Dynamic Euler-Bernoulli Beam Equation: Classification and Reductions

Abstract

We study a dynamic fourth-order Euler-Bernoulli partial differential equation having a constant elastic modulus and area moment of inertia, a variable lineal mass density g(x), and the applied load denoted by f(u), a function of transverse displacement u(t,x). The complete Lie group classification is obtained for different forms of the variable lineal mass density g(x) and applied load f(u). The equivalence transformations are constructed to simplify the determining equations for the symmetries. The principal algebra is one-dimensional and it extends to two- and three-dimensional algebras for an arbitrary applied load, general power-law, exponential, and log type of applied loads for different forms of g(x). For the linear applied load case, we obtain an infinite-dimensional Lie algebra. We recover the Lie symmetry classification results discussed in the literature when g(x) is constant with variable applied load f(u). For the general power-law and exponential case the group invariant solutions are derived. The similarity transformations reduce the fourth-order partial differential equation to a fourth-order ordinary differential equation. For the power-law applied load case a compatible initial-boundary value problem for the clamped and free end beam cases is formulated. We deduce the fourth-order ordinary differential equation with appropriate initial and boundary conditions.

R. Naz 1 and F. M. Mahomed 2

Academic Editor:Bin Jiang

1, Centre for Mathematics and Statistical Sciences, Lahore School of Economics, Lahore 53200, Pakistan
2, DST-NRF Centre of Excellence in Mathematical and Statistical Sciences, School of Computer Science and Applied Mathematics, University of the Witwatersrand, Johannesburg, Wits 2050, South Africa

Received 8 May 2015; Revised 20 August 2015; Accepted 23 August 2015; 30 August 2015

This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

Daniel Bernoulli and Leonard Euler developed the theory of the Euler-Bernoulli beam problem. Let [figure omitted; refer to PDF] be the transverse displacement at time [figure omitted; refer to PDF] and position [figure omitted; refer to PDF] from one end of the beam taken as the origin, [figure omitted; refer to PDF] the flexural rigidity, and [figure omitted; refer to PDF] the lineal mass. The transverse motion of an unloaded thin beam is represented by the following fourth-order partial differential equation (PDE): [figure omitted; refer to PDF]

Euler-Bernoulli beam equation (1) has been frequently studied in the literature. Gottlieb [1] studied the isospectral properties of this equation and its nonhomogeneous variants with [figure omitted; refer to PDF] and [figure omitted; refer to PDF] . Soh [2] considered the equivalence problem for an Euler-Bernoulli beam utilizing the Lie symmetry approach. Later on Morozov and Soh [3] attempted the problem with the aid of Cartan's equivalence method. Recently, Ndogmo [4] obtained the complete equivalence transformations of the Euler-Bernoulli equation which were initially considered in the work [3] in terms of some undetermined set of functions. Özkaya and Pakdemirli [5], using the symmetry method, investigated the transverse vibrations of a beam moving with time-dependent axial velocity and obtained approximate solutions for an exponentially decaying and harmonically varying problem.

Now let [figure omitted; refer to PDF] be the elastic modulus, let [figure omitted; refer to PDF] be the area of inertia, let [figure omitted; refer to PDF] be the mass per unit length, let [figure omitted; refer to PDF] be the transverse displacement at time [figure omitted; refer to PDF] and position [figure omitted; refer to PDF] , and let [figure omitted; refer to PDF] be the applied load. The transverse motion of a loaded thin elastic beam is governed by the following dynamic beam fourth-order PDE [6]: [figure omitted; refer to PDF] where the applied load [figure omitted; refer to PDF] is a function of [figure omitted; refer to PDF] . Bokhari et al. [7] studied the following dynamic Euler-beam equation from the symmetry viewpoint with [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] as constants and [figure omitted; refer to PDF] dependent on [figure omitted; refer to PDF] : [figure omitted; refer to PDF] A complete group classification was obtained for (3). The symmetry reductions were derived to reduce the fourth-order PDE to fourth-order ordinary differential equations (ODEs). For the power-law load function, compatible initial-boundary value problems corresponding to clamped end and free end beams were formulated and the reduced fourth-order ODEs were determined. The static beam problem was discussed by Bokhari et al. [8].

