INTRODUCTION
Liao (1992) employed the basic ideas of the homotopy
in topology to propose a general analytic method for nonlinear problems, namely
the homotopy analysis method and then modified it, step by step (Liao,
2003, 2004). This method does not need small/large
parameters and has been successfully applied to solve many types of nonlinear
problems in solid and fluid mechanics (Cheng et al.,
2005; Rahimpour et al., 2008;
Kimiaeifar, 2008; Kimiaeifar and Saidai, 2008; Sajid
et al., 2008).
Recently, considerable attention has been directed towards analytical solutions
for nonlinear equations based on homotopy technique. Homotopy theory becomes
a powerful mathematical tool, when it is successfully coupled with the perturbation
theory (He, 1998, 2000; Hillermeier,
2001; Kimiaeifar, 2008). He’s ParameterExpanding
Method (PEM) is one of the most effective and convenient method for analytical
solving of nonlinear differential equations. PEM has been shown to effectively,
easily and accurately solve a large class of linear and nonlinear problems with
components converging rapidly to accurate solutions. PEM was first proposed
by He (2006) and was successfully applied to various engineering
problems. It is worth mentioning that there are a few works on using parameterexpanding
method in the literature; Xu (2007) suggested He’s
parameterexpanding methods for strongly nonlinear oscillators. Tao
(2008) proposed frequencyamplitude relationship of nonlinear oscillators
using PEM.
Prestressed cable structures and their continuous counterparts in membrane or fabric structures, are often perceived as architecturally elegant structural forms; particularly for large clear span coverings. The extremely low weight to plan area ratio of such structures and the associated curved surfaces, often resent cable and fabric structures as refreshing alternatives to the more common bulky rectangular forms. The use of prestressed mechanisms as structural forms also tends to give clients and the general public the impression of utilizing the most modern of available technology. However, the same three properties of lowweight, unusual curved surfaces and nonlinear response to load, combine to form challenging problems to the structural engineer charged with ensuring a cable or fabric structure has safe dynamic characteristics, especially under wind loading. Nonlinear vibration has several phenomena not found in linear vibration and, in particular, any displacementtime relationship is dependent on initial conditions. Thus different values of socalled natural frequencies can be obtained for a given system simply by altering the initial velocity or displacement.
In this study He’s parameterexpanding method and homotopy analysis method
are used to calculate the displacement functions of geometrically nonlinear
prestressed cable structures. It is shown that the HAM solution is very accurate
for whole domain and for all effective parameters by using high number of series
solutions. In PEM solution only one term in series expansions is sufficient
to obtain an accurate solution but increasing the coefficients of nonlinear
term, the error of PEM solution increases.
GOVERNING EQUATION OF VIBRATION OF TWOLINK STRUCTURE
The symmetrical prestressed twolink structure is shown in Fig.
1 which has a single degree of freedom. It can be shown (Kwan,
1998) that a central load P for this structure is related to its corresponding
static deflection x by:
where, EA is the axial stiffness, t_{0} is the initial pretension and
L_{0} is the original undeformed length, of the twolinks. Consider
now the vibration of the two links such that they remain straight throughout
in which case the acceleration of a small element of length dx at a distance
x from the support is xy/L_{0}, where, y is the acceleration of joint
B. The D’Alembert forces D for one link are thus given by:
where, ρ is the mass per unit length of the links. If we isolate the portion
BC and take free body moment of the portion BC about B, it is obtained:
and
where, R is the vertical dynamic reaction at the supports. Substitution of
Eq. 13 into the overall vertical equilibrium
relationship,
leads to:

Fig. 1: 
Geometry of problem: The symmetrical twolink structure (Bars
AB and BC Joint at point B) 
which is the equation describing the free undamped vibration of the prestressed
twolink. If the twolink structure had a concentrated mass M at B, then Eq.
5 would be altered slightly to:
By definition:
Eq. 6 reduce to:
APPLICATION OF HAM
The governing equation for the nonlinear prestressed cable structures is expressed
by Eq.7. Nonlinear operator is defined as follow:
where, q∈[0,1] is the embedding parameter. As the embedding parameter
increases from 0 to 1, U(t; q), varies from the initial guess, U_{0}(t),
to the exact solution, U(t):
Expanding x(t;q) in Taylor series with respect to q results in:
Where:
Homotopy analysis method can be expressed by many different base functions
(Liao, 2003), according to the governing equations; it is
straightforward to use a base function in the form of:
where, b_{kpm} are the coefficients to be determined. When the base
function is selected, the auxiliary functions H(t), initial approximations U_{0}(t)
and the auxiliary linear operators L must be chosen in such a way that the corresponding
highorder deformation equations have solutions with the functional form similar
to the base functions. This method referred to as the rule of solution expression
(Liao, 2003).
The linear operator L is chosen as:
From Eq. 15 and 16 results in:
where, c_{1} to c_{2} are the integral constants. According
to the rule of solution expression and the initial conditions, the initial approximations,
U_{0}(t) as well as the integral constants, c_{1} to c_{2}
are formed as:
The zeroth order deformation equation for U(t) is:
According to the rule of solution expression and from Eq. 12,
the auxiliary function H(t) can be chosen as follows:
Differentiating Eq. 16, m times, with respect to the embedding
parameter q and then setting q = 0 in the final expression and dividing it by
m!, it is reduced to:
Equation 19 is the mth order deformation equation for x(t),
where:
and
As a result, the first and second terms of the solution’s series are as
follows:
The solution’s series U(t) is developed up to 12th order of approximation.
CONVERGENCE OF HAM SOLUTION
The analytical solution should converge. It should be noted that the auxiliary
parameter ħcontrols the convergence and accuracy of the solution series
(Liao, 2003). The analytical solution represented by Eq.
12 contains the auxiliary parameter ħ, which gives the convergence
region and rate of approximation for the homotopy analysis method In order to
define a region such that the solution series is independent on ħ, a multiple
of ħcurves are plotted. The region where the distribution of x’ and
x versus ħis a horizontal line is known as the convergence region for the
corresponding function. The common region among the x(t) and its derivatives
are known as the overall convergence region.
To study the influence of ħon the convergence of solution, the ħcurves
of x’ and x(1) are plotted for different values of constant parameters,
as shown in Fig. 2. Moreover, increasing the order of approximation
increases the range of the convergence region (Fig. 3).

Fig. 2: 
The ħ curves to indicate the convergence region, EA =
556 kN, L_{0} = 1.143 m, ρ = 4.6416x10^{2} kg m^{1}
and M = 100 kg m^{1} and M = 100 kg: (a) t_{0} = 3558.6
N, (b) t_{0} = 4448 N 

Fig. 3: 
The effect of order of approximation on convergence region,
EA = 556 kN, L_{0} = 1.143 m, ρ = 4.6416x10^{2} kg
m^{1} and M = 100 kg m^{1} and t_{0} = 4448 N:
(a) x (1), (b) x’ (1) 
PEM FOR SOLVING THE PROBLEM
According to the PEM (He, 2006), Eq. 7
can be rewritten as:
and the initial conditions are as follows:
The form of solution and the constants one in Eq. 25 can
be expanded as:
Substituting Eq. 2729 into Eq.
25 and processing as the standard perturbation method, we have:
The solution of Eq. 30:
Substituting x_{0}(t) from the above equation into Eq.
31 results in:
But from Eq. 28 and 29:
Based on trigonometric functions properties we have:
Replacing Eq. 32 into 31 and eliminating
the secular terms yields:
Set p = 1 then two roots of this particular equation can be obtained as:
Replacing ω from Eq. 37 into 32 yields:
Finally, x(t) is the answer of above problem.
RESULTS AND DISCUSSION
In this study, the usefulness of the presented parameterexpanding method and
homotopy analysis method are investigated by considering above problem. To validate
the results, convergence studies are carried out and the results are compared
with those obtained using numerical results base on fourth order Runge Kutta
method (Hoffman, 1992) and shown in Table
1 and 2 in the case of EA = 556 kN, L_{0} = 1.143
m, ρ = 4.6416x10^{2} kg m^{1}, M = 500 kg and t_{0}
= 3558.6 N. It is worth mentioning that the relative error is defined as follows:
The effect of constant parameters has been studied in Fig. 4
and 5 that are compared with the numerical results. Also,
in the Fig. 7 the percentage of relative error has been shown
to indicate the accuracy of the procedure. In addition in the Fig.
7 it has been shown that the maximum error is about 2.5%, that it is very
small error for PEM solution.
Table 1: 
Compression between results of x(t) predicted by PEM, HAM
and numerical method 


