Arch Appl Mech (2008) 78: 321–328 DOI 10.1007/s00419-007-0164-7
O R I G I NA L
Li-Qun Chen · Hu Ding
Two nonlinear models of a transversely vibrating string
Received: 19 January 2007 / Accepted: 31 July 2007 / Published online: 28 August 2007 © Springer-Verlag 2007
Abstract Modeling transverse vibration of nonlinear strings is investigated via numerical solutions of partialdifferential equations and an integro-partial-differential equation. By averaging the tension along the deflected string, the classic nonlinear model of a transversely vibrating string, Kirchhoff’s equation, is derived from another nonlinear model, a partial-differential equation. The partial-differential equation is obtained via neglecting longitudinal terms in a governing equation for coupled planar vibration. The finite difference schemes are developed to solve numerically those equations. An index is proposed to compare the transverse responses calculated from the two models with the transverse component calculated from the coupled equation. A steel string and a rubber string are treated as examples to demonstrate the differences between the two models of transverse vibration and their deviation from the full model of coupled vibration. The numerical results indicate that the differences increase with the amplitude of vibration. Both models yield satisfactory results of almost the same precision for vibration of small amplitudes. For large amplitudes, the Kirchhoff equation gives better results. Keywords Nonlinear string · Transverse vibration · Modeling · Finite difference scheme 1 Introduction The linear model is a representation of small-amplitude transverse vibration of a string that consists of linear elastic material. If the large amplitude vibration is concerned, the geometric nonlinearity should be taken into consideration. Kirchhoff (1876) [9] proposed the first nonlinear model for transverse vibration of a string, which is a nonlinear integro-partial-differential equation. Actually, he considered the effect of significant string tension temporal variation due to the transverse displacement, while the linear model assumes the tension to take a constant value, because the tension in the static equilibrium configuration is large and the transverse displacement is small. Carrier (1945, 1949) [7,8] developed a more rigorous approach to model transverse vibration via the coupled governing equation of planar vibration and recovered the nonlinear integro-partialdifferential equation, without quoting Kirchhoff. Narasimha (1968) [18] also obtained the equation using another approach. The equation is called the Kirchhoff string equation in the literature. The Kirchhoff string equation is accepted as a good first approximation to nonlinear behavior in the transverse direction of a string. The Kirchhoff string equation is studied and applied extensively. Oplinger (1960) [20] compared numerical and experimental results of frequency response. Molteno and Tufillaro (2004) [15] checked qualitatively the agreement between the analytical results obtained via the truncated Kirchhoff L.-Q. Chen (B) Department of Mechanics, Shanghai University, 99 Shang Da Road, Shanghai 200444, China E-mail:
[email protected] L.-Q. Chen · H. Ding Shanghai Institute of Applied Mathematics and Mechanics, Shanghai 200072, China
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string equation and the experimental results. The various numerical schemes, such as the Galerkin method [6] and finite difference methods [2,14,21], were developed to simulate the nonlinear Kirchhoff string model. The mathematical analysis of the Kirchhoff string equation pertains to the topics such as the well-posedness [1] and the existence and uniqueness [12]. Besides, the Kirchhoff string equation is applied to design the controllers for nonlinear strings [11,22]. Although the Kirchhoff string equation has been extensively studied and applied, there is another model, a nonlinear partial-differential equation, proposed by Murthy and Ramakrishna [17]. In spite of very limited application to stationary strings, the partial-differential equation, with a suitable modification of the time derivative, has been widely used to analyze transverse vibration of axially moving strings, summarized in [3], after the work of Mote [16]. Only recently, attention was paid to the difference of the two models, namely, the integro-partial-differential equation and the partial-differential equation [4]. However, so far it has not been clear which model yields the better outcomes. To highlight the differences between the two models and to determine the superiority in the sense of better approximating the coupled governing equation of planar vibration, the dynamic responses calculated from the two models are contrasted with the results based on the coupled equations of planar vibration. A finite difference