Vibration of damped uniform beams with general end conditions under moving loads

Engineering Structures 126 (2016) 40–52

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Vibration of damped uniform beams with general end conditions under moving loads C. Svedholm a,c,⇑, A. Zangeneh a,c, C. Pacoste a,c, S. François b, R. Karoumi a a

Division of Structural Engineering and Bridges, KTH Royal Institute of Technology, 100 44 Stockholm, Sweden Department of Civil Engineering, K.U.Leuven, Kasteelpark Arenberg 40, B-3001 Leuven, Belgium c ELU Konsult AB, Stockholm, Sweden b

a r t i c l e

i n f o

Article history: Received 28 January 2016 Revised 16 June 2016 Accepted 22 July 2016

Keywords: Moving load Non-proportional damping Bernoulli–Euler beam Complex mode

a b s t r a c t In this paper, an analytical solution for evaluating the dynamic behaviour of a non-proportionally damped Bernoulli–Euler beam under a moving load is derived. The novelty of this paper, when compared with other publications along this line of work is that general boundary conditions are assumed throughout the derivation. Proper orthogonality conditions are then derived and a closed form solution for the dynamical response for a given eigenmode is developed. Based on this, the dynamical response of the system to any load can be determined by mode superposition. The proposed method is particularly useful for studying various types of damping mechanisms in bridges, such as soil–structure interaction, external dampers, and material damping. Several numerical examples are presented to validate the proposed method and provide insight into the problem of non-proportionally damped systems. The numerical examples also allow for some interesting observations concerning the behaviour of modal damping for closely spaced modes (with respect to undamped natural frequencies). Ó 2016 Elsevier Ltd. All rights reserved.

1. Introduction The problem of vibration of a Bernoulli–Euler beam under a moving load has already been intensively investigated for many years. In fact, the first work on this subject was reported as early as 1849, by Willis and Stokes, after the collapse of the Chester Railway Bridge [1]. Based on the first analytical solution, researchers have continued to further study the problem in more detail by considering various support conditions, continuity, damping and the presence of vehicles [2,3]. An extensive summary of the developments in this field over the last 150 years can be found in the books by Fryba [4,5]. The one thing that all of these solutions have in common is that the equation of motion is decoupled by assuming that the damping is either negligible or proportional to the mass or stiffness. This ensures synchronized motion (standing waves) and thus orthogonal eigenmodes. In practice, such a simplification is often justified by the uncertainty associated with the damping mechanisms. In spite of this, this simplification is not correct and leads to a loss of accuracy for problems involving travelling waves. This occurs, for example, for structures with discrete dampers or when radiation damping in the soil is present. ⇑ Corresponding author at: Division of Structural Engineering and Bridges, KTH Royal Institute of Technology, 100 44 Stockholm, Sweden. E-mail address: [email protected] (C. Svedholm). http://dx.doi.org/10.1016/j.engstruct.2016.07.037 0141-0296/Ó 2016 Elsevier Ltd. All rights reserved.

The primary purpose of this paper is therefore to study nonproportionally damped systems and provide an analytical solution for these cases. In 1958, Foss [6] published his classic work on the uncoupling of the equation of motion of a damped linear system with lumped parameters. Perhaps the most important contribution of Foss’s work is his proof of orthogonality for non-proportionally damped systems. Over the years, there have been several important developments of the original method. A few years after Foss’s paper, other researchers applied the same theories to determine the frequency domain equation/ equilibrium equation of continuous systems [7–13]. The paper by Al-Jumaily and Faulkner [7] was the first to analytically formulate the equilibrium equation for a non-proportionally damped beam in the frequency domain. A few years later, Zarek and Gibbs [8] obtained the natural frequencies and mode shapes for a damped beam with general end conditions. In their solution, they used a numerical procedure to solve the eigenvalue equation. Another important paper was published in 1990 by Prater and Singh [9], who suggested a step-by-step procedure to solve the complex eigenvalue equation. A slightly different approach to obtain the complex eigenvalues was suggested by Chang et al. [10], who obtained a closed-form solution after applying a Laplace transform to the equation of motion to determine the eigenvalues of a simply

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C. Svedholm et al. / Engineering Structures 126 (2016) 40–52

Nomenclature c cs cw ch E I I0 J kw kh L m

external damping per unit length of beam (N s/m2) internal damping per unit length of beam (N s/m2) damping coefficient of vertical viscous damper (N s/m) damping coefficient of rotational viscous damper (Nm s/ rad) Young’s modulus of elasticity (Pa) area moment of inertia (m4) mass moment of inertia (kg m2) Impulse (N s) vertical spring stiffness (N/m) rotational spring stiffness (N m/rad) length of beam (m) mass per unit length of beam (kg/m)

supported beam carrying a general number of miscellaneous attachments. It was not until 1994 that Hull [14] managed to derive an analytical solution for the vibration of a longitudinal bar with a viscous boundary condition subjected to point loading. Note that all previous work that applied an analytical methodology only treated natural frequencies and mode shapes. The trick that allowed Hull to solve the equation of motion was to redefine the interval of the spatial coordinate, and then apply the reflection property of integrals. This work was followed by that of Oliveto et al. [15], who studied a vibrating beam with two rotational viscous dampers at its ends. These authors established a set of orthogonality conditions in order to decouple the equation of motion. However, the resulting equations were derived under the assumption that translations were prevented at the supports. Greco and Santini [16] extended the work of Oliveto et al. to also consider moving loads. By 1998, Gürgöze and co-workers had also begun to publish in this field. Their first two papers focused on the frequency equation for a damped cantilever beam carrying a mass on its tip [17,18]. This was then followed by two papers on the dynamic response of non-proportionally damped systems [19,20]. In the latter references, Gürgöze and Erol followed the procedure developed by Oliveto to obtain a set of equations to decouple the equation of motion. Around the same time, Sorrentino et al. [21] obtained the complex modes by using a state-space form, applied in conjunction with a transfer matrix technique. Furthermore, by using the inner product, Sorrentino et al. [22,23] managed to obtain a set of orthogonality conditions. A drawback of the results presented in those papers is that numerical results were provided only for a cantilever beam carrying a mass on its tip, even though the authors claimed that their method was applicable to any dynamical system. In addition, [24–29] should be mentioned, since these publications also treat this problem. The novelty of this paper is that it derives a set of equations to solve the continuous dynamical system shown in Fig. 1. Of course this problem could be solved very efficiently with a commercial finite element package using direct time integration techniques; however, an analytical solution presents the following advantages: 1. Physical insight into the dynamical behaviour of these systems. For instance the relevance of closely spaced eigenmodes on the damping is outlined in the numerical examples. 2. Accurate results for high frequencies. 3. A computationally efficient method suitable for Monte Carlo simulations [2] or real-time control systems. The present paper can be viewed as a generalization of the work of Greco and Santini [16] with respect to the boundary conditions,

