beam interaction

J Braz. Soc. Mech. Sci. Eng. DOI 10.1007/s40430-014-0277-1

TECHNICAL PAPER

On the computation of moving mass/beam interaction utilizing a semi‑analytical method Ali Nikkhoo · Ali Farazandeh · Mohsen Ebrahimzadeh Hassanabadi 

Received: 29 August 2014 / Accepted: 11 November 2014 © The Brazilian Society of Mechanical Sciences and Engineering 2014

Abstract  This article proposes a new orthogonal basis for the spatial discretization tracking the dynamic response of shear deformable beams. The corresponding initial boundary value differential equations of motion are dealt with focusing on Timoshenko and Reddy–Bickford beam theories. The presented technique takes advantage of a compatible trigonometric set of functions defining the rotation field in combination with orthogonal splines for the transverse deformation. A broad survey is conducted gaining the beam natural frequencies in free vibration; the forced vibration of the beam is attacked considering a traveling inertial body interacting with the base beam. For different beam end-fixity cases, the orthogonal basis could be straightforwardly constructed representing a rapid convergence. In this regard, a remedy is achieved for the inconvenient procedure of creating the shape functions in other competing methodologies. Keywords  Orthogonal basis · Spatial discretization · Shear deformable beams · Free vibration · Traveling inertial body

Technical Editor: Fernando Alves Rochinha. A. Nikkhoo · A. Farazandeh  Department of Civil Engineering, University of Science and Culture, Tehran, Iran e-mail: [email protected] A. Farazandeh e-mail: [email protected] M. Ebrahimzadeh Hassanabadi (*)  Department of Structural Engineering, Building and Housing Research Center (BHRC), Postal Box: 13145‑1696, Tehran, Iran e-mail: [email protected]; [email protected]

1 Introduction A plenty of industrial and engineering structures are permanently subjected to the moving loads such as bridges, railways and guide ways. Therefore, amongst the ever increasing published studies on the forced vibration of continuums, those concentrating the moving load dynamic problems have attained a significant interest of the researchers [1–4]. In this regard, Akin and Mofid [5] employed an analytical–numerical method computing the dynamic response of a beam with arbitrary boundary conditions carrying a moving mass. An analytical method has been proposed by Yang et al. [6] scrutinizing a simple beam vibration subjected to the moving loads. They considered the inertial effects of the moving vehicles. Rao [7] investigated the influence of a traveling inertial object on the dynamic behavior of an Euler–Bernoulli beam (EB) with simple end conditions. Yavari et al. [8] attacked the moving load problem regarding a Timoshenko beam (TB) excited by a traveling mass by adopting DET (discrete element technique). Nikkhoo and Kananipour [9] dealt with the motion equations of curved beams by making recourse to the differential quadrature method. Utilization of shear deformation beam theories is proven to be inevitable aiming at adequately precise estimations of thick beam’s dynamic behavior. The effect of shear deformation and rotary inertia of the base beam has been underlined in some recently published articles considering a Timoshenko beam theory (TBT). Kargarnovin et al. [10] simulated a delaminated TB traversed by an oscillatory mass taking advantage of orthogonal modal series expansion technique. Using dynamic green function, Ghannadiasl and Mofid [11] achieved an analytical solution to the dynamic response of uniform TBs with arbitrary boundary

13



conditions. Eftekhar Azam et al. [12] contributed to the moving load problem by performing a comprehensive parametric exploration of a simply supported TB subjected to a moving sprung mass. They emphasized the fast converging rate of the eigenfunction expansion method in the discretization of the spatial domain. Thanking the high converging rate and satisfactory computing performance, orthogonal series-based solutions have been frequently employed in the vibration analyses of beamtype structures [2, 3, 13]; in this manner, favorable computational procedures of analysis could be established by the engineers and the specialists. A notable volume of published contributions currently exists addressing the free vibration of beams [14–16]; The EEM (eigenfunction expansion method) is one of the most well-known series-based solutions to the free/forced vibration of beams [1, 3, 12, 15]. In such solution techniques, the governing equations of the base beam motion are reduced to time domain alone by assuming the beam degrees of freedom in modal coordinates. Therefore, a major concern would be the attempt required to reveal the beam closed form natural shape functions and frequencies to conduct the computations. It should be noted that, excepting the specific case of simply supported beams, the complexity of obtaining the analytical mode shapes grows; by regarding shear deformation and also increasing the order of the shear deformation beam theory, achieving the analytical eigensolution of the base beam gets much more complex and challenging. This mainly initiates from the coupled form of the related partial differential equations which demands rather convoluted analytical steps to determine the roots of the characteristic equation. In this paper, a semi-analytical procedure aimed at finding the dynamic response of the shear deformable beams, is developed. To this end, the governing motion equations of the EB, TB and Reddy–Bickford beam (RB) due to a moving mass are derived through employing Hamilton’s principle. Then, the characteristic orthogonal polynomials (COPs) and trigonometric functions compatible with the boundary conditions are utilized for spatial discretization, and the matrix-exponential method is used for the proper time discretization of the problem. Actually, the proposed framework takes the advantage of both analytical and numerical techniques; i.e., accuracy and convenience. Moreover, a survey is performed to evaluate the associated convergence rate estimating the free vibration frequencies of the beam with different boundary conditions and beam theories. Finally, the dynamic responses of the beams acted upon by a moving mass are determined and discussed. The results issued from all three theories, demonstrate a high accuracy, as well as a good coincidence with those extracted from the purely numerical methods; despite using the lower number of the shape functions.

