Добротність 2D коротаційного сплайну променя порівняно з геометрично нелінійно пружним променем
The goal of this paper is to further investigate the properties and advantages of corotational beam spline, CBS, as suggested recently. Emphasis is placed on the relatively simple task of drawing the spline between two endpoints with prescribed tangents. In the capacity of “goodness” of spline, the...
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| author | Orynyak, Igor Yablonskyi, Petro Koltsov, Dmytro Chertov, Oleg Mazuryk, Roman |
| author_facet | Orynyak, Igor Yablonskyi, Petro Koltsov, Dmytro Chertov, Oleg Mazuryk, Roman |
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| description | The goal of this paper is to further investigate the properties and advantages of corotational beam spline, CBS, as suggested recently. Emphasis is placed on the relatively simple task of drawing the spline between two endpoints with prescribed tangents. In the capacity of “goodness” of spline, the well-known notion of “fairness” is chosen, which presents itself as the integral from the squared curvature of spline over its length and originates from the elastic beam theory as the minimum of energy of deformation. The comparison is performed with possible variants of the cubic Bezier curve, BC, and geometrically nonlinear beam, GNB, with varying lengths. It was shown that CBS was much more effective than BC, where any attempt to provide better fairness of BC by varying the distances from endpoints to two intermediate points generally leads to lower fairness results than CBS. On the other hand, GNB, or in other words, the elastica curve, can give slightly better values of fairness for optimal lengths of the inserted beam. It can be explained by the more sophisticated scientific background of GNB, which employs 6 degrees of freedom in each section, compared with CBS, which operates only by 4 DoF. |
| doi_str_mv | 10.20535/SRIT.2308-8893.2024.3.07 |
| first_indexed | 2025-07-17T10:28:27Z |
| format | Article |
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Publisher IASA at the Igor Sikorsky Kyiv Polytechnic Institute, 2024
Системні дослідження та інформаційні технології, 2024, № 3 107
UDC 519.652
DOI: 10.20535/SRIT.2308-8893.2024.3.07
FAIRNESS OF 2D COROTATIONAL BEAM SPLINE
AS COMPARED WITH GEOMETRICALLY
NONLINEAR ELASTIC BEAM
I. ORYNYAK, P. YABLONSKYI, D. KOLTSOV, O. CHERTOV, R. MAZURYK
Abstract. The goal of this paper is to further investigate the properties and advan-
tages of corotational beam spline, CBS, as suggested recently. Emphasis is placed
on the relatively simple task of drawing the spline between two endpoints with pre-
scribed tangents. In the capacity of “goodness” of spline, the well-known notion of
“fairness” is chosen, which presents itself as the integral from the squared curvature
of spline over its length and originates from the elastic beam theory as the minimum
of energy of deformation. The comparison is performed with possible variants of the
cubic Bezier curve, BC, and geometrically nonlinear beam, GNB, with varying
lengths. It was shown that CBS was much more effective than BC, where any at-
tempt to provide better fairness of BC by varying the distances from endpoints to
two intermediate points generally leads to lower fairness results than CBS. On the
other hand, GNB, or in other words, the elastica curve, can give slightly better val-
ues of fairness for optimal lengths of the inserted beam. It can be explained by the
more sophisticated scientific background of GNB, which employs 6 degrees of free-
dom in each section, compared with CBS, which operates only by 4 DoF.
Keywords: corotational beam spline, geometrically nonlinear beam, 2D, Bezier
curve, fairness, transfer matrix method.
INTRODUCTION
In this paper we analyze the aesthetical quality, or in other words the fairness of
the newly proposed Corotational Beam Spline, CBS [1]. For a few examples, we
will also compare the CBS results with those obtained by an accurate Geometri-
cally Nonlinear Beam, GNB, approach [2]. It is done intentionally because some-
times there is confusion as to the difference between the real beam and the beam
splines. For the case of small displacements, both approaches are the same – we
mean cases of explicit presentation )(xyy , where for example, y is the verti-
cal coordinate of the point, and x is the horizontal one. Yet in the case of large
displacements GNB operates by 6 parameters and presents itself as the solution of
the differential equation of the 6th order, whereas the explicit beam spline is al-
ways the solution of the 4th order equation.
Historically, the beams have generated splines both as technical tools ini-
tially, and later as the mathematical model [3]. For many years, starting from
early AD Roman times the elastic beams (long and thin strips of wood) have been
used by draftsmen to fair in a smooth curve between specified points for ship-
building [4]. Mathematical cubic spline approximation in its present form was
suggested in 1957 by Holladay [5]. He noted that for curves with modest slopes,
I. Orynyak, P. Yablonskyi, D. Koltsov, O. Chertov, R. Mazuryk
ISSN 1681–6048 System Research & Information Technologies, 2024, № 3 108
the cubic spline became identical to the bending of a straight beam. The latter op-
erates by four degrees of freedom – displacement, angle of rotation (first deriva-
tive), bending moment (second one), and transverse force (third one). To apply
the beam spline the positions of consecutive points should be known and one ad-
ditional boundary condition should be specified at each end. The beam spine is
called the natural one if the second derivative is taken as zero. In beam theory,
this corresponds to simply supported end, when displacement is fixed, and mo-
ment is zero.
The language and technique of the beam theory were fruitfully employed
later. Mention only a few ideas. First of all, in analogy to beam the different end
conditions can be considered [3; 6]. They are: free end conditions, when second
and third derivatives (moment and force) are equal to zero and the position is un-
known; the clamped one, when the position and angle (first derivative) are pre-
scribed; zero force condition (third derivative is additionally equal to zero). An-
other finding of the beam theory is the employment of the tensed beam model to
use in beam spline – this is to make the spline straighter [7] at the expense of in-
creased curvature at the prescribed positions (control points). Instead of the 3rd-
order polynomial the 1st-order polynomial and two exponential functions,
kxexp , are used, where coefficient k is proportional to the square root of the
prescribed axial force (used in addition to the usual transversal one) [8]. Note, that
if 0k we get the usual beam spline. Also note, that 4 parameters tensed beam
model is a simplification of 6 parameters planar beam problem (analog of the
elastica).
Mention some additional beam features suggested during the “golden age”
of the beam domination in the spline development. Very effective is the applica-
tion of the beam on the elastic support model instead of the usual rigid ones,
where the curve was suspended by springs attached to its control points for
smoothing the errors of measurements [9]. The spring stiffness controls how
closely the beam interpolates these points [10]. Asker [11] introduces several ap-
proaches to overcome wiggles [12]. The variable beam stiffness is in a piecewise
constant fashion and in a piecewise linear fashion, which allows to change locally
the spline behavior while keeping the same control points. As noted in [12] these
methods are equivalent to the weighted spline of Salkauskas [13].
The main drawback of ‘classical’ beam splines is that they are suitable only
for interpolation to plane curves, which turn through an angle of less than 180°
[14]. The reason is the explicit presentation of the form )(xfy , which is axis
dependent and is not able to represent multiple-valued functions, and cannot be
used where a constraint involves an infinite derivative [15].