M n N P q t v w x k K /

lumped mass (kg) number of modes (–) number of axles (–) concentrated load (N) generalized coordinate time (s) speed of moving load (m/s) transverse displacement (m) spatial coordinate (m) separation constant eigenvalue spatial eigenfunction

w kθ

cθ

M, I 0

cw

kw

v P

x EI, m, c, cs I L

Fig. 1. Structural system for a beam with general end conditions.

which requires a new set of orthogonality conditions. Moreover this study also includes modes with real valued eigenvalues, a problem which has not been addressed by Greco and Santini. 2. Outline of the paper This study examines the Bernoulli–Euler beam shown in Fig. 1. The partial differential equation describing the transverse displacement wðx; tÞ, which is a function of both the spatial coordinate x and the time t, is:

m


 @2w @w @ 4 @w þ þ EIw ¼ Pdðx  v tÞ þ c c I s @t @x4 @t @t 2

ð1Þ

Here, the bending stiffness EI and mass per unit length m are assumed constant over the beam length L. Energy dissipation is accounted for by continuous and discrete dampers, where the former include both external damping mechanisms, represented by c, and internal damping mechanisms, represented by cs . In Eq. (1), the external damping gives the familiar term cð@w=@tÞ, while the internal damping (material damping), which gives a term cs Ið@ 5 w=@t @x4 Þ, is caused by an internal force which reacts against the variation of the curvature with time. Both ends of the beam are supported symmetrically by a translational and a rotational support with stiffnesses kw and kh and discrete damping cw and ch , respectively. To decouple the equation of motion into an infinite set of modal coordinates qðtÞ and mode shapes /ðxÞ, the displacement is written by making use of separation of variables:

wðx; tÞ ¼ qðtÞ/ðxÞ

ð2Þ

Eq. (2) is then substituted into Eq. (1) and P is set to zero to obtain the free vibration. The equation is then rearranged by grouping the functions depending on t on the left-hand side and those depending on x on the right-hand side: 0000

€ðtÞ þ ðc=mÞqðtÞ _ q / ðxÞ ¼ _ ðcs I=mÞqðtÞ /ðxÞ þ ðEI=mÞqðtÞ

ð3Þ

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C. Svedholm et al. / Engineering Structures 126 (2016) 40–52

Eq. (3) is now reduced to a pair of ordinary differential equations by setting each side of the equation equal to a constant, i.e. k4 : 0000

/ ðxÞ  k4 /ðxÞ ¼ 0
 c cI EI _ þ k4 s qðtÞ _ €ðtÞ þ qðtÞ þ qðtÞ ¼ 0 q m m m

Eq. (8). Now, substituting Eq. (5a) into Eq. (7) and arranging the four equations into matrix form gives:

ð4bÞ

gþ1 g1 gi 6 ðg  1ÞeLk ðg þ 1ÞeLk ðg þ iÞeLki 6

CðkÞJ ¼ 6 4

n þ 1 nþ1 ðn  1ÞeLk ðn þ 1ÞeLk 2 3 2 3 0 A1 6A 7 607 6 27 6 7 6 7¼6 7 4 A3 5 4 0 5

The solutions to the above equations are assumed to be of the form:

/ðxÞ ¼ A1 ekx þ A2 ekx þ A3 eikx þ A4 eikx

ð5aÞ

qðtÞ ¼ BeKt

ð5bÞ

where the eigenvalue K, the shape coefficients A1 –A4 and the coefficient B are determined from the initial and boundary conditions. The separation variable k can be obtained after substituting Eq. (5b) into Eq. (4b):

k4 ¼ 

mK2 þ cK EI þ cs IK

ð6Þ

3. Natural frequencies and mode shapes The classical procedure to obtain the complex natural frequencies and mode shapes is to impose boundary conditions on Eq. (2). It is interesting to note that for proportionally damped systems, all derivatives of wðx; tÞ are taken with respect to x and therefore, a classical modal analysis is restricted to finding the (spatial) mode shapes. This is not the case, however, for the structural system shown in Fig. 1, due to the damper, lumped mass and internal damping; see Fig. 2. The forces in the above figure represent the boundary conditions. From consideration of equilibrium of the vertical and rotational forces acting on the free body in Fig. 2, a set of equations is obtained:

D/000 ð0Þ ¼ Z w ðKÞ/ð0Þ

ð7aÞ

D/00 ð0Þ ¼ Z h ðKÞ/0 ð0Þ D/000 ðLÞ ¼ Z w ðKÞ/ðLÞ

ð7bÞ ð7cÞ

D/00 ðLÞ ¼ Z h ðKÞ/0 ðLÞ

ð7dÞ

where the term D ¼ EI þ Kcs I accounts for the contribution of internal damping to the bending stiffness. In Eq. (7), the constraints imposed by the supports are described by impedances Z, where Z w and Z h are the impedances to shear and bending, respectively:

Z w ð KÞ ¼ M K2 þ c w K þ k w

ð8aÞ

Z h ðKÞ ¼ I0 K2 þ ch K þ kh

ð8bÞ

Note that the impedances can easily be modified to consider more complex boundary conditions, with any desired frequency dependence; however, the orthogonality derived later is still limited to

kθ

D

7 5 ni  1 Lki ðni þ 1Þe ð9Þ

where g ¼ Z w =k3 D and n ¼ Z h =kD. The eigenvalues are calculated by solving DetðCðkÞÞ ¼ 0. Due to the nature of complex eigenvalues, it is sufficient to search in the domain RðKÞ 6 0 and IðKÞ P 0. One should, however, be aware of the fact that there exists a complex conjugate K for all IðKÞ > 0. The Nelder–Mead simplex direct search method as implemented in MATLAB was used for the eigenvalue extraction. The sequence of starting points is obtained by a bisection method. The eigenvector corresponding to the eigenvalue Kn for the nth mode is obtained by applying singular value decomposition to the matrix CðKn Þ. Once the eigenvalues have been estimated, it is possible to calculate the undamped angular natural frequencies xn and the modal damping ratios fn :

xn ¼ jKn j fn ¼

ð10aÞ

RðKn Þ

ð10bÞ

xn

The undamped angular natural frequency is given by the familiar equation xn ¼ 2pf n , where f n is the undamped natural frequency. Moreover, if one instead is interested in the damped angular natural frequency this can be obtained as the absolute value of the imaginary part of the eigenvalue (xd;n ¼ jIðKn Þj). Eigenvalues for non-proportionally damped systems are either real-valued or complex-valued depending on the nature of the problem [30]. If an eigenvalue is complex, then the mode is underdamped and the free response is a decaying oscillatory motion. However, if an eigenvalue is real, then the response is a pure exponential decay. Based on the above definition of undamped and damped natural frequency, it is interesting to note that for a mode with a real eigenvalue the damped natural frequency is always zero but the undamped natural frequency can be different from zero. 4. Orthogonality conditions In order to solve the equation of motion analytically, the response can be expressed as a linear combination of eigenmodes. However, to be able to do this, one needs to show that the

kθ

∂ 2 w(0, t ) ∂x 2

∂ 3 w(L, t ) ∂ 2 w(L, t ) ∂w(L, t ) + cθ + I0 ∂x∂t 2 ∂x∂t ∂x

D

∂ 2 w(L, t ) ∂x 2

∂ 3 w(0, t ) ∂x 3

k w w(0, t ) + cw

ni  1 ðni  1ÞeLki

gþi ðg  iÞeLki 7 7

0

A4

∂ 3 w(0, t ) ∂ 2 w(0, t ) ∂w(0, t ) + cθ + I0 ∂x∂t 2 ∂x∂t ∂x

D

3

2

ð4aÞ

D

∂ 2 w(0, t ) ∂w(0, t ) + M ∂t 2 ∂t

k w w(L, t ) + cw Fig. 2. Free-body diagram (boundary conditions).

∂ 3 w(L, t ) ∂x 3

∂ 2 w(L, t ) ∂w(L, t ) + M ∂t 2 ∂t

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C. Svedholm et al. / Engineering Structures 126 (2016) 40–52

eigenmodes are mutually orthogonal. The following derivation is a generalization of the one given in [15]. The governing equation for the nth mode shape is given in Eq. (4a): 0000

/n ðxÞ  k4n /n ðxÞ ¼ 0

ð11Þ

In Eqs. (18) and (19), Dn ¼ EI þ Kn cs I and Dm ¼ EI þ Km cs I, where Kn and Km are the corresponding eigenvalues for mode n and m. 5. Complex mode superposition method

Multiplying by an arbitrary normal mode /m ðxÞ and integrating over L gives:

5.1. Modal impulse response functions

Z

The response is expanded in terms of a linear combination of the time-dependent generalized coordinates qn ðtÞ and the normal modes /n ðxÞ using the expansion theorem:

L

0

Z

0000

/n ðxÞ/m ðxÞ dx  k4n

L

/n ðxÞ/m ðxÞ dx ¼ 0

ð12Þ

0

Next, making use of integration by parts (twice) of the left term on 0000

the left-hand side of the equation to reduce the order of /n ðxÞ/m ðxÞ to /00n ðxÞ/00m ðxÞ, gives:



L /000 n ðxÞ/m ðxÞ 0 Z  k4n





L /00n ðxÞ/0m ðxÞ 0

Z

L

þ 0

/00n ðxÞ/00m ðxÞ

dx

L

/n ðxÞ/m ðxÞ dx ¼ 0

ð13Þ

0

Introducing the boundary conditions, i.e. Eq. (7) into Eq. (13), and substituting k4n into Eq. (6) results in:

Z

L

0

/00n ðxÞ/00m ðxÞ dx þ

2 n

mK þ cKn Dn

Z

/n ðxÞ/m ðxÞ dx þ

UZ w;n þ U Z h;n

0

Dn

¼0

ð14Þ where the variables U and U0 are defined as:

U ¼ /n ðLÞ/m ðLÞ þ /n ð0Þ/m ð0Þ U0 ¼ /0n ðLÞ/0m ðLÞ þ /0n ð0Þ/0m ð0Þ

ð15aÞ ð15bÞ

Eq. (14) is then reformulated by interchanging the n’s and the m’s and subtracted from its original form, which gives:

0

þ cs IðM Km Kn  kw Þ þ U0 ½EIðch þ I0 Km þ I0 Kn Þ þ cs IðI0 Km Kn  kh Þg ¼ 0 ð16Þ It then follows from Eq. (16) that for all Km – Kn ,

Z

L

/n ðxÞ/m ðxÞ dx 0

þ U0 ½EIðch þ I0 Km þ I0 Kn Þ þ cs IðI0 Km Kn  kh Þ ¼ 0

ð17Þ

which corresponds to the first orthogonality condition. The second orthogonality condition is obtained by first multiplying Eq. (14) by Km and then subtracting the equation obtained by interchanging the n’s and the m’s:

D m D n ð Km  Kn Þ 0

Z

L

/00n ðxÞ/00m ðxÞ dx  Km Kn ðKm  Kn ÞðEIm  cs IcÞ

L

0

þ cs Iðkw Km þ kw Kn þ cw Km Kn Þ þ U0 ½EIðkh  I0 Km Kn Þ þ cs Iðkh Km þ kh Kn þ ch Km Kn Þg ¼ 0

ð18Þ

which, for Kn – Km , leads to:

Z

Dm Dn 0

L

_ €ðtÞ ¼ K2 qðtÞ. The The latter equation implies that qðtÞ ¼ KqðtÞ and q generalized coordinates take the form of Eq. (5b) (free vibration) due to the impulsive nature of the excitation. The derivation starts by expressing the equation of motion (Eq. (1)) in terms of qn ðtÞ and /n ðxÞ: 1 h X

0000

ð21Þ

Note that the excitation Pdðx  v tÞ has been replaced by Jdðx  xv ÞdðtÞ to simulate an impulse J at the location xv . Then, multiplying Eq. (21) by /m and integrating over L gives:

 Z L Z L Z L 1  X 0000 €n q m/n /m dx þ q_ n c/n /m dx þ ðq_ n cs I þ qn EIÞ /n /m dx n¼1

Z

0

0

0

L

¼ 0

Jdðx  xv ÞdðtÞ/m dx

ð22Þ

"
# Z L Z 1 X c Dn L 0000 €n m þ q /n /m dx þ 2 /n /m dx Kn 0 Kn 0 n¼1 Z L Jdðx  xv ÞdðtÞ/m dx ¼

ð23Þ

making it possible to separate the time and space variables on the left-hand side. The equation is then modified further by applying integration by parts (twice) and introducing the boundary conditions into the equation:

"
Z L
# Z L 1 X c Dn UZ w;n þ U0 Z h;n 00 00 €n m þ q / / dx þ 2 þ /n /m dx Kn 0 n m Dn Kn 0 n¼1 Z L Jdðx  xv ÞdðtÞ/m dx ð24Þ ¼ 0

Using the second orthogonality relation for free vibration (i.e. Eq. (19)) gives:

(

Z

½EIc þ EImðKm þ Kn Þ þ cs ImKm Kn 

L

/n /m dx 0

þ U½EIðcw þ M Km þ M Kn Þ þ cs IðMKm Kn  kw Þ )

¼ L

0

/n ðxÞ/m ðxÞ dx  U½EIðkw  M Km Kn Þ 0

þ cs Iðkw Km þ kw Kn þ cw Km Kn Þ  U0 ½EIðkh  I0 Km Kn Þ þ cs Iðkh Km þ kh Kn þ ch Km Kn Þ

¼ Jdðx  xv ÞdðtÞ

n¼1

Z

¼ Km Kn ðEIm  cs IcÞ

i

þ U0 ½EIðch þ I0 Km þ I0 Kn Þ þ cs IðI0 Km Kn  kh Þ

/00n ðxÞ/00m ðxÞ dx Z

0000

€n m/n þ q_ n c/n þ q_ n cs I/n þ qn EI/n q

1 X €n q K Dm n n¼1

/n ðxÞ/m ðxÞ dx þ ðKm  Kn ÞfU½EIðkw  MKm Kn Þ



ð20bÞ

0

þ U½EIðcw þ M Km þ M Kn Þ þ cs IðM Km Kn  kw Þ

Z

qn ðtÞ ¼ Bn eKn t

After making use of the relationships between the principal coordinates and their derivatives, Eq. (22) becomes:

ðKn  Km Þ½EIc þ EImðKm þ Kn Þ þ cs ImKm Kn  Z L  /n ðxÞ/m ðxÞ dx þ ðKn  Km ÞfU½EIðcw þ M Km þ M Kn Þ

½EIc þ EImðKm þ Kn Þ þ cs ImKm Kn 

ð20aÞ

n¼1

0

L

1 X qn ðtÞ/n ðxÞ

wðx; tÞ ¼

ð19Þ

L

Jdðx  xv ÞdðtÞ/m dx

ð25Þ

Notice that the braced terms on the left-hand side of Eq. (25) exactly match the first orthogonality relation, Eq. (17). It is therefore obvious that the left-hand side of the equation is equal to zero for all Kn – Km . Then for n ¼ m:

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C. Svedholm et al. / Engineering Structures 126 (2016) 40–52

€n q Kn Dn

( h

EIc þ 2EImKn þ cs ImK2n

iZ

L 0

h

v

/2n ðxÞ dx

 i þ U EIðcw þ 2M Kn Þ þ cs I MK2n  kw ) h  i 0 2 þ U EIðch þ 2I0 Kn Þ þ cs I I0 Kn  kh Z

Jdðx  xv ÞdðtÞ/n dx

0

EI, m L

L

¼

P

cθ

ð26Þ

This uncouples the equation of motion. Finally, the time response is obtained by integrating Eq. (26) over the time interval 0 to 0þ and þ

making use of Eq. (20b) (recall that eK0 ¼ 1):

( iZ L Bn h EIc þ 2EImKn þ cs ImK2n /2n ðxÞ dx Dn 0 h  i þ U EIðcw þ 2M Kn Þ þ cs I MK2n  kw ) h  i 0 2 ¼ J/n ðxv Þ þ U EIðch þ 2I0 Kn Þ þ cs I I0 Kn  kh

ð27Þ

Fig. 4. Structural system for a simply supported beam with two rotational viscous dampers.

where xv ¼ v s moves the impulse over the beam and the integral sum the responses of all impulses to time t  . The time is replaced by t  s to account for the fact that the impulse response function is derived for an impulse at t ¼ 0 and now it varies as the load move over the beam. Based on the principle of superposition, it is now possible to sum the responses to a series of moving loads as follows (see Fig. 3b):

wtrain ¼

N X w ðt  ti Þ i¼1

Using Eq. (27), after some rearrangement, the coefficient B for the nth mode is obtained as:

( h iZ Bn ¼ Dn J/n ðxv Þ EIc þ 2EImKn þ cs ImK2n

L

0

/2n ðxÞ dx

h  i þ U EIðcw þ 2MKn Þ þ cs I M K2n  kw ) h  i 1 0 2 þ U EIðch þ 2I0 Kn Þ þ cs I I0 Kn  kh

ð28Þ

5.2. Moving load Once the analytical expressions for the unit impulse response functions have been found, it is possible, by applying a convolution integral, to obtain the response for a moving load. This can be achieved by defining the relative motion between the bridge and the load according to Fig. 3a. Then, as can be seen from Eqs. (20a) and (20b), the unit impulse-response function hðx; xv ; tÞ to a single impulse force at a location xv and time t ¼ 0 is:

hðx; xv ; tÞ ¼

1 X Bn ðxv ÞeKn t /n ðxÞ

ð29Þ

n¼1

with J ¼ 1 in Eq. (28). By considering a force P that moves simultaneously in the interval ½0; xv  of the spatial coordinate x and in the interval ½0; t   of the time coordinate s, a moving load may be described by the following convolution integral:

w ðt  Þ ¼

Z 0

t

Phðx; v s; t  sÞ ds ¼ P

Z 1 X /n ðxÞ

t 0

n¼1

 Bn ðv sÞeKn ðt sÞ ds

ð30Þ

w ðt Þ ¼ Hðt ÞP

ð31aÞ

( R t  1 X Bn ðv sÞeKn ðt sÞ ds if t  6 T /n ðxÞ R0T  B ðv sÞeKn ðt sÞ ds otherwise n¼1 0 n