13

J Braz. Soc. Mech. Sci. Eng.

2 Mathematical modeling A uniform finite beam traversed by a moving mass, m, having a constant velocity, v, along the beam length, L, and with various boundary conditions is considered. It is further assumed that the moving mass travels in contact with the beam at all times. Moreover, the vertical acceleration component, the complementary acceleration and the centripetal acceleration are applied in the terms of mass transverse acceleration to investigate the inertial effects of the moving mass [1]. The material of the beam is set to be undamped, linear isotropic homogeneous with an elastic modulus of E and a shear modulus of G. The beam has a constant cross-section with a uniform mass distribution, i.e., the cross-sectional area, A, and the beam density, ρ, are uniform throughout the beam length. 2.1 The governing equations of the problem based on various beam theories To describe the governing equations of various beam theories, it is needed to introduce a coordinate system xyz which is fixed to the left hand end of the undeformed beam. The x-, z- and y-axes are taken correspondingly along the length, thickness and width of the beam. Let ux = ux (x, z, t) and uz = uz (x, z, t) denote the longitudinal and transverse deformation components of the beam, respectively. On the other hand, it could be stated that the deformation components are only functions of the x and z coordinates of a point in time t . Here, it is further assumed that the displacement parallel to the y-axis is identically zero. In this part, to specify the system dynamic matrices, the discrete governing equations are distinctly derived by applying the Hamilton’s principle for each theory; i.e., Euler–Bernoulli beam theory (EBT), TBT and Reddy– Bickford beam theory (RBT), while the mass exists throughout the beam length (0 ≤ t ≤ L/v). Obviously, when the mass leaves the beam (t > L/v), the terms related to the moving mass will be set to zero. Accordingly, the total potential energy of the beam, Et , can be written as

Et = Ek − (Es + Ep ),

(1)

where Ek is the kinetic energy; Es is the elastic strain energy, and Ep is the potential energy of the beam subjected to the moving mass. These are generally defined as        ∂uz 2 1 L ∂ux 2 ρ + dAdx, Ek = 2 0 A ∂t ∂t   1 L Es = (σxx εxx + σxz γxz )dAdx , 2 0 A  2   L  2  ∂ uz ∂ 2 uz 2 ∂ uz + v Ep = − m g− + 2v ∂t 2 ∂x∂t ∂x 2 x=vt 0

×(uz )x=vt δ(x − vt) dx ,

(2)

J Braz. Soc. Mech. Sci. Eng.

in which, δ is the Dirac delta function and g is the gravitational acceleration. εxx /σxx and γxz /σxz represent correspondingly the transverse normal and transverse shear strain/stress components. It is worth noting that the superscripts and subscripts E, T and R denote the quantities related to the EBT, TBT and RBT throughout the paper, respectively. 2.1.1 Formulations based on the EBT Since the strain energy associated with the transverse shear strain is zero in the EBT, the deformation field of the beam is only restricted to the vertical deflection of the beam [17]. Thus, the transverse and longitudinal displacements E are given by uzE = wE (x, t) and uxE = −zw  ,x, respectively. E Regarding that σxx = Eεxx and εxx = dux dx, Eq. (2) is rewritten as

0  L 

   E 2 dx , EI w,xx Es = 1 2 0  L    E + v2 wE Ep = − m g− w ¨ E + 2vw˙ ,x ,xx 0

(3) x=vt



wE (vt, t) δ(x − vt) dx ,

wherein I is the second moment inertia of the beam section. It should be noted that, superscripts (˙) and (˙˙) signify correspondingly the first and the second derivatives of the functions with respect to t . Subsequently, by applying the spatial discretization technique, the unknown displacement field wE (x, t) can be defined as n E E E w (x, t) = j=1 ϕj (x)wj (t) where n is the total required number of the shape functions to receive the demand accuracy. ϕjE (x) is the jth assumed shape function which is determined by the provided method in Sect. 2.2. To calculate wjE (t), one should solve the below equation of motion,

