908. Dynamic responses of an elastic beam moving over a simple beam using modal superposition method

Size: px

Start display at page:

Download "908. Dynamic responses of an elastic beam moving over a simple beam using modal superposition method"

Justin Rudolph Parker
5 years ago
Views:

1 98. Dynamic responses of an elastic eam moving over a simple eam using modal superposition method Y. J. Wang, J. Shi, Y. Xia 3, School of Civil Engineering, Beijing Jiaotong University, Beijing 44, China, 3 Department of Civil and Environmental Engineering, Rutgers The State University of New Jersey, NJ 8854, USA yingjiewang@jtu.edu.cn, yw84@rci.rutgers.edu, jshi@jtu.edu.cn, 3 xiaye.cee@rutgers.edu (Received Octoer ; accepted 4 Decemer ) Astract. The dynamic responses of an elastic eam moving over a simple eam are investigated. The elastic eam is modeled as an Euler eam with oth ends free, and connected to the simple eam y two spring units. With modal superposition method, the dynamic responses of these two eams are studied. The virations of the simple eam are almost the same due to the moving elastic eam and rigid eam, even the latter eam ignores the flexural viration of the elastic eam. However, the acceleration of the moving elastic eam is much larger than that of the moving rigid eam, which can e attriuted to the flexile viration of the elastic eam. With various flexural stiffness of the elastic eam, the max accelerations of the elastic eam at midpoint are computed along with different moving velocities. It is oserved that, with an increase of flexural stiffness of the elastic eam, the max acceleration of the elastic eam decreases evidently. Keywords: the elastic eam, the simple eam, flexile viration, flexural stiffness.. Introduction Traffic induced structural viration has een an interesting topic in the field of civil engineering, such as railway and roadway ridge virations. Numerous researches on the dynamic ehaviors of ridge under moving vehicle have een conducted [-]. In these studies, the moving vehicle is regarded as moving load [4, 5], moving mass [6, 7], moving oscillator [8, 9], and moving suspend eam considering pitching effect [, ]. As the demand of lightweight design for car structures, the structural stiffness of car frame will e decreased [, 3]. To consider the flexile viration of the moving structure, Zhang et al. modeled an elastic eam moving over another Euler eam with spring connected at two discrete points and some approximate analytical results were put forward [4]. Cojocaru et al. assumed a moving elastic eam connected to another eam with a series of rigid interfaces and the dynamic responses of this system were solved y means of symolic computation [5, 6]. In the aove mentioned work, some important conclusions have een rought out; while relatively little research attention so far seems to conduct the effect of flexural stiffness of the moving elastic eam on the dynamic responses of the system. In the present paper, the dynamic responses of the elastic eam moving over a simple eam are investigated with modal superposition method. The elastic eam is regarded as an Euler eam with oth ends free, and it is connected to the simple eam y two spring units. Firstly, the dynamic responses of these two eams are compared under the moving elastic and rigid eam. Secondly, with different flexural stiffness of the elastic eam, the max accelerations of the elastic eam at midpoint are computed along with different moving velocities. It is oserved that the increase of flexural stiffness of the elastic eam can lead to a notale decrease of its max acceleration. From practical view, these results are useful for lightweight design of car structures to relief much more flexile viration and improve riding comfort. 84 VIBROENGINEERING. JOURNAL OF VIBROENGINEERING. DECEMBER. VOLUME 4, ISSUE 4. ISSN

2 . Formulations In this paper, an elastic eam travels over a simple eam at a constant speed v, as shown in Fig.. In this model, the elastic eam is modeled as an Euler eam with oth ends free, and connected to the simple eam y spring units at two discrete points. For the initial conditions, the rear spring unit of the elastic eam is located at the left-hand end of the simple eam.. The elastic eam Fig.. An elastic eam moves over a simple eam As shown in Fig., the viration of the moving elastic eam is measured from its static equilirium position, and its viration can e divided into two parts: rigid viration and flexile viration [4]. The following symols are used in Fig. : m v = the total mass, I v = the pitch of inertia, E v I v = the flexural stiffness, C v = the damping per unit length, ρa = the constant mass per unit length, l v = the total length, l t = the distance etween the front and rear spring units, k v = the stiffness of each spring unit. When the elastic eam runs on the simple eam, the equation of motion for the elastic eam can e written as: (, ) (, ) (, ) 4 Z xv t Z xv t Z xv t v v + ρ + 4 v = jδ v j x t t ( ) E I A C P x l () where Z(x v, t) is the vertical displacement of the elastic eam at time t ( x v l v ); j =, represent the front and rear spring units, δ( ) is the Dirac delta function, P are the interaction forces from the spring units: j (, ) (, ) Pj = k v Z l j t Z x j t () where Z (x, t) is the vertical displacment of the simple eam, l j = l v /+( ) j+ l t /, x j is the distance of the front/rear spring units to the left-hand of the simple eam. The viration of the elastic eam can e regarded as the comination of rigid motion and flexile motion. When the rigid modes are included in the flexile modes, the first mode is chosen as the ounce of rigid mode and its shape function is taken as Y (x v ) =. The second mode is the pitch and its shape function is Y (x v ) = x v l v /. When NM v modes are considered, the vertical displacement of the elastic eam can e written as [7]: VIBROENGINEERING. JOURNAL OF VIBROENGINEERING. DECEMBER. VOLUME 4, ISSUE 4. ISSN

