In this study, the iterative harmonic balance method was used to develop analytical solutions of period-one rotations of a pendulum driven horizontally by harmonic excitations. The performance of the method was evaluated by two criteria, one based on the system energy error and the other based on the global residual error. As a comparison, analytical solutions based on the multi-scale method were also considered. Numerical solutions obtained from the Dormand-Prince method (ODE45 in MATLAB ©) were used as the baseline for evaluation. It was found that under lower-level excitations, the multi-scale method performed better than the iterative method. At higher-level excitations, however, the performance of the iterative method was noticeably more accurate.
Under external excitations, a pendulum can exhibit rich dynamical behavior. The dynamical behavior is related to the potential well of pendulum. Captured in the potential well, the motion of a pendulum corresponds to oscillation [
In recent studies, period-one rotating solutions of a vertically excited pendulum have been approximately solved by the multi-scale method [
In this study, the analytical solution for period-one rotating orbits of a horizontally excited pendulum was developed using iterative harmonic balance. The performance of the analytical solutions was evaluated by two criteria, i.e. the system energy error and the global residual error, and was compared with that of the multi-scale method. The numerical calculation obtained from the Dormand-Prince (ODE45 in MATLAB©) was used as the baseline for the performance comparison. It was found that the analytical solutions developed were in excellent agreement with the numerical methods. Moreover, the iterative method performed better than the multi-scale method at higher-level excitations, while the multi-scale method excelled at lower-level excitations.
For a planar pendulum with a point mass of m and massless arm of length of l under a horizontal harmonic base excitation (i.e. the displacement
where θ is the angular displacement of the pendulum, (?) denotes differentiation of the argument with respect to a non-dimensional time variable
For a period-one rotating orbit, the magnitude of the angular velocity of the pendulum is equal to the excitation frequency with a small periodic perturbation with zero mean over one period [
where
Based on Equation (2), the exact solution of period-one rotations can be readily obtained as
where
in which
Substituting Equation (3) into the right-hand side of Equation (1) and expanding
The following iterative process is proposed for the estimation of the parameters in Equations (3) and (4):
Step 1: The iteration starts with the zero-order approximation of
An approximate solution
Step 2: Using the approximate parameters in Equation (5) and truncating the series at a desired order, an approximate equation of motion can be obtained (Equation (12)) and solved using Equations (3) and (4).
…
Step n: Using the solution from the (n-1)th step and follow the same iterative process, the solution for the nth iteration can be obtained.
The solution of Equation (6) is
where
and
in which
Note that
which gives a lower bound on the normalized excitation amplitude, p.
In the second iteration, using the result of the first iteration and considering the first-order approximation, the following modified governing equation is obtained.
where
Equation (12) is solved as
where
and
in which
and
As
Following a similar procedure, higher-order iterations can be derived. The general form of higher-order iterations is reported in Appendix.
In this study, the performance of analytical solutions based on the method considered was evaluated using two different criteria, i.e. the system energy from a physical perspective and the global residual of the governing equation from a mathematical perspective. Error in system energy was defined as the root-mean-square value of the relative error in one period η, i.e.
in which
where
The global residual error was defined as the root-mean-square value of the error in one period Re, i.e.
Error analysis for the analytical solutions was first evaluated. Excitations were fixed at ω = 3. Three levels of excitation, i.e. p = 0.1, p = 1, and p = 10, were considered. Under each level of excitation, the normalized damping ratio, γ was swept from nearly zero to near the threshold value defined in Equation (11). For comparison, the solutions based on the multi-scale method [
The relationship between the performance index from the system energy error, η, and the normalized damping ratio, γ is shown in
The relationship between the performance index based on the global residual error, Re, and the normalized damping ratio, γ is summarized in
The phase portraits over one period are described in Figures 3-5. Three levels of excitations, i.e. p = 0.1, 1, 10, were considered. It can be seen from
The convergence of the coefficients Ck obtained from the iterative method was then investigated. The maximum number of iterations considered was 16. The normalized frequency was fixed at ω = 3 and the damping ratio was fixed at γ = 0.01. Three levels of excitation, i.e. p = 0.1, p = 1, and p = 10, were considered. The effect of the number of iterations on the convergence of the coefficients Ck is summarized in
ones, only the results for C1 to C8 are presented. As can be seen from
The lower bound of the excitation amplitude is shown in
In this study, period-one rotating solution of a horizontally excited pendulum was solved by the iterative harmonic balance method. The general formulas of solutions were derived. The numerical solution obtained from the Dormand-Prince method (ODE45 in MATLAB©) was used as the baseline for performance evaluation. The performance of the solutions was evaluated by two criteria, one based on the system energy error and the other based on the global residual error. The results showed that two iterations were sufficient for a reasonable accuracy. In addition, the solutions based on the multi-scale method were also shown for comparison. Under lower- level excitations, the performance of solution based on the multi-scale method was slightly better than that of the iterative method. Under higher-level excitations, however, the iterative harmonic balance method was more accurate.
This research was partially supported by National Science Foundation (Grant No. CMMI 0758632). Such generous support is greatly acknowledged.
In the nth iteration, the modified governing equation is
where
The solution of Equation (22) can be obtained as
where
and
in which
and
As