The dynamic fourth-order Euler-Bernoulli PDE having a constant elastic modulus and area moment of inertia, a variable lineal mass density [figure omitted; refer to PDF] , and the applied load denoted by [figure omitted; refer to PDF] , a function of transverse displacement [figure omitted; refer to PDF] , is given by [figure omitted; refer to PDF]

In this paper we study dynamic Euler-Bernoulli beam equation (4) from the symmetry point of view.

We give a complete classification of the Lie symmetries for dynamic Euler-Bernoulli beam equation (4). The principal algebra is one-dimensional and it extends to two- and three-dimensional algebras for an arbitrary applied load, general power-law, exponential, and log type applied loads for different forms of [figure omitted; refer to PDF] (see Table 1). For the linear applied load case, we obtain an infinite-dimensional Lie algebra. We recover the Lie symmetry classification results discussed by Bokhari et al. [7] when [figure omitted; refer to PDF] is a constant with variable applied load [figure omitted; refer to PDF] . We derive the group invariant solutions for the general power-law and exponential cases. The fourth-order PDE reduces to a fourth-order ODE with the help of similarity transformations. For the power-law applied load case compatible initial-boundary value problems for the clamped and free end beam cases are formulated. We deduce the corresponding fourth-order ODE with appropriate initial and boundary conditions. We show that the solution fails to satisfy the initial or boundary conditions for the exponential and logarithmic cases.

Table 1: Complete Lie symmetry classification of beam PDE (4).

Equation	Lie symmetries
[figure omitted; refer to PDF]	[figure omitted; refer to PDF]
[figure omitted; refer to PDF]	[figure omitted; refer to PDF]
[figure omitted; refer to PDF]	[figure omitted; refer to PDF]
[figure omitted; refer to PDF]	[figure omitted; refer to PDF]
[figure omitted; refer to PDF]	[figure omitted; refer to PDF]
[figure omitted; refer to PDF]	[figure omitted; refer to PDF]
[figure omitted; refer to PDF]	[figure omitted; refer to PDF]
[figure omitted; refer to PDF]	[figure omitted; refer to PDF]
[figure omitted; refer to PDF]	[figure omitted; refer to PDF]
[figure omitted; refer to PDF]	[figure omitted; refer to PDF]
[figure omitted; refer to PDF]	[figure omitted; refer to PDF]
[figure omitted; refer to PDF]	[figure omitted; refer to PDF]
[figure omitted; refer to PDF]	[figure omitted; refer to PDF]
[figure omitted; refer to PDF] ,	[figure omitted; refer to PDF]
[figure omitted; refer to PDF]	[figure omitted; refer to PDF]
[figure omitted; refer to PDF] or [figure omitted; refer to PDF]
[figure omitted; refer to PDF] ,	[figure omitted; refer to PDF]
[figure omitted; refer to PDF]
[figure omitted; refer to PDF] , or [figure omitted; refer to PDF] ,	[figure omitted; refer to PDF]
[figure omitted; refer to PDF]

The paper is organized as follows. In Section 2, the complete Lie point symmetry classification up to equivalence transformations is presented. The nontrivial symmetry reductions and initial-boundary value problems which correspond to clamped and free end beams are discussed in Section 3. The conclusions are summarized in the last section.

2. Complete Lie Symmetry Classification

We derive the equivalence transformations which are important for the simplification of the determining equations and for obtaining disjoint classes [9]. Equivalence transformations of the PDE (4) are point transformations in the space of independent and dependent variables of the equation and these point transformations leave invariant family (4). That is, the equivalence transformations transform any equation (4) with arbitrary functions [figure omitted; refer to PDF] and [figure omitted; refer to PDF] into the same family (4) with, in general, different functions [figure omitted; refer to PDF] and [figure omitted; refer to PDF] . Equivalence transformations of the PDE (4) are easy to obtain although the computations are tedious. These are [figure omitted; refer to PDF] where [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] are constants and [figure omitted; refer to PDF] .