ME (t) x¨ E (t) + CE (t)˙xE (t) + KE (t)xE (t) = fE (t) ,

(4)

in which, the system dynamic matrices are introduced as

[ME ]ij = ρA



L

0



+ρI

0

ϕiE (x)ϕjE (x)dx

L

E E ϕi,x (x)ϕj,x (x)dx + mϕiE (vt) ϕjE (vt) ,

E (vt) , [CE ]ij = 2mvϕiE (vt) ϕj,x

[KE ]ij = EI



0

L

E E ϕi,xx (x)ϕj,xx (x) dx

[fE ]j = −mg ϕjE (vt),

2.1.2 Formulations based on the TBT Unlike the EBT, in this theory, the deflections due to the transverse strains and rotary inertia are taken into account [17]. Hence, the deformation field of the beam includes a transverse displacement in the vertical direction as uzT = wT (x, t) and a rotation of the beam cross-section over the y-axis as θ T = θ T (x, t). Therefore, the longitudinal displacement could be expressed as uxT = −zθ T . In view of the displacement field, the strain relations are given by T − θ T , whereby the stress compoεxx = −zθ,xT and γxz = w,x nents σxx = Eεxx and σxz = Gγxz are accessible. By substituting the mentioned variables into Eq. (2) it results in the below energy equations,  Ek = 1 2

 L    2  2  E dx , ρA w˙ E + ρI w˙ ,x Ek = 1 2

(5)

[xE ]j = wjE (t).

 L 0

 2  2  dx , ρA w˙ T + ρI θ˙ T

 L      2  T − θ T 2 dx , EI θ,xT + Ks GA w,x Es = 1 2 (6) 0  L     T + v2 wT Ep = − m g− w ¨ T + 2vw˙ ,x wT (vt, t) δ(x − vt) dx , ,xx x=vt

0

where Ks is the shear correction factor that depends on the cross-section geometry of the beam. The displacement field and the rotation  field of the TB may be discretized as wT (x, t) = nj=1 ϕjT (x)wjT (t) and  θ T (x, t) = nj=1 ψjT (x)θjT (t), respectively. The shape func-

tions ϕjT (x) and ψjT (x) would be explored for different end conditions in Sect. 2.2. In addition, one can obtain the values of wjT (t) and θjT (t) by solving the below equation of motion,

MT (t) x¨ T (t) + CT (t) x˙ T (t) + KT (t) xT (t) = fT (t) , in which,  ww MT MT = MTθw  ww KT KT = KTθw

(7)

 ww   MTwθ Cwθ CT T = , C , T MTθθ Cθw Cθθ T T  w  T   wj (t) fT KTwθ , , fT = θ , [xT ]j = fT θjT (t) KTθθ (8)

wherein the non-zero submatrices are expressed as  L  ww  MT ij = ρA ϕiT (x) ϕjT (x) dx + mϕiT (vt) ϕjT (vt) , 0

E + mv2 ϕiE (vt)ϕj,xx (vt) ,



MTθθ



ij

=ρI



0

L

ψiT (x) ψjT (x) dx ,

 ww  T CT ij = 2mvϕiT (vt)ϕj,x (vt) ,

13



J Braz. Soc. Mech. Sci. Eng.

 ww  KT ij = Ks GA

L



0

 wθ  KT ij = −Ks GA 

KTθw



KTθθ



ij



ij

= Ks GA = Ks GA



T T T ϕi,x (x)ϕj,x (x) dx + mv2 ϕiT (vt) ϕj,xx (vt) ,



L 0

L

0

 L 0

MR (t) x¨ R (t) + CR (t) x˙ R (t) + KR (t) xR (t) = fR (t), T ϕi,x (x)ψjT (x)dx ,

T ψi,x (x)ϕjT (x) dx ,

ψiT (x) ψjT (x) dx + EI

 L 0

T (x) ψ T (x)dx , ψi,x j,x

(9)

2.1.3 Formulations based on the RBT

L 0

MR =



MRww MRθ w

KR =



KRww KRθ w

   ww Cwθ MRwθ CR R , , CR = CθRw CθRθ MRθ θ  (13)    w wjR (t) fR KRwθ , fR = θ , [xR ]j = , KRθ θ fR θjR (t)

in which,

 ww  MR ij = I0



L

ϕiR (x)ϕjR (x)dx

0

+α 2 I6

In the higher-order beam theories, the transverse strains are much more sensitive to the transverse stress than TBT. Although, according to [17], third-order theory is recommended to be used, serving sufficient accuracy. Assuming RB, the displacement field is characterized by uzR = wR (x, t) R ) wherein, α = 4/3h2 (h is and uxR = zθ R − αz3 (θ R + w,x the thickness of the beam section) and θ is the deflection angle of the beam cross-section about the y-axis. Finally, by R ) replacing the strain relations εxx = zθ,xR − αz3 (θ,xR + w,xx 2 R R and γxz = (1 − 3αz )(θ + w,x ), and their corresponding stress components into Eq. (2), the energy equations are obtained as follows:



