Exact closed form solution for the analysis of the transverse vibration modes of a Timoshenko beam with multiple concentrated masses

Applied Mathematics and Computation 238 (2014) 342–357

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Exact closed form solution for the analysis of the transverse vibration modes of a Timoshenko beam with multiple concentrated masses Keivan Torabi ⇑, Adel Jafarzadeh Jazi, Ehsan Zafari Department of Mechanical Engineering, University of Kashan, Kashan, Islamic Republic of Iran

a r t i c l e

i n f o

Keywords: Free vibration Timoshenko beam Concentrated masses Dirac’s delta function

a b s t r a c t Concentrated masses on the beams have many industrial applications such as gears on a gearbox shafts, blades and disks on gas and steam turbine shafts, and mounting engines and motors on structures. Transverse vibration of the beam carrying a point mass was studied in many cases by both Euler–Bernoulli and Timoshenko beam theory for a limited number of concentrated masses mounted on a specific place on the beam. This was also investigated for a beam carrying multiple concentrated masses, yet they were solved by numerical methods such as Differential Quadrature (DQ) method. The present study investigated an exact solution for free transverse vibrations of a Timoshenko beam carrying multiple arbitrary concentrated masses anywhere on the beam with various boundary conditions. Using Dirac’s delta in governing equations, the effects of concentrated masses were imposed. After extracting a closed form solution, basic functions were used to reduce the amount of computations. Standard symmetric and asymmetric boundary conditions were enforced for beam; in addition, the effects of value, position, and number of concentrated masses were examined. Generally, while the existence of concentrated masses reduces the natural frequencies, the reduction depends on the parameters of concentrated masses. Finally, there were acquired mode shapes for different boundary conditions and different value, position, and number of concentrated masses. Ó 2014 Elsevier Inc. All rights reserved.

1. Introduction Numerous studies were conducted for the analysis of transverse vibration of elastic beams which carry a restricted number of point masses on a limited position of beam with only one or two boundary conditions. Most of these studies were presented without considering the effects of shear forces and rotating inertia in beams. They utilized Euler–Bernoulli beam theory while Timoshenko beam theory has more accurate results than Euler–Bernoulli theory, and the number of studies using this theory is limited. Considering the influence of masses on a shaft or beam is very important due to the decrease of natural frequencies of the shaft or beam in the presence of concentrated masses. This reduction should be considered in designing and manufacturing of structures, shafts and other applications. As mentioned above, some researchers were studied the vibration of the beams with concentrated masses by Euler– Bernoulli theory. Laura et al. [1] studied an Euler–Bernoulli Cantilever beam with a point mass and disregarded the effect ⇑ Corresponding author. E-mail address: [email protected] (K. Torabi). http://dx.doi.org/10.1016/j.amc.2014.04.019 0096-3003/Ó 2014 Elsevier Inc. All rights reserved.

K. Torabi et al. / Applied Mathematics and Computation 238 (2014) 342–357

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of shear and rotary inertia. He studied only one boundary condition and considered one position for a concentrated mass. The transverse vibration of a beam with an arbitrary placed concentrated mass and elastically restrained-hinged boundary condition at both ends was conducted by Goel [2]. He used Dirac’s delta to impose the effect of one concentrated mass to governing equation and used Laplace transform in his solution. Parnell and Cobble [3] studied lateral displacement of a vibrating Cantilever beam with a concentrated mass with general boundary condition by Laplace transform method. They also considered one position for the point mass. A research on vibration of a Cantilever beam with a concentrated mass and base excitation was carried out by To [4]. He imposed the effect of distance between tip mass center of gravity and point of its attachment to end of the beam. Grant [5] investigated the influence of rotary inertia and shear deformation or Timoshenko beam theory, on the frequency and normal mode of uniform beams carrying a concentrated mass. He used Dirac’s delta function to represent the effects of the concentrated mass on Timoshenko beam and then solved governing equations by Laplace transform method. Bruch and Mitchell [6] studied vibration of a Clamped-Free Timoshenko beam which carries lumped mass-rotary inertia on its free end. He considered the effect of shear force and rotating inertia of lumped mass-rotary inertia and imposed these two effects as a boundary condition at the free end of the beam, then he proved the reduction of first five natural frequencies of beam due to increasing mass or rotating inertia of lumped mass-rotary inertia. Abramovich and Hamburger [7] considered the effect of distance between the tip mass centroid and the point of tip mass attachment on the transverse vibration of a Cantilever beam carrying a tip mass at its free end. This effect causes a moment at the end of beam, and accompanied by effects of shear force and rotating inertia of tip mass. He compared the obtained results with results of Bruch [6]. In another research [8] Abramovich and Hamburger restudied vibration of a uniform Cantilever Timoshenko beam with translational and rotational springs and with a tip mass. In Ref. [9] Rossi et al. investigated free vibrations of Timoshenko beams carrying elastically mounted concentrated masses. He used governing equations of Timoshenko beam, and then compatibility conditions were used to impose the effect of shear force which is caused by mass-spring system on transverse vibration of a Timoshenko beam. In recent years, scientists tried to solve more complex problems related to the effect of concentrated mass on vibration of beams. Salarieh and Ghrashi [10] studied the effect of finite mass on both torsional and transverse vibration of Timoshenko beam. Free vibration analyses of an immersed beam carrying an eccentric tip mass with rotary inertia is performed by Wu and Hsu [11]. Lin and Tsai [12] used Euler–Bernoulli beam theory to analyze a uniform multi-span beam carrying multiple spring-mass systems. They only used Pinned–Pinned boundary condition for the concerned beam. Finally some researchers used numerical procedures to investigate free vibrations of non-uniform beams carrying concentrated mass or masses. Matsuda et al. [13] presented a method for vibration analysis of tapered Timoshenko beam carrying a tip mass at its end and solved governing differential equations by transforming them into integral equations and integrating them numerically. A DQEM (differential quadrature element method) for vibration of non-uniform Timoshenko beam carrying concentrated masses and rotary inertia with elastic supports was presented by Karami et al. [14]. Transfer matrix method used by Wu and Chen [15] for analyzing free vibration of multi-step Timoshenko beam with eccentric lumped masses and rotary inertias. Free vibration of elastically restrained Cantilever tapered beam carrying concentrated mass and damper was performed by De Rosa et al. [16]. They used symbolic Mathematical software in order to find free vibration frequencies of the Euler–Bernoulli beam. The disadvantages of analytical research which investigated the effect of point masses on the vibration of Timoshenko beam were their limitation on the number of studied point masses, the number of considered boundary conditions, and the position of point masses. Also, they have not presented a general closed form solution for this issue. In this paper an exact closed form solution for the analysis of the transverse vibration modes of Timoshenko beam carrying multiple concentrated masses is presented. As indicated in results, the number of concentrated masses is not limited, and they can be positioned anywhere on the beam with different boundary conditions. 2. Constitutive correlations A uniform Timoshenko beam is depicted in Fig. 1. As it can be observed in Fig. 1, only transverse vibration is taken account. In this figure, y(x, t) is transverse displacement of beam, f(x, t) is lateral force imposed to the beam, E is Young’s module, I is cross section moment of inertia about beam’s neutral axis, q and A are density and cross section area respectively, and L is length of the beam. The concentrated mass Mi, which is located at the position xi, is distributed in dx and  i dx and it can be modeled by Dirac’s delta function as Mid(x  xi). is presented as mi(x), where M i ¼ m Fig. 2 shows the positive sign convention of shear forces and moments in the beam element, where Q(x, t) is shearing force and M(x, t) is bending moment. In Timoshenko beam theory these forces are considered as Eq. (1).