Here N is the number of axles. The Heaviside function Hðt Þ is necessary to fulfil the causality criterion. A variable T ¼ L=v has been introduced to describe the free-vibration part of the signal. An explicit solution of Eq. (31b) is not given here for brevity. However, a closed-form solution can be obtained using any symbolic mathematical program (e.g. Maple or Mathematica). Similar equations for the velocities and accelerations may be obtained by differentiate Eq. €n ðtÞ). (20a) with respect to time (i.e. by replacing qn ðtÞ with q_ n ðtÞ or q 6. Numerical examples Three numerical examples are presented in this section with the aim of showing the versatility of the proposed method. The results are validated by reference to time-domain finite element (FE) simulations. In all of these examples, the numerical simulations were carried out using the commercial code Abaqus 6.11-1. The FE model was built from Bernoulli–Euler beams (B23), using an element size of 100 mm, with lumped spring–mass–damper systems attached to the supports. A moving load or train was simulated by applying time-dependent amplitude functions to the nodal forces along the beam axis. The time integration were performed using the Hilber-Hughes-Taylor scheme with default parameters and a time step of 1/1000 s. Referring to Fig. 1, a beam with a total length L ¼ 42 m, mass per unit length m ¼ 18; 400 kg/m, area moment of inertia I ¼ 0:61 m4 and Young’s modulus of elasticity E ¼ 200 GPa is considered. This values are obtained from a single track composite railway bridge located on the Bothnia line in Sweden. The remaining variables are defined in Sections 6.1–6.3.

t* τ x

P1

P1

PN

ð31bÞ

Pi

P1

vτ

x1 xi

vt*

xN

Fig. 3. Moving loads.

x

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C. Svedholm et al. / Engineering Structures 126 (2016) 40–52

(a)

(b)

∞ 30

3th Bending mode

160

f [Hz]

3th Bending mode

Im (Λ)

140

20 2th Bending mode

120

10

100

0

1st Bending mode Mode B 0.131

0.185

Mode A

0.306

(c)

80

2th Bending mode

1 0.8

ζ [−]

60 40

0.6 0.4

1st Bending mode

20

0.2 Mode A & B

0

0 −150

−100

−50

0

0

0.2

0.4

0.6

μ

Re (Λ)

cθ

Fig. 5. Modal properties for increasing

0.8

1

[−]

lch : (a) root locus, (b) undamped natural frequencies and (c) modal damping ratios.

μcθ [-] 0 1

1:st bending

0.131

0.185

0.306

1

f = 2.2929 Hz

2.3752 Hz

2.4789 Hz

3.4848 Hz

5.1261 Hz

9.1718 Hz

10.2046 Hz

11.5191 Hz

13.4145 Hz

14.2537 Hz

20.6364 Hz

24.1289 Hz

26.0439 Hz

27.3922 Hz

28.0268 Hz

27.0362 Hz

13.9043 Hz

3.6547 Hz

0.4811 Hz

27.1156 Hz

13.2170 Hz

5.4330 Hz

1.4143 Hz

0 −1

3:th bending

ϕ (x)

2:th bending

Mode A

Mode B 0

0.2

0.4

0.6

0.8

1

x/L

Fig. 6. Mode shapes for increasing lch : —, real part; - - -, imaginary part. Red color = Bending mode maximum damped. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

6.1. Example 1: Two rotational viscous dampers The simply supported beam in Fig. 4 is loaded with a moving load of 100 kN travelling at a constant velocity of 240 km/h in the positive x-direction. Energy dissipation is introduced into the system via two rotational dampers with damping coefficient ch . To achieve simply supported boundary conditions, it would of course be preferable to replace Eqs. (7a) and (7c) with /ð0Þ ¼ 0 and /ðLÞ ¼ 0, respectively. These modifications would, however, require that a new set of equations were derived. Another possibility would be to put kw ¼ 1. The drawback of this alternative is numerical instability caused by computer rounding errors. One practical way of getting around this dilemma, which was applied here, is to include a sufficiently large spring stiffness (kw ¼ 1016 ) to simulate a rigid base. Based on the work of Zarek and Gibbs [8], a non-dimensional variable lch for the rotational damping may be introduced as follows:

c

h ffi lch ¼ pffiffiffiffiffiffiffiffi

mEIL

ð32Þ

The root locus for lch 2 ½0; 1 and the first five eigenvalues are shown in Fig. 5a. A colour change from blue1 (lch ¼ 0) to red (lch ¼ 1) has been introduced to show the direction of the locus line. We recall, from the basic theory of the root locus, that lines of constant undamped natural frequency and damping can be drawn as arcs and as radii from the origin, respectively. With this in mind, it is obvious that both the undamped natural frequencies and the damping ratios vary with lch . The complex mode shape associated with each eigenvalue are shown in Fig. 6. The solid lines show the real part of the spatial eigenfunctions and the dashed lines the imaginary part. It can be seen by examining the figures that three modes can be identified as bending modes, even though both their eigenvalues and mode shapes are complex. However, the two remaining modes (Mode A & B) do not exist for a simply supported beam and should be viewed as an effect of the discrete dampers. In fact these modes do not exist unless the system is non-proportional damped. In

1 For interpretation of color in Fig. 5, the reader is referred to the web version of this article.

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C. Svedholm et al. / Engineering Structures 126 (2016) 40–52

Analytical solution (22 modes) 0.5

Load leave

1st Bending mode

Displacement (mm)

Finite element

Load enter 3th Bending mode

0 2th Bending mode & mode B

−0.5

Mode A

−1 −1.5

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Time (s) Fig. 7. Vertical displacement at midspan under a moving load for

lch ¼ 0:306; P ¼ 100 kN and v ¼ 240 km/h.

