**2. Continuous piecewise linearization method**

### **2.1 Concept of the continuous piecewise linearization method**

The main idea of the CPLM is based on the piecewise linearization of the nonlinear restoring force of a conservative oscillator. The linearization technique used by the CPLM was first applied in another algorithm [6, 54] for the solution of half-space impact models called force indentation linearization method (FILM). The FILM has been applied to formulate theoretical solutions for rigid body motions and local compliance response during nonlinear elastoplastic impact of dissimilar spheres [55]. However, because the FILM is limited to impact excitations that are nonoscillatory, it cannot be applied to nonlinear conservative oscillators. Hence, the CPLM applies the piecewise linearization technique of the FILM to provide a periodic solution for nonlinear conservative oscillators.

Essentially, the linearization technique of the CPLM involves *n* equal discretization of the nonlinear restoring force with respect to displacement (**Figure 1**) and formulating a linear restoring force for each discretization. Therefore, a linear ODE can be derived for each discretization. The solution of the linear ODE approximates the solution of the original nonlinear oscillator for a time-range that is automatically determined by the CPLM and updated continuously from one discretization to the next. Details on the discretization and linearization technique of the CPLM can be found in the following references [6, 8, 54], while the applications of the CPLM to nonlinear conservative systems are presented in [8, 56].

*Periodic Solution of Nonlinear Conservative Systems DOI: http://dx.doi.org/10.5772/intechopen.90282*

Also, methods that rely on a simple harmonic approximation of the oscillation history, such as the energy balance method, amplitude-frequency formulation, Hamiltonian approach, max-min approach, and variational methods, can only give reliable estimate of the frequency-amplitude response. Sometimes these methods perform poorly in predicting the oscillation history during large-amplitude and/or strong nonlinear vibrations. Other methods that usually require high-order approximations, such as Adomian decomposition method, harmonic balance method, and variational iteration method, present algebraic complexities in their determination of higher order solutions and may be impractical for oscillators with highly complex nonlinearities such as the slider-crank mechanism [1] and the bifilar pendulum [52]. Furthermore, it has been observed that higher order estimates do not always improve the solution of the oscillation history [27]. Finally, some nonperturbation methods are heuristic in nature (e.g., energy balance method and variational methods) and require experience to choose the initial trial function and the

The continuous piecewise linearization method (CPLM) is an iterative analytic algorithm that was formulated to overcome most of the above challenges by providing simple and accurate solutions for the oscillation history and frequencyamplitude response of Duffing-type oscillators. In another study [53], the CPLM was modified in order to generalize it so that it can handle more complicated nonlinear conservative oscillators. Interestingly, the CPLM does not require higher order approximations or any small, artificial, or embedded parameter. Also, the algorithm is inherently stable, straightforward, and based on closed-form analytical approximations. This chapter is aimed at presenting the generalized CPLM algorithm as a veritable approach for accurate periodic solution of Hamiltonian and non-Hamiltonian conservative oscillators with complex nonlinearity. As is shown later, the CPLM retains the same simplicity in its implementation irrespective of

condition for error minimization [8].

*Progress in Relativity*

the complexity of the nonlinear conservative system.

**2. Continuous piecewise linearization method**

odic solution for nonlinear conservative oscillators.

**238**

**2.1 Concept of the continuous piecewise linearization method**

The main idea of the CPLM is based on the piecewise linearization of the nonlinear restoring force of a conservative oscillator. The linearization technique used by the CPLM was first applied in another algorithm [6, 54] for the solution of half-space impact models called force indentation linearization method (FILM). The FILM has been applied to formulate theoretical solutions for rigid body motions and local compliance response during nonlinear elastoplastic impact of dissimilar spheres [55]. However, because the FILM is limited to impact excitations that are nonoscillatory, it cannot be applied to nonlinear conservative oscillators. Hence, the CPLM applies the piecewise linearization technique of the FILM to provide a peri-

Essentially, the linearization technique of the CPLM involves *n* equal discretization of the nonlinear restoring force with respect to displacement (**Figure 1**) and formulating a linear restoring force for each discretization. Therefore, a linear ODE can be derived for each discretization. The solution of the linear ODE approximates the solution of the original nonlinear oscillator for a time-range that is automatically determined by the CPLM and updated continuously from one discretization to the next. Details on the discretization and linearization technique of the CPLM can be found in the following references [6, 8, 54], while the applications of the CPLM to nonlinear conservative systems are presented in [8, 56].