3 NM v lv Z( x, t) = Z ( t) + x θ ( t) + p ( t) Y ( x ) (3) v v v v m m v m= 3 where Z v (t) and θ v (t) are the modal amplitude of the ounce and pitch motion respectively, Y m (x v ) and p m (t) are the modal amplitude and mode shape function of the flexile viration for the elastic eam. Consider a eam with oth ends free, when m >, the mode shape functions of the elastic eam can e given as [8]: cosh λ cosλ Y x x x x x sinhλ sinλ m m ( ) = cosh( β ) + cos( β ) sinh( β ) + sin( β ) m v m v m v m v m v m m (4) where λ m and β m satisfy: cosh λ cos λ =, β = λ / l (5) m m m m v Sustituting Eq. (3) into Eq. (), and multiplying y Y i (x v ) and integrating the resultant equation with respect to x v etween and l v, and considering the orthogonality conditions of the natural viration modes, the equations of motion in terms of the NM v modal displacements can e given as: ( ) m Z ɺɺ t + P = (6) v v j lt j+ vθv( ) ( ) j I ɺɺ t + P = (7) v v v v v i ( ) ( ) ( ) ( ) 4 v i i i i j j v ρ A ρ A m C m E I β mɺɺ p t + pɺ t + p t = Y l P i= 3 NM (8) Let E I β v v ρ A 4 i Cv = ωvi, = ξviωvi, the Eq. (8) can e given as: ρ A ( ) ξ ω ( ) ω ( ) ( ) v i vi vi v i v vi i i j j v mɺɺ p t + m pɺ t + m p t = Y l P i= 3 NM (9) Therefore Eqs. (6), (7) and (9) are expressed for the rigid motion and flexile motion of the elastic eam.. The simple eam As shown in Fig., the equation of motion for the simple eam due to moving elastic eam can e written as: (, ) (, ) (, ) 4 Z x t Z x t Z x t = (, ) E I x m t C t F x t 86 VIBROENGINEERING. JOURNAL OF VIBROENGINEERING. DECEMBER. VOLUME 4, ISSUE 4. ISSN ()

4 where E I is the flexural stiffness, m is the constant mass per unit length, C is the damping per unit length, F (x, t) is the external force acting on the simple eam from the elastic eam: F ( x, t) = m g+ P δ( x x ) () v j j Based on the modal superposition method, the solution of Eq. () can e expressed as: NM ( ) φ ( ) Z ( x, t) = q t x () n n n= where q n (t) is the nth modal amplitude and φ n (x) is the nth mode shape function of the eam. For simple eam, the natural frequency and mode shape functions can e expressed as: nπ ωn = l φ n ( x) sin E I m nπ x = l (3) (4) Sustituting Eqs. () and () into Eq. (), and multiplying y φ k (x) and integrating the resultant equation with respect to x etween and l, and considering the orthogonality conditions of the natural viration modes, the equation of motion of the kth generalized system in terms of the generalized displacement q k (t) is given as: ( ) + ξ ω ( ) + ( ) = ( ) M qɺɺ t M qɺ t K q t F t (5) k k k k k k k k k where ω k, ξ k and M k are the modal frequency, damping ratio, and modal mass of the kth mode, respectively, and Kk( M kωk) generalized force F k (t) is expressed as: k v j k j = means the geneneralized stiffness of the kth mode. The F ( t) = m g+ P φ ( x ) (6) 3. Solution Susequently, sustituting Eqs. (), (3), () into Eqs. (6), (7), (9) and moving the terms with Z, θ, p, q to the left side of the differential equation, the equations of motion for the elastic ( ) v v m n eam can e written as: NM v NM ( ) ( ) ( ) ( ) φ ( ) ( ) m Z ɺɺ t + k Z t + k Y l p t k x q t = (7) v v v v v m j m v n j n m= 3 n= VIBROENGINEERING. JOURNAL OF VIBROENGINEERING. DECEMBER. VOLUME 4, ISSUE 4. ISSN