Analytical solution (22 modes)

Finite element

Load enter

0.2

Load leave

Acceleration (m/s 2 )

1st Bending mode

0.1

3th Bending mode

0 2th Bending mode & mode B

−0.1 Mode A

−0.2

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Time (s) Fig. 8. Vertical acceleration at midspan under a moving load for

the paper this modes will be referred to (for lack of a better term) as non-classical modes. Using Eqs. (10a) and (10b), the undamped natural frequencies (Fig. 5b) and modal damping ratios (Fig. 5c) for lch 2 ½0; 1 are obtained. Several conclusions can be inferred from these figures. The undamped natural frequencies of the complex modes 1–3 (bending modes) increase gradually with increasing lch . Note also that at lch ¼ 0 and lch ¼ 1 the undamped natural frequencies match the values for a simply supported and a clamped–clamped beam, respectively. Modes A and B are present over the range of values lch 20; 1. Unlike the bending modes, the undamped natural frequencies of modes A and B decrease with increasing lch . Examining the damping, it is found that mode A and B are overdamped for all lch except for lch ¼ 0. In contrast, for the bending modes, the damping ratio varies between 0 and 0.65, depending on lch and the mode number. One interesting feature to note from Fig. 5b and c is that the maximum damping occurs, close to the point, where the undamped natural frequency of a bending mode and that of a non-classical mode coincide, or in other words when the norm of the eigenvalues coincide. This observation may be useful, especially when one wishes to calculate the optimal damping coefficient for nonproportionally damped systems. The midspan vertical displacement and acceleration are plotted in Figs. 7 and 8, respectively, for both the proposed model and the FE simulation. Recall that the response of an underdamped mode becomes real-valued when the mode and its conjugate are summed. A value of 0.306 was used for the non-dimensional dashpot coefficient lch to maximize the damping ratio of the first bending mode. This value gives a damping ratio of 0.65 for the 1:st bending mode, hence few vibrations are to be expected. It was assumed that the load entered the beam at t ¼ 0 s and left at t ¼ 0:63 s.

lch ¼ 0:306; P ¼ 100 kN and v ¼ 240 km/h.

Commonly, the first bending mode governs the response of a simply supported uniform beam to a moving load of constant magnitude and velocity (see, for example, [31]). This is not the case, however, for a simply supported beam with two rotational viscous dampers attached to the ends. The results in Figs. 7 and 8 clearly show that mode A is also a substantial contributor to the deflection and acceleration. Another important finding worth emphasizing is that in the first bending mode and mode A, there are discontinuities in the acceleration signal, but with opposite signs, at the times when the load enters and leaves the beam. The discontinuities disappear progressively with increasing number of modes, and are almost absent in the final response curve (obtained by summing 22 modes). From the figures, it is clear that the proposed method converges tightly to the FE values, which validates the derivation of the theoretical formulae presented in this paper. Fig. 9 examines how the 1st bending mode and mode A contribute to the displacement for various lch . As already stated, in the absence of dampers (lch ¼ 0) the response is governed by the 1st bending mode. This was also found in the present study for values of lch < 0:306. However for larger values of lch mode A is the governing one. 6.2. Example 2: Elastically supported beam with two vertical viscous dampers The second example, shown in Fig. 10, is a beam supported by two vertical spring–dashpot systems. As in the first example, the load is assumed to move at a constant speed of 240 km/h (from left to right). This model is particularly useful for studying the dynamic response of a beam supported by a viscoelastic bearing or in the presence of soil–structure interaction. In fact, the spring–dashpot system can be used to represent the dynamic behaviour of a rigid footing on an elastic half-space (see, e.g., [32]).

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C. Svedholm et al. / Engineering Structures 126 (2016) 40–52

2

Analytical solution (22 modes) |w(t)| max (mm)

1.5

1

0.5

1st Bending mode Mode A

0

0

1

0.5

1.5

μ c θ (-) Fig. 9. Maximum vertical displacement at midspan under a moving load for

v P EI, m

kw

cw

L Fig. 10. Structural system for an elastically supported beam with two vertical viscous dampers.

The following non-dimensional variables are introduced:

kw L3 EI cw L ¼ pffiffiffiffiffiffiffiffiffi mEI

lkw ¼ lc w

ð33aÞ ð33bÞ

where lkw is the non-dimensional spring stiffness and lcw is the non-dimensional dashpot coefficient. The validity of these equations was verified in [8]. In order to simplify the visualization of the results, the variable lkw is set to either 0 (free–free) or 333.33 (elastically supported). The variable lcw is varied from 0 to 50. Using these variables, along with Eq. (9), the complex eigenvalues and eigenvectors for the free–free (Fig. 11) and elastically

supported (Fig. 12) beam are calculated. A significant difference between the curves in the root locus plots may be noticed. For the free–free beam, the undamped natural frequencies decrease as lcw increases, but for the elastically supported beam the undamped natural frequencies increase as lcw increases (except for the third bending mode). It should also be noted that the non-classical modes (mode A and B) originate as two rigid-body modes for lkw ¼ 0 and lcw ¼ 0. As lcw increases these modes will also include a deformational component (bending) superimposed on the rigid body one (see, e.g., Fig. 13). Another interesting feature that can be noticed for the elastically supported beam (see Fig. 12) is that the non-classical mode is split into two modes at lcw ¼ 10:31, whereafter the undamped natural frequency for one of the mode approaches zero and the other approaches infinity. Based on the figure, it seems that the split occurs when the modes goes from being underdamped to overdamped. By inspecting the mode shape of the non-classical modes in Fig. 14, one can note that there are only minor differences in the shape before and after the split. The damping ratio, on the other hand, shows a more consistent trend than the undamped natural frequency. Clearly, fn increases as lcw increases. In the case of modes A and B (i.e. the modes with a significant rigid body component), the damping ratio would continue to increase until the modes become overdamped. This is not the case for bending modes 1, 2 and 3. In this case, the results show that there exists a value lcw ;crit such that f decreases for all

∞

(a)

(b)

3.263 7.640

40

160

30

f [Hz]

3th Bending mode

140

3th Bending mode

20 2th Bending mode

10

120 Mode B

Im ( Λ)

lch ¼ v aries; P ¼ 100 kN and v ¼ 240 km/h.

100

1st Bending mode

0 0

5.410

10

20

80

30

40

50

30

40

50

(c)

Mode A

1

2th Bending mode

ζ [−]

60 40

0.5

1st Bending mode

20 Mode A & B

0

0 −150

−100

−50

Re (Λ) Fig. 11. Modal properties of a free–free beam (lkw ¼ 0) for increasing

0

0

10

20

μc [−] w

lcw : (a) root locus, (b) undamped natural frequencies and (c) modal damping ratio.

