached to the point at xi of the beam length. If, the vertical displacement (wi) at xi is equal to zero, the natural frequency of this sprung mass is qffiffiffiffiffiffiffiffiffiffiffiffi ð71Þ oi ¼ ki =mi , i ¼ 1,2,3,:::

ð65Þ

Traditionally, the relation between the determinant of the dynamic stiffness matrix 9KDSM9 and the natural frequencies o should be defined to calculate the required natural frequencies. However, when the dynamic stiffness matrix has obtained from assembling the element stiffness matrices, the relation between the determinant of the dynamic stiffness matrix and the natural frequencies is too irregular and non-algebraic. Furthermore, some natural frequencies may be close together or coincident, or others may be calculated from . So, in each trial and error method which

oj is the jth lowest natural frequency of the sandwich beam carrying sprung mass during free vibration: xij ¼

oj oi

ð72Þ

yij, the displacement of the ith sprung mass in jth natural frequency is defined by substituting Eq. (72) into Eq. (35): yij ¼

1 w 1xij i

no summation

ð73Þ

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S.M.R. Khalili et al. / International Journal of Mechanical Sciences 52 (2010) 1620–1633

As can be seen in Eq. (73), yij is opposite to the direction of wi, when oi o oj and is in the direction of wi when oi 4 oj . In other hand, if oi b oj or xij-0, then yij ¼wi. Therefore, the displacement of the sprung mass is equal to the vertical displacement of the point at which the mass is attached to the beam. In this case, the behavior of sprung mass on the beam is similar to the attached mass(mi). Also if oi 5 oj or xij-N, then yij ¼0. In other words, the displacement of sprung mass is equal to zero. Therefore, in this case, the behavior of sprung mass on the beam is similar to the vertical elastic support(ki).

The results for 250 and 500 elements are shown in Fig. 6 and Tables 1 and 2. The bending stiffness and the total mass of sandwich beam are determined as follows: Ef I L3 mBeam ¼ mL

kBeam ¼

where I ¼ 2If þ

6. Numerical results and discussion In this section, first the verification of the presented theory is examined and then the effect of various parameters such as stiffness and mass of the sprung masses on natural frequencies is studied. Because no researches have been reported on the analysis of sandwich beam carrying sprung masses yet, the present results are compared with the results of the Euler–Bernoulli beam carrying sprung masses [1–4] and with the results of sandwich beam without any sprung masses [11–15]. The results of finite element formulation in each case are obtained by various numbers of elements from 100 to 500, to check the convergence problem. As the results, there are no significant differences between the magnitudes obtained above 250 elements. Therefore, in the FEM analysis, 250 elements are enough to get the convergence and reduce the analysis time.

ð74Þ

Af ðh þhc Þ2 , 2 f

m ¼ 2mf þ mc

ð75Þ

6.1. Verification 6.1.1. Euler–Bernoulli beams carrying sprung masses As an example for verification of the present model, the Euler– Bernoulli beam carrying sprung masses in Ref. [24] is considered. Due to the assumptions made in the present paper, it is not possible to completely consider the sandwich beam as the Euler–Bernoulli beam. Therefore, the effect of the core in the sandwich beam must be reduced for better adaptation of the sandwich beam to the Euler– Bernoulli beam. For this purpose, the mass of the core is neglected and the thickness must approach to zero, i.e., hhc -0, rc ¼ 0. In order f to get the range of variation of shear modulus of the core for better approximation of the presented model to the Euler–Bernoulli beam, the variation of the first natural frequency with the ratio of Gc =Ef is

Fig. 6. Variation of first natural frequency with Gc =Ef . x1 ¼ L : ½m1t ¼ 0:2mBeam , k1t ¼ 10kBeam ; m1b ¼ 0,k1b ¼ 0, hc =hf ¼ 0:001,

rc ¼ 0. *Number of elements.

