**3.3. Approximate rotational solutions when** *<sup>ε</sup>* <sup>≈</sup> <sup>0</sup> **and** *<sup>ω</sup>* <sup>∼</sup> <sup>√</sup>*<sup>ε</sup>*

We assume that values of *<sup>ε</sup>* and *<sup>ω</sup>*<sup>2</sup> are small of the same order of smallness, i.e. *<sup>ε</sup>* <sup>∼</sup> *<sup>ω</sup>*<sup>2</sup> � 1, so we can introduce new parameter *w* = *ω*2/*ε*. One can deduce from (38) and current assumptions that either gravity *g* is small or the frequency of excitation Ω is high with such damping *c* and mass *m* so that damping coefficient *β* ∼ 1. All small terms are in the right-hand side of equation (39). To solve equation (39) we assume that general solution of equation (39) has the form

$$
\theta = -\tau + \theta\_0 + \varepsilon\theta\_1 + \varepsilon^2\theta\_2 + \dots \tag{48}
$$

Next the general solution is substituted into equation (39), where sines are expanded into the Taylor series with respect to *ε*. By grouping together the terms with the same powers of *ε* and equating to zero, the set of differential equations is obtained

$$
\ddot{\theta}\_0 + \mathcal{\beta}\dot{\theta}\_0 + \mu \sin(\theta\_0) = \mathcal{\beta},
\tag{49}
$$

$$\ddot{\theta}\_1 + \beta \dot{\theta}\_1 + \mu \cos(\theta\_0) \,\theta\_1 = \sin(2\tau - \theta\_0) + w \sin(\tau - \theta\_0 - \delta),\tag{50}$$

$$\ddot{\theta}\_2 + \beta \dot{\theta}\_2 + \mu \cos(\theta\_0) \,\theta\_2 = \mu \sin(\theta\_0) \,\theta\_1^2 / 2 - \left(\cos(2\tau - \theta\_0) + w \cos(\tau - \theta\_0 - \delta)\right) \theta\_1. \tag{51}$$

We have already found solution (41) for equation (49) in the previous section. Here we consider the same stable regular rotations 1:1 (with the period equal to the period of excitation) whose zero approximation is given by (45). Hence, *θ*<sup>0</sup> is a constant. Thus, equations (50) and (51) can be written in the following way

$$
\ddot{\theta}\_1 + \theta \dot{\theta}\_1 + \sqrt{\mu^2 - \beta^2} \theta\_1 = \sin(2\tau - \theta\_0) + w \sin(\tau - \theta\_0 - \delta) \tag{52}
$$

$$
\ddot{\theta}\_2 + \beta \dot{\theta}\_2 + \sqrt{\mu^2 - \beta^2} \theta\_2 = \beta \theta\_1^2 / 2 - \left(\cos(2\tau - \theta\_0) + w \cos(\tau - \theta\_0 - \delta)\right) \theta\_1. \tag{53}
$$

where we denote *<sup>μ</sup>* sin(*θ*0) = *<sup>β</sup>* and *<sup>μ</sup>* cos(*θ*0) = *<sup>μ</sup>*<sup>2</sup> <sup>−</sup> *<sup>β</sup>*<sup>2</sup> with the use of relation (42) and the second condition in (44).

### *3.3.1. First order approximation*

14 Will-be-set-by-IN-TECH

sin(*θ*0) = *<sup>β</sup>*

To investigate the stability of these solutions we present the angle *θ* as *θ* = *θ*<sup>0</sup> − *τ* + *η*, where *η* = *η*(*τ*) is a small addition, and substitute it in equation (40). Then linearizing (40) and using

According to the Lyapunov stability theorem based on the linear approximation, solution (41) is asymptotically stable if all eigenvalues of linearized equation (43) have negative real parts.

