## Abstract

The dynamic response of a stationary structure excited by a moving structure is studied in this paper. The stationary structure is in the form of a truss made of a number of rigidly connected Timoshenko beams, while the moving structure consists of two masses linked by a spring and a dashpot (oscillator). To facilitate the mathematical model of the moving-load dynamics of the whole system, the frequencies and modes of the stationary structure are first obtained by the finite-element method and then they are cast in an analytical form within each element through the element shape functions. This is a distinct advantage of this paper. Each component beam of the stationary structure is meshed with an adaptable number of Timoshenko beam elements to allow efficient modelling of the vibration of the structure for a wide range of travelling speeds of the moving oscillator. During the horizontal travel and vertical vibration of the oscillator, it may separate from the vibrating stationary structure and subsequently may reattach to the stationary structure with impact. These two phenomena have been studied in only a few papers for simple moving-load problems in the past and have never been studied for the present problem. It is found through simulated examples that the dynamic response at high speeds can be several times higher than the relevant static response, and separation and reattachment with impact produce a noticeable difference in the dynamic response. Multiple separation and impact events are possible at high speeds. It is also interesting to observe that separation may occur at high subcritical speeds and impact at various values of the coefficient of restitution has mild local influence on the dynamic response after reattachment.

## 1. Introduction

Vibration of a stationary structure excited by another structure moving on the surface of the stationary structure is very common in engineering and science. Various examples include vibration of a variety of bridges (Au *et al*. 2001), railway tracks (Metrikine & Verichev 2001) or roads (Pesterev *et al*. 2002) under travelling vehicles and vibration of the vehicles themselves (Yang *et al*. 2004), vibration of data storage disks (Kim *et al*. 2000), disc-brake squeal (Cao *et al*. 2004) and vibration in centrifugal atomization (Ouyang 2005). In the simplest cases of a vehicle–bridge interaction, the moving structure may be modelled as a constant force, a mass or an oscillator while the stationary structure may be modelled as a single or continuous beam (Akin & Mofid 1989; Chatterjee *et al*. 1994*a*; Fryba 1999; Yang *et al*. 2000). Analytical solutions of simple moving-load problems can be found in Fryba's monograph (1999). For a moving flexible body (Ouyang & Mottershead 2007), numerical methods must be used.

More realistic representations of real structures can be made to the moving structure or the stationary structure or both. In the case of the moving structure, a train was modelled as a two-axle mass–spring–damper system supporting a rigid body in Fryba (1968) and as a system of multiple oscillators in Marchesiello *et al*. (1999). As for the stationary structure, various types of bridges were studied by a number of researchers (Chatterjee *et al*. 1994*b*; Humar & Kashif 1995; Henchi *et al*. 1998; Huang & Wang 1998; Wang & Lin 1998; Cheung *et al*. 1999; Yau & Yang 2004; Lee & Yhim 2005).

Many bridges are made from pin-connected or rigidly connected members in various configurations. The vibration of a bridge in the form of a simple plane frame excited by moving loads was studied by Wang & Lin (1998). A plane truss structure excited by a moving oscillator is the subject of this paper. As shear may be important in many cases, the Timoshenko beam theory (Géradin & Rixen 1997) is used. A mass in contact with the stationary structure and supporting another mass through a spring and a dashpot as a simplistic representation of a vehicle is the model for the moving structure and is referred to as a moving oscillator. At low travelling speeds, the dynamic effect on the truss produced by the moving oscillator is small and hence a small number of modes are excited. At high travelling speeds, on the other hand, higher modes are excited and thus must be involved in the dynamics of the truss. To allow efficient simulation for a wide range of travelling speeds from low to high, each beam component of the truss is meshed with an adaptive element mesh. Numerical modes and frequencies of the truss are obtained using Matlab by the finite-element (FE) method. The modes obtained for the nodes of the mesh are then replaced by analytical forms for mathematical convenience and numerical efficiency in solving this moving-load problem. The equations of motion for the whole system are solved numerically. The computing code is capable of automatically generating a number of typical truss configurations. To the authors' best knowledge, dynamic interaction between a truss structure and a moving oscillator has not been studied in the past.