Table 1 Five lowest natural frequencies for a cantilever beam carrying one sprung mass. hc hf

Methods

x1 (Hz)

x2 (Hz)

x3 (Hz)

x4 (Hz)

x5 (Hz)

1

DSM (sandwich beam) FEM-250 (sandwich beam) FEM-500 (sandwich beam)

25.854 25.854 25.854

95.101 95.101 95.101

234.927 234.932 234.928

627.460 627.489 627.467

1214.083 1214.186 1214.109

0.001

DSM (sandwich beam) FEM-250 (sandwich beam) F.E.M-500 (sandwich beam)

25.860 25.860 25.860

95.129 95.130 95.129

235.498 235.502 235.499

631.211 631.240 631.217

1227.302 1227.408 1227.326

–

[4] (Euler–Bernoulli beam)

25.865

95.141

235.920

634.076

1237.551

Number of elements:

 x1 ¼ L : ½m1t ¼ 0:2mBeam ,k1t ¼ 10kBeam ; m1b ¼ 0,k1b ¼ 0,  Gc ¼ 1:0345  1011 N m2 , rc ¼ 0:

S.M.R. Khalili et al. / International Journal of Mechanical Sciences 52 (2010) 1620–1633

1629

Table 2 Five lowest natural frequencies for a cantilever beam carrying three sprung masses. Methods

x1 (Hz)

x2 (Hz)

x3 (Hz)

x4 (Hz)

x5 (Hz)

1

DSM (sandwich beam) FEM-250 (sandwich beam) FEM-500 (sandwich beam)

16.356 16.356 16.356

30.035 30.035 30.035

39.572 39.572 39.571

55.541 55.541 55.541

226.266 226.271 226.268

0.001

DSM (sandwich beam) FEM-250 (sandwich beam) FEM-500 (sandwich beam)

16.359 16.359 16.359

30.037 30.038 30.038

39.572 39.572 39.573

55.550 55.550 55.549

226.833 226.837 226.834

-

[1] (Euler–Bernoulli beam)

-

[2] (Euler–Bernoulli beam) [3] (Euler–Bernoulli beam)

16.360 16.347 16.361 16.360

30.038 30.042 30.038 30.038

39.575 39.551 39.572 39.572

55.563 55.568 55.563 55.563

227.278 227.268 227.265 227.265

hc hf

FEM ANCM





1 L : m1t ¼ 0:2 mBeam ,k1t ¼ 3 kBeam ; m1b ¼ 0,k1b ¼ 0 ,  x1 ¼ 10





 x2 ¼ 25L : m2t ¼ 0:5 mBeam ,k2t ¼ 4:5 kBeam ; m2b ¼ 0,k2b ¼ 0 , 



 x3 ¼ 45L : m3t ¼ mBeam ,k3t ¼ 6 kBeam ; m3b ¼ 0,k3b ¼ 0 ,  Gc ¼ 1:0345  1011 N m2 , rc ¼ 0:

Table 3 Properties of sandwich beam. Case number

L (mm)

hf (mm)

hc (mm)

b (mm)

Ef (N m  2)

Gc (N m  2)

qf (kg m  3)

qc (kg m  3)

1st case [11] 2nd case[11]

914.4 711.2

0.4572 0.4572

12.7 12.7

25.4 25.4

6.89  1010 6.89  1010

82.68  106 82.68  106

2680 2680

32.8 32.8

considered and compared to the results of Ref. [4] which is an exact method for the Euler–Bernoulli beam. This comparison is shown in Fig. 6. The properties of the Euler–Bernoulli beam are L ¼ 1m,

In Table 2, the cantilever beam is carrying three sprung masses. As can be seen, the differences between the results and those from Refs. [1] to [3] are negligible.

