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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 ii = 1, 2), respectively; obtaining graphical solutions
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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
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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 are50º10º

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
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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 are50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
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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.
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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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