@w @x Q ðx; tÞ ¼ K s GAcðx; tÞ

Mðx; tÞ ¼ EI

ð1Þ

where w is beam rotation, and c is shear strain. Using the second Newton law and Euler law, the motion equations of beam would be extracted as Eqs. (2) and (3).

f ðx; tÞdx þ Q þ

@Q @2y dx  Q ¼ ðqAdx þ mi ðxÞdxÞ 2 @x @t

ð2Þ

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K. Torabi et al. / Applied Mathematics and Computation 238 (2014) 342–357

Fig. 1. Multiple concentrated masses mounted on a vibrant beam.

Fig. 2. Positive sign convention of shear and bending forces in beam element.

Mþ

@M dx @2w dx  M þ Qdx  f ðx; tÞdx ¼ Im 2 @x 2 @t

ð3Þ

In which Im is the mass moment of inertia, and has a relationship with cross section moment of inertia I as Im ¼ qIdx. Also, the total slope of Timoshenko beam could be written as a function of shear deformation and angle of rotation of the beam element due to bending as:

@y @y ¼wþc)c¼ w @x @x

ð4Þ

In the free vibration study, the external force f(x, t) is eliminated, then by substituting Eq. (4) into Eqs. (2) and (3), equations of motion in dimensionless form for a beam carrying multiple concentrated masses are obtained as below:

@2W @f2



! N X @w qL2 @2W  1þ ai dðf  fi Þ ¼0 @f K s G @t2 i¼1

EI @ 2 wðx; t Þ L2

@f2

  @W @ 2 wðx; tÞ þ K s GA  w  qI ¼0 @f @t 2

ð5Þ

ð6Þ

where dimensionless variables are:

f  xL W  yL

0
ð7Þ 1

In Eq. (5) the property d(x  xi) = d[L(f  fi)] = L d(f  fi) is used, and dimensionless concentrated mass parameter ai = Mi/m0 is introduced, where m0  qAL. Both displacement and rotation of beam are functions of dimensionless position f and time t. By separating position and time functions, the displacement and rotation of beam could be assumed as W(f, t) = g(f)g(t) and w(f, t) = l(f)g(t), respectively that g(t) is a harmonic function of time with rotating frequency of x [17]. So the equations of motion without time dependent function can be written as

K. Torabi et al. / Applied Mathematics and Computation 238 (2014) 342–357 2

d gðfÞ df2 2

d

lðfÞ 2

df



! N X dlðfÞ þ k1 1 þ ai dðf  fi Þ gðfÞ ¼ 0 df i¼1

þm

345

ð8Þ

  dgðfÞ  lðfÞ þ k2 lðfÞ ¼ 0 df

ð9Þ

in which 2

2

k1 ¼ qLK sx G

2

2

k2 ¼ qL Ex

2

m ¼ K s GAL EI

ð10Þ

Note that k1 and k2 are related to each other, so k2 could be expressed with respect to k1 . The differential equation which contains displacement function g(f) can be obtained by combination of Eqs. (8) and (9):

giv þ k23 g00  k44 g ¼ BðfÞ

ð11Þ

k23 ¼ k1 þ k2

ð12Þ

k44 ¼ k1 ðk2  mÞ

ð13Þ

where

BðfÞ ¼ k1 "  k1

!

N X

ai dðf  fi Þ

2

d gðfÞ df2 !

i¼1 N X

"  2k1

ai d00 ðf  fi Þ þ k1 ðk2  mÞ

i¼1

!#

N X

0

ai d ðf  fi Þ

i¼1 N X

dgðfÞ df !#

ai dðf  fi Þ

gðfÞ

ð14Þ

i¼1

As seen in the Eq. (14), the function B(f) includes all of Dirac’s delta functions. 3. Extraction of vibration mode equation Eq. (11) is an ordinary nonhomogeneous differential equation which mode shape equations for different boundary conditions are extracted from it. The homogeneous solution of Eq. (11) is renowned Timoshenko’s uniform beam solution, which is combination of hyperbolic and trigonometric functions. Using the method of undetermined coefficient [18], the nonhomogeneous solution must be written in the same form of homogenous solution with variable coefficients as:

gðfÞ ¼ c1 sinhðb1 fÞ þ c2 coshðb1 fÞ þ c3 sinðb2 fÞ þ c4 cosðb2 fÞ ci ¼ ci ðfÞ i ¼ 1; 2; 3; 4

ð15Þ

where

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k23 þ k43 þ 4k44 pffiffiffi b1 ¼ 2 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b2 ¼

k23 þ

ð16Þ

k43 þ 4k44 pffiffiffi 2

Then the solution of Eq. (11) is obtained as:

gðfÞ ¼

1 ðb22 þ b21 Þb1 b2



( N X

  ai k1 ðk2  mÞTðf  fi Þ þ T 00 ðf  fi Þ gðfi Þuðf  fi Þ

) þ DðfÞ

ð17Þ

i¼1

where

DðfÞ ¼ d1 sinhðb1 fÞ þ d2 coshðb1 fÞ þ d3 sinðb2 fÞ þ d4 cosðb2 fÞ

ð18Þ

di(i = 1, 2, 3, 4) constants are integration coefficients, and they would be obtained from relevant boundary conditions. Function T(f) is defined as a term of trigonometric and hyperbolic function to make Eq. (17) more simply.

