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ПРОЕКТИРОВАНИЕ И КОНСТРУИРОВАНИЕ СТРОИТЕЛЬНЫХ СИСТЕМ. СТРОИТЕЛЬНАЯ МЕХАНИКА. ОСНОВАНИЯ И ФУНДАМЕНТЫ, ПОДЗЕМНЫЕ СООРУЖЕНИЯ
УДК 624.31, 534.111 DOI: 10.22227/1997-0935.2019.3.284-292
Analysis of the natural frequencies of oscillations of a planar truss with an arbitrary number of panels
Mikhail N. Kirsanov, Dmitriy V. Tinkov
National Research University "Moscow Power Engineering Institute " (MPEI), 14 Krasnokazarmennaya st., Moscow, 111250, Russian Federation
ABSTRACT
Introduction. Analytical solutions for problems of structural mechanics are not only an alternative approach to solving problems of strength, reliability and dynamics of structures, but also the possibility for simple performance evaluations and optimization of structures. Frequency analysis of planar trusses, most often used in construction and engineering, is an important part of the study of structures.
Objectives — development of a three-parameter induction algorithm for deriving the analytical dependence of the natural oscillation frequencies of the truss on the number of panels.
Materials and methods. A flat, statically definable truss with one additional external link and double braces has been considered. The inertia properties of the truss are modeled by point masses located in the nodes of the lower straight truss belt. Each mass is assumed to have only one vertical degree of freedom. The stiffness of all truss rods is assumed to be the same. The task is to obtain analytical dependences of the oscillation frequencies of the proposed truss model on the number of panels. The derivation of the desired formulas is performed by the method of induction in three stages — according to the 2 ~ numbers of rows and columns of the compliance matrix, calculated using the Maxwell - Mohr formula and the number of
U
9 ®
г г
О О
сч сч
WW
к ш
и
panels. To find common members of the obtained sequences of coefficients, an apparatus was used to compile and solve ^ q the recurrent equations of the Maple computer mathematics system. The task of determining frequencies has been reduced
m £ to the eigenvalue problem of a bisymmetric matrix.
Results. For the elements of the compliance matrix, general formulas have been found, according to which the frequency I- JJ equations are compiled and solved. It is shown that in the frequency spectra of trusses with different numbers of panels there
is always one common frequency (middle frequency) located in the middle of the spectrum. An expression is found for the maximum value of the average oscillation frequency as a function of the height of the truss. 'jjj Conclusions. The proposed truss scheme, despite its external static indeterminacy and the lattice, which does not allow for
^ ^ the calculation of forces by such methods as the method of cutting nodes and the cross section method, allows analytical
= ot solutions for the natural frequencies of loads in the nodes. The obtained formulas have a rather simple form, and some
O u] general properties, such as frequency coincidences for different numbers of panels and the presence of an analytically
o ^ calculated maximum of the average frequency function of the truss height, make this solution convenient for practical
§ O structural evaluations.
4 °
o KEYWORDS: oscillation frequency, truss, induction, Maple, analytical solution
cm £
> FOR CITATION: Kirsanov M.N., Tinkov D.V. Analysis of the natural frequencies of oscillations of a planar truss with an
i= arbitrary number of panels. Vestnik MGSU [Proceedings of Moscow State University of Civil Engineering]. 2019; 14:3:284-
„, 292. DOI: 10.22227/1997-0935.2019.3.284-292
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Анализ собственных частот колебаний плоской фермы
g t с произвольным числом панелей
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$ =§ М.Н. Кирсанов, Д.В. Тиньков
ф Национальный исследовательский университет «МЭИ» (НИУ «МЭИ»),
111250, г. Москва, ул. Красноказарменная, д. 14
0 ={ АННОТАЦИЯ
^ щ Введение. Аналитические решения задач строительной механики — не только альтернативный подход к решению
* Е проблем прочности, надежности и динамики сооружений, но и возможность для простых оценок работоспособности
¡5 х и оптимизации конструкций. Частотный анализ плоских ферм, наиболее часто применяющихся в строительстве и
1 с машиностроении, является важной составной частью исследования сооружений. Цели — разработка алгоритма О (о трехпараметрической индукции для вывода аналитической зависимости собственных частот колебаний фермы от щ ¡¡> числа панелей.