L



0



MRwθ ij 

R R ϕi,x (x)ϕj,x (x)dx + mϕiR (vt) ϕjR (vt),

2

= (α I6 − αI4 )

 θw  MR ij = (α 2 I6 − αI4 )



L



L

0

R ϕi,x (x)ψjR (x)dx ,

R ψi,x (x)ϕjR (x) dx ,

0

 θθ  MR ij = (I2 − 2αI4 + α 2 I6 )



L

0

ψiR (x)ψjR (x) dx ,

 ww  R CR ij = 2mvϕiR (vt) ϕj,x (vt) ,

    2    2  2 R R R ˙R R 2 R R ˙ ˙ ˙ + I2 θ − 2αI4 θ θ + w˙ ,x + α I6 θ + w˙ ,x I0 w˙ dx ,

 L      2 2   2  R R R R 2 R R R R 1 Es = 2 J2 θ,x − 2αJ4 θ,x θ,x + w,xx + α J6 θ,x + w,xx + H θ + w,x dx , 0  L     R R m g− w ¨ R + 2vw˙ ,x + v2 w,xx wR (vt, t) δ(x − vt) dx , Ep = − 0

H = H0 − 6αH2 + 9α 2 H4 ,  Hi = Gzi dA ; i = 0, 2 , 4, A Ji = Ezi dA ; i = 2, 4 , 6, A  ρzi dA ; i = 0, 2, 4 , 6. Ii =

(10)

x=vt

in which

 ww  KR ij = H



L

(11)

 wR (x, t) = nj=1 ϕjR (x)wjR (t) Denoting and  θ R (x, t) = nj=1 ψjR (x)θjR (t), the shape functions ϕjR (x) and ψjR (x) are the same as the TB shape functions and hence, are presented in Sect. 2.2. Moreover, to evaluate the wjR (t)

 wθ  KR ij = H

R R ϕi,x (x)ϕj,x (x)dx

0

+α 2 J6

A

13

(12)

where

 w fT j = −mgϕjT (vt).

 Ek = 1 2

and θjR (t), it is required to solve the below equation of motion,





L 0

R R R ϕi,xx (x)ϕj,xx (x) dx + mv2 ϕiR (vt)ϕj,xx (vt) ,

L R ϕi,x (x)ψjR (x)dx

0

+(α 2 J6 − αJ4 )

 θw  KR ij = −H



0

L



L R R ϕi,xx (x)ψj,x (x)dx ,

0

R ψi,x (x)ϕjR (x)dx

−(α 2 J6 − αJ4 )



0

L R R ψi,xx (x)ϕj,x (x)dx ,

J Braz. Soc. Mech. Sci. Eng. Table 1  Shape functions ϕ1 (x) and ψ1 (x) for various boundary conditions

B.C.

SS

ϕ1 (x)

  x 4.508

ψ1 (x)

 θθ  KR ij = H ×





L

0 L

0

√ L



2 L

CC

L

−2

 x 3 L

+

   cos πx L

L

SC

  x 2 25.1 √

L



2 L

L



−2

 x 3 L

  sin 2πx L

+

 x 4  L

5.758 √ L



2 L

  x L

−3

 x 3 L

   cos πx 2L

+2

 x 4  L

It is important to note that, since the ψ1 (x) is not a polynomial function, it could be transformed into its Maclaurin series (containing the first six terms, in this study) to apply the Gram–Schmidt process. Accordingly, the same process employed for generating ϕn (x), is used to generate the other shape functions of ψn (x).

ψiR (x)ψjR (x)dx + (J2 − 2αJ4 + α 2 J6 ) R R ψi,x (x)ψj,x (x)dx ,

 w fR j = −mgϕjR (vt) .

 x 4 

(14)

2.3 Solving the differential equations in the time domain

2.2 Determining the shape functions As stated earlier, the assumed shape functions must be achieved to find the deformation field of various beams. Herein, it is performed through utilizing a procedure based on the COPs in the Rayleigh–Ritz method [14, 18]. In this connection, to specify the shape functions ϕj (x) and ψj (x), it should be conjectured a primary shape function as ϕ1 (x) and ψ1 (x), satisfying the natural and geometrical boundary conditions. In Table 1, ϕ1 (x) and ψ1 (x) are provided correspondingly as a polynomial function of order four and an appropriate trigonometric function for different beams under common end conditions; i.e., simple–simple (SS), clamped–clamped (CC) and simple–clamped (SC). Based on the COPs, obtaining a more precise solution of the problem is accomplished by increasing the number of shape functions. To achieve this aim, the Gram–Schmidt process is employed to evaluate the other required shape functions as follows [18]

ϕ2 (x) = (x − P2 )ϕ1 (x), ϕn (x) = (x − Pn )ϕn−1 (x) − Qn ϕn−2 (x) ; n = 3, 4, 5, . . .