due to the Routh–Hurwitz conditions. From conditions (44), assumption *μ* > 0 in (38), and

are unstable, where *k* is any integer number. For negative damping, *β* < 0, both these solutions are unstable. From now on we will assume that the following conditions are satisfied

which ensure the existence of stable rotational solution (45) as it is seen from (42) and (44). Indeed, in order to guarantee asymptotic stability *β* should be not only positive, but also strictly less than *μ* because of the second condition in (44), which can be transformed to inequality *<sup>μ</sup>* cos(*θ*0) = *<sup>μ</sup>*<sup>2</sup> <sup>−</sup> *<sup>β</sup>*<sup>2</sup> <sup>&</sup>gt; 0 with the use of the positive root for *<sup>μ</sup>* cos(*θ*0) from

 *β μ* 

> *β μ*

*θ* = *θ*<sup>0</sup> − *τ*, *θ*<sup>0</sup> = arcsin

*θ* = *θ*<sup>0</sup> − *τ*, *θ*<sup>0</sup> = *π* − arcsin

*θ* + *μ* sin(*τ* + *θ*) = 0 (40)

*θ* = *θ*<sup>0</sup> − *τ*, (41)

*η*¨ + *βη*˙ + *μ* cos(*θ*0)*η* = 0 . (43)

*β* > 0, *μ* cos(*θ*0) > 0 (44)

0 < *β* < *μ*, (47)

+ 2*πk* (45)

+ 2*πk* (46)

*<sup>μ</sup>* , (42)

¨ *θ* + *β* ˙

where constants *θ*<sup>0</sup> are defined by the following equality

Which happens when all coefficients in (43) are positive

equality (42), it follows for *β* > 0 that the solutions

are asymptotically stable, while the solutions

equality (42), we obtain the linear equation

which has exact solutions

provided that |*β*| ≤ *μ*.

(42).

In consequence of conditions (47) non-homogeneous linear differential equation (52) can be presented in the following form

$$
\ddot{\theta}\_1 + \theta \dot{\theta}\_1 + \sqrt{\mu^2 - \beta^2} \theta\_1 = A\_1 \cos(\tau) + B\_1 \sin(\tau) + A\_2 \cos(2\tau) + B\_2 \sin(2\tau) \tag{54}
$$

where *A*<sup>1</sup> = −*w* cos(*δ*)*β*/*μ* − *w* sin(*δ*) <sup>1</sup> <sup>−</sup> *<sup>β</sup>*2/*μ*2, *<sup>B</sup>*<sup>1</sup> <sup>=</sup> *<sup>w</sup>* cos(*δ*) <sup>1</sup> <sup>−</sup> *<sup>β</sup>*2/*μ*<sup>2</sup> <sup>−</sup> *<sup>w</sup>* sin(*δ*)*β*/*μ*, *<sup>A</sup>*<sup>2</sup> <sup>=</sup> <sup>−</sup>*β*/*μ*, *<sup>B</sup>*<sup>2</sup> <sup>=</sup> <sup>1</sup> <sup>−</sup> *<sup>β</sup>*2/*μ*2, lower index denotes harmonics number. Equation (54) has a unique periodic solution

$$\theta\_1(\tau) = a\_1 \cos(\tau) + b\_1 \sin(\tau) + a\_2 \cos(2\tau) + b\_2 \sin(2\tau),\tag{55}$$

where *<sup>a</sup>*<sup>1</sup> <sup>=</sup> <sup>−</sup> (1<sup>−</sup> <sup>√</sup>*μ*<sup>2</sup>−*β*<sup>2</sup> )*A*1+*βB*<sup>1</sup> *μ*<sup>2</sup>+1−2 <sup>√</sup>*μ*<sup>2</sup>−*β*<sup>2</sup> , *<sup>b</sup>*<sup>1</sup> <sup>=</sup> <sup>−</sup> <sup>−</sup>*βA*1+(1<sup>−</sup> <sup>√</sup>*μ*<sup>2</sup>−*β*<sup>2</sup> )*B*<sup>1</sup> *μ*<sup>2</sup>+1−2 <sup>√</sup>*μ*<sup>2</sup>−*β*<sup>2</sup> , *<sup>a</sup>*<sup>2</sup> <sup>=</sup> <sup>−</sup> (4<sup>−</sup> <sup>√</sup>*μ*<sup>2</sup>−*β*<sup>2</sup>)*A*2+2*βB*<sup>2</sup> 3*β*<sup>2</sup>+*μ*<sup>2</sup>+4(4−2 <sup>√</sup>*μ*<sup>2</sup>−*β*<sup>2</sup> ) , and *<sup>b</sup>*<sup>2</sup> <sup>=</sup> <sup>−</sup>2*βA*2+(4<sup>−</sup> <sup>√</sup>*μ*<sup>2</sup>−*β*<sup>2</sup> )*B*<sup>2</sup> 3*β*<sup>2</sup>+*μ*<sup>2</sup>+4(4−2 <sup>√</sup>*μ*<sup>2</sup>−*β*<sup>2</sup> ) . Thus, the solution for (39) in the first approximation can be