So, generally, for any geometry, the implicit representations of the form
0),( yxf , and parametrical representation of the form )(tfy and )(tgx ,
where t is an additional parameter, are used for curve splines [15]. As to beam
splines in particular, Ferguson [16] introduces the parametric cubic spline curve
by applying the cubic spline function for each coordinate by employing the inde-
pendent curve parameter t , by prescribing for each consequent vector point
),( yx the non-decreasing value of parameter t . Their drawback is in the arbi-
trariness of the parameter t , the choice of which leads to different configurations
[17], especially in smoothing the sharp corners [18], or in general, in case of large
curvature [19]. Of course, it can be ‘repaired’ by imposing the additional re-
Fairness of 2D corotational beam spline as compared with geometrically…
Системні дослідження та інформаційні технології, 2024, № 3 109
quirement to transverse force (third derivative), by changing the positions of cor-
ner points [18], but it implies additional complications. So, the parametric beam
spline is rarely used nowadays.
Very popular now are splines based on Bezier curves, B-splines, and
NURBS [15]. Their peculiarities are that they are formed by special polynomial
or rational functions, and the sum of them in each control point is equal to 1. Oth-
er kinds of curves are used very often, too [19; 20]. Nevertheless, despite of pre-
sent less usability, the cubic beam splines have tremendous historical significance
and a large impact on the development of the spline theory. We can formulate, at
least, three of their salient contributions.
1. The quantitative notion of the curve aesthetic measure or ‘fairness’. The
number of curves passing through a set of points is infinite, thus the interpolation
problem is by nature ill-posed [21]. So convenient criteria for the best curve must
be formulated. Of course, the notion of a fair curve appeared long before the ori-
gin of cubic beam spline, and various qualitative formulations were abundantly
employed in literature [12; 21]. However, the first mathematically exact definition
was based on the analogy with the elastic beam theory, where the energy of de-
formation E is given by expression [5; 8]:
dllE
L
)(2
0
, (1)
where is the curvature, l is an element of length, and L is the length. So, the
curve is deemed to be the best, if it provides the minimum of energy E . In the
context of CAD, this integral becomes one of the standard criteria for the fairness
of a planar spline curve [22]. Note, that cubic splines give the minimum energy
only in case of small deflections.
The expression (1) for the energy E very often is supplemented by other
components, which also have the ‘beam’ origin. For example, for a 3D beam, it is
common to introduce the ‘stretch’ energy, which is proportional to the
‘elongation’ of the beam and is the integral from the squared first derivative, or
the ‘twist’ energy, which is found as the integral from the squared third derivative
(rotation of the beam) [23]. For approximation spline, the control points are often
considered as the springs [10], and the extension or compression of which makes
the additional contribution to the elastic energy. So, an additional term for each
“spring’ (control) point is considered, which is proportional to the squared
difference between the position of the spring and smoothened points [24]. Such
curves are named minimal energy curves, MEC, [12].
Of course, the beam-based energy criteria are not unique mathematical for-
mulations for defining the best curve. Other formulations are widely used, too, but
they were either inspired by the energy criteria analogy or emerged as a result of
the drawback of MEC for the best curve construction. Explain this. When the
length of the spline is not restricted, the best MEC (mathematically) may be at-
tained for the spline of infinite length and minimal curvature [25].
So, in general, MEC does not correspond to the common requirement of
Farin [26], that a curve’s curvature plot must be almost piecewise linear, continu-
ous, and with only a small number of segments. So, a different functional, which
satisfies Farin’s criteria, the so-called minimum variation curve, MVC, was pro-
posed in [12]. Instead of )(2 l in functional (1) the square of the derivative of
I. Orynyak, P. Yablonskyi, D. Koltsov, O. Chertov, R. Mazuryk
ISSN 1681–6048 System Research & Information Technologies, 2024, № 3 110
curvature
2
)(
dl
ld
is used. In contrast to the MEC which bends as little as possi-
ble, the MVC bends as uniformly or as smoothly as possible [12]. Yet the MVC
has no clear physical sense, it is not as flexible as the MEC to account for other
constraints, for example, for the required proximity to the control points at ap-
proximation.
So, the MEC are still widely used in the usual continuous [22] or discrete
form [27], where the energy is accounted for integrally in the control points as the
sum of squared angular misalignments. The minimization of functional can give
an aesthetically pleasant curve, which is stable for relatively slowly changing an-
gles between neighboring control points. Very often the energy minimization is
applied as a polishing tool for curves, which are derived by other kinds of splines.
For example, in many works, energy minimization is applied to Bezier curves
[28; 29], cubic spline curves [30], biarck splines [31], B-splines [32], for Hermite
splines [33], and many others.
2. Development of the technique of construction of elastica and promotion
of its popularity. The notion of elastica originated at the end of the 17th century
due to the efforts of the Bernoulli brothers [34]. The elastica is the free-form de-
formation of the elastic beam, whose shape is such that its squared curvature (1)
was minimized. It was an interesting mathematical task, and many famous scien-
tists contributed to its solution and application to different problems, to mention
only Euler, Laplace, Kirchhoff, Max Born, Love, etc [34].
The practical resurrection of interest in elastica originated in the works of
Schoenberg [34] in 1946, where the spline was defined as a variational problem
that minimizes the functional (1) but makes the small-deflection approximation.
The basic shortcoming of the works of Schoenberg and Holliday [34] was under-
lined by Birkhoff and de Boor in 1965 [35], where it was noted that linearized
interpolation schemes are not invariant under rigid rotation. So, they suggested
replacing linearized spline curves with non-linear splines (or “elastica”). Further-
more, they obtained the differential equation for curvature functions )(l , which
satisfy to minimum energy requirement (1):
0
2
1
)( 3
¨
l . (2)
This curve was treated as a free elastic curve as it refers to a planar elastic
curve without length constraints. This result was extended in work [36], where it
was shown that equation (2) can be applied for segmented curvature function with
natural end conditions that pass through a prescribed set of control points. Other
generalization of these results consists in the justification of the validity of equa-
tion (2) when the end conditions are given in the form of prescribed tangents [37].
Based on these general results the various algorithms of elastica construction
were proposed. The first work on the numerical construction of elastica was pro-
posed by Glass in [38], where the discrete points on the curve were specified it-
eratively. Technically this algorithm was later improved by Malcolm [39].
Mehlum [40] used circular arc approximation of arbitrary precision. Mehlum uses
these methods in the Autokon system for curve and surface design, which became
the first commercial CAD software, and underlined the tight relation between
Fairness of 2D corotational beam spline as compared with geometrically…
Системні дослідження та інформаційні технології, 2024, № 3 111
free-form shape representations and physically inspired mathematical functions
[41]. Another technique of nonlinear spline construction was suggested in [22],
where the looking for function is presented as a piecewise polynomial curvature
function.
Among the many early works on nonlinear elastica, we see the work of Horn
[42], where it was carefully studied a specific MEC segment, defined by two
points on a baseline with a vertical tangent constraint specified at each point. The
approach consists of a presentation of the looking-for curve as a set of circular
arcs with a minimization of energy. Starting from one arc, then two or more arcs
up to sixty-four were considered, while the resulting curve resembles a croquet
hoop. The resulting energy was as small as 0.913953% of the semicircle. Then the
elliptical curve with minimal energy was constructed and energy was equal to
93.42% of the semicircle, and for the Cornu spiral the energy was as low as
0.9178%. Then Horn computes closed-form expressions for the energy, arc
length, and maximum curvature of his subject curve and the lowest value is equal
to 0.913893. MEC has at least two principal shortcomings: the first cause it to fail
a very desirable property for splines such as roundness [21], and another one, is
that energy depends on an unspecified length of elastica. So, in the works of Kal-
lay [43; 44] the theoretical substantiation and numerical method are elaborated for
computing that shape, given the positions and directions of the endpoints and the
total length. In this case, the notion of energy and the goal of optimization be-
come clearer and are related to the fixed length. In work [45] the elastica is con-
structed by the elastic curve segments which are expressed in a closed form via
the elliptic functions. The method depends on the good initial guess for the ap-
proximating curve with subsequent application of gradient-driven optimization.