5 NM NM k l k l k l I ɺɺ θ t + θ t + Y l p t φ x q t = (8) v v t v t j+ v t j+ ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) v v v m j m n j n m= 3 n= k l mɺɺ p t m p t m p t k Y l Z t Y l t v t j+ ( ) + ξ ω ɺ ( ) + ω ( ) + ( ) ( ) + ( ) ( ) θ ( ) v i vi vi v i vi v i v i j v i j v NM NM v ( ) ( ) ( ) ( ) φ ( ) ( ) + k Y l Y l p t k Y l x q t = v i j m j m v i j n j n m= 3 n= Sustituting Eqs. (), (3), (), (6) into Eq. (5) and moving the terms with Z, θ, p, q to the left side of the differential equation, the equation of motion for the ( ) v v m n simple eam can e written as: (9) NM ( ) ξ ω ɺ ( ) ( ) ( ) ( ) M qɺɺ t + M q t + K q t + k q t Φ k Z t Φ k k k k k k k k v n nk v v k n= NM v kvlt j+ mv g ( ) θ ( t) k p ( t) Y ( l ) Φ Φ = Φ v k v m m j k k m= 3 () Φ = and Φ = φ ( x ). where nk φn( x j) φk( x j) k k j As shown in Eqs. (7), (8), (9) and (), the elastic eam and the simple eam are coupled and interacting with each other. By comining Eqs. (7), (8), (9) and () together, the equations of motion in modal space are given in a matrix form as: [ M]{ U} + [ C]{ U} + [ K]{ U} = { F} ɺɺ ɺ () where [M], [C], [K] denote the mass, damping and stiffness matrices; ({ U }{, U },{ U }) ɺ ɺɺ are the vectors of displacement, velocity, and acceleration, respectively; and {F} represents the vector of exciting forces applied to the dynamic system. To compute oth the dynamic responses of the simple eam and the elastic eam, the equations of motion as given in Eq. () will e solved using a step-y-step integration method, i.e. Newmark-β method [9]. The integration scheme of Newmark-β method consists of the following equations: { U } = a ( { U } { U } t t t) a { U } a 3{ U } t+ t + t t { U} = { U} + a6{ U} + a7{ U} ɺɺ ɺ ɺɺ () ɺ ɺ ɺɺ ɺɺ (3) t+ t t t t+ t where the coefficients are: γ α =, α =, α =, α3 =, β t β t β t β γ t γ α4 =, α5 = ( ), α6 = t( γ ), α7 = γ t. β β (4) 88 VIBROENGINEERING. JOURNAL OF VIBROENGINEERING. DECEMBER. VOLUME 4, ISSUE 4. ISSN

6 In this study, β = /4 and γ = / are selected, which implies a constant acceleration with unconditional numerical staility. 4. Numerical investigation Fig. shows an elastic eam crossing a simple eam at a constant speed v, and the properties of these two eams are listed in Tale and Tale, where ω v and ω denote the first natural frequencies of the elastic eam and the simple eam, respectively. In the illustrative example, a time step of. s and ending time of t end = (l l t )/v are employed to compute the dynamic responses of the elastic eam and the simple eam. Tale. Properties of the elastic eam m v (kg/m) l v (m) l t (m) E v I v (kn m ) I v (kg m ) k v (N m - ) ξ v ω v (Hz) % 9.8 Tale. Properties of the simple eam m (kg/m) l (m) E I (kn m ) ξ ω (Hz) % 5.59 To compute the dynamic response of the simple eam under the moving elastic eam, a sufficient numer of modes in Eqs. (3) and () are required for accuracy of the response computed from Eq. (). For the simple eam, modes are sufficient to determine the dynamic response of the simple eam [], which are also used in this paper. For the elastic eam, in order to verify that a sufficient numer of modes has een used in the analysis, we first compute the acceleration response of the elastic eam at midpoint using either 8, 4, or modes (NM v =, 6, or ) with the moving speed at 5 m/s. From the convergent verification of computed results in Fig., the first modes (NM v = ) are sufficient to compute the acceleration response of the elastic eam moving over the simple eam. For this reason, the same numer of modes will e considered in the following examples modes 4 modes modes a vmid (m/s ) vt/l Fig.. Test of convergence 4. Comparison of the moving elastic and rigid eam Firstly, the moving eam is regarded as an elastic eam and a rigid eam, respectively. When the moving eam runs over the simple eam at 5 m/s, the displacement responses of the simple eam at midpoint are computed, as shown in Fig. 3. It can e noticed that the displacement responses of the simple eam at midpoint under the moving elastic and rigid eam are almost the VIBROENGINEERING. JOURNAL OF VIBROENGINEERING. DECEMBER. VOLUME 4, ISSUE 4. ISSN