#### 16 Will-be-set-by-IN-TECH 84 Nonlinearity, Bifurcation and Chaos – Theory and Applications Dynamics of a Pendulum of Variable Length and Similar Problems <sup>17</sup>

written as follows

$$\theta = -\tau + \theta\_0 - \varepsilon \frac{2\beta\cos(2\tau - \theta\_0) + \left(4 - \sqrt{\mu^2 - \beta^2}\right)\sin(2\tau - \theta\_0)}{3\beta^2 + \mu^2 + 8(2 - \sqrt{\mu^2 - \beta^2})}$$

$$-\omega^2 \frac{\beta\cos(\tau - \theta\_0 - \delta) + \left(1 - \sqrt{\mu^2 - \beta^2}\right)\sin(\tau - \theta\_0 - \delta)}{\mu^2 + 1 - 2\sqrt{\mu^2 - \beta^2}} + o(\varepsilon),\tag{56}$$

where constant *θ*<sup>0</sup> is defined in (45).

### *3.3.2. Second order approximation*

Equation (54) takes the following form

$$
\ddot{\theta}\_2 + \dot{\theta}\_2 + \sqrt{\mu^2 - \beta^2}\theta\_2 = \frac{A\_0'}{2} + \sum\_{n=1}^4 \left( A\_n' \cos(n\pi) + B\_n' \sin(n\pi) \right),
\tag{57}
$$

0 2\*pi

numerical first approx. second approx.

Dynamics of a Pendulum of Variable Length and Similar Problems 85

*θ* calculated from the first order approximate solution (56), second order

<sup>2</sup>*<sup>τ</sup>* <sup>−</sup> <sup>√</sup>*εϑ*

 + *εμ ϑ*3

*q*

<sup>+</sup> <sup>√</sup>*ε<sup>w</sup>* sin(*<sup>τ</sup>* <sup>−</sup> <sup>√</sup>*εϑ* <sup>−</sup> *<sup>δ</sup>*) (61)

(62)

<sup>√</sup>*<sup>ε</sup>* and as in the previous section *<sup>w</sup>* <sup>=</sup>

<sup>√</sup>*<sup>μ</sup> <sup>f</sup>*(*τ*, *<sup>q</sup>*, *<sup>ψ</sup>*), (64)

<sup>6</sup> <sup>+</sup> *<sup>o</sup>*(*ε*),

<sup>+</sup> *<sup>q</sup>* cos(*ψ*), *<sup>ϑ</sup>*˙ <sup>=</sup> <sup>−</sup>√*μ<sup>q</sup>* sin(*ψ*). (63)

time τ

approximate solution (60), and results of numerical simulation, when damping coefficient *β* is not small.

One can see in (38) that assumptions *<sup>ω</sup>* <sup>∼</sup> *<sup>β</sup>* <sup>∼</sup> <sup>√</sup>*<sup>ε</sup>* are valid for the high frequency of excitation <sup>Ω</sup> <sup>∼</sup> 1/√*<sup>ε</sup>* with other parameters being of order 1. Another option is small gravity *<sup>g</sup>* <sup>∼</sup> *<sup>ε</sup>* along

