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 Moatimid13.14, 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

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


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 Moatimid13.14the recognized natural frequency (omega^{2}) can be expanded as follows:

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


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


Allow Laplace transforms (LJ) mixtures of Eqs. (11-13), we find

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


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

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


Subsequently, the nonlinear part can be formulated as follows:

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



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


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:





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

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


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


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

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


Therefore, the appropriate solution (theta_{2}


Given the HPM, the approximate bounded solution of the most advanced equation as given in Equation. (10) can be entered as follows:



where (theta_{0}


Required stability standards require that (Omega^{2}) be real and positive. Numerical calculation showed that Eq. (25) has only two true positive roots as follows: (Omega = 0.735216) and (Omega = 0.009315).

The desired stability requires that (Omega^{2}) is both real and positive. To confirm the updated HPM, the approximate analytical and numerical solutions are presented in a single diagram for more opportunity. Therefore, for a randomly chosen system where (r = 1) and (k = 1), the diagram below is drawn, according to the Mathematica software (, where Eq. (25) just has a real root like (Omega = 0.735216).

The approximate solution (AS) as given in Eq. (24) received a lot of attention, especially when (k = 1) and (r = 1). Therefore, the part (a) in Fig. 2 shows the time evolution of the obtained solution (theta) against time (t)meanwhile, part (b) reveals the phase plane diagram of this solution as a function of its first derivative (theta^{prime}) at the same considered values ​​of (k) and (r). A closer look at the curves drawn in this figure reveals that we have obtained a symmetric periodic wave, where its amplitude and wavelength remain stationary. The conclusion that can be made here is that the accomplished solution has a stable behavior, where the partially symmetric closed curve (b) affirms this assertion.

Figure 2

Displays the AS at (k = 1) and (r = 1).

Moreover, the numerical solution (NS) of Eq. (10) is performed using the RK4 approach and it is plotted with the same mentioned values ​​of (k) and (r)see Figs. 2. The curves displayed in parts (a) and (b) of this figure show the variation of (theta) against time (t) and the corresponding phase plane is graphically represented, respectively. It is clear that the characterized wave is represented periodically to emphasize its stability during the period of time studied. This behavior is plotted against its first derivative to give a closed symmetric curve as in the part (b). An inspection of the curves of Figs. 2 and 3 show a great coherence between them, which reveals the good precision of the two solutions.

picture 3
picture 3

Reveal the NS at (k = 1) and (r = 1).

The curves sketched in FIG. 4 indicate the variation of (Omega) Going through (omega) for different values ​​of (r) when (k = 2). These trajectories are obviously symmetric around the horizontal axis, which is in good agreement with Eq. (25).

Figure 4
number 4

Illustrates the variation of (r) in the plane (omega_{c} Omega) at (k = 1).