**3.2. Exact rotational solution when** *ε* = 0 **and** *ω* = 0

Conditions *ε* = *ω* = 0 mean that we find the mode of rotation for the circular excitation *X* = *Y* with absence of gravity *g* = 0. In this case, we call equation (39) the *unperturbed equation*

<sup>1</sup> Note that this formula excludes the generalization *y* = *Y* cos(Ω*t* + Φ) which is considered e.g. in the model of unbalanced rotor [31–33]. Instead of Φ we introduce the angle *δ* of deviation of gravitational acceleration *g* from the vertical direction.

14 Will-be-set-by-IN-TECH 82 Nonlinearity, Bifurcation and Chaos – Theory and Applications Dynamics of a Pendulum of Variable Length and Similar Problems <sup>15</sup>

$$
\ddot{\theta} + \beta \dot{\theta} + \mu \sin(\tau + \theta) = 0 \tag{40}
$$

which has exact solutions

$$
\theta = \theta\_0 - \tau\_\prime \tag{41}
$$

**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>*

equating to zero, the set of differential equations is obtained

*θ*<sup>2</sup> + *μ* cos(*θ*0) *θ*<sup>2</sup> = *μ* sin(*θ*0) *θ*<sup>2</sup>

...

*<sup>μ</sup>*<sup>2</sup> <sup>−</sup> *<sup>β</sup>*<sup>2</sup> *<sup>θ</sup>*<sup>2</sup> <sup>=</sup> *βθ*<sup>2</sup>

(51) can be written in the following way

*θ*<sup>1</sup> + 

*θ*<sup>2</sup> + 

has the form

¨ *θ*<sup>0</sup> + *β* ˙

> ¨ *θ*<sup>1</sup> + *β* ˙

> ¨ *θ*<sup>2</sup> + *β* ˙

¨ *θ*<sup>1</sup> + *β* ˙

where *<sup>a</sup>*<sup>1</sup> <sup>=</sup> <sup>−</sup> (1<sup>−</sup>

and *<sup>b</sup>*<sup>2</sup> <sup>=</sup> <sup>−</sup>2*βA*2+(4<sup>−</sup>

unique periodic solution

the second condition in (44).

*3.3.1. First order approximation*

presented in the following form

*θ*<sup>1</sup> + 

where *A*<sup>1</sup> = −*w* cos(*δ*)*β*/*μ* − *w* sin(*δ*)

*μ*<sup>2</sup>+1−2

3*β*<sup>2</sup>+*μ*<sup>2</sup>+4(4−2

<sup>√</sup>*μ*<sup>2</sup>−*β*<sup>2</sup> )*A*1+*βB*<sup>1</sup>

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

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

¨ *θ*<sup>1</sup> + *β* ˙

¨ *θ*<sup>2</sup> + *β* ˙

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)

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

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

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

In consequence of conditions (47) non-homogeneous linear differential equation (52) can be

*<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

*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>

*θ*<sup>0</sup> + *μ* sin(*θ*0) = *β*, (49)

*<sup>μ</sup>*<sup>2</sup> − *<sup>β</sup>*<sup>2</sup> *<sup>θ</sup>*<sup>1</sup> = sin(2*<sup>τ</sup>* − *<sup>θ</sup>*0) + *<sup>w</sup>* sin(*<sup>τ</sup>* − *<sup>θ</sup>*<sup>0</sup> − *<sup>δ</sup>*) (52)

*<sup>μ</sup>*<sup>2</sup> − *<sup>β</sup>*<sup>2</sup> *<sup>θ</sup>*<sup>1</sup> = *<sup>A</sup>*<sup>1</sup> cos(*τ*) + *<sup>B</sup>*<sup>1</sup> sin(*τ*) + *<sup>A</sup>*<sup>2</sup> cos(2*τ*) + *<sup>B</sup>*<sup>2</sup> sin(2*τ*) (54)

*θ*1(*τ*) = *a*<sup>1</sup> cos(*τ*) + *b*<sup>1</sup> sin(*τ*) + *a*<sup>2</sup> cos(2*τ*) + *b*<sup>2</sup> sin(2*τ*), (55)

<sup>√</sup>*μ*<sup>2</sup>−*β*<sup>2</sup> )*B*<sup>1</sup>

<sup>√</sup>*μ*<sup>2</sup>−*β*<sup>2</sup> , *<sup>a</sup>*<sup>2</sup> <sup>=</sup> <sup>−</sup> (4<sup>−</sup>

. Thus, the solution for (39) in the first approximation can be

<sup>1</sup> <sup>−</sup> *<sup>β</sup>*2/*μ*2, *<sup>B</sup>*<sup>1</sup> <sup>=</sup> *<sup>w</sup>* cos(*δ*)

*θ*<sup>1</sup> + *μ* cos(*θ*0) *θ*<sup>1</sup> = sin(2*τ* − *θ*0) + *w* sin(*τ* − *θ*<sup>0</sup> − *δ*), (50)

<sup>2</sup>*θ*<sup>2</sup> + ... (48)

<sup>1</sup>/2 − (cos(2*τ* − *θ*0) + *w* cos(*τ* − *θ*<sup>0</sup> − *δ*)) *θ*1, (51)

Dynamics of a Pendulum of Variable Length and Similar Problems 83

<sup>1</sup>/2 − (cos(2*τ* − *θ*0) + *w* cos(*τ* − *θ*<sup>0</sup> − *δ*)) *θ*1, (53)

<sup>1</sup> <sup>−</sup> *<sup>β</sup>*2/*μ*<sup>2</sup> <sup>−</sup> *<sup>w</sup>* sin(*δ*)*β*/*μ*,

3*β*<sup>2</sup>+*μ*<sup>2</sup>+4(4−2

<sup>√</sup>*μ*<sup>2</sup>−*β*<sup>2</sup>)*A*2+2*βB*<sup>2</sup>

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

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

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

$$\sin(\theta\_0) = \frac{\beta}{\mu},\tag{42}$$

provided that |*β*| ≤ *μ*.

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 equality (42), we obtain the linear equation

$$
\ddot{\eta} + \beta \dot{\eta} + \mu \cos(\theta\_0) \eta = 0. \tag{43}
$$

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. Which happens when all coefficients in (43) are positive

$$
\beta > 0, \quad \mu \cos(\theta\_0) > 0 \tag{44}
$$

due to the Routh–Hurwitz conditions. From conditions (44), assumption *μ* > 0 in (38), and equality (42), it follows for *β* > 0 that the solutions

$$
\theta = \theta\_0 - \tau, \quad \theta\_0 = \arcsin\left(\frac{\beta}{\mu}\right) + 2\pi k \tag{45}
$$

are asymptotically stable, while the solutions

$$
\theta = \theta\_0 - \tau, \quad \theta\_0 = \pi - \arcsin\left(\frac{\beta}{\mu}\right) + 2\pi k \tag{46}
$$

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

$$0 < \beta < \mu,\tag{47}$$

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 (42).