284
© Mikhail N. Kirsanov, Dmitriy V. Tinkov, 2019
Материалы и методы. Рассмотрена плоская статически определимая ферма с одной дополнительной внешней связью и сдвоенными раскосами. Инерционные свойства фермы моделируются точечными массами, расположенными в узлах нижнего прямолинейного пояса фермы. У каждой массы предполагается наличие только одной вертикальной степени свободы. Жесткость всех стержней фермы принимается одинаковой. Ставится задача получения аналитических зависимостей частот колебаний предложенной модели фермы от числа панелей. Вывод искомых формул производится методом индукции в три этапа — по номерам строк и столбцов матрицы податливости, вычисленной по формуле Максвелла - Мора и по числу панелей. Для нахождения общих членов полученных последовательностей коэффициентов применялся аппарат составления и решения рекуррентных уравнений системы компьютерной математики Maple. Задача определения частот свелась к задаче на собственные значения бисимметричной матрицы. Результаты. Для элементов матрицы податливости найдены общие формулы, по которым составлены и решены частотные уравнения. Показано, что в спектрах частот ферм с различным числом панелей всегда присутствует одна общая частота (средняя частота), располагающаяся в середине спектра. Найдено выражение для максимального значения средней частоты колебаний как функции высоты фермы.
Выводы. Предложенная схема фермы, несмотря на свою внешнюю статическую неопределимость и решетку, не позволяющую применять для расчета усилий такие методы, как метод вырезания узлов и метод сечений, допускает аналитические решения для частот собственных колебаний грузов в узлах. Полученные формулы имеют достаточно простой вид, а некоторые общие свойства, такие как совпадения частот для разных чисел панелей и наличие аналитически рассчитываемого максимума функции средней частоты от высоты фермы, делают это решение удобным для практических оценок конструкций.
КЛЮЧЕВЫЕ СЛОВА: частота колебаний, ферма, индукция, Maple, аналитическое решение
ДЛЯ ЦИТИРОВАНИЯ: Кирсанов М.Н., Тиньков Д.В. Analysis of the natural frequencies of oscillations of a planar truss with an arbitrary number of panels // Вестник МГСУ. 2019. Т. 14. Вып. 3. С. 284-292. DOI: 10.22227/1997-0935.2019.3.284-292
INTRODUCTION
Modern computer systems of symbolic mathematics make it possible to find analytical solutions to problems of structural mechanics as an alternative approach to solving the problems of strength, reliability, and dynamics of structures [1-7]. In [8-13], the induction method involving the operators of the Maple system obtained formulas for the dependence of the deflection of planar trusses on the number of panels. Analytical solutions of problems on the oscillation of a load with one degree on a truss with an arbitrary number of panels were obtained in [14-17]. A more accurate picture of the dynamics of trusses can be given by analyzing a truss model with a distributed mass, or at least with a mass distributed over the nodes of the lower belt. The main difficulty in obtaining such solutions is to determine the rigidity of the structure. In the elastic stage of the truss rods with small oscillations to find the compliance matrix inverse to the stiffness matrix, a very convenient method is to use the Maxwell - Mohr formula. The forces in the truss rods included in this formula in solving the problems of the stiffness of the arches [18-23], lattice [24-30] and spatial trusses [31-33] were determined on the basis of the program [8-11] written in Maple language on basis of the cutting knots method. The main limitation for the analytical method, designed for the analysis of systems with an arbitrary number of panels, is the regularity of the truss schemes [34, 35]. If there are periodically repeating structures in the structure, for example, panels, then the induction method is applicable to such trusses. R.G. Hutchinson
and N.A. Fleck [36, 37] dealt with the problems of the existence of regular statically definable schemes, and methods of their calculation. Some particular problems of periodic trusses are considered in [38].
MATERIALS AND METHODS
Consider a truss with double braces and an additional horizontal external link on the left support (Fig. 1). The truss has 2n panels and ns = 16n + 4 rods, including four rods, modeling the supports. It is assumed that all rods have the same stiffness EF. An analytical solution of the problem of the deflection of this truss for an arbitrary number of panels is given in [39]. Solutions for the case of uniform load over the nodes of the upper and lower belts are obtained by generalizing a number of solutions for trusses with the number n of panels in half span from 1 to 10:
A = PCn (a3 + 2bh + c3)/(h2EF),
where c = V a2 + h1 is the length of the brace; Cn is a coefficient depending on the type of load. The Maxwell - Mohr formula was used to calculate the deflection.