On the other hand, the analytical solution for Euler’s Elastica motivated
within the structural mechanical community the development and application of
one-dimensional theories for the deformation of elastic slender bodies [46] and
especially the elaboration of the comprehensive and efficient numerical formula-
tion [47]. Geometrically nonlinear computational models of the beam under finite
rotation are obtained from three basic approaches: total Lagrangian, updated La-
grangian, and Co-rotational [47]. It is beyond the goal of the paper to discern
them in detail.
We only mention that technically they often are reduced to solution by the
transfer matrix method, TMM, either within the Lagrangian approach [48] or in
the corotational formulation [49]. The transfer matrix, which relates the set of
governing parameters at any point of the element including its end with those at
the beginning of the element, is called the field transfer matrix, FTM. It is derived
by the solution of physical differential equations. The continuity relations be-
tween the parameters of two neighboring elements at the border between them are
given by the point transfer matrix, PTM. Sometimes the method is called in litera-
ture as a method of initial parameters [50]. The transfer matrix method is a very
effective tool, which allows to eliminate the intermediate unknowns of inner ele-
ments, thus while keeping a large number of degrees of freedom, technically it
reduces the ultimate matrix to the size determined only by the number of real
points [51].
3. Employment of local coordinates system for each element. In structural
mechanics, this approach is called a corotational approach [52; 53]. Here, the total
I. Orynyak, P. Yablonskyi, D. Koltsov, O. Chertov, R. Mazuryk
ISSN 1681–6048 System Research & Information Technologies, 2024, № 3 112
configuration of a beam is presented as the sum of two components: straight ele-
ments position and pure deformational displacement of points. The deformation is
measured from a rotating frame attached to the straight element, and linearized
formulation solutions are employed in the numeric incremental procedure. The
nonlinearity is accounted for by the rotation matrix between the elements. So, the
discontinuous angles of rotation between elements are the key parameters of the
corotational scheme. Note that the geometrically nonlinear beam model of work
[2] is the enhanced corotational approach, where the reference geometry is part of
a circle and already contains build-in deformation (basic solution), which is sup-
plemented by smoothing solution derived by integration of governing linear dif-
ferential equations written in curvilinear (polar) coordinates.
Within the computer graphics world, the looking for curves are usually pre-
sented as continuous ones at every stage (iteration) of computation. The only
known exception is the Fowler–Wilson method [54], which was based on usual
cubic beam splines in a local two-dimensional coordinate system. The Fowler-
Wilson scheme is a transition from the explicit presentation of spline )(xfy to
the implicit one 0),( yxg . It was very popular till the beginning of the 21st cen-
tury and was used in many industries around the world [55]. So, initially, the
spline is given by a set of straight sections, which determine the tangent vector
and normal vector. Local sections of the spline are calculated along the normal to
the section. The main requirement is to provide the continuity of slope and curva-
ture at the borders between points. The nonlinear equations of continuity are ob-
tained and iteratively solved.
Now return to the goal of this paper. The corotational beam spline of our
work [1], among other ideas, uses the idea of straight initial sections drawn be-
tween control points, which determine the local system of coordinates. So, the
purpose of splines is to smooth out the so-called misalignment (gap) angles. This
resembles the idea of Fowler–Wilson. As to the task of interpolation, the main
differences in our work [1] consist of two peculiarities. First, from the very be-
ginning, our spline is constructed in a linearized statement, which is usual for the
theory of beam:
sintg , (3)
where is the calculated angle of rotation, this noticeably simplifies the calcula-
tions [56; 57]. Second, to suppress the errors induced by (3) we introduce the no-
tion of auxiliary ‘imaginary’ points. So, we consider that control points are of two
kinds. Points of the first kind are of the real kind, where the outer constraints are
explicitly given. Points of the second kind are imaginary ones, which are arbitrar-
ily placed between the real ones, their positions are not specified and naturally
refined during the calculation process. They are intended to: a) make the length of
the straight section approximately equal to the length of the spline section; b) de-
crease the maximal calculated angle within each section to provide better accu-
racy of (3). Another enhancement of the method is of technical significance and
consists of the employment of the transfer matrix method, which allows keeping
the resulting matrix for spline with imaginary points of the same dimension as
without them. Besides, the geometrically exact definition of curvature is used.
The goal of the paper is to analyze the aesthetical quality or energy (1) of the
CBS and compare it with GNB for example a simple task defined by two end-
Fairness of 2D corotational beam spline as compared with geometrically…
Системні дослідження та інформаційні технології, 2024, № 3 113
points with a tangent constraint specified at each point. This choice is taken be-
cause: 1) similar tasks were considered in old theoretical investigations of elastica
[42; 44]; 2) the fairness of such curve can be easily assessed visually; 3) it is a
practical task for thin deformable wire held at each end by a robotic gripper [58]
and for applications of robotic hot-blade cutting [45; 59].
THEORETICAL FOUNDATION OF CBS AND GNB
Short introduction to CBS
Note, that very simplified logic, designations, and equations of more general pa-
per [60] are outlined here. The main reason for simplification is that here we con-
sider only the task of interpolation, so many enhancements related to consistency
with the task of approximation will be omitted. Note, that the solution process is
organized according to the transfer matrix method, TMM, methodology.
Let we have enumerated consequently both the exactly measured (real)
points and the inserted between them (in any number) imaginary points,
),( mmm YXA , where m is the point number, and mX , mY are their Cartesian co-
ordinates in the absolute coordinate system. Usually, we do not discern between
them and name them as the control points, because referring to the transfer matrix
method, TMM, the field transfer matrix, FTM, is the same for any element placed
between any two neighboring points. The difference exists only for the point
transfer matrix, PTM, and depends on whether the considered point is a real or
imaginary one. So, in this case, the points will be discerned by using the addi-
tional lower indexes: “r” for real points, and “i” for imaginary ones. For example,
rmA , means that point mA is the real, and imA , is the imaginary one.
Connect the consequent points mA by straight lines and get the open or
closed polygon. Consider the particular straight beam section, named as m sec-
tion, which is placed between control points 1mA and mA , Fig 1. Introduce the
notion of iteration number, k . The real points retain the same position at each
iteration; however, the imaginary points change their position. Furthermore, the
algorithm envisages that new imaginary points might be inserted during the itera-
tion process. This is controlled by the maximal value of calculated angle , as to
condition (3). If the angle is large enough, say, larger than
30
, we insert a new
imaginary point. This provides the accuracy as to (3) within 0.2%. So, the inser-
tion of a new imaginary point may change the general enumeration. Thus, the
numeration is iteration dependent, and control points should be presented as
),( k
m
k
m
k
m YXA , where k is the iteration number. Nevertheless, in most cases, the
upper index k will be omitted.