(15)

wherein, 2 L  0 x ϕn−1 (x) W (x) dx ; n = 2, 3, 4, . . . Pn =  L  2 0 ϕn−1 (x) W (x) dx L x ϕn−1 (x) ϕn−2 (x) W (x) dx Qn = 0  L  ; n = 3, 4, 5, . . . 2 ϕ (x) W (x) dx n−2 0 (16)

in which, W (x) is the weight function which is taken as 1, in this study. It is noteworthy to say that the polynomial functions generated by this process are orthogonal; i.e.,

L 0

ϕa (x) ϕb (x) W (x) dx



= 0 if a �= b . �= 0 if a = b

(17)

Once the required shape functions are obtained, Eqs. (4), (7) and (12) can be solved to acquire the deformation amplitude of the beams. In this regard, to ease the solving process, the above-mentioned equations could be demonstrated in the state space as given below: [19]

˙ X(t) = A(t)X(t) + F(t) ,

(18)

where in,     0 I x(t) X(t) = , A(t) = , x˙ (t) k×1 −M−1 K −M−1 C k×k   0 F(t) = , (19) M−1 f k×1 in which, the system dynamic matrices of M, C, K and f have distinctly defined by the corresponding beam theory in Sect. 2.1. Additionally, k is considered as  2n if EBT k= . (20) 4n if TBT or RBT The approximate solution of the Eq. (18) could be found via the matrix-exponential method presented in [20]. Based on this approach, the deformation amplitude is procurable through the below equation,

¯ i )X(ti ) + F(t ¯ i ), X(ti+1 ) = A(t

(21)

where,

¯ i) ∼ A(t = eA(ti )�(ti ) ,   ¯ i ) − I A−1 (ti )F(ti ), ¯ i) ∼ F(t = A(t

(22)

in which, �(ti ) = ti+1 − ti is an assumed as time interval. It should be stated that the mentioned approach is very convenient to carry out through utilizing some kinds of the computer algebra system. The efficiency of this method in such problems has been proved, previously [21].

13



J Braz. Soc. Mech. Sci. Eng.

Table 2  Convergence test of the first five nondimensionalized frequencies of the EB for the CC and SS end conditions (h L = 0.01)

Table 3  Convergence test of the first five nondimensionalized frequencies of the TB for the CC and SS end conditions (h L = 0.01)

Table 4  Convergence test of the first five nondimensionalized frequencies of the RB for the CC and SS end conditions (h L = 0.01)

Table 5  Verification of the first five non-dimensionalized frequencies of the TB with the SS end condition for different  values of h L

a

  LS (Lee and Schultz) [22]

13

λi

Number of shape functions CC

SS

5

7

10

13

15

5

7

10

13

15

λ1 λ2 λ3 λ4

4.73 7.854 11.004 14.891

4.73 7.852 10.993 14.177

4.73 7.852 10.993 14.132

4.73 7.852 10.993 14.132

4.73 7.852 10.993 14.132

3.142 6.285 9.457 15.631

3.142 6.283 9.423 12.716

3.142 6.283 9.423 12.562

3.142 6.283 9.423 12.562

3.142 6.283 9.423 12.562

λ5

18.79

17.404

17.274

17.269

17.269

21.306

16.132

15.719

15.7

15.7

λi

Number of shape functions CC

SS

5

7

10

13

15

5

7

10

13

15

λ1 λ2 λ3 λ4

4.729 7.863 11.154 15.874

4.728 7.847 10.985 14.254

4.728 7.847 10.981 14.107

4.728 7.847 10.980 14.107

4.728 7.847 10.980 14.107

3.141 6.290 9.547 36.931

3.141 6.281 9.419 13.208

3.141 6.281 9.418 12.551

3.141 6.281 9.418 12.549

3.141 6.281 9.418 12.549

λ5

55.294

18.386

17.235

17.225

17.225

47.597

17.675

15.697

15.676

15.676

λi

Number of shape functions CC

SS

5

7

10

13

15

5

7

10

13

15

λ1 λ2 λ3 λ4

4.728 7.859 11.154 15.490

4.728 7.847 10.985 14.220

4.728 7.847 10.981 14.107

4.728 7.847 10.980 14.107

4.728 7.847 10.980 14.107

3.141 6.288 9.511 33.100

3.141 6.281 9.419 13.017

3.141 6.281 9.418 12.550

3.141 6.281 9.418 12.549

3.141 6.281 9.418 12.549

λ5

49.466

18.384

17.233

17.226

17.225

42.651

17.113

15.683

15.676

15.676

λi

h/L 0.005 a

LS

0.01 Present work

LS

0.05 Present work

LS

0.1 Present work

LS

Present work

λ1

3.142

3.142

3.141

3.141

3.135

3.138

3.116

3.128

λ2

6.283

6.283

6.281

6.282

6.231

6.256

6.091

6.118

λ3

9.423

9.423

9.418

9.421

9.255

9.332

8.841

8.912

λ4

12.562

12.562

12.549

12.558

12.181

12.287

11.343

11.450

λ5

15.700

15.700

15.675

15.690

14.993

15.141

13.613

13.749

J Braz. Soc. Mech. Sci. Eng. Table 6  Verification of the first five non-dimensionalized frequencies of the TB with the CC end condition for different  values of h L