7 same. It also means that the effect of flexiility of the moving eam on the viration of the simple eam can e negligile, and this is ecause the forces acting on the simple eam have little variation from the gravity of the moving eam []..9.8 Rigid eam Elastic eam.7.6 D mid (mm) vt/l Fig. 3. Displacement responses of the simple eam at midpoint When the moving eam runs over the simple eam at 5 m/s, the acceleration responses of the elastic and rigid eam at midpoint are computed, as shown in Fig. 4. It can e seen that the accleeration response of the elastic eam at midpoint is a little larger than that of the rigid eam, which can e attrituted to the flexile viration of the elastic eam excited y ridge viration. This phenomean is confirmed y the acceleration spetruem analysis, as shown in Fig. 5. The first dominant frequencies of the moving elastic and rigid eam are 5 Hz, which is close to the natural frequency of the simple eam (See Tale ). Besides the first dominant frequency at 5 Hz, there is another dominant frequency for the moving elastic eam at Hz, which is consitant with the natural frequency of the elastic eam (See Tale )...5 Rigid eam Elastic eam. a vmid (m/s ) vt/l Fig. 4. Acceleration responses of the elastic eam at midpoint 4. Effect of different flexural stiffness Acctually, the flexile viration of the elastic eam is excited y ridge viration as indicated in Section 4., and the effect of different flexural stiffness on the viration of the elastic eam will e performed y considering various flexural stiffness, i.e., E v I v /, E v I v, and E v I v. Then, the max accerelations of the elastic eam at midpoint are plotted against the moving velocities at 5,, 5,, and 5 m/s in Fig. 6. And the results are also compared with the moving rigid eam. 83 VIBROENGINEERING. JOURNAL OF VIBROENGINEERING. DECEMBER. VOLUME 4, ISSUE 4. ISSN

8 .8.7 Rigid eam Elastic eam.6 5 Hz a vmid (m/s ) Hz f (Hz) Fig. 5. Accelerations spectrum of the elastic eam at midpoint.3.5. Rigid eam Flexile eam with E v I v / Flexile eam with E v I v Flexile eam with E v I v a v,max (m/s ) v (m/s) Fig. 6. Max accelerations of the elastic eam at midpoint vs moving velocities It can e seen from the Fig. 6 that, with the increase of flexural stiffness of the elastic eam, the max acceleration of the elastic eam at midpoint decreases oservaly. In this numerical example, when the flexural stiffness of the elastic eam is eaqual to E v I v, resulting in the natural frequency of 6.65 Hz, the max acceleration of the elastic eam is almost the same with the rigid eam. Therefore, to supress the flexile viration of the elastic eam, its flexural stiffness should e improved at a certain value, which is important for the lightweight design of car structures. 5. Conclusions In this paper, the dynamic responses of an elastic eam moving over a simple eam are investigated using modal superposition method. In this model, the elastic eam is modeled as an Euler eam with oth ends free, and connected to the simple eam y two spring units. On the ase of this study, the following conclusions can e drawn: () The displacements of the simple eam at midpoint are almost the same under the moving elastic and rigid eam, whereas the acceleration of the moving elastic eam at midpoint is a little larger than that of the moving rigid eam. This can e attriuted to the flexile viration of the elastic eam excited y simple eam viration. () With various flexural stiffness of the elastic eam, the max accelerations of the elastic eam at midpoint are computed along with different moving velocities. It is oserved that an increase of the flexural stiffness of the elastic eam can lead to a notale decrease of its max acceleration. From VIBROENGINEERING. JOURNAL OF VIBROENGINEERING. DECEMBER. VOLUME 4, ISSUE 4. ISSN