E ¼ 2:069  1011 N m2

kBeam ¼ 6:34761  104 Nm1 , mBeam ¼ 15:3875 kg

ð76Þ

For this purpose, the thickness ratio hc/hf ¼ 0.001. As it is obvious, the range of Gc =Ef should be 0:1 r Gc =Ef r 10 to get good correspondence between the results with Ref. [4] and to model the presented sandwich beam as the Euler–Bernoulli beam. As shown in Fig. 6, when Gc =Ef r 0:1, the core rigidity decreases and the axial displacements of the layers increase, on the other hand, when Gc =Ef Z10, the core rigidity increases and the axial displacements of the layers decrease that are caused inaccuracy between the exact results of Ref. [4] and the model of the presented sandwich beam as the Euler–Bernoulli beam. In the range between 0.1 and 10, the assumption of the properties of the core is similar to the uniform beam, i.e., the variation of displacement of the core is nearly same as the variation of the displacement of the uniform beam considered in Ref. [4]. The verification of the results obtained by the presented theory is obtained by comparison with the results obtained in Refs. [4,1–3] and for the first five natural frequencies. These comparisons are shown in Tables 1 and 2, respectively. In Table 1, the Euler–Bernoulli beam is cantilever and carrying one sprung mass at free end. The presented results are obtained for two thickness ratios, i.e., 0.001 and 1. The maximum discrepancy between the presented results and the exact solution of Ref. [4] is 1.8% and it is for the 5th natural frequency and the thickness ratio of 1. Hence, the results are in good agreement with the exact solution of Ref. [4].

6.1.2. Sandwich beam without any sprung masses As the second example for verification, a sandwich beam with the properties given in Table 3 with two lengths without any sprung masses worked by Refs. [11–15] is considered and the results obtained are compared to the results of the presented theory.

6.1.2.1. First case. The results of the exact solution as well as the dynamic stiffness method (DSM) and the FEM of the present theory are compared with the results of Refs. [11–13,15], in which the sandwich beam of Table 3 is simply supported. The comparison is shown in Table 4. As it is observed, the results of the presented theory used the dynamic stiffness method and the exact solutions are the same. Also, the results of the FEM, shows a good agreement with the DSM or exact solution of the presented theory. The results of FEM are obtained with 250 or 500 elements. All the results obtained by different approaches in presented theory are in good agreement with the results obtained in previous references. The maximum discrepancy is 3%.

6.1.2.2. Second case. In Table 5, the results of five lowest natural frequencies obtained in Refs. [11–15] for a cantilever sandwich beam with physical and geometrical properties in Table 3 are compared to the DSM and FEM approaches in presented theory. Again, the results are in good agreement and the maximum discrepancy is occurred in the 4th natural frequency between the results of the presented theory and Banerjee’s approach [14] is

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Table 4 Lowest five natural frequencies for a simply supported sandwich beam. Methods

x1 (Hz)

x2 (Hz)

x3 (Hz)

x4 (Hz)

x5 (Hz)

Present (exact) Present (DSM) Present (FEM-250) Present (FEM-500) Ahmed [11] Ahmed [12] Sakiyama et al. [13] Howson and Zare [15]

57.1234 57.1234 57.1238 57.1235 57.5 55.5 56.159 57.1358

219.4227 219.4227 219.4279 219.4240 – – 215.82 219.585

464.5652 464.5652 464.5871 464.5707 467 451 457.22 465.172

766.8505 766.8505 766.9060 766.8644 – – 755.05 768.177

1104.5269 1104.5269 1104.6334 1104.5535 1111 1073 1087.9 1106.68

x5 (Hz)

Table 5 Five lowest natural frequencies for a cantilever sandwich beam. Methods

x1 (Hz)

x2 (Hz)

x3 (Hz)

x4 (Hz)

Present (DSM) Present (FEM-250) Present (FEM-500) Ahmed [11] Ahmed [12] Sakiyama et al. [13] Banerjee [14] Howson and Zare [15]

33.7455 33.7457 33.7456 33.97 32.79 33.146 31.46 33.7513

198.7881 198.7932 198.7892 200.5 193.5 195.96 193.7 198.992

511.3739 511.4013 511.3794 517 499 503.43 529.2 512.307

905.1199 905.1938 905.1347 918 886 893.28 1006 907.299

1346.0600 1346.2006 1346.0882 1368 1320 1328.5 1349.65

Fig. 7. Variation of 1st and 2nd natural frequencies with stiffness of sprung mass (simply supported sandwich beam): (a) 1st natural frequency and (b) 2nd natural frequency. x1 ¼ 0:5L : ½m1t ¼ mBeam , k1t ¼ 10d kBeam ; m1b ¼ 0,k1b ¼ 0

approximately 9%. This difference is due to neglecting the mass of core by Ref. [14].