Since both the stationary and moving structures vibrate vertically, the moving structure may be bounced off the stationary structure and may then descend on the stationary structure. An impact may result at the instant of this reattachment. The possibility of separation in moving-load problems was mentioned by Yang *et al*. (2004) and Fryba (1999) but not studied. Lee (1998) seems to be the first researcher who studied it in the very simple problem of a mass moving on a beam. However, he simplistically assumed that the vertical velocity of the mass took the value of the vertical velocity of the beam at the reattachment point with no velocity change. Cheng *et al*. (1999) considered separation and put forward a sophisticated method for determining the velocity of the oscillator and the beam after reattachment with impact. Stancioiu *et al*. (2008) suggested a simpler approach for dealing with reattachment with impact for a beam subjected to a moving oscillator. Separation and reattachment with impact are investigated for the more realistic model of a bridge as a truss structure in this paper. It should be pointed out that the works of Lee (1998), Cheng *et al*. (1999) and Stancioiu *et al*. (2008) seem to be the only ones that studied separation and reattachment among hundreds of papers on moving-load problems.

## 2. Vertical motion of the oscillator

The equations of vertical motion of the sprung and unsprung masses of the oscillator, as shown in figure 1, are(2.1)(2.2)where(2.3)and *W*_{z}=−*m*_{z}*g* and *W*_{y}=−*m*_{y}*g* are the weights of the two masses, in which *g* is the gravitational constant. *f*_{c} is the dynamic contact force at the oscillator–truss contact point, i.e. the moving load from the oscillator acting onto the stationary structure.

It is assumed that when the oscillator is in contact with the truss, the vertical motion *y*(*t*) of the unsprung mass and the transverse motion *w*(*x*, *t*) of the beam at the point of instantaneous contact are equal. The transverse vibration of the beam can be expressed as a modal expansion of(2.4)where *ψ*_{w}(*x*) is the vector of the modes for the transverse-displacement components, i.e. *w*, of the stationary structure and ** q**(

*t*) is the corresponding modal coordinates. The superscript ‘T’ stands for the transpose of a vector or matrix. The contact condition dictates that(2.5)and therefore(2.6)(2.7)where

*s*(

*t*) is the horizontal motion of the moving oscillator. In this paper, the overhead dot and prime represent the derivative with respect to time

*t*and space coordinate

*x*, respectively.

## 3. Vibration of the stationary structure

The stationary structure in the form of a truss structure made of a number of rigidly connected Timoshenko beams, such as the one shown in figure 2, is studied. The equation of motion for the free transverse vibration of an individual beam is (Géradin & Rixen 1997)(3.1)where *w* and *θ* are the transverse displacement and the rotation of the cross section of the beam, respectively; *ρ* is the density; *A* and *I* are the area and the second moment of area of the beam's cross section, respectively; *G* and *E* are the shear modulus and Young's modulus, respectively; *γ* is the shear factor; *δ* is the Dirac delta function; and *f*_{c} is the dynamic contact force (moving load) from the oscillator.

For a truss made of a number of Timoshenko beams as shown in figure 2, the equations for the modes and frequencies are simultaneous transcendental equations that are difficult to solve accurately. From this consideration, the FE method should be used. For a moving-load problem, however, there is a clear advantage of expressing the modes and hence the vibration of the stationary structure in an analytical manner in that the vertical motion of the moving structure can be easily related mathematically to the deflection of the supporting structure (e.g. equations (2.5)–(2.7)). On the other hand, if the vibration of the supporting structure is in a discrete form, for example, as in an FE model, it is difficult to track the position of the moving structure and relate its motion to that of the FE nodal displacement vector of the stationary structure as the moving structure traverses different element domains of the discretized stationary structure. However, in real structures such as bridges consisting of a number of distinct components an analytical expression of their motion is either non-existent or is exceedingly complicated. To enable investigations into the vibration of structures more complicated than a single beam and retain the mathematical convenience and hence computing efficiency, the stationary structure is first discretized with finite elements to give modes and frequencies and then an analytical form of the FE modes is obtained inside each element by interpolation using the FE modal solution and the element shape functions.