approach is proposed for those equations. It should be mentioned that a string here refers a one-dimensional continuum without bending stiffness and with a straight line equilibrium configuration. The modeling of a loose string (a cable) with sag [10] will not be treated here. The present paper is organized as follows. Section 2 revisits the governing equation of an elastic string undergoing coupled planar vibration, and derives two mathematical models for transverse vibration from the governing equation. Section 3 develops finite difference schemes to solve those equations obtained in Sect. 2 numerically and proposes some indexes to compare the outcomes of two models for transverse vibration with the transverse component of coupled vibration. Section 4 presents two numerical examples, a steel string and a rubber string, and discusses the numerical results. Section 5 ends the paper with some concluding remarks. 2 Problem formulations Consider a string of density ρ, area of cross-section A, and initial tension P0 . Assume that the deformation of the string is confined to the vertical plane. At axial spatial coordinate x and time t, the in-plane vibration of the string is specified by the longitudinal displacement u(x, t) and the transverse displacement v(x, t) related to a spatial frame. For the string, Newton’s second law yields equation of motion ρ Au,tt = (Pu ),x ,
ρ Av,tt = (Pv ),x ,
(1)
where Pu and Pv are, respectively, the longitudinal and transverse components of the tension in the string. Let σ (x, t) denote the additional stress in the string due to its deflection. An element of initial length d s is deformed into an element of length (1 + u,x )2 + v,2x ds by the longitudinal and transverse displacements. Therefore longitudinal and transverse components of the tension in the string are respectively (P0 + Aσ )(1 + u,x ) , Pu = (1 + u,x )2 + v,2x Substitution of Eq. (2) into (1) leads to ⎡ ⎤ ∂ ⎣ (P0 + Aσ )(1 + u,x ) ⎦ ρ Au,tt = , ∂x (1 + u, )2 + v,2 x
x
(P0 + Aσ )v,x Pv = . (1 + u,x )2 + v,2x
(2)
⎡ ⎤ ∂ ⎣ (P0 + Aσ )v,x ⎦ ρ Av,tt = . ∂x (1 + u, )2 + v,2 x
(3)
x
For a linear elastic string, the constitutive relation is expressed by Hooke’s law σ = Eε
(4)
where E is the modulus of elasticity, and ε is the strain disturbance of the string given by the strain–displacement relation ε = (1 + u,x )2 + v,2x − 1. (5)
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Substitution of Eqs. (4) and (5) into Eq. (3) yields, after some calculation ρ Au,tt −E Au,x x −(E A − P0 )
(1 + u,x )v,x v,x x −u,x x v,2x = 0, 3 (1 + u,x )2 + v,2x
(1 + u,x )2 v,x x −(1 + u,x )u,x x v,x ρ Av,tt −E Av,x x +(E A − P0 ) = 0. 3 2 2 (1 + u,x ) + v,x
(6)
Equation (6) is the governing equation of in-plane vibration of an elastic string, which was first derived by Carrier [6,7]. Although there is the more sophisticated model [13], Eq. (6) is accurate enough to serve as a reference for the comparison of two models of transverse vibration. Consider the problem of finite but small stretching problem treated in literature on nonlinear oscillations [19]. In this case, only a few lower-order nonlinear terms need to be retained so that the corresponding dynamic model will be obtained. Omitting higher-order nonlinear terms in Eq. (3) yields
∂ 1 ∂ ρ Au,tt − (7) = 0, ρ Av,tt − [(P0 + Aσ )v,x (1 − u,x )] = 0. (P0 + Aσ ) 1 − v,2x ∂x 2 ∂x Correspondingly, Eq. (5) reduces to the strain-displacement relation of the von Karman theory of plates with finite deflection 1 εK = u,x + v,2x . 2
(8)
For a linear elastic string with small but finite stretching, keeping different order nonlinear terms in Eq. (6) will lead to some known approximations of the governing equation for planar vibration of a string [5]. Although the transverse vibration is generally coupled with the longitudinal vibration, it is a weak smallamplitude vibration. Therefore, for finite but small stretching problems, one can restrict oneself to the transverse vibration. In this case, only the lowest-order nonlinear terms need to be retained. Inserting u = 0 into Eqs. (7) and (8), and then omitting higher-order nonlinear terms give a governing equation of transverse vibration ρ Av,tt −P0 v,x x −
∂ (Aσ v,x ) = 0 ∂x
(9)
and the Lagrangian strain εL =
1 2 v, . 2 x
(10)
Substituting Eq. (10) into (4) and the resulting equation into Eq. (9) gives a dynamic model of nonlinear strings in the form 3 ρ Av,tt −P0 v,x x − E Av,2x v,x x = 0. 2
(11)
On the other hand, if the spatial variation of the tension is rather small, then one can use the averaged value l of the tension disturbance 1l 0 Aσ dx to replace the exact value Aσ , where l is the length of the string. In this case, Eq. (9) is modified to v,x x ρ Av,tt −P0 v,x x − l
l Aσ dx = 0,
(12)
0
which leads to another dynamic model of nonlinear strings E Av,x x ρ Av,tt −P0 v,x x − 2l
l v,2x dx = 0. 0
Equation (13) is the Kirchhoff string equation.