<sup>−</sup> <sup>√</sup>*εβ*˜*ϑ*˙ <sup>+</sup> <sup>√</sup>*<sup>ε</sup>* sin

*<sup>ω</sup>*2/*ε*. With zero right-hand side equation (61) *<sup>ϑ</sup>*¨ <sup>+</sup> *μϑ* <sup>−</sup> *<sup>β</sup>*˜ <sup>=</sup> 0 would describe harmonic oscillations about *β*˜/*μ* value with frequency √*μ*. After Taylor's expansion of sines in the

which describes oscillator with both basic and parametric excitations. To solve equation (62) we use the method of averaging [9, 18, 19]. For that purpose we write (62) in the *standard form* of first order differential equations with small right-hand sides. First, we use *Poincaré variables <sup>q</sup>* and *<sup>ψ</sup>* defined via the following solution of *generating system <sup>ϑ</sup>*¨ <sup>+</sup> *μϑ* <sup>−</sup> *<sup>β</sup>*˜ <sup>=</sup> 0 which is (62)

In Poincaré variables equation (62) becomes a system of first order differential equations

<sup>√</sup>*<sup>μ</sup> <sup>f</sup>*(*τ*, *<sup>q</sup>*, *<sup>ψ</sup>*), *<sup>ψ</sup>*˙ <sup>=</sup> <sup>√</sup>*<sup>μ</sup>* <sup>−</sup> cos *<sup>ψ</sup>*

**3.4. Approximate rotational solutions when** *<sup>ε</sup>* <sup>≈</sup> <sup>0</sup>**,** *<sup>ω</sup>* <sup>∼</sup> <sup>√</sup>*ε***, and** *<sup>β</sup>* <sup>∼</sup> <sup>√</sup>*<sup>ε</sup>*

After change of variable *<sup>θ</sup>* <sup>=</sup> <sup>−</sup>*<sup>τ</sup>* <sup>+</sup> <sup>√</sup>*εϑ* equation (39) takes the following form

*<sup>ϑ</sup>*¨ <sup>+</sup> (*<sup>μ</sup>* <sup>+</sup> *<sup>ε</sup>* cos(2*τ*) <sup>+</sup> *<sup>ε</sup><sup>w</sup>* cos(*<sup>τ</sup>* <sup>−</sup> *<sup>δ</sup>*)) *<sup>ϑ</sup>* <sup>−</sup> *<sup>β</sup>*˜

sin(2*τ*) <sup>+</sup> *<sup>w</sup>* sin(*<sup>τ</sup>* <sup>−</sup> *<sup>δ</sup>*) <sup>−</sup> *<sup>β</sup>*˜*ϑ*˙

right-hand side of (61) about *ϑ* = 0 we obtain the following equation

−1.2

−1.1

−1

−0.9

angular velocity dθ / dτ

Parameters: *δ* = 0, *μ* = 1, *ω* = 0.3, *ε* = 0.2, *β* = 0.5.

**Figure 10.** Angular velocities ˙

with small ratio *<sup>c</sup>*/*<sup>m</sup>* <sup>∼</sup> <sup>√</sup>*ε*.

*<sup>ϑ</sup>* <sup>−</sup> sin<sup>√</sup>*εϑ* √*ε*

with small right-hand side, where we denote *β*˜ = *β*/

<sup>=</sup> <sup>√</sup>*<sup>ε</sup>* 

> *<sup>ϑ</sup>* <sup>=</sup> *<sup>β</sup>*˜ *μ*

*<sup>q</sup>*˙ <sup>=</sup> <sup>−</sup>sin *<sup>ψ</sup>*

*<sup>ϑ</sup>*¨ <sup>+</sup> *μϑ* <sup>−</sup> *<sup>β</sup>*˜ <sup>=</sup> *<sup>μ</sup>*