Vi S. s, l, A = P ' ' '
j=1
EF
< П
is
kK
о
0 CD CD
1 n ю
СЛ
CD CD
О 3 о Сл)
S (
S P
where lj and Sj is the length and force in the j th rod from the action of the load; sj is the force from a single vertical force applied to the central node in the lower belt. The forces in the rods were determined by cutting the nodes from the system of linear equilibrium equations
r z
1-й
>< о
f -
CD
i s
v Q
П о
i i
n n
CD CD CD
n
a ■ . DO
■ т
s 3
s у с о <D D WW
2 2 О О л —ь
(О (О
Fig. 1. The truss scheme, n = 3
CO
ra
compiled for all the nodes of the structure as a whole, which made it possible to overcome the external static indeterminacy of the truss. The solution of the system of linear equations was in symbolic form according to a program written in the language of computer mathematics Maple.
To derive a formula for the dependence of the frequency of oscillations of loads located in the nodes of the lower belt on the number of panels and the geometry of the structure, we will use the same method. The equations of vertical oscillations of cargo we write in the form
№ 9 r r O O N N
WW
* (V U 3
> in
E (A
on *
ÎÎ
<D <U
cz ç 1=
O w
o ^
o =ï
CD O CD
4 °
o
CO
CM <»
[Mn ]Y + [Dn ]Y = 0,
(1)
M n =
m 0
0
0 m ... 0
00
The compliance matrix [BJ, the inverse stiffness matrix [D ], has the following elements:
b, j = tSt) Sij ) 4/ ( EF ),
(2)
« I
CO O CO "
CD
*Î5
Z CT OT £= in T3 — <u <u o o
■8
Ï!
Ü in
№
X = 1/ (mo2).
(3)
Thus, to solve the problem, it is necessary to obtain analytical expressions for the matrix members [BJ. This matrix is symmetric not only with respect to the main diagonal (due to symmetry (2) with respect to i and j), but also with respect to the secondary diagonal. The last property is related to the symmetry of the structure. The vertical displacement of the node k from the action of a unit load at the node 2n - k is equal to
the displacement of the node 2n - k from the unit force at the node k.
Bisymmetric matrices were studied in [40]. When n = 3 we have the following form of the matrix
[ * ] = n
205 308 315 250 137
* 520 558 452 *
* * 657 * *
where [MJ is the matrix of inertia; Y is the vector of vertical displacements of masses; [DJ is the stiffness
matrix; Y is the vector of accelerations. If the masses of the loads are the same, then the inertia matrix is diagonal:
where S(k'} is the force in the rod k from the action of a single vertical force at node '; lk is the length of the rod. Multiplying (1) from the left by the matrix [BJ, we get the equation m[ Bn ]Y + Y = 0. The vector of vertical displacements will be represented as a periodic function Y = A sin(rai + 90). From here, taking into account the
relation Y = -ra2Y, we obtain an eigenvalue problem
[ Bn ]Y = XY, where
where denoted n = (a3 + 2bh + c3 )/(3h2EF), and the * symbol denotes elements whose values follow from the properties of the matrix symmetry. This kind of result allows, in the decision process, to calculate by the formula not for all values of ', j = 1, 2, ..., 2n - 1, but only for j = 1, ..., n, i = j ... 2n - j, which significantly reduces conversion time. To obtain the common members of the sequences in the rows of the matrix [Bn], we use the rff_findrecur operator of the Maple system's gen-func package, which returns a recurrent equation that is satisfied by the sequence members. Then the rsolve operator gives a solution to the equation defining the common term of the sequence. The result can be obtained if the sequence under investigation has a sufficient length. This task requires a sequence of at least eight. Therefore, all calculations must be started from the trusses, the number of panels is more than four. For the first row (j = 1) of the matrix with n = 5, consisting of elements 657, 1128, 1407, 1518, 1485, 1332, 1083, 762, 393, we have the equation
Ki = 4b1,i-1 - 6b1,i-2 + 4b1,i-3 - Ki-4' i = 1
2n -1.