0.005

0.01

LS

Present work

LS

Present work

LS

Present work

4.730

4.728

4.729

4.690

4.696

4.580

4.599

7.852

7.852

7.847

7.848

7.704

7.738

7.331

7.369

λ3

10.992

10.992

10.980

10.985

10.640

10.734

9.856

9.925

λ4

14.129

14.129

14.106

14.117

13.461

13.586

12.145

12.246

λ5

17.265

17.265

17.225

17.244

16.159

16.318

14.232

14.372

(b)

TBT EBT RBT

0.5

1.5 1.0 0.5 0.0

0

2

4

6

8

-0.5

10

0

2

4

t/T1

1.0

1.0

0.0

0.0

0

1

2

3

4

-2.0

5

0

1

2

3

4

5

3

4

5

t/T1

2.0

(f) 2.0

1.0

1.0

0.0

0.0

WN

WN

10

-1.0

t/T1

-1.0 -2.0

8

2.0

WN

WN

(d)

-1.0

(e)

6

t/T1

2.0

-2.0

Present work

4.730

0.0

(c)

LS

λ1

1.5

-0.5

0.1

λ2

1.0

WN

0.05

WN

(a)

h/L

λi

-1.0 0

1

2

3

4

5

t/T1

-2.0

0

1

2

t/T1

Fig. 1  Time history of the normalized deflection of an SS beam due to a moving mass with M N = 0.15: a V N = 0.1, S = 20; b V N = 0.1, S = 60; c V N = 0.5, S = 20; d V N = 0.5, S = 60; e V N = 1.0, S = 20; f V N = 1.0, S = 60

3 Numerical studies 3.1 Estimating the convergence rate and verification of the non‑dimensionalized frequencies To check the proficiency of the proposed method in predicting the natural frequencies of the structure, the achieved

results are compared with those of Lee and Schultz [22]. In this regard, consider an SS and a CC single-span beam with various theories. The material and geometrical properties of the assumed beams are taken as E = 2.1 × 1011 N m−2, G = 8.0769 × 1010 N m−2, ρ = 7800 kg m−3 and a rectangular cross-section with Ks = 0.833. Moreover, the nondimensionalized frequency related to the ith shape function

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J Braz. Soc. Mech. Sci. Eng.

1.5 1.0 0.5

WN

(b)

TBT EBT RBT

1.5 1.0 0.5

WN

(a)

0.0 -0.5

0.0 0

5

10

15

-0.5

20

0

5

10

2.0

1.5

1.5

1.0

1.0

WN

WN

(d)

2.0

0.5

0.5

0.0 -0.5

0.0 0

1

2

3

4

-0.5

5

0

1

2

t/T1 3.0

4

5

3

4

5

3.0

(f)

1.5

1.5

0.0

0.0

-1.5 -3.0

3

t/T1

WN

WN

(e)

20

t/T1

t/T1

(c)

15

-1.5 0

1

2

3

4

5

t/T1

-3.0

0

1

2

t/T1

Fig. 2  Time history of the normalized deflection of a CC beam due to a moving mass with M N = 0.15: a V N = 0.1, S = 20; b V N = 0.1, S = 60; c V N = 0.5, S = 20; d V N = 0.5, S = 60; e V N = 1.0, S = 20; f V N = 1.0, S = 60

 is defined as i = L (ωi2 ρA EI)1/4 wherein, ωi is the ith natural frequency of the beam corresponding to the i. A preliminary analysis is carried out to estimate the convergence rate of the i for the EB, TB and RB with thickness-tolength ratio h L = 0.01, and the results are given in Tables 2, 3 and 4, respectively. The number of shape functions varies from 5 to 15. Obviously, these results show the fast convergence rate of the proposed method that requires less than13 shape functions for the first five non-dimensionalized frequencies to converge to the first three significant digits.  Besides, the results of the TB for different values of h L, obtained via employing 13 COPs, are verified with those issued from the pseudo-spectral method [22] in Tables 5 and 6. As it could be seen, the computed results are in a reasonable agreement with those achieved through using 35 terms  of Chebyshev polynomial [22]. Hence, in all cases of h L, the predicted values of the first five non-dimensionalized frequencies correspond to those of Lee and Schultz

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[22] with the relative error lower than 1 %; in both SS and CC end conditions. 3.2 Assessing the dynamic responses of the beams subjected to the moving mass Assessing the dynamic behavior of the single-span EBs, TBs and RBs caused by a moving mass excitation, it would be proper to introduce some normalized parameters. For purpose, the normalized parameter  this st is considered wherein, W dyn and W st W N = W dyn Wmax max denote correspondingly the dynamic deflection of the beam due to the moving mass and the maximum static deflection of an EB subjected to a point  load, mg. An  appropri3 48EI , MgL 3 192EI and ate static analysis gives MgL  st related to SS, CC and SC 0.0098124MgL 3 EI for Wmax boundary conditions, respectively. Furthermore, the normalized slenderness, velocity and mass parameters are

J Braz. Soc. Mech. Sci. Eng.

