Abstract
In theoretical mechanics field, solution methods for nonlinear differential equations are very important because many problems are modelled using such equations. In particular, large deflection of a cantilever beam under a terminal follower force and nonlinear pendulum problem can be described by the same nonlinear differential equation. Therefore, in this work, we propose some approximate solutions for both problems using nonlinearities distribution homotopy perturbation method, homotopy perturbation method, and combinations with Laplace-Padé posttreatment. We will show the high accuracy of the proposed cantilever solutions, which are in good agreement with other reported solutions. Finally, for the pendulum case, the proposed approximation was useful to predict, accurately, the period for an angle up to yielding a relative error of 0.01222747.
1. Introduction
Solving nonlinear differential equations is an important issue in science because many physical phenomena are modelled using such equations. During last century, the approximate solution of such equations was a task performed by hand using methods like the perturbation method. Nowadays, symbolic software like Maple or Mathematica allow researchers to calculate highly accurate approximate solutions using new methods like the homotopy perturbation method (HPM) [1–38]. The HPM method is one of the most famous analytic techniques for nonlinear differential equations, which is widely applied in science and engineering.
In [28, 39] was reported the nonlinearities distribution homotopy perturbation method (NDHPM) as an extended version of HPM that allows the distribution of the nonlinearities between the different iterations of HPM method. The main advantage of this process is the increase in the number of possible iterations by constructing easier to solve linear equations for HPM procedure. Additionally, several works reported that a Laplace-Padé posttreatment [40–49] of power series solutions obtained by HPM can improve accuracy and convergence to the exact solution. Therefore, in this work, we propose the use of the aforementioned methods to obtain approximate solutions for two nonlinear problems, related by to same nonlinear differential equation. The first one is the large deflection of a cantilever beam under a terminal follower force [50–55]. This kind of structures arises in marine riser and other practical applications of civil engineering. The mathematical formulation of such a problem yields a nonlinear two-point boundary-value problem, which usually can be solved by numerical methods [50]. The second one is the nonlinear pendulum problem [56–58], which is modelled by the same equation of the Cantilever’s case study. The pendulum or systems containing the pendulums are well-studied complex phenomen [59]. For both problems, the results exhibited high accuracy using our proposed solutions, which are in good agreement to the results reported by other works. In particular, for the pendulum case, we succeeded to predict, accurately, the period until an initial angle up to achieving a relative error of .
This paper is organized as follows. In Section 2, the basic idea of the HPM method is provided. We give an introduction to NDHPM method in Section 3. In Section 4, the basic concept of Padé approximants is explained. The Laplace-Padé coupling with NDHPM and HPM methods is presented in Section 5. In Sections 6 and 7, the cantilever and pendulum problems are solved, respectively. In addition, a discussion on the results is presented in Section 8. Finally, a brief conclusion is given in Section 9.
2. Basic Concept of HPM
It can be considered that a nonlinear differential equation can be expressed as having boundary condition as where is a general differential operator, is a known analytic function, is a boundary operator, is the boundary of domain , and denotes differentiation along the normal drawn outwards from [50]. The operator, generally, can be divided into two operators, and , which are linear and nonlinear operators, respectively. Hence, (1) can be rewritten as
Now, a possible homotopy formulation is where is the initial approximation for (3) which satisfies the boundary conditions and is known as the perturbation homotopy parameter.
For the HPM method [4–7], we assume that the solution for (4) can be written as a power series of
Considering that , results that the approximate solution of (1) is
The series (6) is convergent on most cases as reported in [4, 7, 20, 21].
3. HPM Method with Nonlinearities Distribution
Recent reports [28, 39] have introduced the NDHPM method, which eases the searching process of solutions for (3) and reduces the complexity of solving differential equations. As first step, a modified homotopy is introduced:
It can be noticed that the homotopy function (7) is essentially the same as (4), except for the nonlinear operator and the nonhomogeneous function , which embeds the homotopy parameter . The arbitrary introduction of within the differential equation is a strategy to redistribute the nonlinearities between the successive iterations of the HPM method and, thus, increase the probabilities of finding the sought solution.
Again, we establish that
Considering that turns out that the approximate solution for (1) is
The convergence of the NDHPM method is exposed in [28].