9 practical view, the results are useful for lightweight design of car structures to relief much more flexile viration and improve riding comfort. Acknowledgment The research reported herein is supported y National Natural Science Foundation of China (Grants No. 588). This financial support is gratefully acknowledged. References [] Ceon D. Handook of Vehicle-Road Interaction. Lisse: Swets & Zeitlinger Pulishers, 999. [] Frýa L. Viration of Solids and Structures under Moving Loads. Groningen: Noordhoff International Pulishing, 97. [3] Yang Y. B., Yau J. D., Wu Y. S. Vehicle-Bridge Interaction Dynamics: with Applications to High- Speed Railways. Singapore: World Scientific, 4. [4] Gonzalez A., OBrien E. J., Cantero D., Li Y. Y., Dowling J., Znidaric A. Critical speed for the dynamics of truck events on ridges with a smooth road surface. Journal of Sound and Viration, Vol. 39, Issue,, p [5] Gladysz M., Sniady P. Spectral density of the ridge eam response with uncertain parameters under a random train of moving forces. Archives of Civil and Mechanical Engineering, Vol. 9, Issue 3, 9, p [6] Lee U. Revisiting the moving mass prolem: onset of separation etween the mass and eam. Journal of Viration and Acoustics, Vol. 8, Issue 3, 996, p [7] Ouyang H. Moving-load dynamic prolems: A tutorial (with a rief overview). Mechanical Systems and Signal Processing, Vol. 5, Issue 6,, p [8] Baeza L., Ouyang H. Viration of a truss structure excited y a moving oscillator. Journal of Sound and Viration, Vol. 3, Issue 3-5, 9, p [9] Wang Y. J., Wei Q. C., Shi J., Long X. Y. Resonance characteristics of two-span continuous eam under moving high speed trains. Latin American Journal of Solids and Structures, Vol. 7, Issue,, p [] Lou P. A vehicle-track-ridge interaction element considering vehicle's pitching effect. Finite Elements in Analysis and Design, Vol. 4, Issue 4, 5, p [] Yang Y. B., Chang C. H., Yau J. D. An element for analysing vehicle-ridge systems considering vehicle's pitching effect. International Journal for Numerical Methods in Engineering, Vol. 46, Issue 7, 999, p [] Pan F., Zhu P. Lightweight design of vehicle front-end structure: contriutions of multiple surrogates. International Journal of Vehicle Design, Vol. 57, Issue -3,, p [3] Mangino E., Carruthers J., Pitarresi G. The future use of structural composite materials in the automotive industry. International Journal of Vehicle Design, Vol. 44, Issue 3-4, 7, p. -3. [4] Zhang T., Zheng G. T. Viration analysis of an elastic eam sujected to a moving eam with flexile connections. Journal of Engineering Mechanics, Vol. 36, Issue,, p. -3. [5] Cojocaru E. C., Irschik H., Gattringer H. Dynamic response of an elastic ridge due to a moving elastic eam. Computers & Structures, Vol. 8, Issue -, 4, p [6] Cojocaru E. C., Irschik H. Dynamic response of an elastic ridge loaded y a moving elastic eam with a finite length. Interaction and Multiscale Mechanics, Vol. 3, Issue 4,, p [7] Zhou J., Goodall R., Ren L., Zhang H. Influences of car ody vertical flexiility on ride quality of passenger railway vehicles. Proceedings of the Institution of Mechanical Engineers, Part F: Journal of Rail and Rapid Transit, Vol. 3, Issue 5, 9, p [8] Humar J. L. Dynamics of Structures. Third Edition, London: CRC Press,. [9] Newmark N. M. A method of computation for structural dynamics. Journal of the Engineering Mechanics, Vol. 85, Issue 3, 959, p [] Yau J. D. Response of a train moving on multi-span railway ridges undergoing ground settlement. Engineering Structures, Vol. 3, Issue 9, 9, p. 5-. [] Cheng Y. S., Au F. T. K., Cheung Y. K. Viration of railway ridges under a moving train y using ridge-track-vehicle element. Engineering Structures, Vol. 3, Issue,, p VIBROENGINEERING. JOURNAL OF VIBROENGINEERING. DECEMBER. VOLUME 4, ISSUE 4. ISSN

Resonance characteristics of two-span continuous beam under moving high speed trains

7(200) 85 99 Resonance characteristics of two-span continuous beam under moving high speed trains Abstract The resonance characteristics of a two-span continuous beam traversed by moving high speed trains