The Lie point symmetry generator [figure omitted; refer to PDF] of dynamic Euler-Bernoulli equation (4), is derived by solving [figure omitted; refer to PDF] where [figure omitted; refer to PDF] is the fourth prolongation of the operator [figure omitted; refer to PDF] . The fourth prolongation of the generator [figure omitted; refer to PDF] is defined as [figure omitted; refer to PDF] where [figure omitted; refer to PDF] can be determined from [figure omitted; refer to PDF] in which [figure omitted; refer to PDF] is the total derivative operator. Equation (7) is separated according to the derivatives of [figure omitted; refer to PDF] and an overdetermined system of partial differential equations for the unknown coefficients [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] is obtained. The determining equations finally yield [figure omitted; refer to PDF] and the following classification equations for [figure omitted; refer to PDF] and [figure omitted; refer to PDF] : [figure omitted; refer to PDF] where [figure omitted; refer to PDF] are constants. For the case, [figure omitted; refer to PDF] arbitrary in [figure omitted; refer to PDF] and [figure omitted; refer to PDF] arbitrary in [figure omitted; refer to PDF] , the only symmetry is [figure omitted; refer to PDF] which constitutes the one-dimensional principal algebra of (4). Now we investigate all the possibilities of [figure omitted; refer to PDF] and [figure omitted; refer to PDF] for which an extension of the principal algebra is possible.

Differentiating classification equation (12) with respect to [figure omitted; refer to PDF] , we have [figure omitted; refer to PDF] Further differentiating (15) twice with respect to [figure omitted; refer to PDF] yields [figure omitted; refer to PDF] Equation (17) after using [figure omitted; refer to PDF] from (16) becomes [figure omitted; refer to PDF] Now differentiating (15) with respect to [figure omitted; refer to PDF] , we have [figure omitted; refer to PDF] We at once look at the possible cases of [figure omitted; refer to PDF] from (18) and (19).

Case 1 ( [figure omitted; refer to PDF] ).

If [figure omitted; refer to PDF] then (15) gives [figure omitted; refer to PDF] and we have [figure omitted; refer to PDF] where [figure omitted; refer to PDF] is arbitrary and (13) gives [figure omitted; refer to PDF] .

Case 2 ( [figure omitted; refer to PDF] and [figure omitted; refer to PDF] ).

If [figure omitted; refer to PDF] and [figure omitted; refer to PDF] then (18) yields the following possible forms of [figure omitted; refer to PDF] :

(i) [figure omitted; refer to PDF] for [figure omitted; refer to PDF] , [figure omitted; refer to PDF] ;

(ii) [figure omitted; refer to PDF] , [figure omitted; refer to PDF] ;

(iii): [figure omitted; refer to PDF] , [figure omitted; refer to PDF] ,

where [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] are constants and from (19), [figure omitted; refer to PDF] . It is worth mentioning here that the principal algebra extends for these forms of [figure omitted; refer to PDF] .

After equivalence transformations, the simplified forms for [figure omitted; refer to PDF] are

(i) [figure omitted; refer to PDF] for [figure omitted; refer to PDF] ,

(ii) [figure omitted; refer to PDF] ,

(iii): [figure omitted; refer to PDF] .

Case 2.1 ( [figure omitted; refer to PDF] ) . Equation (12) for [figure omitted; refer to PDF] with [figure omitted; refer to PDF] results in the following operators: [figure omitted; refer to PDF] with [figure omitted; refer to PDF] from (13) satisfying [figure omitted; refer to PDF] The solution of (22) is [figure omitted; refer to PDF] . When [figure omitted; refer to PDF] then [figure omitted; refer to PDF] in (21) and (22).

Case 2.2 ( [figure omitted; refer to PDF] ) . Equation (12) for [figure omitted; refer to PDF] with [figure omitted; refer to PDF] results in the following operators: [figure omitted; refer to PDF] with [figure omitted; refer to PDF] satisfying (22). When [figure omitted; refer to PDF] then [figure omitted; refer to PDF] in (22) and (23).

Case 2.3 ( [figure omitted; refer to PDF] ) . In this case [figure omitted; refer to PDF] , we have [figure omitted; refer to PDF] with [figure omitted; refer to PDF] from (13) satisfying [figure omitted; refer to PDF] The solution of (25) is [figure omitted; refer to PDF] . We deduce the same results for both cases [figure omitted; refer to PDF] and [figure omitted; refer to PDF] .

Case 3 ( [figure omitted; refer to PDF] ).

For the case [figure omitted; refer to PDF] , the equivalence transformations yield [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] , where [figure omitted; refer to PDF] .