TðfÞ ¼ b2 sinh b1 ðfÞ þ b1 sin b2 ðfÞ

ð19Þ

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K. Torabi et al. / Applied Mathematics and Computation 238 (2014) 342–357

u(f  fi) is unit step (Heaviside) function which is distributional of Dirac’s delta and indicates the effect of mass discontinuity in the position of point mass fi(i = 1, 2, . . ., N). By applying distributional product with Dirac’s delta to Eq. (17), g(fj) is obtained.

Z

þ1

gðfÞdðf  fj Þdf ¼

1

( N X

1

 ai k1 ðb22 þ b21 Þb1 b2 i¼1 Z þ1 DðfÞdðf  fj Þdf þ

Z

þ1



00

)



ðk2  mÞTðf  fi Þ þ T ðf  fi Þ gðfi Þuðf  fi Þdðf  fj Þdf

1

ð20Þ

1

and in an easy form:

( j1 X

1

gðfj Þ ¼

ðb22 þ b21 Þb1 b2



)



00

ai k1 ðk2  mÞTðfj  fi Þ þ T ðfj  fi Þ gðfi Þ þ Dðfj Þ

ð21Þ

i¼1

For making Eq. (17) to a more useful shape, the Timoshenko beam basic functions are used. These functions are:

2 ½ g 1 ðfÞ g 2 ðfÞ g 3 ðfÞ g 4 ðfÞ  ¼

1 b21

þ

b22

0

6 6 b2 m2  ½ sinhðb1 fÞ coshðb1 fÞ sinðb2 fÞ cosðb2 fÞ 6 6 0 4 b1 m1

 b1 bk21m2

b1

0

0

1

0

b2

0

b1 b2 m1 k1

0

1

0

3 7 7 7 7 5 ð22Þ

where

m1 ¼ b1 þ

k1 ; b1

m2 ¼ b2 

k1 b2

ð23Þ

D(f) in Eq. (18) could be written as Eq. (24) by using basic functions.

DðfÞ ¼ g0 g 1 ðfÞ þ l0 g 2 ðfÞ þ M 0 g 3 ðfÞ þ Q 0 g 4 ðfÞ

ð24Þ

where g0, l0, M0, Q0 are amplitude of dimensionless displacement, rotation, moment and shear force at the first end of beam respectively. According to the fourth basic function, Eq. (17) could be rewritten as

gðfÞ ¼

N X

ai g 4 ðf  fi Þðk1 Þgðfi Þuðf  fi Þ þ DðfÞ

ð25Þ

i¼1

As seen in Eq. (21), g(fj) contains the value g(fi) for i = 1, 2, 3, . . ., j  1. The recursive expression of bending deformation at the point mass g(fi), for i = 1, . . ., N in which given by Eq. (21) can be rewritten as the following explicit form:

gðfi Þ ¼ g0 ai þ l0 bi þ M0 ei þ Q 0 fi

ð26Þ

where recursive coefficients are obtained by substituting Eq. (26) into Eq. (21) as the following:

aj ¼

j1 X

ai ðk1 Þg 4 ðfj  fi Þai þ g 1 ðfj Þ

i¼1

bj ¼

j1 X

ai ðk1 Þg 4 ðfj  fi Þbi þ g 2 ðfj Þ

i¼1

ej ¼

ð27Þ

j1 X

ai ðk1 Þg 4 ðfj  fi Þei þ g 3 ðfj Þ

i¼1

fj ¼

j1 X

ai ðk1 Þg 4 ðfj  fi Þfi þ g 4 ðfj Þ

i¼1

Finally the solution of Eigen-mode using basic functions is obtained by introducing Eq. (26) into Eq. (25).

gðfÞ ¼ g0

" N X

#

ai ðk1 Þg 4 ðf  fi Þai uðf  fi Þ þ g 1 ðfÞ þ l0

i¼1

þ M0

" N X

#

" N X i¼1

ai ðk1 Þg 4 ðf  fi Þei uðf  fi Þ þ g 3 ðfÞ þ Q 0

i¼1

#

ai ðk1 Þg 4 ðf  fi Þbi uðf  fi Þ þ g 2 ðfÞ

" N X

ai ðk1 Þg 4 ðf  fi Þfi uðf  fi Þ þ g 4 ðfÞ

# ð28Þ

i¼1

By obtaining the desire form of displacement equation, the other parameters such as rotation, moment and shear force are achieved as below.

K. Torabi et al. / Applied Mathematics and Computation 238 (2014) 342–357

347

lðfÞ ¼

" # 3 1 dgðfÞ d gðfÞ þ ðm þ k1 Þ ðm  k2 Þ df df3

MðfÞ ¼

dlðfÞ d gðfÞ ¼ þ k1 gðfÞ df df2

ð30Þ

Q ðfÞ ¼

dgðfÞ  lðfÞ df

ð31Þ

ð29Þ

2

4. The frequency equation for different boundary conditions The multiple concentrated mass beam equation is extracted by imposing the standard boundary conditions to Eqs. (28)–(31). The closed form solution, which is presented by these equations, is used for simply supported (Pinned–Pinned, PP), Cantilever (Clamped-Free, CF), Clamped–Clamped (CC), Free–Free (FF), and Clamped-Pinned (CP) Timoshenko beam boundary conditions. Some of them are symmetric and the others are asymmetric boundary condition. The frequency equations are derived, then by substituting numerical values, the effects of different point masses parameters on the natural frequencies and mode shapes are displayed. For more convenient the below symbols are used for last end of beam in order to reduce the size of relations.

gð1Þ ¼ g0 A11 þ l0 A12 þ M0 A13 þ Q 0 A14 lð1Þ ¼ g0 A21 þ l0 A22 þ M0 A23 þ Q 0 A24 Mð1Þ ¼ g0 A31 þ l0 A32 þ M 0 A33 þ Q 0 A34 Q ð1Þ ¼ g0 A41 þ l0 A42 þ M 0 A43 þ Q 0 A44

ð32Þ

4.1. Simply supported beam The boundary conditions of simply supported beam are presented as the following:

gð0Þ ¼ 0 Mð0Þ ¼ 0 gð1Þ ¼ 0 Mð1Þ ¼ 0

ð33Þ

Concerning properties of basic functions at the first end of beam, some coefficients equals zero.

gð0Þ ¼ 0 ) g0 ¼ 0

ð34Þ

Mð0Þ ¼ 0 ) M 0 ¼ 0 By imposing the boundary conditions at the end of the beam, other coefficients can be obtained.

gð1Þ ¼ 0 ) l0 A12 þ Q 0 A14 ¼ 0 Mð1Þ ¼ 0 ) l0 A32 þ Q 0 A34 ¼ 0

ð35Þ

The matrix form of Eq. (35) is expressed as



A12 A32

A14 A34



l0



  0 0

¼

Q0

ð36Þ

The frequency equation of simply support beam carrying multiple concentrated masses is obtained by computing second order determinant of Eq. (36).

