# Analytical solution for the movement of a pendulum with rolling wheel: stability analysis

As stated earlier, the fundamental equation of motion (10) is nonlinear. In reality, it has no bearing on an exact closed-form solution. Subsequently, it must be scrutinized by a disturbance technique. Consistent with our previous work, as reported by Moatimid^{13.14}, traditional HPM generates physically inconvenient secular terms. Accordingly, a nonlinear broadened frequency adaptation to the HPM is proposed. In this case, the homotopy equation could be written as follows to achieve this goal

$$ddot{theta } + omega^{2} theta + rho left( { – alpha theta^{2} ddot{theta } – alpha theta dot{theta } ^{2} + frac{alpha }{6}theta^{3} dot{theta }^{2} – beta theta^{3} } right) = 0,,, ,,rho in left[ {0,,,1} right].$$

(11)

To reach a perturbed solution, for more relevance, we can accept that the following initial conditions: (,,theta (0) = 1) and (point{theta}(0) = 0).

By the technique of the previous complete work of Moatimid^{13.14}the recognized natural frequency (omega^{2}) can be expanded as follows:

$$Omega^{2} = omega^{2} + sumlimits_{i = 1}^{n} {rho^{i} } omega_{i} .$$

(12)

According to HPM procedures, the time-dependent function (theta

(13)

Allow Laplace transforms (*L*_{J}) mixtures of Eqs. (11-13), we find

$$L_{T} left{ {theta (t;rho )} right} = frac{s}{{s^{2} + Omega^{2} }} – frac{ rho }{{s^{2} + Omega^{2} }}L_{T} left{ { – alpha theta^{2} ddot{theta } – alpha theta dot {theta }^{2} + frac{alpha }{6}theta^{3} dot{theta }^{2} – beta theta^{3} – left( {omega_ {1} + rho omega_{2} } right)} right}.$$

(14)

Keeping the inverse Laplace transforms in Eq. (19), we get

$$theta (t;rho ) = cos ,Omega t – rho L_{T}^{ – 1} left[ {frac{1}{{s^{2} + Omega^{2} }}L_{T} left{ { – alpha theta^{2} ddot{theta } – alpha theta dot{theta }^{2} + frac{alpha }{6}theta^{3} dot{theta }^{2} – beta theta^{3} – left( {omega_{1} + rho omega_{2} } right)} right}} right].$$

(15)

Subsequently, the nonlinear part can be formulated as follows:

$$Nleft( {sumlimits_{i}^{n} {rho^{i} theta_{i} } } right) = N_{0} (theta_{0}) + rho N_{1} (theta_{0} ,,theta_{1} ) + rho^{2} N_{2} (theta_{0} ,,theta_{1} ,,theta_{ 2} ) + cdots + rho^{n} N_{n} (theta_{0} ,,theta_{1} ,,theta_{2} ,, ldots ,theta_{n} ),$$

(16)

where

$$N_{n} (theta_{0} ,,theta_{1} ,,theta_{2} ,, ldots ,theta_{n} ) = frac{1}{n !} mathop {lim }limits_{rho to 0} frac{partial }{{partial rho^{n} }}Nleft( {sumlimits_{i}^{n} { rho^{i} theta_{i} } } right).$$

(17)

Using Time Dependent Function Expansion (theta (t;rho )) as given by Eq. (13), then equating the coefficient of similar powers (rho) on both sides, we obtain the following orders:

$$rho^{0}:theta_{0}

(18)

$$rho:,theta_{1}

(19)

By replacing Eq. (18) in eq. (19) to obtain a uniform development it is necessary to eliminate the secular terms. Basically, the coefficient of the circular function should disappear. Moreover, the elimination of the coefficient of the function (cos,Omega t) product

$$omega_{1} = frac{1}{48}left( { – 36beta + 25alpha Omega^{2} } right).$$

(20)

It follows that the uniform solution of (theta_{1}

(21)

Again, the substitution of Eqs. (20) and (21) to the second order of Eq. (15), it can be seen that the deletion of the secular term requires

$$omega_{2} = frac{1}{{18432Omega^{2} }}left( { – 864beta^{2} + 1644alpha beta Omega^{2} + 71 alpha^{2} Omega^{4} – 576beta omega_{1} + 1120alpha omega_{1} Omega^{2} } right).$$

(22)

Therefore, the appropriate solution (theta_{2}

(23)