(13)
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Equation (13) can be obtained through decoupling the governing equation for coupled longitudinal and transverse vibration under the quasi-static stretch assumption [19]. The assumption means that the dynamic tension component is a function of time alone, on the condition that E A >> P0 . In that traditional derivation, Eq. (13) seems to be more precise than Eq. (9), because it is the model of transverse vibration in which the longitudinal displacement is taken into account, although it is finally removed from the equation. However, the derivation here indicates that Eq. (9) can be reduced to Eq. (13) based on the replacement of the tension disturbance by its averaged value. As a result, the Kirchhoff string Eq. (13) can be interpreted as an approximate model of Eq. (12). Therefore, the superiority of one of the two models cannot be determined via their derivations. Besides, the condition that E A >> P0 is no longer necessary in the present derivation of Eq. (13).
3 Method of solution The finite difference method will be employed to solve numerically Eqs. (6), (11) and (13). Some indexes will be introduced to compare the solutions. In the following investigations, the fixed boundary conditions u(0, t) = u(l, t) = 0, v(0, t) = v(l, t) = 0
(14) (15)
will be used for Eq. (6), and only condition (15) for Eqs. (11) and (13). With the emphasis on transverse vibration, the initial conditions for Eq. (6) are u(x, 0) = 0, u,t (x, 0) = 0, v(x, 0) = α(x), v,t (x, 0) = β(x),
(16) (17)
where α(x) and β(x) are prescribed functions. Of course, Eqs. (11) and (13) need condition (17) only. Introduce the equispaced mesh grid x j = j h, j = 0, 1, 2, . . . , L tn = nτ, n = 0, 1, 2, . . . , T.
(l = h L),
(18) (19)
Denote the function values u(x, t) and v(x, t) at (x j , tn ) as u nj and v nj , respectively. Then the grid values u nj and v nj are used in the finite difference schemes as an approximation to the continuous solutions u(x, t) and v(x, t) to Eq. (6). Application of centered difference approximations [23] to both the time and space derivatives in Eq. (6) leads to u n+1 − 2u nj + u n−1 j j
n n n E u j+1 − 2u j + u j−1 P0 − E A − · 2 2 τ ρ h ρA 2 2 v nj+1 − 2v nj + v nj−1 v nj+1 − v nj−1 2h + u nj+1 − u nj−1 − 4 u nj+1 − 2u nj + u nj−1 v nj+1 − v nj−1 , · 2 2 23 h 2h + u nj+1 − u nj−1 + v nj+1 − v nj−1 (20) n n n − 2v nj + v n−1 v n+1 E v j+1 − 2v j + v j−1 P0 − E A j j = + · τ2 ρ h2 ρA 2 2 v nj+1 −2v nj +v nj−1 2h +u nj+1 −u nj−1 −2 v nj+1 − v nj−1 u nj+1 −2u nj +u nj−1 2h +u nj+1 −u nj−1 · . 2 2 23 h 2h + u nj+1 − u nj−1 + v nj+1 − v nj−1
=
With the boundary conditions and the initial conditions, u nj and v nj can be solved from Eq. (20) successively for n = 1, 2, . . . , T and j = 1, 2, . . . , L.
