Elsevier

Applied Mathematical Modelling

Volume 100, December 2021, Pages 192-217
Applied Mathematical Modelling

Dynamic interaction and instability of two moving proximate masses on a beam on a Pasternak viscoelastic foundation

https://doi.org/10.1016/j.apm.2021.07.022Get rights and content

Highlights

  • It is proven that instability of two proximate moving masses occurs at the critical velocity in the absence of damping.

  • Instability of two proximate moving masses can be shift deeply into the subcritical velocity range in damped cases.

  • The critical distance between mases in dimensionless parameters is only slightly dependent on the damping ratio.

  • Moving mass ratio for which subcritical velocity can induce instability is derived as a function of damping ratio.

  • Enhancement of modal expansion method for computational time saving proposed and implemented for results validation.

Abstract

The study analyzes the dynamic interaction of proximate masses using a new form of semi-analytical results for the moving mass problem developed by the author. This paper presents the results for a particular case of two moving masses of equal value acted upon by constant forces of equal values. The interaction level was demonstrated to be substantially high, making it impossible to superimpose the results obtained for one mass unless the masses were located at a significantly large distance apart. In the undamped case, the onset of instability occurred at the critical velocity, as with one moving mass; however, the onset of instability could be shifted significantly into the subcritical velocity range in the damped case. The critical distance between the moving masses was determined as the value at which the exponential increase in the amplitudes was severe. The implementation of dimensionless parameters revealed that this distance was slightly dependent on the damping. Moreover, the limiting moving mass ratio for such unexpected instability was derived.

The results were validated by eigenmode expansion analysis on long finite beams, for which computational time savings were proposed by the rearrangement of the terms involved. Excellent agreement between the results was obtained, validating the new formulae. Furthermore, extension to more general cases and additional moving masses can easily be achieved.

Introduction

Numerous engineering fields, particularly those relating to road and rail transport, include studies on structures subjected to moving loads. The increasing demands on rail networks and EU directives to shift from road to rail transport systems have renewed the need for an improved understanding of the dynamic phenomena pertaining to train-to-ground interactions. Thus, moving load problems remain active areas of current investigations.

Advances in symbolic software and high-precision calculations with an adaptable number of digits have promoted interest in semi-analytical solutions offering the benefits of closed-form solutions, thereby providing rapid and highly accurate results that can be directly supplemented by sensitivity and parametric analysis. In addition, the presentation of results involving dimensionless parameters enables a general view of the physical phenomena for a wide range of input data involved in the problem. Moreover, these results can only be evaluated in areas under focus, without a full-time history.

The moving load problem is a fundamental problem in structural dynamics; however, in its traditionally used designation, it is unclear whether the inertial effects in the moving object are considered. Thus, the problem specification should clearly state whether moving forces or masses, or oscillators, are considered [1], which can make a fundamental difference in solution methods and undesirable effects that may appear.

The moving force problem is much simpler than that with moving masses. When the force moves at a constant velocity over a homogeneous structure, analytical solutions for finite and infinite beams on a Pasternak viscoelastic foundation are available. These are conveniently summarized, for example, in a monograph [2]. The superposition of results is possible when the structure exhibits linear properties, and the initial instant is not vital for the analysis. For constant velocity, a steady-state stage can readily be achieved. A major concern relates to the resonant behavior for identifying the critical velocity, while questions regarding instability are irrelevant. Therefore, many works have analyzed only the steady-state regime. Among the recent works on analytical solutions, [3,4] are notable.

The moving mass problem is complex because it is inherently nonlinear and does not allow for the superposition of results, typically exploited in moving force problems. By applying the eigenvalue expansion method to homogeneous finite beams, it can be demonstrated that the governing equations in modal space are coupled, thereby requiring a complementary numerical solution or simplifications to ensure an acceptable approximate solution. The moving mass problem was initially solved using finite beams as early as 1929 by Jeffcott [5], using the method of successive approximations. The first solution obtained by eigenfunction expansion was probably published in [6]. The authors considered an undamped, simply-supported Euler–Bernoulli beam; however, they omitted the Coriolis and centrifugal forces in the analysis. In [7], the so-called modal elements were developed. Methods apart from the finite element method include the spectral element method, dynamic Green's function [8], and reproducing kernel particle method [9]. The extension of the modal expansion method to a Timoshenko beam was presented in [10], whereas other issues such as control and support excitation were considered in [11,12]. A further extension to the accelerating mass by modal expansion was conducted in 1996 [13], and subsequently using the spectral element method [14] and with the aid of a newly developed finite element [15]. Large deflections were analyzed using finite elements in [16], and the numerical part was solved by perturbation analysis in [17]. Nevertheless, as this research aims to present the results in an analytical form as far as possible, the present discussion on state-of-the-art focuses on works with an analytical basis in the solution method, implying that numerous works using the finite element method are not mentioned.

