**5. Concluding remarks**

Therefore, the restoring force is *f u*ð Þ¼ *<sup>Λ</sup>* <sup>1</sup> <sup>þ</sup> <sup>4</sup>*q*2*A*<sup>2</sup> � �*u<sup>=</sup>* <sup>1</sup> <sup>þ</sup> <sup>4</sup>*q*2*u*<sup>2</sup> ð Þ<sup>2</sup>

�, and the velocity was evaluated as *u*\_ ¼ �

*Tex* <sup>¼</sup> 4 1 <sup>þ</sup> <sup>4</sup>*q*<sup>2</sup>

<sup>0</sup> <sup>1</sup> � *<sup>k</sup>*<sup>2</sup> sin <sup>2</sup> *ϕ* � �1*=*<sup>2</sup>

numerically using the NDSolve function in Mathematica™.

<sup>1</sup>*<sup>=</sup>* <sup>1</sup> <sup>þ</sup> <sup>4</sup>*q*<sup>2</sup>*u*<sup>2</sup>

**Figure 10.**

**Figure 11.**

**248**

*Lines; exact solution—Markers.*

as follows [4]:

*r* � �<sup>2</sup>

*Progress in Relativity*

where *E k*<sup>2</sup> � � <sup>¼</sup> <sup>Ð</sup> *<sup>π</sup>=*<sup>2</sup>

second kind and *<sup>k</sup>*<sup>2</sup> <sup>¼</sup> <sup>4</sup>*q*2*A*<sup>2</sup>

shows that *f u*ð Þ is linear at small amplitudes and strongly nonlinear at large amplitudes. The main CPLM constant was calculated as *Krs* <sup>¼</sup> *<sup>Λ</sup>* <sup>1</sup> <sup>þ</sup> <sup>4</sup>*q*<sup>2</sup>*A*<sup>2</sup> � � <sup>1</sup>*<sup>=</sup>* <sup>1</sup> <sup>þ</sup> <sup>4</sup>*q*<sup>2</sup>*u*<sup>2</sup>

The exact time period for this oscillator can be derived in terms of elliptic function

*A*<sup>2</sup> � �*=Λ* � �1*=*<sup>2</sup>

as *ωex* ¼ 2*π=Tex*, while the exact oscillation history was obtained by solving Eq. (25)

*(a) Frequency-amplitude response for particle on a rotating parabola. (b) CPLM error analysis.*

*Oscillation history of particle on a rotating parabola for (a) A* ¼ *0:50 and (b) A* ¼ *2:0. CPLM estimate—*

A comparison of CPLM frequency estimate and the exact frequency is shown in **Figure 10**, while the oscillation histories for *A* ¼ 0*:*50 and *A* ¼ 2*:*0 are shown in **Figure 11**. As demonstrated in [4], periodic solutions for this system exist only for *Λ*> 0. Hence, the simulations in **Figures 10** and **11** were conducted for *Λ* ¼ 10 and *q* ¼ 1*:*0. An excellent agreement is observed between the CPLM estimates and the exact results. For 0< *A* ≤20, the maximum error in the CPLM estimate of the frequency-amplitude response is 0.642% for *n* ¼ 10 and 0.101% for *n* ¼ 20, both of which are well below 1.0%. Also, the CPLM solution gives an accurate prediction of the strong anharmonic response in the oscillation history as shown in **Figure 11**.

. **Figure 9b**

h

*E k*<sup>2</sup> � �, (26)

q

*dϕ* is the complete elliptic integral of the

*<sup>=</sup>* <sup>1</sup> <sup>þ</sup> <sup>4</sup>*q*2*A*<sup>2</sup> � �. Then, the exact frequency was computed

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *<sup>Λ</sup> <sup>A</sup>*<sup>2</sup> � *<sup>u</sup>*<sup>2</sup> � �*<sup>=</sup>* <sup>1</sup> <sup>þ</sup> <sup>4</sup>*q*2*u*<sup>2</sup> ð Þ

*s* � �<sup>2</sup>

�

.

Conservative oscillators generally exhibit nonlinear response, and they form a large class of natural and artificially vibrating systems. Hence, the study of the dynamic response of nonlinear conservative systems is important for understanding many physical phenomena and the design of systems. The main challenge in the theoretical analysis of nonlinear conservative systems is that exact solutions are normally not available except for a few special cases where exact solutions are derived in terms of special functions.

To date, many approximate analytical methods have been formulated for the periodic solution of nonlinear conservative oscillators. This chapter provides a brief survey of the recent advances in the formulation of approximate analytical schemes and then introduced a recent approximate analytical algorithm called the continuous piecewise linearization method. The CPLM has been shown to overcome the challenges of solution accuracy and simplicity usually encountered in using most of the existing approximate analytical methods. The CPLM combines major desirable features of solution schemes such as inherent stability, accuracy, and simplicity. It is simple enough to be introduced at the undergraduate level and is capable of handling conservative oscillators with very complex nonlinearity. Conservative systems of broad interest were used to demonstrate the wide applicability of the CPLM algorithm. As demonstrated above, an accuracy of less than 1.0% relative error can be achieved for most oscillators using few discretizations, say *n*≤20, except for oscillators with trigonometric nonlinearity where such accuracy is achieved with many discretizations. This chapter has been designed to stimulate interest in the use of CPLM for analyzing various types of conservative systems, especially those with complex nonlinearity.