The element mass and stiffness matrices are available in Przemieniecki (1968). A Matlab code is written for a truss structure with an arbitrary number of beams and an arbitrary number of Timoshenko beam elements for each component beam. In each beam element, the local coordinate *ξ* is defined for the interval [−1,1] (figure 3), thus *x*=*x*_{i}+(*L*_{e}/2)(1+*ξ*), where *x*_{i} is the *x*-coordinate of the l.h.s. node of the element and *L*_{e} is the length of a beam element. The *n*th *w*-mode shape function for the *i*th element can be expressed through the local coordinate as follows:(3.2)From the *w*- and *θ*-values of the FE modes at left and right nodes of each element, the coefficients in equation (3.2), *A*_{wn}, *B*_{wn}, *C*_{wn} and *D*_{wn} can be determined and then the analytical form of the modes of the *w*-components of the stationary structure, *ψ*_{w}(*x*), can be obtained. It should be pointed out that the above methodology applies to trusses and frames of any configuration.

## 4. Dynamics of the whole system

The equation of motion of the truss in modal coordinates can be expressed as follows:(4.1)where diag[*ω*^{2}] is a diagonal matrix of appropriate dimension whose diagonal elements are the natural frequencies squared of the truss structure ranked in ascending order.

From equations (2.2) and (4.1), it follows that(4.2)By substituting equation (2.7) into equation (4.2), one can obtain(4.3)where ** I** is the identity matrix of appropriate dimension. Equations (2.1)–(2.3) and (4.3) are solved simultaneously to obtain the vibration of the truss and the oscillator.

If the oscillator travels at constant speed *v*, equation (4.3) reduces to(4.4)

## 5. Separation and reattachment

Under certain conditions, the moving oscillator may separate from the beam during its travel and vibration. The most influential parameter seems to be the speed *v* (Stancioiu *et al*. 2008). It is also probable that the oscillator may reattach to the beam with an impact after separation (Cheng *et al*. 1999; Stancioiu *et al*. 2008).

Separation takes place whenever *f*_{c}(*t*) drops to zero (it is defined as positive when it is compressive). During the numerical integration, the value of *f*_{c}(*t*) is checked constantly. Once it becomes negative, the time instant goes backward by one step and a new time step is predicted so that at the end of this new time step *f*_{c}(*t*) becomes zero within a small acceptable error. Then, a new set of equations (free vibration of the oscillator and the truss) will be solved with appropriate initial conditions. These new equations are(5.1)(5.2)

After separation, *y*(*t*) and *w*(*s*, *t*) are monitored. When *y*(*t*)=*w*(*s*, *t*), the oscillator reattaches to the top deck of the truss. If the vertical velocity of the oscillator is different from that of the beam at the reattachment point, there is an impact. Cheng *et al*. (1999) put forward a sophisticated method for dealing with the impact and determining the velocity after the impact. Here, the simple approach proposed by Stancioiu *et al*. (2008) is used.

Suppose that the impact (reattachment) takes place at time instant *t*_{r} and the impulse is *p*. For that particular beam onto which the oscillator descends, the equation of motion is(5.3)Using the same modal approach employed in §4, equation (5.3) may be written in the modal coordinate vector as(5.4)Integration in the time domain containing the instant of impact gives(5.5)where and are the instants just before and just after the impact. From equations (2.4) and (5.5), one can obtain(5.6)

For the moving oscillator, similarly (Stancioiu *et al*. 2008),(5.7)Combination of equations (5.5) and (5.7) leads to(5.8)

Two simplest types of impact, perfectly elastic and perfectly plastic, are first studied in §5*a*,*b*. In general, an impact will be neither perfectly elastic nor perfectly plastic, but somewhere in-between. Interested readers may refer to Goldsmith (1960) for more general cases of impact. Next, in §5*c*, these general cases are approximately represented by the coefficient of restitution.

### (a) Perfectly elastic impact

If the impact is perfectly elastic, the total energy before and after the impact are the same. The difference between these two energy values is then(5.9)Please note that just before and just after the impact, the values of the beam's potential energy are the same and the values of the kinetic energy of the sprung mass are the same too, and therefore these terms do not appear in equation (5.9).