A12 A34  A14 A32 ¼ 0

ð37Þ n

The zeros of this equation indicate the frequency parameters k1 ; which nth natural frequency could be extracted from it. By substituting frequency parameter into boundary condition system of Eq. (36), the value of integration coefficients providing vibration mode shape of simply supported multi concentrated mass beam are obtained as:

l0 Q0

n

¼ n

A14 A12

hP

Q 0 ¼ 1 ) l 0 ¼  hP

N n i¼1 i ð k1 Þg 4 ð1

a

 fi Þfi þ g 4 ð1Þ

i

i N n i¼1 ai ð k1 Þg 4 ð1  fi Þbi þ g 2 ð1Þ

ð38Þ

348

K. Torabi et al. / Applied Mathematics and Computation 238 (2014) 342–357

Substituting the obtained coefficients by Eqs. (34) and (38) into Eq. (28), the closed form expression of the vibration mode shapes of simply supported beam carrying multiple concentrated masses for nth natural frequency are achieved as the following. n

" # " # N N X A14 X n n gðfÞ ¼  n ai ð k1 Þg 4 ðf  fi Þbi uðf  fi Þ þ g 2 ðfÞ þ ai ð k1 Þg 4 ðf  fi Þfi uðf  fi Þ þ g 4 ðfÞ A12 i¼1 i¼1 n

ð39Þ

The most important benefit of using basic functions is that in each boundary condition, two unknown coefficients are equal zero. 4.2. Cantilever beam The boundary conditions of a Cantilever (Clamped-Free) beam are expressed as below:

gð0Þ ¼ 0 ) g0 ¼ 0 lð0Þ ¼ 0 ) l0 ¼ 0

Mð1Þ ¼ 0 Q ð1Þ ¼ 0

ð40Þ

and for the last end of the beam:

Mð1Þ ¼ 0 ) M0 A33 þ Q 0 A34 ¼ 0 Qð1Þ ¼ 0 ) M 0 A43 þ Q 0 A44 ¼ 0

 )

A33

A34

A43

A44



M0



Q0

¼

  0

ð41Þ

0

The frequency equation of Cantilever beam carrying multi concentrated masses is attained by evaluating the second order determinant of Eq. (41).

A33 A44  A34 A43 ¼ 0

ð42Þ

The roots of Eq. (42) are values of frequency parameter n k1 : By substituting frequency parameter into first equation of boundary condition system in Eq. (41) the value of integration coefficients which provide vibration mode shape equation of Cantilever beam carrying multiple concentrated mass can be obtained. n M0 A34 ¼ n Q0 A33 Q0 ¼ 1 ) nP N

o

n 00 i¼1 i ð k1 Þ½g 4 ð1

 fi Þ þ ðn k1 Þg 4 ð1  fi Þfi þ ½g 004 ð1Þ þ ðn k1 Þg 4 ð1Þ

N 00 n i¼1 i ð k1 Þ½g 4 ð1

 fi Þ þ ðn k1 Þg 4 ð1  fi Þei þ ½g 003 ð1Þ þ ðn k1 Þg 3 ð1Þ

M 0 ¼  nP

a

a

ð43Þ

o

Substituting the obtained coefficients of Eqs. (40) and (43) into Eq. (28), the closed form expression of the vibration mode shapes of Cantilever beam carrying multiple concentrated masses is acquired. n

" # N   A34 X n gðfÞ ¼  n ai ð k1 Þg 4 ðf  fi Þei uðf  fi Þ þ g 3 ðfÞ þ sumNi¼1 ai ðn k1 Þg 4 ðf  fi Þfi uðf  fi Þ þ g 4 ðfÞ A33 i¼1 n

ð44Þ

4.3. Clamped–Clamped beam The boundary conditions of a Clamped–Clamped beam are expressed as below:

gð0Þ ¼ 0 ) g0 ¼ 0 lð0Þ ¼ 0 ) l0 ¼ 0

gð1Þ ¼ 0 lð1Þ ¼ 0

ð45Þ

so



gð1Þ ¼ 0 ) M0 A13 þ Q 0 A14 ¼ 0 A13 A14 ) lð1Þ ¼ 0 ) M0 A23 þ Q 0 A24 ¼ 0 A23 A24



M0 Q0

 ¼

  0 0

ð46Þ

The frequency equation of Clamped–Clamped beam carrying multiple concentrated masses are gained by evaluating the second order determinant of Eq. (46).

A13 A24  A14 A23 ¼ 0

ð47Þ n

The zeros of this equation indicate the frequency parameters k1 . By substituting frequency parameter into first equation of boundary condition system of Eq. (46), we would have:

K. Torabi et al. / Applied Mathematics and Computation 238 (2014) 342–357 n M0 A14 ¼ n Q0 A13

nP

o

N n i¼1 i ð k1 Þg 4 ð1

 fi Þfi þ g 4 ð1Þ

N n i¼1 i ð k1 Þg 4 ð1

 fi Þei þ g 3 ð1Þ

Q 0 ¼ 1 ) M 0 ¼  nP

a

a

349

ð48Þ

o

Finally, such as previous boundary conditions, the vibration mode shapes equation of Clamped–Clamped beam carrying multiple concentrated masses is achieved as: n

n

gðfÞ ¼  n

" # " # N N X A14 X ai ðn k1 Þg 4 ðf  fi Þei uðf  fi Þ þ g 3 ðfÞ þ ai ðn k1 Þg 4 ðf  fi Þfi uðf  fi Þ þ g 4 ðfÞ A13 i¼1 i¼1