Case 3.1 ( [figure omitted; refer to PDF] ) . Equations (12) and (13) yield [figure omitted; refer to PDF] where [figure omitted; refer to PDF] and [figure omitted; refer to PDF] satisfy [figure omitted; refer to PDF]

Case 3.2 ( [figure omitted; refer to PDF] ) . From (12) and (13), we have [figure omitted; refer to PDF] with [figure omitted; refer to PDF] , [figure omitted; refer to PDF] satisfying [figure omitted; refer to PDF]

Case 3.3 ( [figure omitted; refer to PDF] ) . For this case the symmetry generator is the same as that given in (28) with [figure omitted; refer to PDF] satisfying (29) and [figure omitted; refer to PDF] : [figure omitted; refer to PDF]

Remark 1.

The Lie algebras for all the nonlinear cases in Table 1 are easily seen by inspection. For the linear cases, they are infinite-dimensional and well known (see also [9]). Also the solution [figure omitted; refer to PDF] of the linear equation in the final linear case of Table 1 is easy to obtain and is not given as we do not use it here.

Now we work out the equivalence transformations for different forms of [figure omitted; refer to PDF] arising from Cases 2 and 3. After equivalence transformations [figure omitted; refer to PDF] becomes as follows:

(i) [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , becomes [figure omitted; refer to PDF] for [figure omitted; refer to PDF] , [figure omitted; refer to PDF] ; if [figure omitted; refer to PDF] , then [figure omitted; refer to PDF] ;

(ii) [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , becomes [figure omitted; refer to PDF] , [figure omitted; refer to PDF] ; if [figure omitted; refer to PDF] , then [figure omitted; refer to PDF] with [figure omitted; refer to PDF] ;

(iii): if [figure omitted; refer to PDF] solves [figure omitted; refer to PDF]

: then after equivalence transformations it reduces to [figure omitted; refer to PDF] with solution [figure omitted; refer to PDF]

(iv) if [figure omitted; refer to PDF] satisfies [figure omitted; refer to PDF]

: then after equivalence transformations it becomes [figure omitted; refer to PDF]

: where [figure omitted; refer to PDF] is arbitrary.

The Lie symmetries for the simplified forms of [figure omitted; refer to PDF] and [figure omitted; refer to PDF] are presented in Table 1 and all cases discussed in [7] are recovered for [figure omitted; refer to PDF] .

3. Symmetry Reductions and Boundary Value Problems

Now we find the symmetry reductions. The initial conditions are [figure omitted; refer to PDF] The four types of boundary conditions (see [7]) are as follows:

: hinged end: [figure omitted; refer to PDF]

: clamped end: [figure omitted; refer to PDF]

: free end: [figure omitted; refer to PDF]

: sliding end: [figure omitted; refer to PDF]

: consider the power-law case: [figure omitted; refer to PDF]

If we take a linear combination of the symmetries [figure omitted; refer to PDF] , the initial condition [figure omitted; refer to PDF] and boundary condition [figure omitted; refer to PDF] are left invariant only by the scaling symmetry [figure omitted; refer to PDF] . The group invariant solution corresponding to [figure omitted; refer to PDF] is [figure omitted; refer to PDF] where [figure omitted; refer to PDF] is the similarity variable. The substitution of (42) into (41) yields the following ODE: [figure omitted; refer to PDF]

[figure omitted; refer to PDF]

For the clamped end beam, the initial and boundary conditions (36) and (38) yield [figure omitted; refer to PDF] where [figure omitted; refer to PDF] . Note that the boundary condition at [figure omitted; refer to PDF] added is due to the solution of the ODE.

We solve (43) subject to conditions (44) and then (42) forms the solution of the clamped end beam (41).

For the free end beam, the initial and boundary conditions (36) and (39) yield [figure omitted; refer to PDF]

For the hinged end beam, the ODE (43) should be solved subject to boundary conditions (44) as well as [figure omitted; refer to PDF] . For the case of the sliding end beam one has to solve ODE (43) subject to (45).

For the following power-law case, the PDE is [figure omitted; refer to PDF] and reduction via [figure omitted; refer to PDF] yields the invariant solution [figure omitted; refer to PDF] Here the second boundary condition [figure omitted; refer to PDF] for the clamped end is not satisfied.