The vectorial length of each beam section is designated as ml
:
jYYiXXAAl mmmmmmm
)()( 111 . (4)
For each straight section introduce the local coordinate system ( mm ws , ) and
basic vectors mt
and mn
. The tangent local vector nt , is derived from (4):
I. Orynyak, P. Yablonskyi, D. Koltsov, O. Chertov, R. Mazuryk
ISSN 1681–6048 System Research & Information Technologies, 2024, № 3 114
jbia
l
l
t mm
m
m
m
,
where
2
1
2
1 )()( mmmmmm YYXXll
.
The normal vector mn
is perpendicular to mt
and rotated clockwise concern-
ing it, Fig 1. Local coordinate system ( ws, ) is related to the basic vectors, where
s is counted from mA
in the direction of t
, and w is directed as n
. Vector mn
is presented as:
jdicn mmm
,
where
m
m
m
m
m
m
b
a
b
a
d
c
01
10
)2/(cos)2/(sin
)2/(sin)2/(cos
.
Important in the model is the misalignment angle between two adjacent
straight beam sections: m and 1m , denoted as m , Fig. 1. It is counted clock-
wise from vector 1mt
to vector mt
. The angle of misalignment 1m is found
from the scalar and vector products of these two vectors:
mmm tt
*)(sin 1 , mmm tt
1)(cos . (5)
Application of both rules is needed to establish the correct angle quadrant.
Now describe the calculation model. Consider the simplest beam model for
an initially straight beam. Each straight beam section is characterized by the vec-
tor of state )(sZ
, which is formed by 4 scalar functions of length coordinate s :
)}();();();({column)( sQsMssWsZ
,
where following the beam traditions we operate by four physical values: )(sW is
displacement directed along the local normal vector n
; )(s is the angle of (de-
Am
Am-1
Am+1
tm-1
nm-1
ψm-1
ψm+1
tmnm
i
j
ψm
Fig. 1. The global cartesian vectors and local corotational basis for each element
Fairness of 2D corotational beam spline as compared with geometrically…
Системні дослідження та інформаційні технології, 2024, № 3 115
formational) rotation of the beam axis concerning initial vector t
, directed
clockwise; )(sM is the bending moment; )(sQ is the transverse force. The direc-
tion of the two latter parameters is chosen so, that the following differential de-
pendencies between all parameters are positive [1]:
0
)(
);(
)(
;
)()(
);(
)(
ds
sdQ
sQ
ds
sdM
EI
sM
ds
sd
s
ds
sdW
, (6)
where EI is the constant of the beam, taken below as 1. The solution of the sys-
tem (6) can be presented in matrix form suitable for the application of TMM:
)()]([))(( 0,, mjim ZspsZ
, (7)
where )0(0 sZZ
is the vector of state in the initial point of the section con-
sidered, and the coefficients of the transfer matrix are the following:
1000
100
2
10
62
1
)]([
2
32
,
s
s
s
ss
s
sp jl .
Note, that equations (7) can give the values of each parameter at the end-
point of each element through the initial parameters at this element by letting
mls , i.e.:
)()]([)())(( 0,,1, mmjimmm ZlpZlZ
,
where lower indexes “0” and “1” mean the beginning and the end of a section,
correspondently.
To formulate the calculation scheme, we need to supplement the FTM (7)
with PTM equations, which relate the vector of state at the border between the
end of the previous and the beginning of the next sections. For real control point,
we have the following PTM relations:
1,10, mm WW , (8)
mmm 1,10, , (9)
1,10, mm MM , (10)
mmm PQQ 1,10, , (11)
00, mW . (12)
Here mP is an unknown force in the beam support (real control point). Gen-
erally speaking, this force is determined from condition (11), or put more cor-
rectly, the introduction of additional unknown mP requires one additional condi-
tion (12). When compiling the system of governing the equation (11) (and
unknown mP ) is not used. Condition (8) means that displacement (deviation of
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position from the initial straight line) should be the same. Condition (9) is to pro-
vide the tangent continuity of the deformed contour, where the deformational an-
gles compensate for the initial misalignment angle. Condition (10) is the equality
of approximate beam curvatures. Condition (12) requires that the position of the
considered control point should not change during the iteration, i.e., it is fixed.
The point transfer matrix in the case of the imaginary point is slightly differ-
ent. They require continuity up to 3rd order, i.e., including the transverse force. So,
we have:
1,10, mm WW ,
mmm 1,10, ,
1,10, mm MM ,
1,10, mm QQ .
Or in matrix form
immm CZIZ ,1,10, )(][)(
,
where iC
:
}0;0;;0{column, mimC
.
In this case, the PTM does not contain any additional unknowns.
Let’s go to the organization of the calculation process by TMM. It is conven-
ient to start with an introduction of four unknown parameters for both the begin-
ning and end of each element. It means we have 8 unknowns for each element. If
the number of elements is M , then the number of unknowns is M8 . There are 4
FTM equations for each element, thus at the whole, there are M4 FTM equa-
tions. On the other hand, there are 1M borders between elements for open pol-
ygon, for which 14 M PTM can be written. So, for an open polygon, 48 M
equations should be supplemented by 2+2 boundary conditions on each boundary.
One of them is the condition of zero displacement, while another is either the re-
quirement to the angle value, or requirement to curvature (moment), or to trans-
verse force. When the contour (polygon) is closed we have no boundary condi-
tions, but instead, one additional PTM (four conditions) is to be written at the
point where the last section meets the first one.
At first glance, accounting for possibly a large number of imaginary points
in this CBS technique requires too many unknowns and can be very slow. First,
show that imaginary points and related unknowns actually can be removed from
consideration. Consider two adjacent sections which are separated by imaginary
points. According to the procedure of elimination [51] write three transfer ma-
trixes between them:
)()]([)( 0,,1, mmjim ZlpZ
, (13)
immm CZIZ ,1,0,1 )(][)(
, (14)
)()]([)( 0,11,1,1 mmjim ZlpZ
. (15)
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Substituting (14) into (15) and later (13) in the resulting equation we for-
mally can get the matrix equation:
immmmjim DZllpZ ,1,1,1,1 )()],([)(
, (16)
where elements of matrix ),( 1, mmji llp , and free terms vector imD ,
are easily
calculated from (13)–(15). The matrix equation (16) is FTM for the combined
element, which starts at the point mA
and ends at point 2mA
, thus eliminating
the imaginary point 1mA
. So, two FTM and one PTM are substituted by one
FTM. Second, for the remaining real points the number of unknowns can be reduced
to unknown moments in them only, as it is usually given in the textbooks [8].
Note, that calculated values of Mc have dimension of curvature and the
meaning of curvature in beam theory formulation. Yet they are not exact geomet-
rical curvatures. So, the additional procedure of refining the values of curvature
based on exact differential geometry formulation was established in [1] and will
be used in the presentation of the results.
Main equations of GNB
The principal distinction of GNB from spline is that it operates by a real object
with real properties and, especially it has a given length, . L In calculations, it is
broken into a necessary number of elements. For each element, m , the notions of
the Basic, )(sBm
, and Smoothing, )(sSm
, solutions are introduced [2; 61], where
s is a curvilinear abscissa of any point of element. Then, the looking for Ultimate
solution, )(sU k
, is the sum of these two constituents:
kkk SBU
1 . (17)
Here the upper index k means the iteration number. So, as follows from
presentation (17) the basic solution is a result of the previous iteration 1i . Note,
that, where possible, the lower and upper indexes m and k will be omitted.