**Figure 1.** *Discretization of the restoring force of a typical nonlinear oscillator.*

#### **2.2 Mathematical formulation of the continuous piecewise linearization method**

The standard form for representation of a nonlinear conservative oscillator moving in the *u*-direction is given as:

$$
\ddot{u} + f(u) = \mathbf{0},
\tag{1}
$$

where *f u*ð Þ is the nonlinear restoring force as shown in **Figure 1**. In **Figure 1**, the numbering on the horizontal axis represents the boundary points of the discretization. The *s th* discretization represents a general discretization with start point at *r* and endpoint at *s* ¼ *r* þ 1. **Figure 1** shows that for each discretization, the slope of the linear approximation of the restoring force can either be positive or negative. To account for this possibility, the linearized force for the *s th* discretization can be expressed as:

$$F\_{\pi}(u) = \pm |K\_{\pi}|(u - u\_r) + F\_r,\tag{2}$$

where *Krs* ¼ *f u*ð Þ�*<sup>s</sup> f u*ð Þ*<sup>r</sup>* ½ �*=*ð Þ *us* � *ur* is the linear slope of *Frs*ð Þ *u* and *Fr* ¼ *f u*ð Þ*<sup>r</sup>* . Since *Frs*ð Þ *u* is an approximation of *f u*ð Þ for the *s th* discretization, then substituting Eq. (2) in (1) gives the approximate equation of motion for each discretization as follows:

$$
\ddot{\boldsymbol{\mu}} \pm |\boldsymbol{K}\_{rs}| \boldsymbol{\mu} = \pm |\boldsymbol{K}\_{rs}| \boldsymbol{\mu}\_r - \boldsymbol{F}\_r. \tag{3}
$$

Eq. (10) is a nonhomogeneous linear ODE and its solution depends on whether the sign is positive or negative.

#### *2.2.1 Solution for positive linearized stiffness*

If *Krs* > 0, the solution for the displacement and velocity can be expressed as:

$$u(t) = R\_{\pi} \sin \left(\alpha\_{\pi} t + \Phi\_{\pi} \right) + C\_{\pi} \tag{4a}$$

$$
\dot{u}(t) = a\rho\_{\text{tr}}R\_{\text{tr}}\cos\left(a\rho\_{\text{tr}}t + \Phi\_{\text{tr}}\right),
\tag{4b}
$$

where *<sup>ω</sup>rs* <sup>¼</sup> ffiffiffiffiffiffi *Krs* <sup>p</sup> , *Crs* <sup>¼</sup> *ur* � *Fr=Krs*, and *Rrs* <sup>¼</sup> ð Þ *ur* � *Crs* <sup>2</sup> <sup>þ</sup> *<sup>u</sup>*\_ ð Þ *<sup>r</sup>=ωrs* <sup>2</sup> h i1*=*<sup>2</sup> . The initial conditions and other parameters are determined based on the oscillation stage. For the oscillation stage that moves from þ*A* to �*A*, the initial conditions for each discretization are *ur* ¼ *ur*ð Þ¼ 0 *A* � *rΔu* and *u*\_*<sup>r</sup>* ¼ *u*\_*r*ð Þ¼� 0 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 Ð *ur <sup>A</sup>* � *f u*ð Þ*du* � � � � <sup>q</sup> , where *Δu* ¼ *A=n*, and the other parameters are calculated as:

$$\Phi\_{\rm tr} = \begin{cases} 0.5\pi & \dot{u}\_r = 0\\ \pi + \tan^{-1}[o\_{\rm tr}(u\_r - \mathcal{C}\_{\rm tr})/\dot{u}\_r] & \dot{u}\_r < 0 \end{cases} \tag{5a}$$