with *ε* = 0

−0.8

−0.7

where coefficients in the right-hand side are the following

*A*� <sup>0</sup> <sup>=</sup> *b*2 <sup>2</sup> <sup>+</sup> *<sup>b</sup>*<sup>2</sup> <sup>1</sup> <sup>+</sup> *<sup>a</sup>*<sup>2</sup> <sup>2</sup> <sup>+</sup> *<sup>a</sup>*<sup>2</sup> 1 *β* + (*A*1*b*<sup>1</sup> + *A*2*b*<sup>2</sup> − *B*2*a*<sup>2</sup> − *B*1*a*1) *A*� <sup>1</sup> = (*a*1*a*<sup>2</sup> + *b*1*b*2) *β* + (*A*1*b*<sup>2</sup> + *A*2*b*<sup>1</sup> − *B*1*a*<sup>2</sup> − *B*2*a*1)/2 *A*� <sup>2</sup> <sup>=</sup> *a*2 <sup>1</sup> <sup>−</sup> *<sup>b</sup>*<sup>2</sup> 1 *β*/2 − (*A*1*b*<sup>1</sup> + *B*1*a*1)/2 *A*� <sup>3</sup> = (*a*1*a*<sup>2</sup> − *b*1*b*2) *β* − (*A*2*b*<sup>1</sup> + *A*1*b*<sup>2</sup> + *B*1*a*<sup>2</sup> + *B*2*a*1)/2 *A*� <sup>4</sup> <sup>=</sup> *a*2 <sup>2</sup> <sup>−</sup> *<sup>b</sup>*<sup>2</sup> 2 *β*/2 − (*A*2*b*<sup>2</sup> + *B*2*a*2)/2 *B*� <sup>1</sup> = (*a*1*b*<sup>2</sup> − *b*1*a*2) *β* − (*A*1*a*<sup>2</sup> − *A*2*a*<sup>1</sup> + *B*1*b*<sup>2</sup> − *B*2*b*1)/2 *B*� <sup>2</sup> = *β a*1*b*<sup>1</sup> + (*A*1*a*<sup>1</sup> − *B*1*b*1)/2 *B*� <sup>3</sup> = (*a*1*b*<sup>2</sup> + *b*1*a*2) *β* + (*A*1*a*<sup>2</sup> + *A*2*a*<sup>1</sup> − *B*1*b*<sup>2</sup> − *B*2*b*1)/2 *B*� <sup>4</sup> = *β a*2*b*<sup>2</sup> + (*A*2*a*<sup>2</sup> − *B*2*b*2)/2. (58)

Periodic solution for equation (57) has the form

$$\theta\_2(\tau) = \frac{A\_0'}{2\sqrt{\mu^2 - \beta^2}} - \sum\_{n=1}^4 \frac{(n^2 - \sqrt{\mu^2 - \beta^2})A\_n' + n\beta B\_n'}{(n^2 - 1)\beta^2 + \mu^2 + n^2(n^2 - 2\sqrt{\mu^2 - \beta^2})} \cos(n\tau)$$

$$- \sum\_{n=1}^4 \frac{-n\beta A\_n' + (n^2 - \sqrt{\mu^2 - \beta^2})B\_n'}{(n^2 - 1)\beta^2 + \mu^2 + n^2(n^2 - 2\sqrt{\mu^2 - \beta^2})} \sin(n\tau) \,. \tag{59}$$

Thus, second order approximate solution can be shortly written in the following form

$$\theta = -\tau + \theta\_0 + \varepsilon \theta\_1(\tau) + \varepsilon^2 \theta\_2(\tau) + o(\varepsilon^2), \tag{60}$$

where constant *θ*<sup>0</sup> is defined in (45), function *θ*<sup>1</sup> in (55), and function *θ*<sup>2</sup> in (59). In Fig. 10 it is shown how first and second order approximate solutions approach the numerical solution.

In this section the straightforward asymptotic method works since all solutions in each approximation converge to corresponding unique periodic solutions because damping *β* is not small. If damping *β* is small the analysis becomes more complicated since equation (49) takes the form ¨ *θ*<sup>0</sup> + *μ* sin(*θ*0) = 0 and has a solution expressed in elliptic functions. One can simplify the analysis assuming that *<sup>θ</sup>* <sup>=</sup> <sup>−</sup>*<sup>τ</sup>* <sup>+</sup> *<sup>ε</sup>*1/2*θ*<sup>0</sup> <sup>+</sup> *εθ*1(*τ*) + *<sup>ε</sup>*3/2*θ*2(*τ*) + . . . instead of (48). We will use this assumption in the next section to apply the classical averaging technique to the problem of small damping *<sup>β</sup>* <sup>∼</sup> <sup>√</sup>*ε*.

