Completed version of this article can be found in Hindawi Publishing Corporation (vol 2006, article ID 25193)
Numeralytics: Investigation of a Spatial Double Pendulum:
an Engineering Approach
S. Bendersky and B. Sandler
1. Abstract
The behavior of a spatial double pendulum (SDP), comprising two pendulums that swing in different planes, was investigated. Movement equations (i.e., mathematical model) were derived for this SDP, and oscillations of the system were computed and compared with experimental results. Matlab® computer programs were used for solving the nonlinear differential equations by the Runge-Kutta method. Fourier transformation was used to obtain the frequency spectra for analyses of the oscillations of the two pendulums. Solutions for free oscillations of the pendulums and graphic descriptions of changes in the frequency spectra were used for the dynamic investigation of the pendulums for different initial conditions of motion.
Fig. 1. Model and computation layout Fig. 2. Photograph of the SDP.
of a SDP device.
2. Introduction
The question that we set out to answer in this study is: What is the influence of the angle Y on the behavior of the SDP? The following steps were taken: formulating the dynamic model by applying the Lagrange method and finding solutions for small and non-small values of the angles ji (i = 1, 2), respectively; obtaining graphical solutions for free oscillations and graphical descriptions of the changes in the frequency spectra by using Fourier transformations for different parameter values and initial conditions.
2.1 General Model
The following equations for free oscillations without damping (obtained by the Lagrange method) govern the behavior of the SDP:
(1)
here T and P are the kinetic and potential energy, respectively, as given by:
(2)
(3)
here 
and 
are the deviation angles of pendulums 1 and 2, respectively, and g is acceleration due to gravity. For deriving the movement equations for our SDP system, we defined:
The equations governing motion may then be expressed as follows:
(4)
There are a number of problems associated with the numerical and analytical solutions to this system of nonlinear equations (4). Iterative techniques are traditionally used to obtain numerical solutions, but nearly all iterative methods are sensitive to the initial solutions. Solutions to the linear model of SDP and to the pure nonlinear case were discussed. The perturbation expansion method for small parameters – widely used to analyze simple nonlinear problems– is not really effective for our mechanism.
3. Application of Numeralytics Solutions
The general nonlinear and linear forms of a mathematical model (movement equations) for the SDP were shown in the previous section. For the linear case, we accepted the conventional analytical expression for the natural frequencies and limitations of the amplitude ratios influencing the angle Y. For the nonlinear case, we took a numeralytical approach by applying the MATLAB® programs package.
Fig. 3. (a) "Phase portraits" for the two pendulums; (b) swinging angles j1(t) and j2(t); (c) oscillation spectrum distributions for the two pendulums. Parameters are
m1 = m2 = 1.57 kg, L1 = L2 = 0.3 m, Y = 0?. Initial conditions are j1 = 50?, j2 = 10?
Since it is not possible to answer the question – the influence of the angle Y on the behavior of the SDP – directly from the given equation system (4), we use the Runge-Kutta method for solving the nonlinear equations and Fourier transformation for investigation of the frequency spectra of the two pendulums. The solutions are represented graphically for a wide range of initial conditions, values of the parameters, and different values of the angle Y. Solutions were found for Y = 0?, Y = 15?, Y = 30?, Y = 60? and Y = 90?. Graphical solutions for the general model of the movement equation (4), for the described parameters, and for different initial conditions are shown in Fig. 3 and 4. (We omit here intermediate cases).
Fig. 4. (a) "Phase portraits" for both pendulums; (b) swinging angles j1(t) and j2(t); (c) oscillation spectrum distributions for the two pendulums. Parameters are
m1 = m2= 1.57 kg, L1= L2= 0.3 m, Y = 30?. Initial conditions are j1 = 50°, j2 = 50°
Let us now consider the case shown in Fig. 4(a) and (c) in comparison with that from Fig. 3. In this case, at least three and four frequencies govern the behavior of the first and second pendulums, respectively. The “phase portraits”, especially those of the second pendulum, are far from to be elliptic: the oscillations are close to non-linear. As opposed to linear case solutions, the nonlinear case is identified by “accidental–like” ("chaotic-like") number and values of frequencies for both pendulums of the system. Here the engineering numeralytical approach, based on statistical investigation of numerical (or/and graphical) solution data, is helpful. We deal here with processes, which formally are predictable, however, practically for some data become “unpredictable”.
4. Behavior of the SDP
In this investigation, we study the influence of the value Y on the frequential content of the pendulum oscillations. Figures 6 and 7 are presenting peak distribution curves for pendulums (PDCP) for a range of parameters and initial conditions, when the number of frequency peaks is plotted versus the angle Y for the first and second pendulums. In addition, the coefficients of those curves equation are shown:
(5)
Various initial conditions and parameter values for the SDP were considered in the search for analytical answers to the question formulated at the outset of the study: What is the influence of the angle Y on the free oscillations of the mathematical model of this system? Using the number of frequency peaks comprising the frequency spectra of the pendulum oscillations as a criterion, we can identify and estimate the behavior of the system and find the coefficients describing the equation (5).
Fig.6. PDCP graphics in cases m1 = m2 when 0.5 kg<m1<4 kg and 5L1<L2<15L1 when 0.1m<L1<5m for: ) 2j2<j1<6j2 when 2?<j2<15?. Solid curve is for pendulum 1, and doted curve for the pendulum 2.
Fig. 7. Peak distribution curves for pendulums (PDCP) for cases in which m1 = m2 when 0.5 kg < m1 < 4 kg and L1 = L2 when 0.1 m < L1 < 5 m for: j1 = j2 when 5?<j1<40?. Solid curve is for pendulum 1, and doted curve for the pendulum 2.