4. Padé Approximants
A rational approximation to on is the quotient of two polynomials and of degrees and , respectively. We use the notation to denote this quotient. The Padé approximations to a function are given by [49, 60]
The method of Padé requires that and its derivative should be continuous at . The polynomials used in (10) are
The polynomials in (11) are constructed so that and agree at and their derivatives up to agree at . For the case where , the approximation is just the Maclaurin expansion for . For a fixed value of the error is the smallest when and have the same degree or when has a degree higher than .
Notice that the constant coefficient of is . This is permissible because it can be noted that and are not changed when both and are divided by the same constant. Hence, the rational function has unknown coefficients. Assume that is analytic and has the Maclaurin expansion
And, from the difference
The lower index in the summation at the right side of (13) is chosen because the first derivatives of and should agree at .
When the left side of (13) is multiplied out and coefficients of the powers of are set equal to zero for , this results in a system of linear equations:
Notice that in each equation the sum of the subscripts on the factors of each product is the same, and this sum increases consecutively from to . The equations in (15) involve only the unknowns and must be solved first. Then, the equations in (14) are successively used to find [49].
5. Laplace-Padé Posttreatment of Power Series Solutions
The coupling of Laplace transform and Padé approximant [40] is used to deal with the lost information of truncated power series [41–49]. The process can be summarized as follows.(1)First, Laplace transformation is applied to a power series solution obtained by HPM or NDHPM methods. (2)Next, is written in place of . (3)Afterwards, we convert the transformed series into a meromorphic function by forming its Padé approximant of order . and are arbitrarily choses, but they should be smaller than the power series order. In this step, the Padé approximant extends the domain of the truncated series solution to obtain better accuracy and convergence. (4)Then, is written in place of . (5)Finally, by using the inverse Laplace transformation, we obtain the modified approximate solution.
The Laplace-Padé posttreatment of HPM and NDHPM methods will be referred throughout the rest of this paper as LPHPM and LPNDHPM, respectively.
6. Solution of Cantilever Problem
Beams subjected to follower load arise in many problems of structural engineering like lumbar spine in biomechanics, smart structures applications, and loads applied to structural components by cables in tension, among others. The governing equations of such structures involve the effects of nonlinearities due to large deformations and material properties [50–55].
In this work, we study the large-deflection problem of a cantilever beam under a tip-concentrated follower force as depicted in Figure 1. The angle of inclination of the force with respect to the deformed axis of the beam remains unchanged during deformation. We assume that the center of the Cartesian system is located at the fixed end and is the local angle of inclination of the centroidal axis beam. Therefore, the mathematical formulation of this problem yields the following nonlinear two-point boundary-value problem: where prime denotes differentiation with respect to . Additionally, the related dimensionless quantities are defined as where represents the length of the originally straight elastic cantilever beam, is the flexural rigidity, is the arc length measured from the tip, and is the terminal follower force with constant angle .
In [50] is reported that (16) can be reduced to an initial-value problem by the following transformation: resulting in the following initial value problem: with the extra condition: where the slope of can be obtained by using (18) and (20), resulting in the following:
Finally, the deformed curve of beam axis can be calculated from solution of the slope as
6.1. Solution Obtained by Using NDHPM Method
In order to circumvent difficulties due to the nonlinear term of (19), we propose to use a seventh-order Taylor series expansion, resulting in the following: and by using (7) and (23), we establish the following homotopy equation: where homotopy parameter has been arbitrarily embedded into the power series of Taylor’s expansion, and linear operator is and trial function is
Substituting (8) into (24) and reordering and equating terms with the same powers, we obtain the following linear set of differential equations: By solving (27), we obtain where .
Substituting solutions (28) into (8) and calculating the limit when results in the third-order approximation:
6.2. Solution Calculated by Using HPM and LPHPM Methods
By using (4) and (23), we establish the following homotopy equation: where linear operator is and trial function is
Substituting (5) into (30) and reordering and equating terms with the same powers, we obtain the following set of linear differential equations: By solving (33), we obtain where the subsequent iterations are calculated using Maple CAS software.