ð49Þ

4.4. Free–Free beam The boundary conditions for both end of a Free–Free beam are:

Mð0Þ ¼ 0 ) M0 ¼ 0

Mð1Þ ¼ 0

Q ð0Þ ¼ 0 ) Q 0 ¼ 0

Qð1Þ ¼ 0

ð50Þ

Conditions stated in Eq. (50) are used to obtain another two unknown coefficient of Eq. (28) as below:

Mð1Þ ¼ 0 ) g0 A31 þ l0 A32 ¼ 0 Q ð1Þ ¼ 0 ) g0 A41 þ l0 A42 ¼ 0

 )

A31

A32

A41

A42



g0 l0

 ¼

  0

ð51Þ

0

By setting the determinant of second order matrix in Eq. (51) equals zero, the frequency equation of a Free–Free beam carrying multiple concentrated mass is achieved.

A31 A42  A32 A41 ¼ 0

ð52Þ n

Extracting frequency parameter k1 from Eq. (52) and substituting it into first equation of Eq. (51), the coefficients that provide vibration mode shapes equation of Free–Free beam carrying multiple concentrated masses are obtained. n g0 A32 ¼ n l0 A31

nP N

o

i¼1

l0 ¼ 1 ) g0 ¼  nPN

ai ðn k1 Þ½g 004 ð1  fi Þ þ ðn k1 Þg 4 ð1  fi Þbi þ ½g 002 ð1Þ þ ðn k1 Þg 2 ð1Þ a

00 n i¼1 i ð k1 Þ½g 4 ð1

 fi Þ þ ðn k1 Þg 4 ð1  fi Þai þ ½g 001 ð1Þ þ ðn k1 Þg 1 ð1Þ

ð53Þ

o

Finally, mode shapes equation of vibration for this boundary condition is obtained as: n

" # " # N N X A32 X n n gðfÞ ¼  n ai ð k1 Þg 4 ðf  fi Þai uðf  fi Þ þ g 1 ðfÞ þ ai ð k1 Þg 4 ðf  fi Þbi uðf  fi Þ þ g 2 ðfÞ A31 i¼1 i¼1 n

ð54Þ

4.5. Clamped-Pinned beam The boundary conditions of a Clamped-Pinned (which is an asymmetric boundary condition) beam are expressed as below:

gð0Þ ¼ 0 ) g0 ¼ 0 lð0Þ ¼ 0 ) l0 ¼ 0

gð1Þ ¼ 0

ð55Þ

Mð1Þ ¼ 0

According to Eqs. (28)–(30) and above conditions, other two coefficients of Eq. (28) are found as below.

gð1Þ ¼ 0 ) M0 A13 þ Q 0 A14 ¼ 0 Mð1Þ ¼ 0 ) M0 A33 þ Q 0 A34 ¼ 0

 )

A13

A14

A33

A34



M0 Q0

 ¼

  0 0

ð56Þ

By setting the determinant of second order matrix in system equation Eq. (56) equals zero, the frequency equation is obtained.

A13 A34  A14 A33 ¼ 0

ð57Þ n

Extracting frequency parameter k1 from Eq. (57) and substituting it into first equation of Eq. (56), the coefficients M0 and Q0 providing vibration mode shapes of Clamped-Pinned beam carrying multiple concentrated masses are attained.

350

K. Torabi et al. / Applied Mathematics and Computation 238 (2014) 342–357

(a)

(b)

τ1 for Pinned-Pinned Beam (PP)

τ1 for Clamped-Free Beam (CF)

1

1

0.9

0.9

0.8

0.8

0.7

0.7

0.6

0.6

0.5

0.5

τ1

τ1

N=1, 2, 3, 5

0.4

0.4 N=1, 2, 3, 5

0.3

0.3

0.2

0.2

0.1

0.1

0

0

1

2

3

4

5

6

7

8

9

0

10

0

1

2

3

4

α

(c)

5

6

7

8

9

7

8

9

10

α

(d)

τ1 for Clamped-Clamped Beam (CC)

τ1 for Free-Free Beam (FF)

1

1

0.9

0.9

0.8

0.8

0.7

0.7

0.6

0.6

0.5

0.5

τ1

τ1

N=1, 2, 3, 5

0.4

0.4 N=1, 2, 3, 5

0.3

0.3

0.2

0.2

0.1

0.1

0

0

1

2

3

4

5

6

7

8

9

0

10

0

1

2

3

α

4

5

6

10

α

(e)

τ1 for Clamped-Pinned (CP) 1 0.9 0.8 0.7

τ1

0.6 0.5 0.4

N=1, 2, 3, 5

0.3 0.2 0.1 0

0

1

2

3

4

5

6

7

8

9

10

α Fig. 3. Investigation of the effect of concentrated mass intensity and number of concentrated masses on first frequency ratio for different boundary conditions: (a) simply support or Pinned–Pinned, (b) Clamped-Free or Cantilever, (c) Clamped–Clamped, (d) Free–Free and (e) Clamped-Pinned.

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K. Torabi et al. / Applied Mathematics and Computation 238 (2014) 342–357

(b)

Frequency Ratios for one concentrated mass and for PP

Frequency Ratios for one concentrated mass and for CC

1

1

0.5

0.5

0

τ1

τ1

(a)

0

0.2

0.1

0.3

0.4

0.5

0.6

0.7

0.8

0

1

0.9

0

0.1

0.2

0.3

0.4

ζ0

τ2

α =1

0

0.2

0.1

0.3

0.4

0.5

0.6

0.7

0.8

0.2

1

0.9

0.6

0.7

0.8

0.9

1

0.6

0.7

0.8

0.9

1

0.6

0.7

0.8

0.9

1

α = 10

0

0.1

0.2

0.3

0.4

0.5

1

0.7

τ3

τ3

1

ζ0

1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0.6 0.2

1

ζ0 1

1

0.7

0.7

0

0.1

0.2

0.3

0.4

0.5

0

0.1

0.2

0.3

0.4

0.5

ζ0

τ4

τ4

0.9

α=1

ζ0

0.4

0.8

α = 0.1

0.6

α = 10

0.3

0.7

1 α = 0.1

0.6 0.2

0.6

ζ0

1

τ2

0.5

0.6

0.7

0.8

0.9

0.4

1

ζ0

0

0.1

0.2

0.3

0.4

0.5

ζ0

Fig. 4. Investigation of the effect of one concentrated mass position on first four frequency ratio for three values of dimensionless mass parameter and symmetric boundary conditions: (a) Simply supported or Pinned–Pinned and (b) Clamped–Clamped.