Fewer works have been published on infinite beams. A pioneering study on the moving mass problem was presented in [18], where integral transforms were exploited, although the inverse transform was computed numerically. Furthermore, several solutions using integral transforms moved directly to the steady-state regime, thereby removing the inertial effect in the moving object and hiding the possibility of instability [19]. Other works exploited the moving element method [20], dynamic Green's function, and other methods. A review can be found in [21].

Instability issues resulting from anomalous Doppler waves induced in the beam may occur while the inertial objects move on a finite structure, but in general, published works on instability issues have been dedicated to infinite structures. In [22,23], the instability conditions were determined by the D-composition method; however, the full deflection shapes of the beam were not presented. It was also assumed that the unsprung mass was in permanent contact with the beam. In [24,25], a nonlinear contact spring was implemented. The deflection shapes were determined numerically, and the instability was analyzed by the D-composition method. In [26], the foundation model was extended to a double beam.

Regarding a sequence of moving masses, a pioneering solution on finite beams was presented in [27,28]. Double Fourier transform in which the possibility of instability was hidden was implemented in [29]. Among recent works on multi-span beams, [30] can be mentioned. A sequence of moving oscillators on an infinite beam was dealt with in [31] using Green's function method and the D-composition method to determine the instability conditions. The D-composition method was also used in recent work [32]. Whereas a contact spring was used in [31], in [32], it was again assumed that the unsprung mass was in permanent contact with the beam. An infinite string was considered in [33], and a numerical solution based on the differential quadrature method was presented in [34], but the masses were stationary.

The problem of several moving masses on a uniform homogeneous structure requires further attention from an analytical perspective. It is crucial to determine the conditions under which superposition is possible and whether the dynamic interaction leads to faster instability than expected. This can be achieved by exploiting the new form of result presentation published by the author [35–37]. The new form of results also enables the evaluation of the severity of the unstable behavior, which is vital for mitigation measures.

The solution presented in this paper is conceptually different from that in [31,32] because the full vibration shapes are determined semi-analytically and not numerically, and the instability is identified directly from the so-called mass-induced frequency. This term highlights the difference from the natural system frequencies because the mass-induced frequency, or simply induced frequency, is induced by the mass movement, and therefore dependent on its velocity. The final results are presented using dimensionless parameters allowing a wide range of realistic scenarios to be covered. Unlike in [31], where only one specific case was presented, readers of this paper can obtain the solutions to different problems without requiring recalculations. As a new result, it is demonstrated that damping can act oppositely and can shift the onset of instability into the subcritical region. Moreover, the so-called critical distance between masses is derived as the distance for which the lowest value of the imaginary part of an induced frequency is achieved.

In [35], the new presentation of the semi-analytical solution was derived under the assumption of homogeneous initial conditions. In [36], the solution was extended to non-homogeneous initial conditions, and in [37], further details on the method and an analysis of moving one- or two-mass oscillators were presented.

In this study, the problem of two moving masses was solved on an infinite homogeneous beam on a homogeneous viscoelastic Pasternak foundation and validated on an equivalent, sufficiently long finite beam. The analysis is limited to the tight contact assumption, commonly used in several works, such as [22–23,32]. Nevertheless, the possible loss of contact can be examined a posteriori. In actual situations relating to vehicle motion, the loss of contact generally originates from severe inhomogeneities in the supporting structure, which is not the case in the present analysis. The other limitation is the massless foundation, also used by several other researchers. However, the analysis of this work can be extended to account for the wave propagation in the foundation, following [38,39], which will be the subject of further research.