The main aim of the basic solution, BS, is to principally account for all non-
linearities, while the smoothing solution, SS, is a linearized analytical correction
to BS. Another purpose of BS is that it gives the system of local curvilinear coor-
dinates and directions concerning which the SS is derived. On the other hand, BS
is permanently refined from iteration to iteration as a result of accounting for the
present SS. New BS is refined according to the following rule:
kkk SgBB
1 , (18)
where g , 10 g is the so-called retardation coefficient, which restricts the ab-
solute change of BS and accounts for whether the process of solution is conver-
gent ( m can be increased) or divergent ( m should be decreased). The rule (18) is
schematic because not all components of SS are used in BS. For the 2D case, BS
geometrically presents itself the part of a perfect circle, the radius of which k
mR ,
(or curvature k
m
k
m R/1 ) and current length k
ml are related with basic (embedded
in) bending moment and axial force [61].
SS solution is formulated for each element in the local curvilinear system of
coordinates, Fig. 2. It operates by six governing parameters, as opposed to a
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straight beam (cubic spline). They are: two force parameters – transverse force Q
and axial force N , bending moment M , angle of rotation , and two displace-
ment parameters — normal one w , and tangential one u . These parameters are
related by six differential equations:
nP
R
N
Rd
dQ
; tP
R
Q
Rd
dN
; Q
Rd
dM
;
EI
M
Rd
d
;
EF
N
R
w
Rd
du
;
R
u
Rd
dw
.
In the subsequent application below we will not consider the action of outer
distributed forces nP and tP , and neglect the axial elongation, i.e., take 0
1
EF
.
The general solution of (5) for SS can be presented in the form suitable for
the application of TMM [61], and schematically is given below. For FTM it can
be written as [61]:
)()]([))(( 0,, mjim ZspsZ
,
where elements of matrix )(, sp ji are the solution of the differential equations (5),
and the vector of state in any point s is:
)}(), ();();();();({column)( sNsQsMssuswsZ
.
For PTM a similar equation can be written [61]:
immm CZHZ ,1,10, )(][)(
.
In this case, the matrix ][H is not an identity matrix, due to the vectorial es-
sence of two force and two displacement parameters and different local vectorial
basis used. As to the vector of free terms imC ,
, its all components are nonzero due
to the discontinuity of the basic solution (so-called gaps [61]). Note that the GNB
approach requires three boundary conditions at each end of the beam.
The aim of this short introduction to GNB is twofold. First, to show that the
technical solution of GNB can be similarly organized by TMM, with formal
elimination of all points (sections) that do not contain the real constraints. Second,
the variety of tools and possibilities of GNB analysis by numerical TMM is much
richer than in the traditional elastica approach. Hint the few possible
Fig. 2. General scheme of 2D curvilinear beam
Am Am+1
θ
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opportunities. If one wants to make the solution “more tensed”, it can be very
easily done by introducing the normal pressure nP . To “tense” or to “relax” the
particular section it needs to increase or decrease the bending stiffness EI , or to
make it variable. The length of particular sections or the beam as a whole can be
controlled by axial stiffness EF .
Calculated curvatures in each curved section are presented as the sum of basic
curvature (constant) and those induced by calculated bending moment )(sM k
m :
EI
sM
R
s
k
m
k
m
k
m
)(1
)( .
Four points-based Bezier spline
Consider set of four consequent control points ), ( 111 YXA , ), ( 222 YXA ,
), ( 333 YXA , ), ( 444 YXA , or in vectorial form:
jYiXA mmm
; 4,3,2,1m .
They can be used for the construction of a third-order Bezier curve,
),( yxP
[15]:
)()(),(
4
1
4
1
tKYjtKXijPiPyxP mm
m
mm
m
yx
; 10 t ,
where )(tKm are the Bernstein’s functions
3
1 )1()( ttK , 2
2 )1(3)( ttttK , )1(3)( 2
3 tttK , 3
4 )( ttK .
Bezier splines have the following properties [15], important to our task:
1. The first and last points on the curve are coincident with the first and last
points of the control polygon.
2. The tangent vectors at the ends of the curve have the same direction as the
first and last polygon spans, respectively.
So, our subsequent task is to construct a spline, which starts in point 1B
at
the angle 1 with a horizontal axis and ends in point 2B
directed at angle 2 .
Introduce the unitary tangent vectors 1
and 2
in these boundary points. They
can be written as:
111 sincos ji
; 222 sincos ji
.
So, four consequent points of cubic Bezier splines can be chosen as follows:
11 BA
, 24 BA
, 1112
DAA ; 2234
DAA ,
where 1D is an absolute distance from point 2A
to point 1A
, and 2D from 4A
to
point 3A
.
Our next task is to obtain the element of length in each point, )(tds , and the
curvature t . According to differential geometry rules, we can write:
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dtPPds yx
22 )()( ,
3
22 )()(
)(
yx
xyyx
PP
PPPP
t .
These expressions will be used in the calculation of the Brazier spline quality.
EXAMPLES OF CALCULATION
Several similar problems will be calculated and compared here. In some cases, the
well-known Bezier curve will be used too. We will consider the relatively simple
task, which is defined by two endpoints with a tangent constraint specified at each
endpoint.
Example task 1
First point 1B is placed in point )0, 0( YX , second point 2B has coordinates
)0 ,150( YX , the tangent in point 1B is directed vertically, i.e., the angle (in
the clockwise direction) with the horizontal axis is equal 90 , and in point 2B
the angle is equal to 90 . It is the famous Horn task [42], which has demonstrated
that minimization of energy is not always a solution for the best curve, and
eventually led to the appearance of other criteria, say minimization of squared
derivative from curvature [12].
Intuitively, the best curve is the semi-circle of diameter equal to 150. Its
length, 0L is 62.235
2
150
0 L . Calculate the quality (energy) of the ideal
semi-circle. According to (1) it is equal to 041888.0
2
150
150
2
2
0
E .
Construct the splines according to different techniques, Fig. 3. Designation
BZ 110 relates to the Bezier spline, where two intermediate points on the pre-
scribed tangent are placed at a distance of 110 from either endpoint. Similarly, the
BZ 115.5 curve employs one point on distance 115.5 on each tangent. GNB
depends on the prescribed length of the beam, so the designation GNB 230 means
that the length of the beam is equal to 230. Many variants of BZ spline and GNB
can be obtained. As to CBS, it gives only one possible configuration.
Analyze the results. First of all, note that CBS gives the ideal semi-circle.
As to other curves, at first glance, they can approach the ideal figure very well,
and each seemingly is capable of depicting the ideal circle.
With this respect, the more informative are graphs of curvature versus the
length coordinate for each spline shown in Fig. 4. More definite conclusions can
be drawn from it. First, note that CBS is indeed capable of giving the ideal circle.
The wavy character of the graph is a reflection of an insufficient number of
imaginary points — the more points, the smoother the curvature. Second, Bezier
splines give noticeable deviation from the ideal circle for all parameters of
optimization (distances from endpoints). Third, GNB is a very powerful
technique, which depends on the chosen length of the beam. In case, when the
prescribed length of GNB coincides with the length of the ideal circle, it actually
gives this ideal circle.