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Discretization of the temporal and spatial variables in Eqs. (11) and (13) via the centered difference yields ⎡ n 2 ⎤ n + v n−1 n n + vn n − 2v − 2v − v v v v n+1 3 E A P j j j j+1 j j−1 ⎣ 0 j+1 j−1 ⎦, = + (21) τ2 h2 ρA 2 ρA 2h − 2v nj + v n−1 v n+1 j j τ2
⎧ ⎡ 2 2 L−1 0 2 ⎤⎫ ⎬ v j+1 − v 0j−1 v nj+1 − 2v nj + v nj−1 ⎨ P0 h E A 1 v10 − v00 v 0L − v 0L−1 1 ⎣ ⎦ . = + + + ⎩ ρ A 2l ρ A 2 ⎭ h2 h 2 h 2h j=1
(22) With the boundary conditions and the initial conditions, v nj (n = 1, 2, . . . , T and j = 1, 2, . . . , L) can be successively solved from Eqs. (21) and (22). To compare the solutions of Eq. (11) and (13) with the transverse component of the solution to Eq. (6), the following indexes are introduced 2 1 L T n n j=0 n=0 v j − v(q) j TL σ(q) = (23) (q = 1, 2), max v nj j = 0, 1, · · · , L n = 0, 1, · · · T n n where v nj is the numerical solution of Eq.(6), and v(1) j and v(2) j are, respectively, the numerical solutions of Eqs. (11) and (13). Therefore, σ(1) and σ(2) indicate the relative accumulative errors between the transverse vibration models and the coupled vibration model.
4 Results and discussions The two models will be compared using the date of a rubber string and a steel string. In the computations, the data l = 1 m, τ = 10−6 s, and h = 10−3 m (and hence L = 1, 000) are assumed. The deviations of the models will depend on the amplitude the motion, which, in a free vibrating system, is determined by the initial conditions. In the following, the initial conditions (12) and (13) will be used with α(x) = Dx(l − x), β(x) = 0.
(24)
Based on the numerical solutions to Eqs. (6), (11) and (13), indexes defined by Eq. (23) will be evaluated for different values of D and with the other parameters as prescribed. Actually, based on Eq. (24), D/4 is the largest initial displacement, and thus D indicates the amplitude of motion. Although the initial conditions used here are rather specific, there is no loss of generality. Numerical simulations as well as physical considerations indicate that the deviations of the models depend on the initial energy and are independent of the concrete initial conditions with the same energy. Consider a steel string with modulus of elasticity E = 2.1 × 1011 Pa and density ρ = 7, 850 kg/m3 . Suppose that the string area of cross-section A = 3.14 × 10−6 m2 , the length l = 1 m, and initial tension P0 = 6, 594 N. In the finite difference scheme, choose τ = 0.000005 s, h = 0.001 m, T = 200, 000, and L = 1, 000 (except indicated otherwise). In this case, condition E A >> P0 holds. The indexes defined by Eq. (23) are calculated for different values of D. The numerical results are shown in Table 1. The differences of both models increase with the initial disturbance amplitude, which is an expected outcome. What is more, the computations also show that σ(1) is always larger than σ(2) , and hence model (13) is superior to model (11) in the sense that model (13) is closer to the coupled model (6) than model (11). For a steel string, D = 0.1 m−1 is a sufficiently large initial disturbance, corresponding to a largest initial displacement of 0.025 m, which is 2.5% of the string span length. The differences between the models can also be illustrated via the transverse displacement of the string center. The displacements are plotted in Figs. 1, 2, 3, in which the dots, the solid lines and the dashed lines
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Table 1 The deviations of the two models for a steel string D (m−1 )
σ(1)
σ(2)
0.010 0.0250 0.050 0.075 0.100
8.0419 × 10−3 4.7979 × 10−2 1.5998 × 10−1 2.8494 × 10−1 3.5235 × 10−1
1.0691 × 10−4 6.6460 × 10−4 2.2548 × 10−3 5.2140 × 10−3 9.4834 × 10−3
Fig. 1 The transverse center displacement of the steel string calculated from three models for D = 0.010
Fig. 2 The transverse center displacement of the steel string calculated from three models for D = 0.025
Fig. 3 The transverse center displacement of the steel string calculated from three models for D = 0.100
stand for the numerical solutions to Eqs. (6), (13) and (11), respectively. Figure 1 shows that the three solutions coincide when the motion is very small, with the amplitude about 0.25% of the string length. Figure 2 shows that the solution to Eq. (11) deviates from the still coinciding solutions of Eqs. (6) and (13). The amplitude is here about 0.6% of the string length. With the further increase of the motion amplitude 2.5%, Fig. 3 show that the solution to Eq. (11) becomes different from still coinciding solutions to Eqs. (6) and (13). Now consider a rubber string with modulus of elasticity E = 7.8 × 106 Pa and density ρ = 930 kg/m3 . Suppose that the string area of cross-section A = 3.14 × 10−6 m2 , the length l = 1 m, and initial tension P0 = 10 N. In the finite difference scheme, choose τ = 0.00005 s, h = 0.001 m, T = 20, 000, and L = 1, 000.