The complete evolution of the transversal vibrations resulting from proximate moving masses is presented, as in [35–37], as a sum of several terms designated as the: (i) steady-state; (ii) unsteady harmonic; and (iii) truly transient part. The reasoning for the truly transient part has been presented in detail in [37]. Such designations are introduced because of the following reason. The solution should traditionally be separated into the steady-state and transient parts. However, as proven in [35–37], the transient part has one dominant harmonic contribution described by a closed-form formula and another insignificant part. The dominant harmonic part is determined by the induced frequencies and can be formulated using standard harmonic terms. This is the reason that it must be named differently. The remaining part to be distinguished is designated as the truly transient part. All three parts represent the complete solution. By summing the first and second parts, the harmonic solution is obtained because the steady-state solution is also a harmonic term with zero frequency. Although it can be argued that the transient vibrations analysis may not be important because these will be attenuated by damping, any non-homogeneity in the structure will cause transient effects; therefore, it is essential to study such solutions. Additionally, instability cannot be detected without such an analysis.

One primary conclusion of this study is that the external viscous damping can act differently than for one moving mass where such damping always shifts the onset of instability into the supercritical velocity range. With proximate masses, damping can act oppositely and shift the onset of instability significantly into the subcritical range. Dimensionless parameters enable the definition of the so-called critical distance between moving masses. Assuming that the masses are at the critical distance, it is possible to determine the limiting moving mass ratio for which the instability occurs in the subcritical velocity range as a function of the applied damping. In contrast to the damped case, the onset of instability occurs exactly at the critical velocity for one or more moving masses in the undamped case. Furthermore, the analysis of the unsteady harmonic part of the beam vibrations enables the definition of the safe distance between masses, for which it is possible to use result superposition. It should be noted that the critical velocity and the velocity below which instability of the moving mass instability occurs are different factors. Instability velocities are not single velocities but constitute an interval, and the lowest one can be identified as the velocity at which the onset of instability occurs.

The rest of this paper is organized as follows: In Section 2, the problem under consideration is formulated, and the simplifying assumptions are listed. The derivation of the new closed-form formula for proximate moving masses is presented in Section 3. Section 4 is dedicated to examples. In Section 4.1, the range of parameters analyzed is established, and the validation possibility by finite beams analysis is described, along with the rearrangement of the involved terms in the eigenmode expansion method, leading to significant time savings, as in [38,40]. Examples are discussed in Section 4.2 for the undamped case and Section 4.3 for the damped case. This section analyzes the derivation of the critical distance between moving masses and the conditions that determine the onset of instability in the subcritical velocity region. Finally, the main conclusions of this study are summarized in Section 5.

Section snippets

Problem statement

Consider the uniform motion of two constant masses traversing a horizontal infinite homogeneous beam supported by a Pasternak viscoelastic homogeneous foundation. The moving masses are acted on by vertical forces with harmonic components. The objective of the analysis is to determine the evolution of the vertical beam vibrations starting from homogeneous initial conditions. These results will confirm the strong dynamic interaction between the moving inertial objects, and thus, the impossibility

Problem solution

To solve the problem defined by Eqs. (1) and (2) as well as the boundary and initial conditions, it is first necessary to remove the additional unknowns w0j(t), j = 1, 2 and express them in terms of the unknown deflection field w(x, t). Owing to the assumption (v), w0j(t) = w(xj,t),  j = 1, 2 holds and x1 = vt, x2 = vt + d. Therefore, the relevant derivatives using the chain rule are:w01,t(t)=vw,x(x,t)+w,t(x,t)withx=x1w01,tt(t)=v2w,xx(x,t)+2vw,xt(x,t)+w,tt(x,t)withx=x1w02,t(t)=vw,x(x,t)+w,t(x,t)

Definition of case studies and validation

This section validates the new formulae in Eq. (36), examines the vibrations resulting from the dynamic interaction of two proximate masses, and analyzes the onset of instability. The importance of the truly transient part of the solution, that is, the option of neglecting it in the harmonic solution obtained by Eq. (36), is also discussed. Such an observation is important because it clarifies the possibility of using only the harmonic terms in closed-form analytical expressions that can be

Conclusion

In this study, the new form of semi-analytical results to characterize the moving mass problem on infinite beams from [35–37] has been extended to account for the dynamic interaction between two proximate moving masses. The objective of this analysis was to demonstrate that result superposition is generally impossible unless the masses are far apart from one another. The other objective was to confirm that the dynamic interaction can alter the conditions for instability of one moving mass,

Acknowledgments

This work was supported by the Portuguese Foundation for Science and Technology (FCT), through IDMEC, under LAETA, project UIDB/50022/2020.

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