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Fig. 3. Several calculated splines according to the Bezier method (4 points), BCS, and GNB
1
2
3
4
5
6
7
7
3
1, 5 2, 4 6
Fig. 4. The graphs of curvatures obtained by different splines for Task 1
3
1 2
1
2
3
4
5
6
4 65
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Compare the quality (energy) of each depicted curve. The results of its cal-
culation are given in Table 1, where the absolute value of energy, as well as the
reduced value of it (divided by the ideal semi-circle value of 0.41888), are pre-
sented. The result for CBS is slightly different from the ideal circle due to a
smaller number of imaginary points (only 28 points are employed here). As to
Bezier’s results, they are close to 1 in the considered range of chosen distances of
additional points, and the lowest result is attained for the BZ 120 curve.
T a b l e 1 . Calculated energies for different splines for Task 1
Curve BZ 110 BZ 115.5 BZ 120 BCS GNB 230 GNB 240 GNB 410
Quality 0.04305 0.04190 0.04113 0.04190 0.043024 0.041181 0.037927
Reduced
quality 1.02774 1.0003 0.9818 1.0003 1.04475 0.98312 0.90544
Evidently, the notion of energy cannot be the sole criterion of fairness. Note,
that splines, which “embrace” the ideal semi-circle give the lesser values of
energy. Concerning the results of Table 1 it is interesting to recall the results of
Horn [42]. Remind that for this task 1 the “best” value of energy equal to 0.91383
was obtained [42]. So, plot the graph of the energy concerning beam length by
GNB approach, Fig. 5. Interesting to note, that in the vicinity of the ideal semi-
circle configuration ( )6.2350 LL the quality of the curve linearly decreases
with length. Yet in the range of length 340310 L , it attends the local “pla-
teau”. The calculated quality in this range is approximately equal to 0.03828 (at
)320L . Dividing this value by 0.041888 (ideal semi-circle) we get the reduced
value equal to 0.9138, which is very close to Horn’s theoretical value. This testi-
fies to the high efficiency of the GNB approach [2]. Further increase of L beyond
this range leads to a permanent slow decrease of energy, which was not predicted
in Horn analysis [42]. This is related to the outward deviation of the calculated
figure from the vertical lines 0x and 150x , which is evident from Fig. 3 for
GNB 410. For L the energy tends to zero.
Example task 2
This task is very similar to symmetrical Task 1. The only difference is that in first
point 1B the tangent angle is inclined to 60 concerning the horizontal axis, and
in second point 2B the angle is equal to 60 .
Fig. 5. Quality of GNB spline concerning the beam length for Task 1
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Intuitively, the best expected (ideal) curve is the semi-circle. From geometri-
cal consideration, its diameter should satisfy the following relation 30cos2/D
75 , where one can get that 20.173D . So, the length of the ideal figure is
equal to 37.181 . As to energy (1) of the ideal curve, it is equal to 0.02418.
Construct the splines according to different techniques, Fig. 6. As above
CBS gives the ideal semi-circle. Bezier spline gives very close results at a dis-
tance equal to 67. When this distance is smaller, than Bezier curve lies below the
ideal circle, and when the distance is larger it is situated above the ideal curve. A
similar picture is for GNB splines. When their lengths are lesser than the ideal
circle length, it is placed below, and above otherwise.
The informative is a graph of curvatures for each spline, shown in Fig. 7.
The best Bezier spline, BZ 67, is very close to the ideal circle, and its curvature is
almost ideal. The same can be said about the BCS and GNB 185. Compared with
CBS for Task 1 (Fig. 4), the curvature for CBS for Task 2 is much smoother: we
use here as many as 120 imaginary points. Graphs of curvature are very important
to judgment about the quality of different splines.
Compare the quality (energy) of each depicted curve. The results of its cal-
culation for Task 2 are given in Table 2. That results in differently-looking curves
that sometimes are very close. This means, that energy cannot be the sole criterion
of the construction of the curve nor for the assessment of its fairness.
1
2
3
4
5
6
7
8
7
3
1,4,6
2
8
5
Fig. 6. Several calculated splines according to the Bezier method, BCS, and GNB, Task 2
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T a b l e 2 . Calculated energies for different splines for Task 2
Curve BZ 67 BZ 80 BZ 120 BCS 181 GNB 170 GNB 181 GNB 190
Quality 0.0241 0.0246 0.0408 0.0242 0.0303 0.0242 0.0243
Example task 3
This task is a more complicated one and relates to the construction of anti-
symmetrical geometry. Point 1B has coordinates )0, 0( YX , second point
2B has coordinates )150, 150( YX , the tangent in point 1B is directed at an
angle 60 of , while in point 2B the angle is equal to 60 , too.
The best solution cannot be formulated intuitively, so here we will subjec-
tively assess the best solution below.
Construct the splines according to different techniques, Fig. 8. Look on the
CBS, which does not require any auxiliary parameters. The general subjective
impression is that it is visually pleasant, and its calculated length is about 294. So,
chose the auxiliary parameters in other spline methods to approach this spline. It
is not always possible for the Bezier splines. If we take the distance to be very
small – it would resemble the straight line between two endpoints, and, of course,
it should be rejected. If we take the distance in the Bezier spline too large, the
graph will be placed well beyond the vertical range of 150150 y . So, we
chose subjectively the distances equal to 120, 150, and 180 as the candidates for
1
2
3
4
5
6
7
8
7
3
1
2
5
4, 6
8
Fig. 7. The graphs of curvatures obtained by different splines for Task 2
Fairness of 2D corotational beam spline as compared with geometrically…
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the best Bezier curve. Nevertheless, they are not pleasantly looking, and this can
be supported by graphs of curvatures, Fig. 9. As to GNB it completely coincides
with CBS, if we take its length as large as 294, Fig. 8. The increase or decrease of
length leads to more loose or tight geometry, respectively.
More informative are the graphs of curvatures concerning the current length
coordinate, Fig. 9. The curvature of CBS is very smooth, it is a straight line (small
fluctuations are due to a limited number of imaginary points). So, an important
conclusion can be drawn from its visual presentation. The CBS is a Cornu spiral,
and this can be explained by the third differential equation of (6). On each small
straight section constP , so the moments (curvature) change linearly. In case,
if intermediate points are the imaginary ones, the force does not change between
them, so the whole section between any real points is a Cornu spiral.
As one can see, the GNB completely coincides with the CBS, in case its
length is equal to the length of CBS. If GNB is shorter than CBS, then its
curvature is larger than that of CBS. And vice versa, for longer GNB its curvature
is smaller. As to the Bezier curve, it demonstrates the large local curvatures for all
three considered distances chosen. As we see, the Bezier curve is ineffective for
anti-symmetric cases.
Compare the energy for each curve. The results of its calculation for Task 3
are given in Table 3. The results for Bezier curves are very poor. So, the very big
difference in energy can testify to the deficiency of the curve. As to GNB, the
results for it are close to CBS because their lengths are similar. In any case, by
varying the length of GNB the quality of it can always be better than that of CBS.