6.2. Sandwich beam carrying sprung masses After verification of the present theory, the effect of various parameters of sprung mass such as mass and stiffness are studied on the free vibrations of sandwich beams. In this section, the simply supported sandwich beam of 2nd case in Table 3, carrying one sprung mass is analyzed. The variations of 1st and 2nd natural frequencies with stiffness and mass of the sprung mass are shown in Figs. 7 and 8, respectively. The variations of 1st and 2nd shape modes with stiffness and mass of the sprung mass are also shown in Figs. 9 and 10, respectively. As shown in Figs. 7 and 9, the displacement of the beam is zero in the first natural frequency, when k1t o10  4kBeam. In this condition, the first natural frequency of the system is o 1 and the second natural frequency is similar to the first natural frequency of the sandwich beam with no attached sprung mass.

Therefore, the vibration of sprung mass has no effect on free vibrations of sandwich beam. By increasing k1t more than 10  4kBeam, the effect of sprung mass on the natural frequencies is increased, so that the vertical displacements of the beam at 1st and 2nd natural frequencies are increased. By increasing k1t above 104kBeam, the natural frequencies remain constant and the sprung mass has the effect similar to attached mass. As shown in Figs. 8 and 10, by decreasing m1t less than 10  4mBeam, the vertical shape modes of the beam are not changing for the two lowest natural frequencies. In this situation, the two lowest natural frequencies of the system are similar to the two lowest natural frequencies when there is no attached sprung mass. Therefore, as shown in Eqs. (37) and (38), the vertical force F1t that is resulted from vibration of the sprung mass becomes less and has no effect on the free vibrations of sandwich beam. By increasing m1t above 10  4mBeam, the effect of sprung mass is increased, so that the vertical displacements of the beam in 1st and 2nd natural frequencies are changed. By increasing m1t above 104mBeam, the natural frequencies remain constant and the sprung mass has the effect similar to elastic support. In other words, by

S.M.R. Khalili et al. / International Journal of Mechanical Sciences 52 (2010) 1620–1633

1631

Fig. 8. Variation of 1st and 2nd natural frequencies with mass of sprung mass (simply supported sandwich beam): (a) 1st natural frequency and (b) 2nd natural frequency. x1 ¼ 0:5L : ½m1t ¼ 10d mBeam ,k1t ¼ kBeam ; m1b ¼ 0,k1b ¼ 0

Fig. 9. Variation of 1st and 2nd shape modes with stiffness of sprung mass (simply supported sandwich beam): (a) 1st shape mode and (b) 2nd shape mode. x1 ¼ 0:5L : ½m1t ¼ mBeam , k1t ¼ 10d kBeam ; m1b ¼ 0,k1b ¼ 0

increasing m1t above 104mBeam and decreasing k1t under 10  4kBeam, the natural frequencies remain constant and the sprung mass has the effect similar to elastic support. In this situation, the vertical displacements of the beam in first natural frequency which is equal to o1 (natural frequency of sprung mass when w1 ¼0) is approached to zero, but in the second natural frequency, y12 (the displacement of the 1st sprung mass in 2nd natural frequency) is approached to zero and the second natural frequency equal to the first natural

frequency of the sandwich beam when there is a vertical elastic support (k1t).