*u t*ðÞ¼ ð Þ *ur* � *Crs* cosh ð Þþ *ωrst Crs:* (9)

*ur* � *Crs* � �*:* (10)

, (11)

*:* (12)

*<sup>r</sup>*. Hence, the time interval is

cosh �<sup>1</sup> *us* � *Crs*

In very rare situations, we may have *Krs* ¼ 0 for one or two discretization around the turning points or relatively flat regions of the restoring force. This is likely when *Δu* is very small, i.e., for very large *n*, and can be eliminated by increasing or decreasing *n* slightly. However, if we want to account for *Krs* ¼ 0,

*u t*ðÞ¼ *Hrs* <sup>þ</sup> *Grst* � <sup>1</sup>

*G*2

*Fr*

1.From the above presentation of the CPLM formulation, it is obvious that the CPLM algorithm is simple and can be implemented by undergrads without

2.The CPLM is inherently stable and does not have convergence issues [8].

4.When dealing with conservative oscillators with odd nonlinearity, which are symmetrical about the origin, discretization of the restoring force is only required for 0 to *A*. This means that there will be 2*n* discretizations from

5.The CPLM algorithm retains the same simplicity in implementation

of Eq. (1) before applying the CPLM algorithm. Fortunately, this

irrespective of the complexity of the restoring force. Only the *Krs* constant and the integral of the restoring force are to be evaluated anew for any oscillator.

6.The CPLM relies on the explicit expression of restoring force, which means that the model for the oscillator must be expressed in the form of Eq. (1). For Hamiltonian systems, the oscillator model is formulated naturally in the form of Eq. (1). For non-Hamiltonian conservative systems, the oscillator model is not formulated naturally in the form of Eq. (1). Therefore, there is a need to transform the model of non-Hamiltonian conservative systems into the form

3.For few discretization, say *n* ≤10, the CPLM algorithm can be implemented with reasonable accuracy using a pocket calculator. However, the CPLM is better executed using a simple code in any programming language such as MATLAB and Mathematica or using a customized MS Excel spreadsheet.

q

2 *Frt* 2

<sup>2</sup> *Frt* 2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

*rs* þ 2*Fr*ð Þ *Hrs* � *us*

*<sup>Δ</sup><sup>t</sup>* <sup>¼</sup> <sup>1</sup> *ωrs*

Therefore,

then we get [53]:

derived from Eq. (11) as:

difficulty.

�*A* to *A*.

**241**

*2.2.3 Solution for zero linearized stiffness*

*Periodic Solution of Nonlinear Conservative Systems DOI: http://dx.doi.org/10.5772/intechopen.90282*

**2.3 Remarks on the CPLM algorithm**

where *Grs* <sup>¼</sup> *<sup>u</sup>*\_*<sup>r</sup>* <sup>þ</sup> *Frtr* and *Hrs* <sup>¼</sup> *ur* � *<sup>u</sup>*\_*rtr* � <sup>1</sup>

*<sup>Δ</sup><sup>t</sup>* <sup>¼</sup> *Grs* <sup>þ</sup>

$$\Delta t = \begin{cases} (\mathbf{0}.5\pi - \Phi\_{\mathrm{m}})/\alpha\_{\mathrm{m}} & (u\_{\mathrm{s}} - \mathbf{C}\_{\mathrm{m}}) \ge R\_{\mathrm{m}} \\ (\mathbf{0}.5\pi + \cos^{-1}[(u\_{\mathrm{s}} - \mathbf{C}\_{\mathrm{m}})/R\_{\mathrm{m}}] - \Phi\_{\mathrm{m}})/\alpha\_{\mathrm{m}} & (u\_{\mathrm{s}} - \mathbf{C}\_{\mathrm{m}}) < R\_{\mathrm{m}} \end{cases} \tag{5b}$$