16 Will-be-set-by-IN-TECH

<sup>1</sup> <sup>−</sup> *<sup>μ</sup>*<sup>2</sup> <sup>−</sup> *<sup>β</sup>*<sup>2</sup>

<sup>4</sup> <sup>−</sup> *<sup>μ</sup>*<sup>2</sup> <sup>−</sup> *<sup>β</sup>*<sup>2</sup>

*<sup>n</sup>* cos(*nτ*) + *B*�

*β* + (*A*1*b*<sup>1</sup> + *A*2*b*<sup>2</sup> − *B*2*a*<sup>2</sup> − *B*1*a*1)

*<sup>n</sup>* + *nβB*� *n*

*<sup>μ</sup>*<sup>2</sup> <sup>−</sup> *<sup>β</sup>*2)

cos(*nτ*)

sin(*nτ*). (59)

<sup>2</sup>), (60)

<sup>3</sup>*β*<sup>2</sup> <sup>+</sup> *<sup>μ</sup>*<sup>2</sup> <sup>+</sup> <sup>8</sup>(<sup>2</sup> <sup>−</sup> *<sup>μ</sup>*<sup>2</sup> <sup>−</sup> *<sup>β</sup>*2)

sin(*τ* − *θ*<sup>0</sup> − *δ*)

sin(2*τ* − *θ*0)

*<sup>μ</sup>*<sup>2</sup> <sup>−</sup> *<sup>β</sup>*<sup>2</sup> <sup>+</sup> *<sup>o</sup>*(*ε*), (56)

*<sup>n</sup>* sin(*nτ*)

, (57)

(58)

2 *β* cos(2*τ* − *θ*0) +

*<sup>μ</sup>*<sup>2</sup> <sup>−</sup> *<sup>β</sup>*2*θ*<sup>2</sup> <sup>=</sup> *<sup>A</sup>*�

<sup>1</sup> <sup>+</sup> *<sup>a</sup>*<sup>2</sup>

<sup>2</sup> = *β a*1*b*<sup>1</sup> + (*A*1*a*<sup>1</sup> − *B*1*b*1)/2

<sup>4</sup> = *β a*2*b*<sup>2</sup> + (*A*2*a*<sup>2</sup> − *B*2*b*2)/2.

4 ∑ *n*=1

(*n*<sup>2</sup> − <sup>1</sup>)*β*<sup>2</sup> + *<sup>μ</sup>*<sup>2</sup> + *<sup>n</sup>*2(*n*<sup>2</sup> − <sup>2</sup>

<sup>2</sup> <sup>+</sup> *<sup>a</sup>*<sup>2</sup> 1 

*<sup>μ</sup>*<sup>2</sup> + <sup>1</sup> − <sup>2</sup>

0 <sup>2</sup> <sup>+</sup>

4 ∑ *n*=1 *A*�

<sup>1</sup> = (*a*1*a*<sup>2</sup> + *b*1*b*2) *β* + (*A*1*b*<sup>2</sup> + *A*2*b*<sup>1</sup> − *B*1*a*<sup>2</sup> − *B*2*a*1)/2

<sup>3</sup> = (*a*1*a*<sup>2</sup> − *b*1*b*2) *β* − (*A*2*b*<sup>1</sup> + *A*1*b*<sup>2</sup> + *B*1*a*<sup>2</sup> + *B*2*a*1)/2

<sup>1</sup> = (*a*1*b*<sup>2</sup> − *b*1*a*2) *β* − (*A*1*a*<sup>2</sup> − *A*2*a*<sup>1</sup> + *B*1*b*<sup>2</sup> − *B*2*b*1)/2

<sup>3</sup> = (*a*1*b*<sup>2</sup> + *b*1*a*2) *β* + (*A*1*a*<sup>2</sup> + *A*2*a*<sup>1</sup> − *B*1*b*<sup>2</sup> − *B*2*b*1)/2