7. Conclusion Free vibration of three-layered symmetric sandwich beam carrying sprung masses is investigated using the dynamic stiffness

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S.M.R. Khalili et al. / International Journal of Mechanical Sciences 52 (2010) 1620–1633

Fig. 10. Variation of 1st and 2nd shape modes with mass of sprung mass (simply supported sandwich beam): (a) 1st shape mode and (b) 2nd shape mode. x1 ¼ 0:5L : ½m1t ¼ 10d mBeam , k1t ¼ kBeam ; m1b ¼ 0,k1b ¼ 0

and the finite element methods. Validation of the approaches was made and a good agreement between the results obtained. Dynamic stiffness method is a fast and easy computation technique compare to the finite element method. The effect of parameters such as mass and stiffness of the sprung mass is studied on the natural frequencies. If the stiffness of the sprung mass is above a certain value, the behavior of the sprung mass on the natural frequencies is similar to the effect of an attached mass. Also, the effect of increasing the mass above a certain value is similar to the effect of a vertical elastic support. The presented analysis of the structures carrying sprung masses is the complete consideration of the structures with attached masses and elastic supports.

Appendix The members of the element stiffness matrix:

where k1 ¼

2Ef Af , l

k5 ¼

Gc Ac ðhf þ hc Þ2 30lh2c

k2 ¼

2Gc Ac l , 3h2c

k3 ¼

Gc Ac ðhf þ hc Þ , 6h2c

k4 ¼

4Ef If l3

The members of the element mass matrix: m11 ¼ m66 ¼ 2m1 m17 ¼ m67 ¼ 6m2 m22 ¼ m77 ¼ 156m3 þ 36m4

m12 ¼ m26 ¼ 6m2 m18 ¼ m36 ¼ lm2 m23 ¼ lð22m3 þ 3m4 Þ

m13 ¼ m68 ¼ lm2 m16 ¼ m1 m38 ¼ l2 ðm4 m5 Þ

m33 ¼ m88 ¼ l2 ð4m3 þ 4m4 Þ mi4 ¼ 0; i ¼ 1,2,3 mi5 ¼ 0; i ¼ 1,2,3,4 mi9 ¼ 0; i ¼ 1,2,3,4,5,6,7,8 mi10 ¼ 0; i ¼ 1,2,3,4,5,6,7,8,9

m28 ¼ lð13m3 þ 3m4 Þ m37 ¼ lð13m3 3m4 Þ m44 ¼ m0t m99 ¼ mlt

m38 ¼ l2 ð3m3 m4 Þ m78 ¼ lð22m3 3m4 Þ m55 ¼ 0b m1010 ¼ mlb

where k11 ¼ k66 ¼ k1 þ 2k2

k12 ¼ k26 ¼ 6k3

k13 ¼ k68 ¼ lk3

k22 ¼ k0t þ k0b þ 6k4 þ 36k5 k77 ¼ klt þ klb þ 6k4 þ 36k5 k23 ¼ k28 ¼ lð3k4 þ 3k5 Þ

k17 ¼ k67 ¼ 6k3 k38 ¼ l2 ðk4 k5 Þ k27 ¼ 6k4 36k5

k18 ¼ k36 ¼ lk3 ki4 ¼ 0; i ¼ 1,3 ki5 ¼ 0; i ¼ 1,3,4

k33 ¼ k88 ¼ l2 ð2k4 þ 4k5 Þ k16 ¼ k1 þ k2 k37 ¼ k78 ¼ lð3k4 3k5 Þ

k24 ¼ k0t k29 ¼ klt k25 ¼ k0b

k44 ¼ k0t k55 ¼ k0b k99 ¼ klt

ki9 ¼ 0; i ¼ 1,2,3,4,5,6,8 ki10 ¼ 0; i ¼ 1,2,3,4,5,6,8,9

k210 ¼ klb

k1010 ¼ klb

m1 ¼

ð6mf þ mc Þl , 18

m2 ¼

mc hf , 72

m3 ¼

ð2mf þ mc Þl , 420

m4 ¼

mc h2f 360l

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