Substituting equation (5.8) into equation (5.9) and solving for yields(5.10)and then from equation (5.8), it follows that(5.11)Therefore, the modal velocity of the truss and the vertical velocity of the oscillator at can be calculated from the modal velocity of the truss and the vertical velocity of the oscillator at . At the point of impact, there is usually a velocity discontinuity (Stancioiu *et al*. 2008).

After a perfectly elastic impact, the oscillator (unsprung mass) bounces off the beam at a newly gained vertical velocity as its initial velocity and the vertical motion of the unsprung mass is now governed by equation (5.1). It should be pointed out that the displacements (motion) of the oscillator and of the beam are each continuous throughout the entire time duration of interest, and thus the displacements just after and just before the impact remain the same and known. Numerical integration of equations (5.1) allows *y*(*t*) and *z*(*t*) to be found. Whenever *y*(*t*) approaches *w*(*s*, *t*) again at a later time, a new impact takes place.

### (b) Perfectly plastic impact

When the impact is perfectly plastic, the oscillator sticks to the beam after the impact, and hence acquires the same displacement and the same vertical velocity of the beam at . For the latter,(5.12)Substituting equation (5.8) into equation (5.12) and noting that (displacements are always continuous) and effectively , one can derive(5.13)(5.14)

After a perfectly plastic impact, the unsprung mass attaches to the beam and vibrates at the vertical initial velocity , given in equation (5.13). Equations (2.1)–(2.3) and (4.2) must then be solved together by numerical integration.

### (c) General impact represented by the coefficient of restitution

The coefficient of restitution for the current problem is defined as(5.15)Combining equations (5.6)–(5.8) and (5.15) after mathematical manipulation yields(5.16)(5.17)It can be easily verified that equations (5.16) and (5.17) reduces to equations (5.10) and (5.11) when *C*_{R}=1, and reduces to equations (5.13) and (5.14) when *C*_{R}=0 (except the convective term in equations (5.13) and (5.14), which is a special feature of a moving-load formulation).

The value of *f*_{c}(*t*) must be constantly checked for possible separation, which occurs whenever *f*_{c}(*t*) drops to zero. The procedure of checking separation and reattachment was detailed in Stancioiu *et al*. (2008).

## 6. Numerical analysis and examples

The truss shown in figure 2 is analysed. It has the following material and geometric data: *EI*=9.3×10^{6} N m^{2}; *EA*=3.15×10^{9} N; *G*=8.1×10^{10} Pa; *γ*=0.9; *ρA*=1125 kg m^{−2}; *m*_{y}=100 kg; *m*_{z}=1000 kg; and *L*=50 m. There are six beams of equal length in the top deck. The height is 7.217 m. The critical speed of the truss *v*_{cr}=35.34 m s^{−1}, which is defined as the ratio of the fundamental frequency in Hz (*w*_{1}/2*π*) of the truss to the lowest wavenumber of a beam in the top deck. All the dynamic values of the system are non-dimensionalized against the relevant static values, as presented in figures 4–15. The impact at reattachment, if it occurs, is treated as perfectly elastic in the examples given in figures 4–11 and 13–15, except in figure 12*a*,*b*.

The non-dimensionalized contact force between the moving oscillator and the truss in the case of *k*=3.95×10^{6} N m^{−1} and *c*=25 130 N s m^{−1} is shown in figure 4. Separation does not occur as *f*_{c} stays positive (compressive).

When different parameter values of *k*=4.93×10^{6} N m^{−1} and *c*=2800 N s m^{−1} are used (other parameter values remain the same), the non-dimensionalized contact force becomes negative at approximately *vt*/*L*=0.9 (figure 5), when contact is assumed to be maintained. When separation and reattachment are considered, the force is depicted in figure 6. Clearly, the force history is different when separation is allowed.

Actually, separation (and possibly reattachment subsequently) may happen at a range of parameter values. Figure 7 shows a three-dimensional plot of the maximal and minimal values of the contact force for a range of non-dimensionalized *k*-values defined as . In this plot, damping *c* varies such that . Permanent contact is assumed in these results to show that separation is possible (when the minimum values of *f*_{c} are negative). Figure 7 involves forces obtained from 810 000 simulations.