(a)

(b)

Frequency Ratios for one concentrated mass and for CP

Frequency Ratios for one concentrated mass and for CF 1 α = 0.1

0.5

τ1

τ1

1 α=1

0.5

α = 10

0

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0

1

0

0.1

0.2

0.3

0.4

1

0.5

0.5

τ2

τ2

1

0

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0

1

0.6

0.6

τ3

τ3

1

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0.2

1

0.9

1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0.6

0.7

0.8

0.9

1

ζ0 1

τ4

1

τ4

0.8

α=1 α = 10

ζ0

0.6 0.2

0.7

ζ0

1

0

0.6

α = 0.1

ζ0

0.2

0.5

ζ0

ζ0

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

ζ0

0.7

0.3

0

0.1

0.2

0.3

0.4

0.5

ζ0

Fig. 5. Investigation of the effect of one concentrated mass position on first four frequency ratio for three values of dimensionless mass parameter and asymmetric boundary conditions: (a) Cantilever or Clamped-Free and (b) Clamped-Pinned.

n M0 A14 ¼ n Q0 A13

nP

o

N n i¼1 i ð k1 Þg 4 ð1

 fi Þfi þ g 4 ð1Þ

N n i¼1 i ð k1 Þg 4 ð1

 fi Þei þ g 3 ð1Þ

Q 0 ¼ 1 ) M 0 ¼  nP

a

a

ð58Þ

o

Finally, mode shapes equation of vibration for clamped pinned beam carrying multiple concentrated masses is gained by substituting extracted coefficients from Eqs. (55) and (58) into Eq. (28) as:

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K. Torabi et al. / Applied Mathematics and Computation 238 (2014) 342–357

(a)

(b)

τ1 for one concetrated mass

τ1 for α = 0.1

τ1 for α = 0.1

0.9

0.8

0.7

τ1 for two symetric concentrated masses 1

1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0.8

0.6

0.4

1

ζ0 CF

PP

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

ζ0 CC

FF

PP

CP

CF

CC

FF

CP

1

1

τ1 for α =1

τ1 for α =1

0.8 0.6 0.4

0.5

0.2 0

0 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0

0.05

0.1

0.15

0.2

τ1 for α = 10

τ1 for α = 10

0.3

0.35

0.4

0.45

0.5

0.3

0.35

0.4

0.45

0.5

1

1 0.8 0.6 0.4 0.2 0

0.25

ζ0

ζ0

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0.5

0

1

0

0.05

0.1

0.15

0.2

ζ0

0.25

ζ0

Fig. 6. Comparison of beams response with different boundary conditions to concentrated mass position. First frequency ratio versus position of concentrated mass for different value of dimensionless concentrated mass parameter for (a) one concentrated mass and (b) two concentrated masses with different boundary conditions: PP (Pinned–Pinned), CF (Clamped-Free), CC (Clamped–Clamped), FF (Free–Free) and CP (Clamped-Pinned).

PP

CF

CC

FF

CP

η( ζ)

1

0

-1

0

0.1

0.2

0.3

0.4

μ(ζ)

0.6

0.7

0.8

0.9

1

0.6

0.7

0.8

0.9

1

0.6

0.7

0.8

0.9

1

0.6

0.7

0.8

0.9

1

0

-5

0 8 x 10

0.1

0 8 x 10

0.1

0

0.1

2 M ( ζ)

0.5

ζ

5

0.2

0.3

0.4

0.5

ζ

1 0 -1

0.2

0.3

0.4

0.5

ζ

Q( ζ)

2 0 -2 0.2

0.3

0.4

0.5

ζ

Fig. 7. The effect of multiple concentrated masses on different parameters of a vibrant Timoshenko beam. First mode shape, dimensionless rotation of beam, bending moment and shear force in a beam carrying 5 concentrated masses with dimensionless mass parameter a = 10 for different boundary conditions: PP (Pinned–Pinned), CF (Clamped-Free), CC (Clamped–Clamped), FF (Free–Free) and CP (Clamped-Pinned).

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K. Torabi et al. / Applied Mathematics and Computation 238 (2014) 342–357

n

n

gðfÞ ¼  n

" # " # N N X A14 X ai ðn k1 Þg 4 ðf  fi Þei uðf  fi Þ þ g 3 ðfÞ þ ai ðn k1 Þg 4 ðf  fi Þfi uðf  fi Þ þ g 4 ðfÞ A13 i¼1 i¼1

ð59Þ

5. Numerical results and discussion In this section, numerical results about the effects of point masses on the natural frequencies and mode shapes of the beam are presented and discussed. After obtaining frequency equations for each boundary condition, natural frequencies are extracted by using Secant method. First of all, the dimensionless frequency ratio s is defined as below: i

k k1

si ¼ 0i 1 ¼

x2i x20i

ð60Þ

In which 0i in x and k1 means natural frequency and frequency parameter of Timoshenko beam having no concentrated masses respectively, and i shows that these parameters belong to Timoshenko beam carrying concentrated masses. The beam which is studied in this section has length of 3 m, circle cross section area with radius 10 cm, shear coefficient 0.9, density 7860 Kg/m3, Young’s modulus of 200 Gpa and shear modulus of 77 Gpa (ASTM-A36 [19]). Fig. 3 displays first frequency ratio versus dimensionless mass parameter with different number of concentrated masses. 1, 2, 3 and 5 concentrated masses were placed on the beam with the same distance to show the effect of number of concentrated masses on natural frequency. As it can be seen in these diagrams, the first frequency ratio decreases by increasing the number of concentrated masses as it was expected. For a = 0 frequency ratio equals to 1 due to the beam carries no masses. Also these diagrams represented inverse relation between frequency ratio and dimensionless mass parameter which exists

(b)

mode shape of beam with one concentrated mass (α =10) PP

mode shape of beam with one concentrated mass (α =10) CF

1

1

0.8

0.8

0.6

0.6

η(ζ)

η(ζ)

(a)

0.4

0.4

0.2

0.2

0 1

0 1 1

1 0.8

0.5

0.8

0.5

0.6

0.6 0.4

0.4

ζ0

(c)

0

0.2 0

ζ0

ζ

(d)

mode shape of beam with one concentrated mass (α =10) CC

1

0

0.2 0

ζ

mode shape of beam with one concentrated mass (α =10) FF

1

0.8

0.5

η(ζ)

η(ζ)

0.6 0.4

0

-0.5

0.2 0 1

-1 1 1

1

0.8

0.5

0.6

0.8

0.5

0.6

0.4

ζ0

0

0.4

0.2 0

ζ

ζ0

0

0.2 0

ζ

Fig. 8. Investigation of the effect of one concentrated mass position (with a = 10) on the first Timoshenko beam mode shape for: (a) Simply supported or Pinned–Pinned, (b) Cantilever or Clamped-Free, (c) Clamped–Clamped and (d) Free–Free boundary conditions.