For the oscillation stage that moves from �*A* to þ*A*, the initial conditions are *ur* ¼ *ur*ð Þ¼� 0 *A* þ *rΔu* and *u*\_*<sup>r</sup>* ¼ *u*\_*r*ð Þ¼ 0 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 Ð *ur <sup>A</sup>* � *f u*ð Þ*du* � � � � <sup>q</sup> ; the other parameters are calculated as:

$$\Phi\_{\pi} = \begin{cases} -0.5\pi & \dot{u}\_{r} = 0\\ \tan^{-1}[o\_{\pi}(u\_{r} - \mathcal{C}\_{\pi})/\dot{u}\_{r}] & \dot{u}\_{r} < 0 \end{cases} \tag{6a}$$

$$\Delta t = \begin{cases} (\mathbf{0}.5\pi - \Phi\_n)/o\_{\rm tr} & (u\_\rm s - \mathbf{C}\_\rm m) \ge R\_{\rm tr} \\ (\mathbf{0}.5\pi - \cos^{-1}[(u\_\rm s - \mathbf{C}\_\rm m)/R\_{\rm tr}] - \Phi\_\rm tr)/o\_{\rm tr} & (u\_\rm s - \mathbf{C}\_\rm m) < R\_{\rm tr} \end{cases} \tag{6b}$$

The time at the end of each discretization is *ts* ¼ *tr* þ *Δt*, and the end conditions *us* and *u*\_*<sup>s</sup>* are calculated by replacing *r* with *s* in the formulae for the initial conditions.

#### *2.2.2 Solution for negative linearized stiffness*

If *Krs* < 0, the solution for the displacement and velocity can be expressed as follows:

$$u(t) = A\_{tr}e^{\alpha\_{nl}t} + B\_{tr}e^{-\alpha\_{nl}t} + C\_{tr} \tag{7a}$$

$$
\dot{u}(t) = \alpha\_{rs} (A\_{rs} e^{\alpha\_{rl}t} - B\_{rs} e^{-\alpha\_{rl}t}),
\tag{7b}
$$

where *<sup>ω</sup>rs* <sup>¼</sup> ffiffiffiffiffiffiffiffiffi *Krs* j j <sup>p</sup> ; *Crs* <sup>¼</sup> *ur* <sup>þ</sup> *Fr<sup>=</sup> Krs* j j. Applying the initial conditions to Eqs. (7a) and (7b) gives: *Ars* <sup>¼</sup> <sup>1</sup> <sup>2</sup> *ur* <sup>þ</sup> *<sup>u</sup>*\_ ð Þ *<sup>r</sup>=ωrs* � *Crs* ; *Brs* <sup>¼</sup> <sup>1</sup> <sup>2</sup> *ur* � *u*\_ ð Þ *<sup>r</sup>=ωrs* � *Crs* . The initial and end conditions are determined in the same way as for *Krs* > 0 above. The end conditions are applied in Eq. (7a) to get the time interval for each discretization as:

$$\Delta t = \left\{ \frac{1}{\frac{\alpha\_{\rm tr}}{\alpha\_{\rm tr}} \log\_{\epsilon} \left[ \frac{\left(u\_{\rm s} - C\_{\rm tr}\right) \pm \sqrt{\left(u\_{\rm s} - C\_{\rm tr}\right)^{2} - 4A\_{\rm tr}B\_{\rm tr}}}{2A\_{\rm tr}} \right] & (u\_{\rm s} - C\_{\rm tr}) > 2\sqrt{A\_{\rm tr}B\_{\rm tr}} \\ \frac{1}{\alpha\_{\rm tr}} \log\_{\epsilon} \left( \frac{u\_{\rm s} - C\_{\rm tr}}{2A\_{\rm tr}} \right) & (u\_{\rm s} - C\_{\rm tr}) \le 2\sqrt{A\_{\rm tr}B\_{\rm tr}} \end{cases} \right\} \tag{8}$$

The sign before the square root in Eq. (8) is negative for the oscillation stage that moves from þ*A* to �*A* and vice versa. We note that if *u*\_*<sup>r</sup>* ¼ 0, then *Ars* ¼ *Brs* ¼ 1 <sup>2</sup> ð Þ *ur* � *Crs* and