1
2
3
4
5
6
7
7
3
5
2
6
1
4
Fig. 8. Several calculated splines according to the Bezier method, BCS, and GNB, Task 3
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T a b l e 3 . Calculated energies for different splines for Task 3
Curve BZ 120 BZ 150 BZ 180 BCS GNB 280 GNB 294 GNB 310
Quality 0.2080 0.2092 0.2173 0.1286 0.1373 0.1284 0.1225
Example task 4
This task is similar to the previous one but is not an antisymmetric. Point 1B has
coordinates )0, 0( YX , second point 2B has coordinates ( ,150( X
)150 Y , the tangent in point 1B is directed at an angle — 60 , while in point
2B the angle is equal to 0 .
The best solution cannot be formulated intuitively, so here we will subjec-
tively assess the best solution below.
Construct the splines according to different techniques, Fig. 10. As to CBS,
the general subjective impression is that it is visually pleasant, and its calculated
length is about 260. So, chose the auxiliary parameters in other splines as to
approach this spline. As in Task 3, it is not possible for the Bezier splines – they
deflect from CBS for any chosen parameter of distance. So, the results for Bezier
splines are shown for three subjectively chosen distances – 100, 125, and 150.
Nevertheless, they are not pleasantly looking, and this impression can be
1
2
3
4
5
6
7
7
3
5
2
6
4
1
Fig. 9. The graphs of curvatures obtained by different splines for Task 3
Fairness of 2D corotational beam spline as compared with geometrically…
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supported by graphs of curvatures, Fig. 11, which has large local peaks of
curvature, which is prohibited for “fair” spline [12]. As to GNB it completely co-
incides with CBS if we take its length as 260, Fig. 10. The increase or decrease of
length leads to more loose or tight geometry, respectively.
Informative are the graphs of curvatures concerning the current length coor-
dinate, Fig. 11. The curvature of CBS is a straight line (small fluctuations are due
to a limited number of imaginary points), so evidently CBS is a Cornu spiral.
As in above Task 3, note that GNB completely coincides with CBS in case,
its length is equal to the length of CBS and can be more tight or loose depending
on whether the length of GNB is shorter or longer than the length of CBS. Bezier
curve demonstrates the large curvatures for all three distances chosen.
Compare the energy for each curve. The results of its calculation for Task 4
are given in Table 4. The results for Bezier curves are very poor and testify to the
deficiency of the curve. As to GNB, the results for energy are close for CBS
because their lengths are similar. In any case, by varying the length of GNB the
quality of it can always be better than that of CBS.
Fig. 10. Several calculated splines according to the Bezier method, BCS, and GNB, Task 4
1
2
3
4
5
6
7
7
3
54, 6
21
Fig. 10. Several calculated splines according to the Bezier method, BCS, and GNB, Task 4
I. Orynyak, P. Yablonskyi, D. Koltsov, O. Chertov, R. Mazuryk
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T a b l e 4 . Calculated energies for different splines for Task 4
Curve BZ 100 BZ 125 BZ 150 BCS GNB 250 GNB 260 GNB 270
Quality 0.1156 0.1114 0.1153 0.0781 0.0835 0.0778 0.0753
CONCLUSION
The main attention of the paper is paid to the discussion of the advantages of
CBS. On one hand, it is the four degrees of freedom simplified version of GNB,
which operates by 6 degrees of freedom at each point. The beam theory origin of
CBS gives a wide prospect for its modernization and application. On the other
hand, the presented here version of CBS for the task of interpolation is reduced to
the new original technique of construction of the Cornu spiral, which is widely
recognized as one of the most aesthetic curves for the geometrical design purpose
[62]. The application of the methodology of linear TMM makes this technique
very simple and effective.
In detail, the method and its comparison with Brazier spline and GNB is
made on the example of two endpoints that are connected by spline with pre-
scribed tangent values. Several local conclusions can be drawn out.
1. As to the Brazier spline with the employment of four points (two interme-
diate ones can be chosen arbitrarily to optimize the geometry), it generally dem-
onstrates poorer results. The resulting curvatures, especially for geometries, when
1
2
3
4
5
6
7
7
3
1
2
6 5
6
4
Fig. 11. The graphs of curvatures obtained by different splines for Task 4
Fairness of 2D corotational beam spline as compared with geometrically…
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it changes the sign, behave very unsmoothly and exhibit very high local peaks. In
this case, the calculated value of energy (1) is much higher than for CBS.
2. CBS for all 4 tasks considered gives very pleasant results. In all cases, the
curvature is either constant (symmetric cases) or linearly changes with the spline
length coordinate. The only technical requirement for its realization is the inser-
tion of a sufficient number of imaginary points.
3. GNB is the most effective technique for spline construction as well as for
modeling the deformation of real flexible beams. Its drawback for the geometrical
design is that the justified length of the beam should be chosen in advance. The
value obtained by the CBS solution is a good initial approximation for further
GNB application. The accuracy of the GNB technique is demonstrated in the ex-
ample of the well-known Horn task [42].
4. Energy criteria of fairness (1) cannot be considered as a sole criterion for
the spline construction. On the other hand, a very big value of it testifies to the
drawback of the applied technique.
REFERENCES
1. I. Orynyak, D. Koltsov, O. Chertov, and R. Mazuryk, “Application of beam theory
for the construction of twice differentiable closed contours based on discrete noisy
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Received 26.04.2024
INFORMATION ON THE ARTICLE
Igor V. Orynyak, ORCID: 0000-0003-4529-0235, National Technical University of
Ukraine “Igor Sikorsky Kyiv Polytechnic Institute”, Ukraine, e-mail: igor_orinyak@yahoo.com
Petro M. Yablonskyi, ORCID: 0000-0002-1971-5140, National Technical University of
Ukraine “Igor Sikorsky Kyiv Polytechnic Institute”, Ukraine, e-mail: ypn@ukr.net
Dmytro R. Koltsov, ORCID: 0000-0002-0396-7255, National Technical University of
Ukraine “Igor Sikorsky Kyiv Polytechnic Institute”, Ukraine, e-mail: koltsovdd@gmail.com
Oleg R. Chertov, ORCID: 0000-0003-0087-1028, National Technical University of
Ukraine “Igor Sikorsky Kyiv Polytechnic Institute”, Ukraine, e-mail: chertov@i.ua
Roman V. Mazuryk, ORCID: 0000-0003-4309-824X, National Technical University of
Ukraine “Igor Sikorsky Kyiv Polytechnic Institute”, Ukraine, e-mail: r.mazuryk.ua@gmail.com
ДОБРОТНІСТЬ 2D КОРОТАЦІЙНОГО СПЛАЙНУ ПРОМЕНЯ ПОРІВНЯНО
З ГЕОМЕТРИЧНО НЕЛІНІЙНО ПРУЖНИМ ПРОМЕНЕМ / І.В. Ориняк,
П.М. Яблонський, Д.Р. Кольцов, О.Р. Чертов, Р.В. Мазурик
Анотація. Метою статті є подальше дослідження властивостей і переваг не-
щодавно запропонованого коротаційного балкового сплайну (КБС). Акцент
зроблено на розгляді досить простої задачі проведення сплайну між двома кін-
цевими точками із заданими дотичними в них. Як критерій «хорошості»
сплайну обрано відоме поняття «добротності», яке являє собою інтеграл від
квадрата кривини сплайну по його довжині, що походить із теорії пружної
балки як енергія деформації. Порівняння «добротності» КБС виконано з де-
якими варіантами кубічної кривої Безьє (КБ) і геометрично нелінійної балки
(ГНБ) зі змінною довжиною. Показано, що КБС є набагато ефективнішим, ніж
КБ, для якої будь-яка спроба забезпечити кращу «добротність» КБ шляхом
зміни відстані від кінцевих точок до двох проміжних точок, як правило, при-
зводить до гірших результатів порівняно з КБС. З іншого боку, ГНБ, або ін-
шими словами, крива «еластика», здатна давати дещо кращі значення «доброт-
ності» для оптимальної довжини балки. Це можна пояснити більш складною
методологічною основою ГНБ, яка використовує 6 ступенів вільності в кож-
ному перерізі порівняно з 4-ма ступенями вільності в КБС.