C. Svedholm et al. / Engineering Structures 126 (2016) 40–52

(a)

40

3th Bending mode

∞

(b)

35.95

30

f [Hz]

160

8.39 10.31

48

140

Mode splitting

Mode B

20

3th Bending mode

Mode A 2th Bending mode

10

120

0

0

Mode A

0

10

20

30

40

50

30

40

50

(c)

80 14.5

Mode splitting

40 20

13.5

−1

ζ [−]

1st Bending mode

60

1

2th Bending mode

Im ( Λ)

1st Bending mode Mode B

100

0.5

0

0

0 −150

−100

−50

0

0

10

20

μc [−]

Re (Λ)

w

Fig. 12. Modal properties of an elastically supported beam (lkw ¼ 333:33) for increasing lcw : (a) root locus, (b) undamped natural frequencies and (c) modal damping ratio.

μcw [-] 0 1:st bending

3.263

1 f = 5.1978 Hz 0

5.410

15

7.640

3.4953 Hz

2.4785 Hz

2.3750 Hz

2.3126 Hz

14.3280 Hz

13.4177 Hz

11.5149 Hz

10.2025 Hz

9.4001 Hz

28.0887 Hz

27.3944 Hz

26.0402 Hz

24.1237 Hz

21.4746 Hz

0.0000 Hz

3.6264 Hz

13.9266 Hz

27.0815 Hz

104.5453 Hz

0.0000 Hz

5.4214 Hz

13.2409 Hz

27.1613 Hz

104.5456 Hz

−1

3:th bending

ϕ (x)

2:th bending

Mode A

Mode B 0

0.2

0.4

0.6

0.8

1

x/L

Fig. 13. Mode shapes of a free–free beam (lkw ¼ 0) for increasing lcw : —, real part; - - -, imaginary part. Red color = Bending mode maximum damped. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

μcw [-] 0 1

1:st bending

8.39

10.31

35.95

11.00

f = 2.1652 Hz

2.1710 Hz

2.1738 Hz

2.1749 Hz

2.2282 Hz

7.276 Hz

7.9137 Hz

8.1849 Hz

8.2762 Hz

9.0878 Hz

30.9206 Hz

23.5247 Hz

20.9885 Hz

20.9078 Hz

20.6516 Hz

13.7770 Hz

2.3467 Hz

12.7767 Hz

18.553 Hz

21.9344 Hz 36.6874 Hz

596.1784 Hz numerical problem

14.4263 Hz

2.2517 Hz

36.7926 Hz

596.1784 Hz numerical problem

0 −1

3:th bending

ϕ (x)

2:th bending

Mode A

21.2695 Hz

19.4951 Hz

22.5724 Hz

Mode B 0

0.2

0.4

0.6

0.8

1

Fig. 14. Mode shapes of a elastically supported beam (lkw ¼ 333:33) for increasing lcw : —, real part; - - -, imaginary part. Red color = Bending mode maximum damped. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

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C. Svedholm et al. / Engineering Structures 126 (2016) 40–52

Analytical solution (37 modes) 1

Displacement (mm)

Finite element

Load enter

Load leave

Mode A

0.5 0 −0.5

2th Bending mode & mode B

3th Bending mode

−1 1st Bending mode

−1.5 −2 −2.5

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Time (s) lkw ¼ 333:33; lcw ¼ 8:4; P ¼ 100 kN and v ¼ 240 km/h.

Fig. 15. Vertical displacement at midspan under a moving load for

Analytical solution (37 modes)

Acceleration (m/s 2 )

Finite element

Load enter

2

Load leave

3th Bending mode

1st Bending mode

1 0

2th Bending mode & mode B

−1 Mode A

−2

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Time (s) Fig. 16. Vertical acceleration at midspan under a moving load for

lkw ¼ 333:33; lcw ¼ 8:4; P ¼ 100 kN and v ¼ 240 km/h. v

kθ cθ cw

M, I 0 kw

P EI, m, c, cs I L

Fig. 17. Structural system for a damped beam with general end conditions.

lcw > lcw ;crit . The value of lcw ;crit is found to occur, close to the point, when the bending frequency and the frequency of one of the non-classical modes A or B, coincide, and this agrees well with the results in Example 1. After the eigenvalues and eigenvectors have been calculated, Eq. (20a) can be applied once more to calculate the midspan displacement (Fig. 15) and acceleration (Fig. 16). The results presented in these figures are obtained for lkw ¼ 333:33 and lcw ¼ 8:4. Because of the support conditions and the impulsive nature of the loading, higher-frequency modes are also excited. This is especially pronounced in the acceleration signal at the time the load enters or leaves the bridge. Note also that very much as expected the displacement converges much faster than the acceleration. If the purpose of the calculation is to study the displacement, the first bending mode is sufficient to represent the response accurately. However, if the purpose of the study is to predict accelerations, then it might be necessary to include highfrequency modes. Finally, in this example, there is a good match between the proposed method and the FE model as before. 6.3. Example 3: Mixed damping mechanisms under resonant condition The third and final example considers the case of a railway bridge under an HSLM A4 train (high-speed load model A4 [33])

at speeds v ¼ 100–300 km/h (Fig. 17). The train consists of 15 intermediate coaches, a power car and a end coach on either sides of the train. This configuration gives a total number of 44 axles with a load of 190 kN/axle. In this example, the base slab is assumed to be supported by an elastic half-space with a static spring stiffness of kw ¼ 907:2 MN/m and a dashpot coefficient of cw ¼ 5407:9 kN/(m/s). The structural and non-structural mass M of the abutment is 691.5 tonnes. Rotational damping devices are considered at the supports to provide reduced vibration at resonance. Each of these devices is assumed to have a mass moment of inertia I0 ¼ 100 tonne m2, a rotational spring constant kh ¼ 100 MN m/rad and a rotational damping coefficient ch ¼ 20 MN m/(rad/s). These latter values have no physical meaning but were chosen to test the algorithm. Two cases were studied, namely case 1, where the internal damping term and the external damping term were neglected (i.e. cs ¼ 0; c ¼ 0), and case 2, where both the internal damping term, cs ¼ 139 MN s/m2, and the external damping term, c ¼ 2:65 kN s/m2, were considered. In case 2, both the external and the internal damping coefficient were calibrated so that they would each give a damping ratio of 0.5% for the first mode of vibration if no other damping were present. The first six undamped natural frequencies are reported in Table 1 for cases 1 and 2, and the corresponding mode shapes are shown in Fig. 18. Looking at the figure, it is apparent that there