(a) 1.5 0.5

1.0

WN

1.0

WN

(b) 1.5

TBT EBT RBT

0.0 -0.5

0.5 0.0

0

2

4

6

8

10

12

-0.5

14

0

2

4

6

t/T1

(c)

(d)

WN

0.0 -1.5

14

3.0

0.0 -1.5

0

1

2

3

4

-3.0

5

0

1

2

3

4

5

3

4

5

t/T1

t/T1

(e) 3.0

(f)

3.0

1.5

1.5

0.0

0.0

WN

WN

12

1.5

WN

1.5

-1.5

-1.5 -3.0

10

t/T1

3.0

-3.0

8

0

1

2

t/T1

3

4

5

-3.0

0

1

2

t/T1

Fig. 3  Time history of the normalized deflection of an SC beam due to a moving mass with M N = 0.15: a V N = 0.1, S = 20; b V N = 0.1, S = 60; c V N = 0.5, S = 20; d V N = 0.5, S = 60; e V N = 1.0, S = 20; f V N = 1.0, S = 60

  assumed to be correspondinglyS = L  r, M N = m ρAL   and V N = v v′ where v′ = π EI ρA L and r is the gyration radius of the beam cross-section about its neutral axis. The material and geometrical properties of the beams, including E, G, ρ and Ks, are those in the previous example. √ The other parameters are taken as L = 10m, h = r 12 and b = 0.1m in which, b is the width of the beam with a rectangular cross-section. In Figs. 1, 2 and 3, normalized deflection time history of the SS, CC and SC beams under the moving mass excitation with M N = 0.15 and V N = 0.1, 0.5 and 1.0 for different values of S are provided, respectively. It should be noted, the time axis in time history diagrams is normalized by the first period of the EB, T1. As it is observed, increasing the amount of S leads the results of various theories to the same values regardless of the moving mass velocity, expectedly (see Figs. 1, 2 and 3 for S = 60).

In addition, the normalized maximum response of the SS, CC and SC beams for the case of V N = 0.5 are presented in Tables 7, 8 and 9 with those achieved by other researchers [23], respectively. As it is clear, the results obtained through this study correlate well with those from the Reproducing Kernel Particle Method (RKPM) [23]. In the case of the SS and SC beams, the results are very close to those of the RKPM (the differences between the results are correspondingly less than 0.5 and 1 %). So that, by employing the limited numbers of COPs (only three shape functions), much accurate results for all cases (S = 20, 40 and 60) are achieved. These results are slightly different in the CC beam. As it could be observed, the results of the RKPM through the RBT with the CC fixity condition seem to be unreasonable in comparison with the SS and SC boundary conditions for lower values of the beam slenderness (see Tables 7, 8 and 9 for S = 20 or 40). However, the

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J Braz. Soc. Mech. Sci. Eng.

Table 7  Verification of the normalized maximum deflection of an SS beam for various values of the base beam slenderness and different beam theories (V N = 0.5)  EBT TBT RBT S=L r Present work RKPM Present work RKPM Present work RKPMa 20 40

1.796 1.791

1.790 1.791

1.940 1.829

1.930 1.828

1.940 1.830

1.930 1.828

60

1.791

1.791

1.820

1.820

1.820

1.820

a

  Reproducing Kernel Particle Method (RKPM), presented in [23]

Table 8  Verification of the normalized maximum deflection of a CC beam for various values of the base beam slenderness and different beam theories (V N = 0.5)  EBT TBT RBT S=L r Present work RKPM Present work RKPM Present work RKPM 20 40

1.256 1.253

1.240 1.235

1.801 1.383

1.800 1.380

1.809 1.389

2.150 1.510

60

1.251

1.220

1.328

1.320

1.334

1.350

Table 9  Verification of the normalized maximum deflection of an SC beam for various values of the base beam slenderness and different beam theories (V N = 0.5)  EBT TBT RBT S=L r Present work RKPM Present work RKPM Present work RKPM 20 40

1.593 1.590

1.590 1.590

2.019 1.763

2.000 1.750

2.203 1.825

2.180 1.810

60

1.590

1.590

1.672

1.670

1.706

1.700

differences between the pertinent CC results dealing with the EBT and TBT are less than 3 %.

