1 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 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 ii = 1, 2), respectively; obtaining graphical solutions 2 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: d Qi , i 1,2 dt qi qi qi (1) here T and P are the kinetic and potential energy, respectively, as given by: 1 m1 m 2 L1 12 1 m 2 L22 22 1 m 2 L22 12 cos 2 sin 2 2 1 m 2 L22 12 cos 2 2 2 2 2 2 2 m 2 L2 1 2 cos m 2 L1 L2 1 2 cos cos 2 m 2 L1 L2 12 cos 2 2 (2) m1 gL1 1 cos 1 m 2 gL1 1 cos 1 m 2 gL2 (1 cos 1 cos 2 ) m 2 gL2 cos sin 1 sin 2 (3) here 1 and 2 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: const ; Qi 0; q1 1 t ; q 2 2 t ; q1 1 t ; q 2 2 t . The equations governing motion may then be expressed as follows: 1 m1 m 2 L1 m 2 L2 cos 2 sin 2 2 m 2 L2 cos 2 2 2m 2 L1 L2 cos 2 2 2 m 2 L2 cos m 2 L1 L2 cos cos 2 2 2 2 1 2 2m 2 L2 cos 2 cos 2 sin 2 2m 2 L2 cos 2 sin 2 2m 2 L1 L2 sin 2 2 2 2 m 2 L1 L2 cos sin 2 2 m1 m 2 gL1 sin 1 m 2 gL2 cos cos 1 sin 2 sin 1 cos 2 0 2 m 2 L2 1 m 2 L2 cos m 2 L1 L2 cos cos 2 2 2 12 m 2 L2 cos 2 sin 2 m 2 L2 cos 2 sin 2 cos 2 m 2 L1 L2 sin 2 2 2 m 2 gL2 cos sin 1 cos 2 cos 1 sin 2 0 (4) There are a number of problems associated with the numerical and analytical solutions to this system of nonlinear equations (4). Iterative techniques are 3 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 . 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 1(t) and 2(t); (c) oscillation spectrum distributions for the two pendulums. Parameters are m1 = m2 = 1.57 kg, L1 = L2 = 0.3 m, 0º. Initial conditions are50º10º Since it is not possible to answer the question – the influence of the angle 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 4 parameters, and different values of the angle . Solutions were found for = 0˚, = 15˚, = 30˚, = 60˚ and = 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 1(t) and 2(t); (c) oscillation spectrum distributions for the two pendulums. Parameters are m1 = m2= 1.57 kg, L1= L2= 0.3 m, 30º. Initial conditions are5050 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 5 In this investigation, we study the influence of the value 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 for the first and second pendulums. In addition, the coefficients of those curves equation are shown: PDCP1, 2 A 4 B 3 C 2 D E (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 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: ) 2<6 when<15º. Solid curve is for pendulum 1, and doted curve for the pendulum 2. 6 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: when 5º<<40º. Solid curve is for pendulum 1, and doted curve for the pendulum 2. een activation energies of proton- promoted and oxalate-promoted dissolution of kaolinite indicates that the magnitude of the rates of each of these reaction mechanisms is determined by the surface concentration of the various adsorbed species and not necessarily by the rate of the hydrolysis. In order for an adsorbed OM to catalyze or inhibit dissolution it should form a strong complex with a metal ion on the surface. In the case of catalysis, the complexation weakens the strength of the bond between the metal and the bulk mineral, resulting in the release of the OM and the ion to solution. In the case of inhibition, this bond should remain strong, so the OM would remain on the surface and block the ion from being removed. Commonly, the same surface sites and faces on which crystal tends to grow under oversaturation conditions are the sites in which dissolution occurs at undersaturation conditions. It is therefore reasonable to assume that OM which effectively inhibits dissolution would inhibit precipitation as well. For catalysis, this is not the case, since adsorbed OM that weakens the bond strength is not expected to catalyze precipitation. In order for an OM inhibitor to effectively inhibit the precipitation of a mineral, a dimensional stereochemical fit between the polar group of the inhibitor and the intercationic or interanionic distances is needed. When such a structural fit exists, the OM can also act as a template or microsubstrate and promote nucleation rather than inhibit it. Indeed, it was found that some organic compounds that accelerate mineral precipitation when present at low concentrations become an effective inhibitor at higher concentrations. A comprehensive review of the literature indicates that the common effect of OM on a wide range of minerals is to inhibit precipitation rather than to catalyze it. It is possible that this observation represent a bias of the literature and not a real effect. The Earth science literature paid little attention to low temperature precipitation of non-carbonates in general and the effect of OM on precipitation in particular. The engineering literature is motivated mainly by scale deposits and therefore tends to study inhibition of its precipitation and catalysis of its dissolution. Bearing in mind that dissolution and precipitation are affected by OM adsorption, it is surprising that only few studies on the effect of OM on dissolution and precipitation have measured the adsorption isotherm of the OM. Taking into account the numerous published papers on OM adsorption, this seems to be a niggling point. However, the effect of sorption on dissolution and precipitation depends on the sorption mechanism (e.g., physical adsorption does not enhance dissolution rate). Even if the adsorption of a given compound is studied, it is not trivial to characterize the mechanisms and, more important, the relative contribution of each of them to the overall adsorption. Moreover, sorption, dissolution and precipitation depend on both the characteristics of the mineral surface and the environmental conditions, and therefore it is complicated to combine adsorption experiments from one laboratory to dissolution or precipitation experiments that were conducted at another laboratory. Even when the adsorption isotherms are conducted at the same laboratory as the dissolution/precipitation experiments, it is not straight forward to incorporate their results. One important limitation is related to the typical duration of the experiments. As sorption is fast, adsorption experiments are usually much shorter than dissolution experiments. However, few studies showed that the coordination structures of the adsorbed OM may change with time. Therefore, it is possible that OM adsorption during dissolution experiments is different than that during the adsorption experiments. Unfortunately, monitoring sorption during dissolution experiments is not an easy task. One problem is that changes in OM concentration due to decomposition (which may be enhanced in the presence of a mineral) may be more pronounced than that due to adsorption. The kinetics of adsorption is generally considered to be fast in comparison with dissolution reactions. Therefore, the adsorption reactions are usually regarded as being in equilibrium. For nucleation inhibition, adsorption kinetics is important, as adsorption must take place before stable nuclei are formed. The importance of OM-water-rock interactions varied in different processes and environments. Adsorption of OM on mineral surfaces, ternary adsorption of OM and metals and complexation between OM and dissolved metals all have a very important role in controlling the fate of both OM and of metals in the environment. The effect of OM on dissolution of minerals in nature seems to be of less significant. The analysis of literature data indicates that the effect of oxalate on silicate dissolution is generally small (median enhancement factor of 2), and is much less than the between-laboratories agreement factor of 5. The possible effects of OM on precipitation of minerals in nature haven’t been properly studied, and therefore, we can not estimate their importance. For industrial applications, the effect of organic antiscalants is most significant. Effective antiscalnts can decrease the nucleation rates by orders of magnitude and thereby achieving a "total inhibition". OM has important catalytic effects on dissolution of scales as well. However, much less scientific effort was devoted to such application, as it is more economical and efficient to control scale by inhibiting its nucleation rather than by dissolving it. OM may also be important in formation of surface coating that protect buildings and monuments from damaging effects of weathering. However, the stability of such coatings over time is still in question.
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