Fig. 4: 
Displacementtime plot for the twolink structure, predicted
by PEM and HAM, EA = 556 kN, L_{0} = 1.143 m, ρ = 4.6416x10^{2}
kg m^{1}, M = 100 kg and t_{0} = 4448 N 

Fig. 5: 
Velocitytime plot for the twolink structure, predicted by
PEM and HAM, EA = 556 kN, L_{0} = 1.143 m, ρ = 4.6416x10^{2}
kg m^{1}, M = 100 kg and t_{0} = 4448 N 

Fig. 6: 
Displacementtime plot for the twolink structure, predicted
by PEM and HAM, EA = 556 kN, L_{0} = 1.143 m, ρ = 4.6416x10^{2}
kg m^{1}, M = 100 kg and t_{0} = 4448 N 
Table 2: 
Compression between results of x’(t) predicted by PEM,
HAM and numerical method 


Fig.7: 
The percentage of relative error for the twolink structure,
predicted by PEM and HAM, EA = 556 kN, L_{0} = 1.143 m, ρ =
4.6416x10^{2} kg m^{1}, M = 100 kg and t_{0} =
4448 N 
Based on Table 1, 2 and Fig.
46, it can be concluded that only one term in series
expansions is sufficient to obtain a highly accurate solution, which is valid
for the whole solution domain.
CONCLUSION
In this study, homotopy analysis method and a new method called He’s parameterexpanding
method has been studied. In the numerical methods, stability and convergence
should be considered so as to avoid divergence or inappropriate results. In
the analytical perturbation method, we should exert the small parameter in the
equation. Therefore, finding the small parameter and exerting it into the equation
are deficiencies of these methods. Two of the semiexact methods which don’t
need small/large parameters are Homotopy Analysis Method and Parameter Expanding
method. In addition, to comprise the obtained results, the governing equation
was solved numerically by authors based on fourth order Runge Kutta method.
Some remarkable virtues of the methods were studied and their applications for
obtaining the displacement functions of geometrically nonlinear prestressed
cable structures analytically, have been illustrated. The obtained results have
a good agreement with those obtained using numerical method. It is clear HAM
is a generalized Taylor series method, searching for an infinite series solution,
PEM is clearly a new perturbation method, searching an asymptotic solution with
only one term and no convergence theory is needed.
Table 3: 
Compression between results of x(t) predicted by PEM, HAM
and Maple software 

Moreover, increasing the
domain of independent parameter or increasing the coefficients of nonlinear
term, increases the error of PEM solution, whereas, the HAM solution is very
accurate for whole domain of solution, as shown in Table 12. Also, as shown in Table 3 the equation was solved by Maple
software to be convincing about authors’ numeric solution. The results
show that the methods are promising for solving this type of problems and might
find wide applications.
NOMENCLATURE
D 
: 
D’Alembert forces 
EA 
: 
Axial stiffness 
H(t) 
: 
Auxiliary function 
L 
: 
Linear operator 
L_{0} 
: 
Original undeformed length 
N 
: 
Nonlinear operator 
q 
: 
Embedding parameter 
R 
: 
Vertical dynamic 
R_{m} (U_{m1}) 
: 
Reminder term 
t 
: 
Time (Independent dimension less parameter) 
t_{0} 
: 
Initial pretension 
p 
: 
Mass per unit length of the links 
ħ 
: 
Auxiliary parameter 