It should be interesting to know what values of *c* and *k* would allow separation to occur. This information is provided in figure 8 that displays the minimum velocity at which separation between the two structures is found. Frequently, this velocity is close to the critical speed and is a function of the viscoelastic properties, non-dimensionalized *c* (defined as ) and *k*, of the oscillator. In figure 8, the number tagged on a curve is a value of *v*/*v*_{cr} allowing separation to occur.

It turns out that separation takes place more readily at supercritical speeds and when there is no damping in the oscillator. Dynamic contact force and the dynamic responses of the oscillator are given in figures 9–11 for the case of *k*=4.93×10^{6} N m^{−1} and *c*=0 N s m^{−1}, where *w*_{st} is the static deflection of the top deck at the mid-span of the truss produced by the *W*_{y}+*W*_{z} acting at the same location and *z*_{st} is the static deflection of the sprung mass (*W*_{z}/*k*). The difference between the results when separation is considered and ignored becomes greater, for example, if one compares figure 9 with figures 5 and 6. Figure 9 is particularly interesting in that multiple events of separation and impact take place. At each impact, much higher forces are exerted to the stationary structure within shorter time intervals (indicated by a steeper force gradient), which is detrimental to both the stationary structure and the moving structure and would be uncomfortable to passengers if the moving structure is a passenger vehicle.

To assess the influence of different types of impact at the reattachment, an example of *k*=4.93×10^{6} N m^{−1} and *c*=0 N s m^{−1} at three different values of *C*_{R} is simulated. As the influence is limited to the time interval shortly after the impact, the dynamic response in that interval is presented in figure 12. The influence of various values of *C*_{R} on the dynamic response seems mild and local.

Next, the attention is turned onto the vibration of the oscillator at various speeds. Numerical results of *y*(*t*)/*w*_{st}, *z*(*t*)/*z*_{st} and *w*(*L*/2,*t*)/*w*_{st} are given in figures 13–15, respectively (*k*=4.93×10^{6} N m^{−1} and *c*=2800 N s m^{−1}). Owing to the damping of the oscillator, the effect of separation is very small. However, the dynamic effect of the moving load is noticeable. This is clear if one compares the dynamic response of the oscillator at the speed close to the critical speed with the static response (response at near-zero speed). It should also be pointed out that at very high speeds, the dynamic response of the oscillator is lower than the static response. It can be speculated that by a suitable design (adjustment of the parameters of the oscillator), the acceleration of the sprung and unsprung masses may be greatly reduced, which would be good for the vehicle and comfortable to the passenger.

## 7. Conclusions

This paper studies the coupled dynamic response of a structure (oscillator) moving on a stationary structure (truss). The stationary structure may have any number of rigidly connected beams. Numerical modes are obtained by the FE method and then converted into an analytical form by means of the element shape functions. The equations of motion of the moving and the stationary structures are established separately and then coupled at the contact point. When the moving structure separates from the stationary structure, a new set of equations of motion are given. Reattachment with impact after separation is considered. Numerical results for several examples reveal interesting dynamics. It is found that separation can occur at high subcritical speeds and easily occurs at supercritical speeds, and when there is no damping in the moving oscillator. It is also found that the dynamic deflection of the truss can be several times higher than the maximum static deflection and the moving load (dynamic contact force) can also be a few times greater than the static force (self-weight of the oscillator). More importantly, there is a considerable difference between the results when separation is ignored and those when separation is considered. This suggests that the neglect of separation (and subsequent possible reattachment) in most past studies of moving-load problems can lead to serious errors in many situations, in particular at high speeds. On the other hand, different types of impact represented by different values of the coefficient of restitution have only mild local influence on the dynamic response after reattachment. Finally, multiple events of separation and impact (reattachment) can happen at high speeds (still well below the record speed of a fast train) and are detrimental to the stationary and moving structures.

## Acknowledgments

This research is conducted at the Department of Engineering, University of Liverpool and is sponsored by the RSB funding of the Department and EPSRC of the UK (Ref. EP/D057671/01: moving-load distributions in structural dynamics). L.B. is sponsored by projects T79/2006 (Ministerio de Fomento–Metro de Madrid–CDM) and TRA2007-67167 (Ministerio de Educación y Ciencia–FEDER).

## Footnotes

- Received February 8, 2008.
- Accepted April 15, 2008.

- © 2008 The Royal Society