*<sup>n</sup>* + (*n*<sup>2</sup> <sup>−</sup> *<sup>μ</sup>*<sup>2</sup> <sup>−</sup> *<sup>β</sup>*2)*B*�

Thus, second order approximate solution can be shortly written in the following form

where constant *θ*<sup>0</sup> is defined in (45), function *θ*<sup>1</sup> in (55), and function *θ*<sup>2</sup> in (59). In Fig. 10 it is shown how first and second order approximate solutions approach the numerical solution. In this section the straightforward asymptotic method works since all solutions in each approximation converge to corresponding unique periodic solutions because damping *β* is not small. If damping *β* is small the analysis becomes more complicated since equation (49)

simplify the analysis assuming that *<sup>θ</sup>* <sup>=</sup> <sup>−</sup>*<sup>τ</sup>* <sup>+</sup> *<sup>ε</sup>*1/2*θ*<sup>0</sup> <sup>+</sup> *εθ*1(*τ*) + *<sup>ε</sup>*3/2*θ*2(*τ*) + . . . instead of (48). We will use this assumption in the next section to apply the classical averaging technique to

*θ* = −*τ* + *θ*<sup>0</sup> + *εθ*1(*τ*) + *ε*

(*n*<sup>2</sup> <sup>−</sup> *<sup>μ</sup>*<sup>2</sup> <sup>−</sup> *<sup>β</sup>*2)*A*�

*n*

*<sup>μ</sup>*<sup>2</sup> <sup>−</sup> *<sup>β</sup>*2)

*θ*<sup>0</sup> + *μ* sin(*θ*0) = 0 and has a solution expressed in elliptic functions. One can

<sup>2</sup>*θ*2(*τ*) + *o*(*ε*

(*n*<sup>2</sup> − <sup>1</sup>)*β*<sup>2</sup> + *<sup>μ</sup>*<sup>2</sup> + *<sup>n</sup>*2(*n*<sup>2</sup> − <sup>2</sup>

*β*/2 − (*A*1*b*<sup>1</sup> + *B*1*a*1)/2

*β*/2 − (*A*2*b*<sup>2</sup> + *B*2*a*2)/2

written as follows

*θ* = −*τ* + *θ*<sup>0</sup> − *ε*

where constant *θ*<sup>0</sup> is defined in (45).

Equation (54) takes the following form

*A*� <sup>0</sup> <sup>=</sup> *b*2 <sup>2</sup> <sup>+</sup> *<sup>b</sup>*<sup>2</sup>

*A*�

*A*� <sup>2</sup> <sup>=</sup> *a*2 <sup>1</sup> <sup>−</sup> *<sup>b</sup>*<sup>2</sup> 1 

*A*�

*A*� <sup>4</sup> <sup>=</sup> *a*2 <sup>2</sup> <sup>−</sup> *<sup>b</sup>*<sup>2</sup> 2 

*B*�

*B*�

*B*�

*B*�

*<sup>θ</sup>*2(*τ*) = *<sup>A</sup>*�

− 4 ∑ *n*=1

takes the form ¨

2

the problem of small damping *<sup>β</sup>* <sup>∼</sup> <sup>√</sup>*ε*.

Periodic solution for equation (57) has the form

0

*<sup>μ</sup>*<sup>2</sup> <sup>−</sup> *<sup>β</sup>*<sup>2</sup> <sup>−</sup>

−*nβA*�

*3.3.2. Second order approximation*

¨ *θ*<sup>2</sup> + ˙ *θ*<sup>2</sup> + 

<sup>−</sup>*ω*<sup>2</sup> *<sup>β</sup>* cos(*<sup>τ</sup>* <sup>−</sup> *<sup>θ</sup>*<sup>0</sup> <sup>−</sup> *<sup>δ</sup>*) <sup>+</sup>

where coefficients in the right-hand side are the following

**Figure 10.** Angular velocities ˙ *θ* calculated from the first order approximate solution (56), second order approximate solution (60), and results of numerical simulation, when damping coefficient *β* is not small. Parameters: *δ* = 0, *μ* = 1, *ω* = 0.3, *ε* = 0.2, *β* = 0.5.