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C. Svedholm et al. / Engineering Structures 126 (2016) 40–52

the results for case 1. Based on the results in the table, one can draw several conclusions: (1) the natural frequencies of the two systems are almost identical, (2) case 2 shows more damping of high-frequency vibrations due to the internal damping, and (3) modes with dominant rigid-body behaviour tend to be more damped than bending modes. The third conclusion agrees well with the findings in Sections 6.1 and 6.2. Then, from the figure, it can be found that Rð/ðxÞÞ  Ið/ðxÞÞ for all modes, which indicates that the problem is dominated by standing waves. It is known from the literature that, for a proportionally damped structure, the train velocity that causes resonance can be calculated from

Table 1 Undamped natural frequencies and damping ratios. n

Case A

1 2 3 4 5 6

Case B

f n (Hz)

fn

f n (Hz)

fn

2.3054 5.1846 5.6918 9.3631 18.3150 28.4588

0.0196 0.0914 0.0993 0.0330 0.0272 0.0261

2.3053 5.1828 5.6919 9.3661 18.3149 28.4592

0.0284 0.0925 0.1007 0.0524 0.0665 0.0875

exist four modes with dominant bending behaviour and two with dominant rigid body behaviour (Mode A & B). For conciseness, the mode shapes for case 2 have been omitted since they are near to

v cr ¼

f n Dtrain ; l

l ¼ 1; . . . ; 1

ð34Þ

1:st bending (n=1)

Mode A (n=2)

Mode B (n=3)

2:th bending (n=4)

3:th bending (n=5)

4:th bending (n=6)

ϕ (x) n

1 0.5 0 −0.5 −1

0.5 0 −0.5 −1 0.2

0.4

0.6

0.8

1

0

0.2

0.4

x/L

0.6

0.8

1

0

0.2

x/L

Analytical solution (Case 1) Analytical solution (Case 2)

Finite element (Case 1)

20 10 0 −10 −20 −30

0

1

2

3

4

5

6

7

8

9

10

Time (s) Fig. 19. Vertical displacement at midspan under a moving train for

Analytical solution (Case 1) Analytical solution (Case 2)

v ¼ 174 km/h.

Finite element (Case 1)

3 2 1 0 −1 −2 −3

0

1

2

3

4

5

0.4

0.6

x/L

Fig. 18. Mode shapes for case 1: —, real part; - - -, imaginary part.

Displacement (mm)

0

Acceleration (m/s 2 )

ϕ (x) n

1

6

7

8

9

Time (s) Fig. 20. Vertical acceleration at midspan under a moving train for

v ¼ 174 km/h.

10

0.8

1

51

C. Svedholm et al. / Engineering Structures 126 (2016) 40–52

Displacement (mm)

25 Analytical solution (Case 1) Analytical solution (Case 2) Finite element (Case 1)

20

15

10 100

120

140

160

180

200

220

240

260

280

300

v (km/h) Fig. 21. Envelope of the maximum vertical displacement at midspan under a moving train.

Acceleration (m/s 2 )

3 Analytical solution (Case 1) Analytical solution (Case 2) Finite element (Case 1)

2

1

0 100

120

140

160

180

200

220

240

260

280

300

v (km/h) Fig. 22. Envelope of the maximum vertical acceleration at midspan under a moving train.

where the distance between carriages Dtrain is 21 m for the HSLM A4 train. More physically, v cr occurs when the loading frequency is close to the nth natural frequency of the beam. For instance, considering the first natural frequency, and making use of Eq. (34) with l ¼ 1, the critical speed is obtained as v cr ¼ 174 km/h. Fig. 19 shows the time history of the displacement at the midspan when the train is moving at the critical speed. The solid lines and the crosses in the figure show results calculated using the analytical solution (cases 1 and 2) and the FE method (case 1), respectively. From this it can be concluded that the results obtained using the analytical solution agree well with the results obtained using the FE method, at least if no distributed damping is present. Similar agreement was found for the vertical acceleration of the bridge deck, for which the simulated data is shown in Fig. 20. All signals have been filtered via a low-pass filter at 30 Hz to remove high frequency components. Modern high-speed lines are typically designed for a speed of 300 km/h. It is therefore common practice to perform dynamical simulations with train speeds ranging from 100 to 300 km/h; see Figs. 21 and 22. Note from these graphs that the maximum response, for both displacement and acceleration, is obtained at the predicted speed v ¼ v cr . Hence, at first glance it seems that Eq. (34) is still valid in the case of a non-proportionally damped system.

be simply decomposed into real valued eigenmodes. Instead, one has to derive a set of orthogonality conditions valid for the problem at hand. The proposed model has been validated against finite element results obtained using direct time integration. Apart from validating the solution/model presented in the paper, the proposed examples have enabled a better insight into the mechanisms governing the dynamical response of a nonproportionally damped system. Thus, for the examples in the present paper, the following qualitative conclusions hold true: 1. Non-proportionally damped beams tend to contain a pair of non-classical modes of vibration (see, e.g., Fig. 5 Mode A & B). For the examples studied here it is found that these normally are highly damped. 2. Based on the results in Figs. 5, 11 and 12 it seems that bending modes attain maximum damping when their undamped natural frequency is close to that of an non-classical mode. 3. Looking at the displacement in Figs. 7 and 9 it can be inferred that non-classical modes of vibration are important for the response of the structure, especially if the structure is highly damped by discrete dampers. The same can be said about Figs. 8 and 16, with regard to accelerations. This is indeed interesting since these mode has received very little attention from researchers in the past.

7. Conclusions Acknowledgement Over the past century, researchers have made great efforts to develop analytical solutions to predict the vibration of beams under moving loads. A literature survey by Fryba [5] has reviewed and summarized over 400 publications from 1851 to 1998 treating various aspects of this problem. The one thing that all of these methods have in common is that the beam is assumed to be proportionally damped. This article presents an analytical solution for a nonproportionally damped beam with general end conditions under a moving load. For such systems, the equation of motion cannot

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