*Periodic Solution of Nonlinear Conservative Systems DOI: http://dx.doi.org/10.5772/intechopen.90282*

$$u(t) = (u\_r - C\_{rs})\cosh\left(\alpha\_{rl}t\right) + C\_{rs}.\tag{9}$$

Therefore,

where *<sup>ω</sup>rs* <sup>¼</sup> ffiffiffiffiffiffi

*Progress in Relativity*

(

�

where *<sup>ω</sup>rs* <sup>¼</sup> ffiffiffiffiffiffiffiffiffi

1 *ωrs* log *<sup>e</sup>*

1 *ωrs* log *<sup>e</sup>*

and (7b) gives: *Ars* <sup>¼</sup> <sup>1</sup>

8 >>>>><

>>>>>:

<sup>2</sup> ð Þ *ur* � *Crs* and

*Δt* ¼

1

**240**

are calculated as:

conditions.

follows:

*Krs*

*ur* ¼ *ur*ð Þ¼� 0 *A* þ *rΔu* and *u*\_*<sup>r</sup>* ¼ *u*\_*r*ð Þ¼ 0

*2.2.2 Solution for negative linearized stiffness*

<sup>p</sup> , *Crs* <sup>¼</sup> *ur* � *Fr=Krs*, and *Rrs* <sup>¼</sup> ð Þ *ur* � *Crs* <sup>2</sup> <sup>þ</sup> *<sup>u</sup>*\_ ð Þ *<sup>r</sup>=ωrs* <sup>2</sup> h i1*=*<sup>2</sup>

initial conditions and other parameters are determined based on the oscillation stage. For the oscillation stage that moves from þ*A* to �*A*, the initial conditions for

> <sup>Φ</sup>*rs* <sup>¼</sup> <sup>0</sup>*:*5*<sup>π</sup> <sup>u</sup>*\_*<sup>r</sup>* <sup>¼</sup> <sup>0</sup> *<sup>π</sup>* <sup>þ</sup> tan �<sup>1</sup> *<sup>ω</sup>rs*ð Þ *ur* � *Crs <sup>=</sup>u*\_*<sup>r</sup>* ½ � *<sup>u</sup>*\_*<sup>r</sup>* <sup>&</sup>lt;<sup>0</sup>

<sup>0</sup>*:*5*<sup>π</sup>* <sup>þ</sup> cos �<sup>1</sup> ð Þ *us* � *Crs <sup>=</sup>Rrs* ð Þ <sup>½</sup> � � <sup>Φ</sup>*rs <sup>=</sup>ωrs* ð Þ *us* � *Crs* <sup>&</sup>lt;*Rrs*

For the oscillation stage that moves from �*A* to þ*A*, the initial conditions are

<sup>Φ</sup>*rs* <sup>¼</sup> �0*:*5*<sup>π</sup> <sup>u</sup>*\_*<sup>r</sup>* <sup>¼</sup> <sup>0</sup>

The time at the end of each discretization is *ts* ¼ *tr* þ *Δt*, and the end conditions

If *Krs* < 0, the solution for the displacement and velocity can be expressed as

*<sup>ω</sup>rst* <sup>þ</sup> *Brse*

end conditions are determined in the same way as for *Krs* > 0 above. The end conditions are applied in Eq. (7a) to get the time interval for each discretization as:

*<sup>ω</sup>rst* � *Brse*

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð Þ *us* � *Crs* <sup>2</sup> � <sup>4</sup>*ArsBrs*

The sign before the square root in Eq. (8) is negative for the oscillation stage that

moves from þ*A* to �*A* and vice versa. We note that if *u*\_*<sup>r</sup>* ¼ 0, then *Ars* ¼ *Brs* ¼

*Krs* j j <sup>p</sup> ; *Crs* <sup>¼</sup> *ur* <sup>þ</sup> *Fr<sup>=</sup> Krs* j j. Applying the initial conditions to Eqs. (7a)

*<sup>Δ</sup><sup>t</sup>* <sup>¼</sup> ð Þ <sup>0</sup>*:*5*<sup>π</sup>* � <sup>Φ</sup>*rs <sup>=</sup>ωrs* ð Þ *us* � *Crs* <sup>≥</sup>*Rrs* <sup>0</sup>*:*5*<sup>π</sup>* � cos �<sup>1</sup> ð Þ *us* � *Crs <sup>=</sup>Rrs* ð Þ <sup>½</sup> � � <sup>Φ</sup>*rs <sup>=</sup>ωrs* ð Þ *us* � *Crs* <sup>&</sup>lt;*Rrs*