Substituting solutions (34) into (5) and calculating the limit when , we can obtain the 18th-order approximation:
Expression (35) is too long to be written here; nevertheless, it can be simplified if we consider a particular case. For instance, when , it results in the following:
By using (36) and (21) we can calculate, approximately, the deflection angle ; nevertheless, accuracy can be increased applying the Laplace-Padé posttreatment. First, Laplace transformation is applied to (36); then, is written in place of . Afterwards, Padé approximant is applied and is written in place of . Finally, by using the inverse Laplace transformation, we obtain the modified approximate solution. For instance, for , the result of LPHPM is
On one side, for , the exact result is [51]. On the other side, by using (21), we can calculate the deflection angle using (36) and (37) approximations, resulting in and , respectively. Therefore, for this case, the Laplace-Padé posttreatment was relevant in order to deal with the HPM truncated power series, resulting a notorious increase of accuracy (see Table 1).
6.3. Power Series Solution by Using NDHPM and Laplace-Padé Posttreatment
By using (7) and (23), we establish the following homotopy equation: where homotopy parameter has been arbitrarily embedded into the power series of Taylor expansion, linear operator , and trial function are taken from (31) and (32), respectively.
Substituting (8) into (38), reordering and equating terms with the same powers, we obtain the following set of linear differential equations:
Solving (39), we obtain where the subsequent iterations are calculated using Maple CAS software.
Substituting solutions (40) into (8) and calculating the limit when , we can obtain the 18th-order approximation:
Expression (41) is too long to be written here; nevertheless, it can be simplified considering a particular case. For instance, a terminal follower normal angle , resulting
In order to increase accuracy of , (42) is transformed by the Padé approximation and Laplace transformation. First, Laplace transformation is applied to (42) and then is written in place of . Then, Padé approximant is applied and is written in place of . Finally, by using the inverse Laplace transformation, we obtain the modified approximate solution. For example, given , the resultant LPNDHPM approximation is
Now, by using (21), we can calculate for (42) and (43) approximations. Results for NDHPM and LPNDHPM are and , respectively (see Table 1). In fact, for , the exact result is ; therefore, the Laplace-Padé after-treatment was a key factor to increase accuracy of NDHPM truncated power series.
7. Solution of Nonlinear Pendulum
The pendulum [56–58] of Figure 2 can be modelled by (19). Nevertheless, we need to redefine the variables and parameters of (19), resulting in the following: considering where prime denotes differentiation with respect to time , is the length of the pendulum, is acceleration due to gravity, and is the initial angle of displacement .
Taking into account the above considerations and substituting into (29), results that the approximate solution (29) obtained by using NDHPM can be considered, also, as the solution of the nonlinear pendulum problem (44).
7.1. Solution Obtained by Using HPM and Laplace-Padé Posttreatment
Although approximate solutions (35) and (41) possess good accuracy for cantilever problem, we need to increase the order of Taylor expansion for the term in (23) to obtain a highly accurate solution for pendulum problem, compared to other reported solutions [50–55]. Therefore, we propose to use a 22th-order Taylor series expansion of term, resulting in the following: where initial conditions are and .
A homotopy formulation that can generate a power series solution for this problem is where linear operator and trial function are taken from (31) and (32), respectively.
Substituting (5) into (47) and reordering and equating terms with the same -powers, we obtain the following set of linear differential equations: by solving (48), we obtain where the subsequent iterations are calculated by using Maple CAS software.
Substituting solutions (49) into (5) and calculating the limit when , we can obtain the 12th-order approximation:
Expression (50) is too large to be written here; nevertheless, it can be simplified considering particular cases. For example, if we consider and , we get the following results:
Now, (51) is transformed by the Laplace-Padé posttreatment. First, Laplace transformation is applied to (51) and then is written in place of . Then, Padé approximant is applied and is written in place of . Finally, by using the inverse Laplace transformation, we obtain the modified approximate LPHPM solution:
Equation (52) is more compact and computationally efficient than power series solution (51). Thus, (52) can be used when accurate and handy expressions are needed. This process can be repeated for different values of and .
We can express pendulum period in terms of where is the complete elliptic function of the first kind formulated as
For comparison purposes, we will normalize the value of to
Finally, is calculated for some values of in Tables 3 and 4.
8. Numerical Simulation and Discussion
8.1. Cantilever Problem
Structures subjected to follower forces have been reported by several researchers [50, 55]. In this work, we propose the solution for large deflections of a cantilever beam subjected to a follower force reported in [55]. Therefore, by using approximate solution (29) and (21), we show in Figure 3 the relationship between terminal slope and load parameter for some values of deflection angle in the range . Flutter instability occurs, and the maximum value of terminal slope is approximately as reported in [50, 55].