The solution of this equation has the form bj,' = 4'3 -120'2 + 803' - 30. Similarly for other lines
b2,.+1 = 4i -108/2 + 587i + 549, i = 1, ..., 2n-3, b3,i+2 = 4i3 - 96i2 + 403i + 872,i = 1, ..., 2n - 5, b4,+3 = 4i3 - 84i2 + 251i + 987, i = 1, ..., 2n - 7.
In the general case, for arbitrary j, we have an ex-
pression bj t + -1 = 4i - a2 5
i2 +a
15i -a05, where the
coefficients a
2,5'
1,5 '
0,5
are to be determined. The sequence of coefficients with i2 has a fairly obvious common term a2 5 = 12 j -132. For other sequences, the rgf_findrecur and rsolve operators are required:
k=1
a15 = 16 j2 - 264j +1051, a05 = 8j3 -176j2 +1051 j - 913.
Solutions are obtained for n = 5. To generalize the solution to an arbitrary number of panels, it is required to repeat the output for other values of n. Omitting the intermediate results we give the corresponding expressions:
n=6:
a25 = 12 j -156,
a15 = 16 j 2 - 312 j +1451,
a05 = 8 j3 - 208j2 +1451 j -1287,
n=7:
a25 = 12 j -180,
a15 = 16j2 - 360j +1915,
a05 = 8 j3 - 240j 2 +1915 j -1725,
Summarizing these expressions for the general case, we obtain
a2,n = 12( j - 1 - 2n)
aj,n = 16j2 - 24(1 + 2n)j + 32n2 + 48n +11, a0,n = 8j3 -16(1 + 2n) j2 + +(32n2 + 48n +11) j - 32n2 - 22n - 3.
Together with the expression
i = 4i3 - a2 ni2 + a J - a
given elastic and geometric characteristics. For n = 2, we have the matrix
j ,i+j-1
0,n
[ «2 H
75 90 57 90 132 90 57 90 75
The eigenvalues of the matrix are X1 = 9n, X2 3 = 3(22± 1572 Compliance matrix at n = 3:
[ «3 ] = n
Eigenvalues of this matrix:
X1 = 9n, X2 = 42n, X3 = 10n /3, X23 = 6(54 ± 3^V3)n.
(4)
205 308 315 250 137
308 520 558 452 250
315 558 657 558 315
250 452 558 520 308
137 250 315 308 205
(5)
When n = 4, the set of seven eigenvalues consists of three values (4) and
X
4,5
X
= 3(l72± 11872±757 236±406272 = 3(172± 11872 + 757 236±406272
these coefficients constitute the main basic part of the matrix, the reflection of which relative to the main and secondary diagonal gives the full matrix, whose eigenvalues give the solution. For reflection on the main diagonal, use the ratios
b,j = b,j, j = 1, ..., 2n-1, i = j +1, ..., 2n -1.
Elements that are symmetrical with respect to the secondary diagonal are obtained using the relations
bi,2n- j = b2n-i, j , j = 1 ..., 2n - 1 i = j + 1 ..., 2n - 1
RESULTS
The result of induction on the three parameters were the expressions for the elements of the matrix, the eigenvalues of which give the oscillation frequencies of the truss, whose inertial properties are modeled by weights in the nodes of the lower belt, which allow only vertical displacements. The oscillation frequencies are determined by the formula (3) as applied to trusses with
< n
is
kK
o
0 CD CD
1 n (Q Cn
It is noted that for all numbers of panels n in the spectrum of natural frequencies there is a value X = 9n, and for even n the values X23 = 3(22 ± 1572)n are also included in the spectrum. In addition, calculations show that for numbers n multiple of three, the spectrum includes values (5), and for numbers n multiple of four, the spectrum includes values of the spectrum for n = 4. It can be assumed that a more general statement is true: the frequency spectrum of a truss with the number n = k1k2 of panels includes formulas for the frequency spectra of trusses with the number of panels k1 and k2. The assertion is verified for a number of numbers, but in the general case it still requires proof. For n = 5, the curves of frequency versus truss height reveal a maximum (Fig. 2).
