Abstract
In the present study, the response of a flexible string with large amplitude transverse vibration is studied utilizing amplitudefrequency formulation, improved amplitudefrequency formulation and maxmin approach. In order to verify the accuracy of these approaches, obtained results are compared with other methods such as variational approach method, variational iteration method, coupling Newton’s method with the harmonic balance method and Hamiltonian approach. It has been found that for this problem, while amplitudefrequency formulation and maxmin approach give the same results, improved amplitude frequency formulation is not an appropriate choice.
Keywords
nonlinear vibration; string; amplitudefrequency formulation; improved amplitudefrequency formulation and maxmin approach
1 INTRODUCTION
Study of nonlinear vibration of strings with large amplitude has attracted the attention of many researches in the fields of physics and engineering. For instance, long cables used in cranes, ships and bridges might expose to forces that causes large amplitude vibrations ( Omran et al., 2013 Omran, M.P.; Amani, A.; Lemu, H.G.; “Analytical approximation of nonlinear vibration of string with large amplitudes”, Journal of Mechanical Science and Technology, 27(4) 981986, 2013. ). The nonlinear partial differential equation for transverse vibrations of a flexible string under constant tension is presented in equation (1) ( Coulson and Jeffrey, 1997 Coulson, C.A.; Jeffrey, A.; Waves, 2nd edition, London: Longman, 1997. ).
where
By considering the transverse vibration as
where
with initial conditions
Many researches have been conducted to solve equation (3) with different methods such as variational approach method, variational iteration method, coupling Newton’s method with the harmonic balance method and Hamiltonian approach. In the present study, this equation is solved using amplitudefrequency formulation, improved amplitudefrequency formulation and maxmin approach. Furthermore, obtained results are compared with that of the mentioned methods.
2 METHODOLOGY
2.1 Amplitudefrequency formulation (AFF)
To illustrate this approach, a generalized nonlinear oscillator is considered as presented in equation (7)
Used Trial functions are presented in equation (8) and (9) .
The residuals are as shown in equations (10) and (11) .
The original frequencyamplitude formulation is presented in equation (12) .
To find the frequency, He (2008a) He, J.H.; “An elementary introduction to recently developed asymptotic methods and nanomechanics in textile engineering”, International Journal of Modern Physics B, 22(1) 34873578, 2008a. used equation (13) , Geng and Cai (2007) Geng, L.; Cai, X.C. “He’s frequency formulation for nonlinear oscillators”, European Journal of Physics, 61(8) 923931,2007. used another location point which is indicated in equation (14) . By substituting
To solve equation (3) ,
Substituting equations (15) and (16) in equation (12) and considering
Parameter k can be drawn by substituting the natural frequency from equation (17) into the integral equation (18) and assuming
2.2 Improved amplitudefrequency formulation (IAFF)
The accuracy of AFF depends upon the location choice. He (2008b) He, J.H.; “Comment on He's frequency formulation for nonlinear oscillators”, European Journal of Physics, 29, 1922, 2008b. suggested the method of weighted residual to overcome this shortcoming. A generalized nonlinear oscillator is considered as presened in equation (19) .
Respectively, solutions of the following linear oscillator equations are considered as two trial functions
where
In this approach, to overcome the shortcomings, two new residuals are defined as
Square of the frequency is approximately determined in the form of equation (26) ( He, 2008b He, J.H.; “Comment on He's frequency formulation for nonlinear oscillators”, European Journal of Physics, 29, 1922, 2008b. )
To solve equation (3) by this method, trial functions are considered as
By changing variable to
Frequency can be obtained by substituting
2.3 Maxmin Approach (MMA)
In this approach, after finding the maximal and minimal solution thresholds, an approximate solution of the nonlinear equation is deduced using HeChengtian’s interpolation ( He, 2008c He, J.H.; “Maxmin approach to nonlinear oscillators”, international Journal of Nonlinear Sciences and Numerical Simulation, 9(2) 207210, 2008c. ). For instance, a generalized nonlinear oscillator is considered in the form of equation (32) ,
where
where
where
HeChentian ( He, 2008c He, J.H.; “Maxmin approach to nonlinear oscillators”, international Journal of Nonlinear Sciences and Numerical Simulation, 9(2) 207210, 2008c. ) interpolation for the square of the frequency is mentioned in equation (36) .
where
By substituting the frequency from equation (37) into the assumed solution of equation (34) , the residual is obtained as equation (38) .
Solving equation (39) , k is obtained as indicated in equation (40) .
Substituting k from equation (40) into equation (37) , frequency can be obtained ( Bayat et al., 2012 Bayat, M.; Pakar, I.; Domairry, G.; “Recent developments of some asymptotic methods and their applications for nonlinear vibration equations in engineering problems: A review”, Latin American Journal of Solids and Structures, 9(2) 193, 2012. ).