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Table 2 The deviations of the two models for a rubber string D (m−1 )
σ(1)
σ(2)
0.1 0.2 0. 3 0.4 0.5
2.7566 × 10−2 1.0447 × 10−1 2.1698 × 10−1 3.5106 × 10−1 4.7640 × 10−1
9.4258 × 10−3 3.7521 × 10−2 8.4360 × 10−2 1.5231 × 10−1 2.4376 × 10−1
Fig. 4 The transverse center displacement of the rubber string calculated from three models for D = 0.1
Fig. 5 The transverse center displacement of the rubber string calculated from three models for D = 0.3
In this case, condition E A >> P0 no longer holds. In fact, EA is larger than P0 here. The indexes defined by Eq. (23) are calculated based on the numerical solutions to Eqs. (6), (11) and (13). The outcomes are listed in Table 2. Model (13) is superior to model (11) in the sense that model (13) is closer to the coupled model (6) than model (11), and the difference increases with the vibration amplitude represented by D here. It should be remarked that D = 0.5 m−1 means a rather large initial disturbance, corresponding to a maximum displacement of 0.125 m, which is 12.5% of the span. The differences between the models can also be illustrated via the transverse displacement of the string center. The displacements are plotted in Figs. 6–10, in which the dots, the solid lines and the dashed lines stand for the numerical solutions to Eqs. (6), (13) and (11). Figure 4 shows that the three solutions coincide when the motion is rather small, with the amplitude about 2.5% of the string length. Figure 5 shows that, when the motion amplitude is moderate, about 8% of the string length, solutions to Eqs. (6) and (13) are still almost the same while the solution to Eq. (11) is different from them. For the rather large motion, with the amplitude about 10% of the string length, Fig. 6 show that all three solutions are different while the solution to Eq. (13) is much closer to that of Eq. (11). Numerical results also show that the differences between Eqs. (11) and (13) are more significant in the case of a steel string. The high value of Young’s modulus of the steel string makes the effects of nonlinear terms in Eq. (11) and (13) more influential even for rather small amplitudes. Hence a larger value of Young’s modulus leads to a larger deviation of the model.
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Fig. 6 The transverse center displacement of the rubber string calculated from three models for D = 0.5
5 Conclusions Two nonlinear models of transverse vibration are compared based on the numerical solutions. The first model, a partial-differential equation, is derived from the governing equation for coupled planar vibrations by omitting its longitudinal terms. The model can be reduced to an integro-partial-differential equation, the Kirchhoff string equation, by averaging of the disturbance of the string tension. The numerical schemes are presented for these equations via the finite difference. An index is proposed to compare the transverse responses calculated from the two models with the transverse component calculated from the coupled equation. A steel string and a rubber string are employed to investigate the model errors. The numerical results lead to the following conclusions. (1) The errors of the two models are both relatively small even for reasonable large vibration amplitude. (2) The model errors increase with the vibration amplitude. (3) The Kirchhoff string equation yields better results, regardless of the validity of E A >> P0 . Acknowledgments The research is supported by the National Natural Science Foundation of China (Project no. 10672092), Natural Science Foundation of Shanghai Municipality (Project no. 04ZR14058), Shanghai Municipal Education Commission Scientific Research Project (No. 07ZZ07), and Shanghai Leading Academic Discipline Project (Project no. Y0103).
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