*us* and *u*\_*<sup>s</sup>* are calculated by replacing *r* with *s* in the formulae for the initial

*u t*ðÞ¼ *Arse*

*u t* \_ðÞ¼ *ωrs Arse*

<sup>2</sup> *ur* <sup>þ</sup> *<sup>u</sup>*\_ ð Þ *<sup>r</sup>=ωrs* � *Crs* ; *Brs* <sup>¼</sup> <sup>1</sup>

q

2*Ars*

ð Þ� *us* � *Crs*

*us* � *Crs* 2*Ars* � �

2 4

2 Ð *ur <sup>A</sup>* � *f u*ð Þ*du* � � � � <sup>q</sup>

tan �<sup>1</sup> *<sup>ω</sup>rs*ð Þ *ur* � *Crs <sup>=</sup>u*\_*<sup>r</sup>* ½ � *<sup>u</sup>*\_*<sup>r</sup>* <sup>&</sup>lt;<sup>0</sup>

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

*<sup>Δ</sup><sup>t</sup>* <sup>¼</sup> ð Þ <sup>0</sup>*:*5*<sup>π</sup>* � <sup>Φ</sup>*rs <sup>=</sup>ωrs* ð Þ *us* � *Crs* <sup>≥</sup>*Rrs*

each discretization are *ur* ¼ *ur*ð Þ¼ 0 *A* � *rΔu* and *u*\_*<sup>r</sup>* ¼ *u*\_*r*ð Þ¼� 0

where *Δu* ¼ *A=n*, and the other parameters are calculated as:

�

�

. The

(5a)

(6a)

*:* (6b)

*:* (5b)

,

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2 Ð *ur <sup>A</sup>* � *f u*ð Þ*du* � � � � <sup>q</sup>

; the other parameters

�*ωrst* <sup>þ</sup> *Crs* (7a)

<sup>2</sup> *ur* � *u*\_ ð Þ *<sup>r</sup>=ωrs* � *Crs* . The initial and

<sup>5</sup> ð Þ *us* � *Crs* <sup>&</sup>gt;<sup>2</sup> ffiffiffiffiffiffiffiffiffiffiffiffi

ð Þ *us* � *Crs* <sup>≤</sup><sup>2</sup> ffiffiffiffiffiffiffiffiffiffiffiffi

*ArsBrs* p

*:*

(8)

*ArsBrs* p

�*ωrst* ð Þ, (7b)

3

$$
\Delta t = \frac{1}{a\nu\_m} \cosh^{-1} \left(\frac{u\_s - C\_m}{u\_r - C\_m}\right). \tag{10}
$$

#### *2.2.3 Solution for zero linearized stiffness*

In very rare situations, we may have *Krs* ¼ 0 for one or two discretization around the turning points or relatively flat regions of the restoring force. This is likely when *Δu* is very small, i.e., for very large *n*, and can be eliminated by increasing or decreasing *n* slightly. However, if we want to account for *Krs* ¼ 0, then we get [53]:

$$u(t) = H\_m + G\_{rt}t - \frac{1}{2}F\_rt^2,\tag{11}$$

where *Grs* <sup>¼</sup> *<sup>u</sup>*\_*<sup>r</sup>* <sup>þ</sup> *Frtr* and *Hrs* <sup>¼</sup> *ur* � *<sup>u</sup>*\_*rtr* � <sup>1</sup> <sup>2</sup> *Frt* 2 *<sup>r</sup>*. Hence, the time interval is derived from Eq. (11) as:

$$
\Delta t = \frac{G\_m + \sqrt{G\_m^2 + 2F\_r(H\_m - u\_s)}}{F\_r}.\tag{12}
$$

#### **2.3 Remarks on the CPLM algorithm**


transformation only requires a simple algebraic manipulation as demonstrated in Section 4.