Ключові слова: коротаційний балковий сплайн, геометрично нелінійна балка,
плоска задача, крива Безьє, добротність, метод початкових параметрів.
|
| id | journaliasakpiua-article-298940 |
| institution | System research and information technologies |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2025-07-17T10:28:27Z |
| publishDate | 2024 |
| publisher | The National Technical University of Ukraine "Igor Sikorsky Kyiv Polytechnic Institute" |
| record_format | ojs |
| resource_txt_mv | journaliasakpiua/c6/14278c74d67090dbc0cb25c8a451c9c6.pdf |
| spelling | journaliasakpiua-article-2989402024-11-16T18:06:34Z Fairness of 2D corotational beam spline as compared with geometrically nonlinear elastic beam Добротність 2D коротаційного сплайну променя порівняно з геометрично нелінійно пружним променем Orynyak, Igor Yablonskyi, Petro Koltsov, Dmytro Chertov, Oleg Mazuryk, Roman коротаційний балковий сплайн геометрично нелінійна балка плоска задача крива Безьє добротність метод початкових параметрів corotational beam spline geometrically nonlinear beam 2D Bezier curve fairness transfer matrix method The goal of this paper is to further investigate the properties and advantages of corotational beam spline, CBS, as suggested recently. Emphasis is placed on the relatively simple task of drawing the spline between two endpoints with prescribed tangents. In the capacity of “goodness” of spline, the well-known notion of “fairness” is chosen, which presents itself as the integral from the squared curvature of spline over its length and originates from the elastic beam theory as the minimum of energy of deformation. The comparison is performed with possible variants of the cubic Bezier curve, BC, and geometrically nonlinear beam, GNB, with varying lengths. It was shown that CBS was much more effective than BC, where any attempt to provide better fairness of BC by varying the distances from endpoints to two intermediate points generally leads to lower fairness results than CBS. On the other hand, GNB, or in other words, the elastica curve, can give slightly better values of fairness for optimal lengths of the inserted beam. It can be explained by the more sophisticated scientific background of GNB, which employs 6 degrees of freedom in each section, compared with CBS, which operates only by 4 DoF. Метою статті є подальше дослідження властивостей і переваг нещодавно запропонованого коротаційного балкового сплайну (КБС). Акцент зроблено на розгляді досить простої задачі проведення сплайну між двома кінцевими точками із заданими дотичними в них. Як критерій “хорошості” сплайну обрано відоме поняття “добротності”, яке являє собою інтеграл від квадрата кривини сплайну по його довжині, що походить із теорії пружної балки як енергія деформації. Порівняння “добротності” КБС виконано з деякими варіантами кубічної кривої Безьє (КБ) і геометрично нелінійної балки (ГНБ) зі змінною довжиною. Показано, що КБС є набагато ефективнішим, ніж КБ, для якої будь-яка спроба забезпечити кращу “добротність” КБ шляхом зміни відстані від кінцевих точок до двох проміжних точок, як правило, призводить до гірших результатів порівняно з КБС. З іншого боку, ГНБ, або іншими словами, крива “еластика”, здатна давати дещо кращі значення “добротності” для оптимальної довжини балки. Це можна пояснити більш складною методологічною основою ГНБ, яка використовує 6 ступенів вільності в кожному перерізі порівняно з 4-ма ступенями вільності в КБС. The National Technical University of Ukraine "Igor Sikorsky Kyiv Polytechnic Institute" 2024-09-28 Article Article application/pdf https://journal.iasa.kpi.ua/article/view/298940 10.20535/SRIT.2308-8893.2024.3.07 System research and information technologies; No. 3 (2024); 107-132 Системные исследования и информационные технологии; № 3 (2024); 107-132 Системні дослідження та інформаційні технології; № 3 (2024); 107-132 2308-8893 1681-6048 en https://journal.iasa.kpi.ua/article/view/298940/306081 |
| spellingShingle | коротаційний балковий сплайн геометрично нелінійна балка плоска задача крива Безьє добротність метод початкових параметрів Orynyak, Igor Yablonskyi, Petro Koltsov, Dmytro Chertov, Oleg Mazuryk, Roman Добротність 2D коротаційного сплайну променя порівняно з геометрично нелінійно пружним променем |
| title | Добротність 2D коротаційного сплайну променя порівняно з геометрично нелінійно пружним променем |
| title_alt | Fairness of 2D corotational beam spline as compared with geometrically nonlinear elastic beam |
| title_full | Добротність 2D коротаційного сплайну променя порівняно з геометрично нелінійно пружним променем |
| title_fullStr | Добротність 2D коротаційного сплайну променя порівняно з геометрично нелінійно пружним променем |
| title_full_unstemmed | Добротність 2D коротаційного сплайну променя порівняно з геометрично нелінійно пружним променем |
| title_short | Добротність 2D коротаційного сплайну променя порівняно з геометрично нелінійно пружним променем |
| title_sort | добротність 2d коротаційного сплайну променя порівняно з геометрично нелінійно пружним променем |
| topic | коротаційний балковий сплайн геометрично нелінійна балка плоска задача крива Безьє добротність метод початкових параметрів |
| topic_facet | коротаційний балковий сплайн геометрично нелінійна балка плоска задача крива Безьє добротність метод початкових параметрів corotational beam spline geometrically nonlinear beam 2D Bezier curve fairness transfer matrix method |
| url | https://journal.iasa.kpi.ua/article/view/298940 |
| work_keys_str_mv | AT orynyakigor fairnessof2dcorotationalbeamsplineascomparedwithgeometricallynonlinearelasticbeam AT yablonskyipetro fairnessof2dcorotationalbeamsplineascomparedwithgeometricallynonlinearelasticbeam AT koltsovdmytro fairnessof2dcorotationalbeamsplineascomparedwithgeometricallynonlinearelasticbeam AT chertovoleg fairnessof2dcorotationalbeamsplineascomparedwithgeometricallynonlinearelasticbeam AT mazurykroman fairnessof2dcorotationalbeamsplineascomparedwithgeometricallynonlinearelasticbeam AT orynyakigor dobrotnístʹ2dkorotacíjnogosplajnupromenâporívnânozgeometričnonelíníjnopružnimpromenem AT yablonskyipetro dobrotnístʹ2dkorotacíjnogosplajnupromenâporívnânozgeometričnonelíníjnopružnimpromenem AT koltsovdmytro dobrotnístʹ2dkorotacíjnogosplajnupromenâporívnânozgeometričnonelíníjnopružnimpromenem AT chertovoleg dobrotnístʹ2dkorotacíjnogosplajnupromenâporívnânozgeometričnonelíníjnopružnimpromenem AT mazurykroman dobrotnístʹ2dkorotacíjnogosplajnupromenâporívnânozgeometričnonelíníjnopružnimpromenem |