4 Conclusions In this paper, a handy semi-analytical method aimed at studying the vibrations and dynamics of thin/thick beams was proposed. To this end, the COPs and trigonometric shape functions based on the end conditions of the beams were utilized. The Rayleigh–Ritz method was then used to obtain the natural frequencies and dynamic response of various beams under the excitation of a moving mass. It was seen that the proposed model makes a satisfactory compromise between the analytical and numerical approaches, by providing an accurate yet simple solution. Additionally, it was revealed that the obtained results for the non-dimensionalized frequencies of the beam in all three theories provide a good convergence rate and a high accuracy in comparison with those obtained via the pseudo-spectral method. Furthermore, the estimated maximum dynamic response of the beams excited by a moving

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mass was in a close agreement with those issued from the RKPM. Moreover, it was shown that a high level of accuracy could be gained by utilizing the low numbers of shape functions. Briefly, it could be pointed out that the proposed method is a simple and precise one to evaluate the dynamic behavior of the beams with various theories under different boundary conditions and loadings. References 1. Frýba L (1999) Vibration of solids and structures under moving loads, 3rd edn. Thomas Telford, London 2. Olsson M (1991) On the fundamental moving load problem. J Sound Vib 145:299–307 3. Ouyang H (2011) Moving-load dynamic problems: A tutorial (with a brief overview). Mech Syst Signal Process 25:2039–2060 4. Eftekhari SA, Jafari AA (2014) A variational formulation for vibration problem of beams in contact with a bounded compressible fluid and subjected to a traveling mass. Arab J Sci Eng 39:5153–5170 5. Akin JE, Mofid M (1989) Numerical solution for response of beams with moving mass. J Struct Eng 115:120–131 6. Yang YB, Yau JD, Hsu LC (1997) Vibration of simple beams due to trains moving at high speeds. Eng Struct 19:936–944

J Braz. Soc. Mech. Sci. Eng. 7. Rao GV (2000) Linear dynamics of an elastic beam under moving loads. ASME J Vib Acoust 122:281–289 8. Yavari A, Nouri M, Mofid M (2002) Discrete element analysis of dynamic response of Timoshenko beams under moving mass. Adv Eng Softw 33:143–153 9. Nikkhoo A, Kananipour H (2014) Numerical solution for dynamic analysis of semicircular curved beams acted upon by moving loads. Proc IMechE Part C J Mech Eng Sci. doi:10.1177/0954406213518908 10. Kargarnovin MH, Ahmadian MT, Talookolaei RAJ (2012) Dynamics of a delaminated Timoshenko beam subjected to a moving oscillatory mass. Mech Based Des Struct Mach Int J 40:218–240 11. Ghannadiasl A, Mofid M (2014) Dynamic Green Function for response of Timoshenko beam with arbitrary boundary conditions. Mech Based Des Struct Mach Int J 42:97–110 12. Eftekhar Azam S, Mofid M, Afghani Khoraskani R (2013) Dynamic response of Timoshenko beam under moving mass. Scientia Iranica, Transaction A: Civil Engineering. doi:10.1016/j.scient.2012.11.003 13. Ahmadi M, Nikkhoo A (2014) Utilization of characteristic polynomials in vibration analysis of non-uniform beams under a moving mass excitation. Appl Math Modell 38:2130–2140 14. Bhat R (1986) Transverse vibrations of a rotating uniform cantilever beam with tip mass as predicted by using beam characteristic

orthogonal polynomials in the Rayleigh-Ritz method. J Sound Vib 105:199–210 15. Leissa AW, Qatu MS (2011) Vibration of continuous systems. McGraw Hill, New York 16. Ebrahimzadeh Hassanabadi M, Nikkhoo A, Vaseghi Amiri J, Mehri B (2013) A new orthonormal polynomial series expansion method in vibration analysis of thin beams with non-uniform thickness. Appl Math Modell 37:8543–8556 17. Wang C, Reddy JN, Lee K (2000) Shear deformable beams and plates: relationships with classical solutions. Elsevier BV 18. Chakraverty S (2010) Vibration of plates. CRC press 19. Brogan WL (1991) Modern control theory. Prentice-Hall, New Jersey 20. Nikkhoo A, Rofooei F, Shadnam M (2007) Dynamic behavior and modal control of beams under moving mass. J Sound Vib 306:712–724 21. Nikkhoo A (2014) Investigating the behavior of smart thin beams with piezoelectric actuators under dynamic loads. Mech Syst Signal Process 45:513–530 22. Lee J, Schultz WW (2004) Eigenvalue analysis of Timoshenko beams and axisymmetric Mindlin plates by the pseudospectral method. J Sound Vib 269:609–621 23. Kiani K, Nikkhoo A, Mehri B (2009) Prediction capabilities of classical and shear deformable beam models excited by a moving mass. J Sound Vib 320:632–648

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