This solution was obtained for mass m = 100 kg, stiffness EF = 2,0 -108N, panel length a = 3 m and height of struts b = 1 m. The following regularity is noted in the graphs: the frequency obtained from the eigenvalue = 9n present in the solutions for any n is located in the middle of the spectrum. This is confirmed by graphs plotted for other values of n. Analytical representation of the solution allows finding the exact values of the extremal point. From the condition d ra */dh = 0 where
0 CD
c g
8 3
y (
t r
t Ij
y O
is
r O
1 3
0 0
f ^
CD
1
v 0
O O
no j i n n
CD CD CD
n
a ■
. DO ■
s □
s y c o
D D ««
2 2 O O a a
<0 <0
rn 9 r r O O N N
« co * <u
u 3
> in
E (A
on *
51
160-
140-
120-
100-
80-
60-
40-
20-
-
CO*
/
/
1 1 1 1 1 1
1 2 3 4 5
Fig. 2. The natural frequencies of the truss (rad /s), depending on the height h with n = 5
<u <u
CZ £= 1=
o£
o ^ o
CD O CD
4 °
o
CO CM
CO
O in
m ( a
3\m (a3 + 2bh2 + c3)
co
<D >
Sef
■ := CB CL 00
« I
CO O 05 ™
9 g
CD
"o
Z CT
OT £=
OT T3
— <U
<u
O O
that the maximum frequency ra *max = — ' = is
3yj m(3a + 2b)
reached when the height value h = a CONCLUSIONS
Methods of symbolic mathematics made it possible to find not only exact expressions for the elements of the matrix that defines the eigenfrequencies of free oscillations of loads in the truss nodes, but also to obtain analytical expressions for the frequencies. In a numerical analysis of the results obtained, it was also found that, regardless of the number of panels, the design under consideration has the same oscillation frequency located in the middle of the spectrum. The comparative simplicity of the solution also allowed us to find the exact expression for the extreme point on the graph of the dependence of the average frequency
on the height of the truss. A significant simplification in the derivation of the desired formulas turned out to be a technique based on the bisymmetric properties of the compliance matrix, which reduces the calculation of all elements of the matrix to the calculation of only the elements of its basic triangle with the subsequent reflection of elements relative to the main and secondary diagonal. Certainly, the experience of the authors in solving the problems of deflection of statically definable flat trusses in analytical form by the method of induction [8-11] was useful for successful work. Compared to these tasks, the solved problem of the oscillation frequencies of a system with many degrees of freedom is significantly more difficult due to the three levels of induction in rows and columns of the matrix and in the number of panels. So if in the simple problem of deflection of the generalization of the result to an arbitrary number of panels, it is necessary in analytical form to solve on average k problems about the forces in the rods and the deflection of the truss, then with triple induction of such problems already k3.
Verification of the results obtained numerically.
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< DO
iiï kK
o
0 CD CD
1 n (Q Cn
CD CD 7
6 3 o CO
s (
S P
r s
100 o
f -
CD
i S v Q
n o
i i
n n
CD CD CD
n
a ■ . DO
■ T
s □
s y c o <D D WW
M 2
O O
a -A
<0 <0
rn о
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29. Arutyunyan V.B. Double induction for deriving a formula for deflecting a frame truss with an arbitrary number of panels. Postulate. 2018; 7(33):5.
30. Voropai R.A. Analysis of the deflection of the regular truss with cross type lattice. Science Almanac. 2016; 4-3(18):238-240. DOI: 10.17117 / na.2016.04.03.03.238
31. Domanov E.V. The analytical dependence of the deflection spatial console triangular profile of the number of panels. Science Almanac. 2016; 6-2(19):214-217. DOI: 10.17117/na.2016.06.02.214 (rus.).
32. Larichev S.A. Inductive analysis of the influence of the building rise on the stiffness of a spatial beam farm. Trends in Applied Mechanics and Mecha-tronics. Moscow, Infra-M, 2015; 1:4-8. (rus.).
33. Kirsanov M.N. The deflection of spatial coatings with periodic structure. Magazine of Civil Engineering. 2017; 8(76):58-66. DOI: 10.18720/MCE.76.6
34. Galishnikova V.V., Ignatiev V.A. Regular rod systems : (theory and methods of calculation). Volgograd, Volgograd State University of Architecture and Civil Engineering Publ., 2006; 551. (rus.).