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K. Torabi et al. / Applied Mathematics and Computation 238 (2014) 342–357

in frequency equations. Frequency ratio has more reduction by increasing number of concentrated masses in boundary conditions which have free end. Figs. 4 and 5 depict the effect of position of one concentrated mass on four frequency ratios for different values of dimensionless mass parameter. As it has envisaged, concentrated mass causes reduction of frequency ratios, but this effect varies from the position of concentrated mass. As a general result, the concentrated mass has more influence on a frequency ratio in positions that corresponding modes shape has larger displacement. In some positions, the concentrated mass does not have any effects on the natural frequency. These positions are the position of bases or position of mode shape nodes having no displacement. According to Figs. 4 and 5, in each position of concentrated mass, frequency ratio decreases with increasing the value of dimensionless mass parameter. Fig. 6 demonstrates the influence of one and two concentrated masses on the first frequency ratio for different boundary conditions and three values of dimensionless concentrated masses (a = 0.1, 1, 10). As frequency ratios are illustrated in this figure, the diagrams for each of boundary condition have a general behavior for different value of dimensionless mass parameter. Only the value of each diagram decreases with enhancing the value of dimensionless mass parameter. For each value of dimensionless mass parameter and for a specific concentrated mass position, the decline in frequency ratio is greater in boundary condition in which the beam has larger displacement. Fig. 7 indicates first mode shape, dimensionless rotation of beam, moment and shear force in the beam for different boundary conditions. Five concentrated masses are located on the beam with dimensionless mass parameter a = 10. Mode shapes are normalized by their maximum value of displacement. Then other parameters are calculated with respect to normalized mode shapes. As it is obvious in these diagrams, existence of concentrated masses cause deflection in the moment curves and discontinuity in the shear force curves.

(a)

(b)

mode shape of beam with one concetrated mass at ζ = 0.2 0

mode shape of beam with one concetrated mass at ζ = 0.2 0 CF

1

1

0.8

0.8

0.6

0.6

η(ζ)

η(ζ)

PP

0.4

0.4

0.2

0.2

0 10

0 10 1

1 0.8

5

0.8

5

0.6

0.6 0.4

0.4

α

(c)

0

0.2 0

α

ζ

(d)

mode shape of beam with one concetrated mass at ζ = 0.2 0

0

0.2 0

ζ

mode shape of beam with one concetrated mass at ζ = 0.2 0 FF

CC

1

1 0.8

0.5

η(ζ)

η(ζ)

0.6 0

0.4 -0.5

0.2 -1 10

0 10

1

1 0.8

5

0.6

0.8

5

0.6 0.4

0.4

α

0

0.2 0

ζ

α

0

0.2 0

ζ

Fig. 9. Investigation of the effect of one concentrated mass intensity on the first Timoshenko beam mode shape. First vibration mode shape of beam carrying one concentrated mass located at f0 = 0.2 with different value of dimensionless mass parameter for (a) Simply supported or Pinned–Pinned, (b) Cantilever or Clamped-Free, (c) Clamped–Clamped and (d) Free–Free boundary conditions.

K. Torabi et al. / Applied Mathematics and Computation 238 (2014) 342–357

355

Fig. 8 reveals the effect of concentrated mass position on beam’s first mode shape for different boundary conditions. Dimensionless mass parameter of concentrated mass is a = 10. This effect is evident for Free–Free (FF) and Clamped– Clamped (CC) boundary conditions. Fig. 9 displays the effect of concentrated mass intensity on the first mode shape of beam carrying one concentrated mass at f0 = 0.2. Concentrated mass causes less displacement at position of concentrated mass and this effect is perceived obviously in Free–Free boundary condition in both figures. 6. Validation To endorse and validate the present study, Ref. [5] is utilized. In this reference, the effects of rotary inertia and shear deformation on the frequency and normal mode equations of uniform beams carrying a concentrated mass are investigated. Natural frequency of uniform Euler–Bernoulli beam is used to normalizing natural frequency. This parameter for a uniform beam is defined as below: n

sffiffiffiffiffiffiffi np EI x0 ¼ L qA

ð61Þ

In which n presents the number of natural frequency. To normalize the obtained first natural frequency from a uniform Timoshenko beam carrying a concentrated mass, it was divided by first natural frequency of Euler–Bernoulli beam without concentrated mass. Therefore below parameter is introduced as:

r2 ¼

I AL2

ð62Þ

A uniform Pinned–Pinned Timoshenko beam carrying one concentrated mass with properties: 25.4 (mm) square cross section, Young’s module of 207 (Gpa), specific weight c = 76.8 (KN/m3), shear coefficient k = p2/12 and shear module G = 3E/8. Moreover the length is considered variable and consequently the parameter r. In Ref. [5], two diagrams are drawn to illustrate the effect of a concentrated mass on the natural frequency of Timoshenko beam. These diagrams are shown in Fig. 10, and same results are obtained by results of the current paper and drawn in Fig. 11. Fig. 11 demonstrates the similarity between two figures of Fig. 10 and the figures in Fig. 11. Comparing these figures leads the present results and calculations to be validated by Ref. [5].

Fig. 10. The effect of one concentrated mass on natural frequency of Timoshenko beam for different value of dimensionless mass parameter a = 0, 0.01, 0.1, 1, 10, 100 with (a) f0 = 0.5 and (b) f0 = 0.25 (as provided in Ref. [5]).