An asymptotic solution was found by Bokhari et al. [7] corresponding to the clamped or free end case for the following power-law case [figure omitted; refer to PDF]

For the remaining cases the solution does not satisfy the initial or boundary conditions. We take one example below.

Consider the following exponential case: [figure omitted; refer to PDF] The group invariant solution of (49) is of the form [figure omitted; refer to PDF] The substitution of (50) into (49) yields the following fourth-order ODE: [figure omitted; refer to PDF] and this ODE fails to satisfy the boundary conditions at [figure omitted; refer to PDF] .

Similarly for the rest of the cases, the initial or boundary conditions are not satisfied when the load is of exponential or logarithmic form. We have ignored the linear cases as much attention has been already focused on these cases.

4. Concluding Remarks

We have performed the complete Lie symmetry classification of the dynamic fourth-order Euler-Bernoulli PDE having a constant elastic modulus and area moment of inertia, a variable lineal mass density [figure omitted; refer to PDF] , and the applied load denoted by [figure omitted; refer to PDF] , a function of transverse displacement [figure omitted; refer to PDF] . The equivalence transformations are constructed to simplify the determining equations for the symmetries. The simplified forms of lineal mass density [figure omitted; refer to PDF] and applied load [figure omitted; refer to PDF] are constructed via equivalence transformations. The principal algebra is one-dimensional for arbitrary [figure omitted; refer to PDF] and [figure omitted; refer to PDF] . The principal algebra is extended to a two- and three-dimensional algebra for arbitrary applied load, general power-law, exponential, and log type of applied loads for different forms of [figure omitted; refer to PDF] whereas an infinite-dimensional algebra is obtained for the linear applied load case (see Table 1). We recover the Lie symmetry classification results discussed in the literature when [figure omitted; refer to PDF] is constant with variable applied load [figure omitted; refer to PDF] . The similarity transformations reduce the fourth-order PDE to a fourth-order ODE. Only for one case with the applied load power-law, compatible initial-boundary value problems for the clamped and free end beam cases are formulated. We deduce the fourth-order ODE with appropriate initial and boundary conditions.

Acknowledgments

F. M. Mahomed thanks the National Research Foundation (NRF) of South Africa for a research grant that has facilitated this research through the Unique Grant no. 92857, 2014. R. Naz thanks Lahore School of Economics for providing funds to complete this work.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

References

[1] H. P. W. Gottlieb, "Isospectral Euler-Bernoulli beams with continuous density and rigidity functions," Proceedings of the Royal Society of London Series A: Mathematical, Physical and Engineering Sciences , vol. 413, no. 1844, pp. 235-250, 1987.

[2] C. W. Soh, "Euler-Bernoulli beams from a symmetry standpoint-characterization of equivalent equations," Journal of Mathematical Analysis and Applications , vol. 345, no. 1, pp. 387-395, 2008.

[3] O. I. Morozov, C. W. Soh, "The equivalence problem for the Euler-Bernoulli beam equation via Cartan's method," Journal of Physics A: Mathematical and Theoretical , vol. 41, no. 13, pp. 135-206, 2008.

[4] J. C. Ndogmo, "Equivalence transformations of the Euler-Bernoulli equation," Nonlinear Analysis: Real World Applications , vol. 13, no. 5, pp. 2172-2177, 2012.

[5] E. Özkaya, M. Pakdemirli, "Group-theoretic approach to axially accelerating beam problem," Acta Mechanica , vol. 155, no. 1-2, pp. 111-123, 2002.

[6] A. E. H. Love A Treatise on the Mathematical Theory of Elasticity , Dover Publications, New York, NY, USA, 1944., 4th.

[7] A. H. Bokhari, F. M. Mahomed, F. D. Zaman, "Invariant boundary value problems for a fourth-order dynamic Euler-Bernoulli beam equation," Journal of Mathematical Physics , vol. 53, no. 4, 2012.

[8] A. H. Bokhari, F. M. Mahomed, F. D. Zaman, "Symmetries and integrability of a fourth-order Euler-Bernoulli beam equation," Journal of Mathematical Physics , vol. 51, no. 5, 2010.

[9] N. H. Ibragimov CRC Handbook of Lie Group Analysis of Differential Equations , vol. 1-3, Chemical Rubber Company, Boca Raton, Fla, USA, 1994-1996.