On one side, considering a fixed value for , we show in Figure 4 the deformed curves of a beam under different values of load parameter . First, the cantilever beam is a straight line, and it is deformed gradually as the amplitude of the end-concentrated follower force increases, until the curvature of the beam is notorious for . Additionally, when the load parameter is larger than the flutter load, the beam deforms towards the opposite direction as reported in [50]. On the other side, we consider a fixed value for load parameter and different values of as shown in Figure 5, resulting that the deflection and terminal slope are influenced by the angle of follower force producing a notorious bending as increases reaching the extreme case at .
Considering a normal angle , we show in Table 1 the terminal slope (using (21)) versus different values of load parameter . In such table, we compare the proposed solutions (29), (35), (41), LPHPM/LPNDHPM approximations (see (37), and (43)) to exact solution [51], a shooting method solution [53], and HPM solution reported in [50]. Our proposed approximate solutions are in good agreement with the other reported approximations. In particular, (29) and LPNDHPM approximations (see (43)) exhibited the best accuracy of all proposed solutions of this work, with a similar accuracy compared to the results reported in [50]. Furthermore, we showed in Table 2, the terminal slope and coordinates (see (22)) of cantilever beam under normal follower forces (), resulting that our proposed solution (29) exhibited an accuracy in good agreement with the values obtained by a direct method (D.M.) [55] and the results reported in [50]. For this work, the cantilever beam is considered elastic and ideal; then, Figures 4 and 5 present an ideally deformed beam. Further work should be performed in order to include nonideal effects due to material characteristics of cantilever.
From above, we can conclude that the deflection and terminal slope are governed by the deflection angle and magnitude of the follower force. This means that the bending of the beam becomes more notorious as the deflection angle or the magnitude of the follower force increases [50].
8.2. Pendulum Problem
On one hand, in [56, 58] the authors achieve approximate solutions for pendulum period, with relative error <2% for , and <1% for , respectively. On the other hand, in Table 3 and Table 4 we show a comparison between , calculated using (29) and (50) and approximations calculated by LPHPM procedure explained in Section 7.1 (considering ). The value is obtained, applying Newton-Raphson method to equation , that is, calculating the first crossing of by zero since the initial angle . In Table 4, we show the relative error (RE) for the range of , resulting that our proposed solutions are in good agreement with exact results. Furthermore, LPHPM solution can predict the for exhibiting a RE of , which is highly accurate compared to the results obtained in [56, 58].
The coupling of Laplace-Padé and HPM method was a key factor to increase the accuracy range until value is really close to . Laplace-Padé posttreatment was able to deal with the truncate power series, in order to obtain better accuracy for angles close to although the behaviour was different for small values of . However, in general terms, the Laplace-Padé posttreatment generates handy and computationally more efficient expressions (see (51) versus (52)). Furthermore, we can appreciate from Tables 3 and 4 that HPM and NDHPM methods do not present useful values for and , due to numerical noise of the proposed approximations. Fortunately, LPHPM approximations achieve a good accuracy within the range . From the above discussion we can conclude that, as the initial angle of displacement approximates to the vicinity , the pendulum period increases notoriously.
Finally, further research should be done in order to obtain approximate solutions of the pendulum problem involving additional nonlinear effects like friction and mass, among others.
9. Conclusions
In this paper we presented the solution for two nonlinear problems related by the same differential equation. The first one was the large deflection of a cantilever beam under a terminal follower force. The second one was the nonlinear pendulum problem. We proposed a series of solutions obtained by NDHPM, HPM, and combinations with Laplace-Padé posttreatment. For both problems, results exhibited high accuracy, which are in good agreement with the results reported by other works. In particular, for the pendulum case, we succeeded to predict, accurately, the period for an initial angle up to . By means of two case studies, we confirmed that NDHPM, HPM, LPNDHPM, and LPHPM are powerful and useful methods; all of them capable of generating highly accurate approximations for nonlinear problems.
Acknowledgments
The authors gratefully acknowledge the financial support provided by the National Council for Science and Technology of Mexico (CONACyT) through Grant CB-2010-01 no. 157024. The authors would like to express their gratitude to Rogelio-Alejandro Callejas-Molina and Roberto Ruiz-Gomez for their contribution to this paper.