To apply this method for the nonlinear partial differential equation for transverse vibrations of a flexible string, equation (1) is rewritten in the form of equation (41).
where
Substituting equations (42) and (43) into equation (37) , square of the frequency is obtained as indicated in equation (44).
Substituting equation (44) into the equation (41) , the residual is presented in equation (45) .
By using equation (46) and equation (44) , the frequency can be attained. The obtained results are compared in the next section.
3 Results and discussion
3.1 Free vibration response of the cable
The response of the cable under free vibration is compared by different methods, including VAM, Fourth order RungeKutta, IAFF, AFF and MaxMin, for different amplitudes in Figure 1 to Figure 5 .
Obtained solutions using AFF, IAFF, MMA and RungeKutta are compared to the results of variational Iteration Method (VIM) ( Taghipour et al., 2014 Taghipour, R.; Akhlaghi, T. Nikkar, A.; “Explicit solution of the large amplitude transverse vibrations of a flexible string under constant tension”, Latin American Journal of Solids and Structures, 11(3) 445455, 2014. ), Hamiltonian Approach (HA) ( Taghipour et al., 2014 Taghipour, R.; Akhlaghi, T. Nikkar, A.; “Explicit solution of the large amplitude transverse vibrations of a flexible string under constant tension”, Latin American Journal of Solids and Structures, 11(3) 445455, 2014. ) and Variational approach method (VAM) which is reproduced using the derived equations in ( Omran et al., 2013 Omran, M.P.; Amani, A.; Lemu, H.G.; “Analytical approximation of nonlinear vibration of string with large amplitudes”, Journal of Mechanical Science and Technology, 27(4) 981986, 2013. ) in Table 1 and Table 2 .
Comparing the results in tables 1 and 2 , the computed response of the cable under the large amplitude vibration using VAM, AFF and MMA are near eachother, while those of the IAFF is far apart from other methods. Moreover, the obtained results by MMA and AFF are the same for this problem.
3.2 Period parameter
The obtained period parameter
Comparing the results in table 3 , one can conclude that for small values of a, obtained results by MMA and AFF is in agreement with the exact solution, and by increasing the value of a , the results deviate from the exact solution. Moreover, utilizing IAFF for this problem is not appropriate since its error is more than other mentioned methods.
4 CONCLUSION
In this paper, AFF, IAFF and MMA are employed to derive the analytical approximate solution for large amplitude nonlinear vibration of a string. The obtained response from these methods were compared to other methods such as VAM, VIM, HA and HB. It is shown that the obtained response from MMA and AFF are the same for this problem. On the other hand, IAFF is not appropriate method for this problem since it is not accurate enough in comparison to other mentioned methods.
References
 Bayat, M.; Pakar, I.; Domairry, G.; “Recent developments of some asymptotic methods and their applications for nonlinear vibration equations in engineering problems: A review”, Latin American Journal of Solids and Structures, 9(2) 193, 2012.
 Coulson, C.A.; Jeffrey, A.; Waves, 2nd edition, London: Longman, 1997.
 Geng, L.; Cai, X.C. “He’s frequency formulation for nonlinear oscillators”, European Journal of Physics, 61(8) 923931,2007.
 Gottlieb, H.P.W.; “Nonlinear vibration of a constanttension string”, Journal of sound and vibration, 143(3) 455460, 1990.
 He, J.H.; “An elementary introduction to recently developed asymptotic methods and nanomechanics in textile engineering”, International Journal of Modern Physics B, 22(1) 34873578, 2008a.
 He, J.H.; “Comment on He's frequency formulation for nonlinear oscillators”, European Journal of Physics, 29, 1922, 2008b.
 He, J.H.; “Maxmin approach to nonlinear oscillators”, international Journal of Nonlinear Sciences and Numerical Simulation, 9(2) 207210, 2008c.
 Lai, S.K.; Xiang, Y.; Lim, C.W.; He, X.F.; Zeng, Q.C.; “Higherorder approximate solutions for nonlinear vibration of a constanttension string”, Journal of Sound and Vibration, 317(35) 440448, 2008.
 Omran, M.P.; Amani, A.; Lemu, H.G.; “Analytical approximation of nonlinear vibration of string with large amplitudes”, Journal of Mechanical Science and Technology, 27(4) 981986, 2013.
 Taghipour, R.; Akhlaghi, T. Nikkar, A.; “Explicit solution of the large amplitude transverse vibrations of a flexible string under constant tension”, Latin American Journal of Solids and Structures, 11(3) 445455, 2014.
Publication Dates

Publication in this collection
04 June 2018 
Date of issue
2018
History

Received
15 June 2017 
Reviewed
20 Nov 2017 
Accepted
26 Nov 2018