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36. Hutchinson R.G., Fleck N.A. Microarchitec-tured cellular solids — the hunt for statically determinate periodic trusses. Zeitschrift für Angewandte Mathematik und Mechanik. 2005; 85(9):607-617. DOI: 10.1002/zamm.200410208
37. Hutchinson R.G., Fleck N.A. The structural performance of the periodic truss. Journal of the Mechanics and Physics of Solids. 2006; 54(4):756-782. DOI: 10.1016/j.jmps.2005.10.008
38. Zok F.W., Latture R.M., Begley M.R. Periodic truss structures. Journal of the Mechanics and Physics of Solids. 2016; 96:184-203. DOI: 10.1016/j. jmps.2016.07.007
39. Kirsanov M.N. Analytical calculation of the deflection of a girder truss with double braces. Structural Mechanics of Engineering Constructions and Buildings. 2018; 14(2):105-111. DOI: 10.22363/1815-5235-201814-2-105-11 (rus.).
40. Vorobiev O., Kirsanov M., Cherepanov S. About some bissymmetric matrix of regular type. Science and education in the XXI century : a collection of scientific papers based on the materials of the International Scientific and Practical Conference. Tambov, Izd-vo TROO «Biznes-Nauka-Obshchestvo», 2013; 23:9-10. URL: https://kpfu.ru/staff_files/F614427580/2013_Tambov.pdf
Received December 21, 2018.
Adopted in a modified form January 25, 2019.
Approved for publication February 27, 2019.
About the authors: Mikhail N. Kirsanov — Doctor of Physical and Mathematical Sciences, Professor of Department of robotics, mechatronics, dynamics and strength of machines, National Research University "Moscow Power Engineering Institute" (MPEI), 14 Krasnokazarmennaya st., Moscow, 111250, Russian Federation, c216@ya.ru;
Dmitriy V. Tinkov — postgraduate of Department of robotics, mechatronics, dynamics and strength of machines, National Research University "Moscow Power Engineering Institute" (MPEI), 14 Krasnokazarmennaya st., Moscow, 111250, Russian Federation, c216@ya.ru.
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36. Hutchinson R.G., FleckN.A. Microarchitec-tured cellular solids — the hunt for statically deter-
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37. Hutchinson R.G., Fleck N.A. The structural performance of the periodic truss // Journal of the Mechanics and Physics of Solids. 2006. Vol. 54. No. 4. Pp. 756-782. DOI: 10.1016/j.jmps.2005.10.008
38. ZokF.W., LattureR.M., BegleyM.R. Periodic truss structures // Journal of the Mechanics and Physics of Solids. 2016. Vol. 96. Pp. 184-203. DOI: 10.1016/j. jmps.2016.07.007
39. Кирсанов М.Н. Аналитический расчет прогиба балочной фермы с двойными раскосами // Строительная механика инженерных конструкций и сооружений. 2018. Т. 14. № 2. С. 105-111. DOI: 10.22363/1815-5235-2018-14-2-105-11
40. Vorobiev O., Kirsanov M., Cherepanov S. About some bissymmetric matrix of regular type // Наука и образование в XXI веке : сб. тр. по мат. Междунар. науч.-практ. конф. 30 сентября 2013. Тамбов: Изд-во ТРОО «Бизнес-Наука-Общество», 2013. Т. 23. С. 9-10. URL: https://kpfu.ru/staff_files/F614427580/2013_Tambov.pdf
Поступила в редакцию 21 декабря 2018 г. Принята в доработанном виде 25 января 2019 г. Одобрена для публикации 27 февраля 2019 г.
Об авторах: Кирсанов Михаил Николаевич — доктор физико-математических наук, профессор кафедры робототехники, мехатроники, динамики и прочности машин, Национальный исследовательский университет «МЭИ» (НИУ «МЭИ»), 111250, г Москва, ул. Красноказарменная, д. 14, c216@ya.ru;
Тиньков Дмитрий Владимирович — аспирант кафедры робототехники, мехатроники, динамики и прочности машин, Национальный исследовательский университет «МЭИ» (НИУ «МЭИ»), 111250, гч сч г. Москва, ул. Красноказарменная, д. 14, c216@ya.ru.
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