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K. Torabi et al. / Applied Mathematics and Computation 238 (2014) 342–357

ζ0= 0.5

(b)

1

ζ = 0.25 0

1

0.9

0.9

0.8

0.8

0.7

0.7

0.6

0.6

0.5

α α α α α α

0.4

0.3

ω / ω0

ω / ω0

(a)

=0 = 0.01

0.5 0.4

= 0.1 =1 = 10

= 0.01 = 0.1 =1 = 10 = 100

= 100 0.2

0.1

0.1

0.02

=0

0.3

0.2

0

α α α α α α

0.04

0.06

r

0.08

0.1

0

0.02

0.04

0.06

0.08

0.1

r

Fig. 11. The effect of one concentrated mass on natural frequency of Timoshenko beam for different value of dimensionless mass parameter a = 0, 0.01, 0.1, 1, 10, 100 with (a) f0 = 0.5 and (b) f0 = 0.25 (as extracted from the calculation of present study).

7. Conclusions In this study, the exact closed form equations for the analysis of the transverse vibration modes of a Timoshenko beam carrying multiple concentrated masses are obtained by using Dirac’s delta function for modeling concentrated masses. After imposing Newton’s second law and Euler’s law on the Timoshenko’s beam element, equations of motion are attained. Then time variable function is separated from equations of motion. Later, two coupled equations of motion are combined, a nonhomogeneous differential equation is extracted with respect to transverse displacement, and undefined coefficients method is used to solve differential equation. Mode shape equation (dimensionless-time invariant equation of transverse displacement) is achieved by solving the differential equation using a specific procedure. This equation has four integrating coefficients extracted from boundary conditions. A new base whose coefficients are defined as physical parameters of beam is considered for mode shape equation; what is more, basic functions are used to rewrite mode shape equation of motion in a new form. After imposing boundary condition at first and end of the beam, a 2  2 matrix is gained whose determinant defines frequency equation. Therefore, minimum mathematical operations are applied to extract frequency and mode shape equations for standard boundary conditions. A numerical study of multiple concentrated masses on Timoshenko beams was conducted to investigate numbers, positions, and mass intensity on the beam natural frequencies and transverse mode shapes in different symmetric and asymmetric boundary conditions. Increasing the number of concentrated masses or increasing concentrated mass intensity caused a decline in natural frequencies. If a concentrated mass is positioned at node of a mode shape, the concentrated mass has no effect on relevant natural frequency. If a concentrated mass is placed at position with more displacement, the natural frequency has more reduction. If one or two concentrated masses have the same position in different boundary conditions, they have more effect on natural frequency of boundary condition which has more displacement at the position of concentrated masses. The concentrated masses caused deflection in moment curve and discontinuity in shear force curve in the beam. Finally, a concentrated mass caused less displacement in a mode shape at its position, and this reduction would become greater if concentrated mass placed at position had more displacement in the mode shape. References [1] P.A.A. Laura, J.A. Pombo, E.A. Susemihl, A note on the vibration of a clamped-free beam with a mass at the free end, J. Sound Vib. 37 (1974) 161–168. [2] R.P. Goel, Free vibrations of a beam-mass system with elastically restrained ends, J. Sound Vib. 47 (1976) 9–14. [3] L.A. Parnell, M.H. Cobble, Lateral displacements of a vibrating cantilever beam with a concentrated mass, J. Sound Vib. 44 (1976) 499–511.

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[4] C.W.S. To, Vibration of a cantilever beam with a base excitation and tip mass, J. Sound Vib. 83 (1982) 445–460. [5] D.A. Grant, The effect of rotary inertia and shear deformation on the frequency and normal mode equations of uniform beams carrying a concentrated mass, J. Sound Vib. 57 (1978) 357–365. [6] J.C. Brunch Jr, T.P. Mitchell, Vibrations of a mass-loaded clamped-free Timoshenko beam, J. Sound Vib. 114 (1987) 341–345. [7] H. Abramovich, O. Hamburger, Vibration of a cantilever Timoshenko beam with a tip mass, J. Sound Vib. 148 (1991) 162–170. [8] H. Abramovich, O. Hamburger, Vibration of a uniform cantilever Timoshenko beam with translational and rotational springs and with a tip mass, J. Sound Vib. 154 (1992) 67–80. [9] R.E. Rossi, P.A.A. Laura, D.R. Avalos, H. Larrondo, Free vibrations of Timoshenko beams carrying elastically mounted concentrated masses, J. Sound Vib. 165 (1993) 209–223. [10] H. Salarieh, M. Ghorashi, Free vibration of Timoshenko beam with finite mass rigid tip load and flexural-torsional coupling, Int. J. Mech. Sci. 48 (2006) 763–779. [11] J.S. Wu, S.H. Hsu, The discrete methods for free vibration analyses of an immersed beam carrying an eccentric tip mass with rotary inertia, Ocean Eng. 34 (2007) 54–68. [12] H.Y. Lin, Y.C. Tsai, Free vibration analysis of a uniform multi-span carrying multiple spring-mass systems, J. Sound Vib. 302 (2007) 442–456. [13] H. Matsuda, C. Morita, T. Sakiyama, A method for vibration analysis of a tapered Timoshenko beam with constraint at any points and carrying a heavy tip body, J. Sound Vib. 158 (1992) 331–339. [14] G. Karami, P. Malekzadeh, S.A. Shahpari, A DQEM for vibration of shear deformable nonuniform beams with general boundary conditions, Eng. Struct. 25 (2003) 1169–1178. [15] J.S. Wu, C.T. Chen, A lumped-mass TMM for free vibration analysis of a multi-step Timoshenko beam carrying eccentric lumped masses with rotary inertias, J. Sound Vib. 301 (2007) 878–897. [16] M.A. De Rosa, M. Lippiello, M.J. Maurizi, H.D. Martin, Free vibration of elastically restrained cantilever tapered beams with concentrated viscous damping and mass, Mech. Res. Commun. 37 (2010) 261–264. [17] C.W. De Silva, Vibration: Fundamentals and Practice, CRC Press, USA, 2000. [18] W.E. Boyce, R.C. DiPrima, Elementary Differential Equations and Boundary Value Problems, tenth ed., John Wiley & Sons, 2012. [19] F.P. Beer, E.R. Johnston, J.T. DeWolf, Mechanics of Materials, fourth ed. in SI Units., McGraw-Hill, New York, 2006.