Free vibration analysis of rod systems using the finite element force method with extended shape functions Текст научной статьи по специальности «Медицинские технологии»

СТРОИТЕЛЬНАЯ МЕХАНИКА

Научная статья УДК 624.04, 534.015

https://doi.org/10.24866/2227-6858/2024-2/127-141

Анализ свободных колебаний стержневых систем с помощью метода конечных элементов в усилиях с расширенными функциями формы

Хыу Хиеу Нго1И, Владимир Владимирович Лалин1, Ты Куанг Чунг Ле2

^анкт-Петербургский политехнический университет Петра Великого, Санкт-Петербург, Российская Федерация 2Вьетнамский институт атомной энергии, Ханой, Вьетнам И [email protected]

Аннотация. Для анализа свободных колебаний стержневых систем предложена формулировка метода конечных элементов в усилиях с расширенными функциями формы, полученными с использованием форм колебаний неподвижно закрепленного стержня. Приведены примеры, иллюстрирующие точность численного решения для трех типов конечных элементов. Использование предложенного подхода для пространственной рамы дало различия между численными решениями для восьми конечных элементов и точными результатами менее 5% для первых пяти собственных частот. Ключевые слова: метод конечных элементов в усилиях, расширенные функции формы, метод сил, матрица податливости, обратная матрица масс

Для цитирования: Нго Х.Х., Лалин В.В., Ле Т.К.Ч. Анализ свободных колебаний стержневых систем с помощью метода конечных элементов в усилиях с расширенными функциями формы // Вестник Инженерной школы Дальневосточного федерального университета. 2024. № 2(59). С. 127-141.

CONSTRUCTION MECHANICS

Original article

Free vibration analysis of rod systems using the finite element force method with extended shape functions

Hieu H. Ngo1E, Vladimir V. Lalin1, Trung T.Q. Le2

:Peter the Great St. Petersburg Polytechnic University, Saint Petersburg, Russian Federation 2Vietnam Atomic Energy Institute, Hanoi, Vietnam El [email protected]

Abstract. For free vibration analysis of rod systems formulation of the finite element force method with extended shape functions obtained by using the mode shapes of a fixed-fixed rod is proposed. Examples illustrating the accuracy of numerical solutions for three types of finite elements are given. The use of proposed approach for the space frame gave differences between numerical solutions dealt with 8 finite elements and exact results less than 5% for the first five natural frequencies.

Keywords: finite element force method, extended shape functions, force method, flexibility matrix, inverse mass matrix

For citation: Ngo H.H., Lalin V.V., Le T.Q.T. Free vibration analysis of rod systems using the finite element force method with extended shape functions. FEFU: School of Engineering Bulletin, 2024, no. 2(59), pp. 127-141.

Introduction

In architecture and construction, rod systems play a significant role and are widely used for various structures, including airport roofs, bridges, electric poles, and many others [1,2]. For struc-

tural analysis and design, the displacement method has been traditionally more commonly used. However, in recent decades, there has been significant development and research on the force method. The process of developing the force method can be divided into two main stages:

Stage 1: Performing structural analysis calculations manually by hand. From the beginning of the 19th century, J.A. Eytelwein1, C.L.M. Navier2, B.P.E. Clapeyron3, J.A.C. Bress4 developed the system of force equilibrium equations for a statically determinate beam system (or a statically inde-terminate beam system using Maxwell's theorem). The main unknowns in this analysis are the forces. This approach is known by different names, for example, the method of consistent deformation, unit load method, flexibility method, and the superposition equations method, but most often it is known as the force method [3].

Stage 2: Automating the force method using a matrix approach [4-7] involves constructing the flexibility matrix for a structure [8-17], and constructing equivalent numerical models using the finite element force method [18-29] is known by several names, such as: the integrated force method [21-24], graph-theoretical force method [25,26], loop resultant method [27], generalized flexibility method [28], and base force element method [29]. The use of matrix algebra and matrix calculus as computational tools greatly simplifies the solution of discrete equations in numerical models [3032]. Instead of solving a problem through manual calculations, digital computers are used primarily to solve equations.

The purpose of this article is to consider three types of finite elements with extended shape forms by the mode shapes of a fixed-fixed rod, and their flexibility and inverse mass matrices respectively. The modes of the following form are used [33]: In the axial vibration:

wa (x, t) = Rn (t )u: (x), (a)

nnx

where the axial mode shapes Ua(x) = sin(-), n = 1,2,3,...; and the corresponding time function

le

Rn (t).

In the bending vibration:

Wnbe (x, t) = Sn (t)Ub; (x) , (b)

where Ube(x) = {sm(^)-sh(&)- Sm(An)-sh(^n) [cos(^)-ch(&-)]} are the bending mode

le le COs(\) - ch(K) le le

shapes corresponding to the parameters X1 = 4.7300, X2 = 7.8532, X3 = 10.9956, X4 = 14.1372, X5 =

2n _j_ ^

17.2788, \ = —-— n (n > 5); and the corresponding time function Sn (t).

Formulations: The differential and the functional forms of vibration in terms of force unknowns

In Fig. 1, an element-rod is subjected to an external distributed longitudinal force q(x) per unit length. The displacement u(x) and internal axial force N(x) represent the response of this ele-ment-rod due to a known applied load.

1 Eytelwein J.A. Handbuch der Statik fester Körper, Berlin: Realschulbuchhandlung, 1808. (In Ger.).

2 Navier C.L.M. Résumé des leçons données à l'École royale des ponts et chaussées sur l'application de la mécanique à l'établissement des constructions et des machines, Paris: Firmin Didot père et fils, etc., 1826. (In Fr.).

3 Clapeyron B.P.E. Calcul d'une pouter élastique reposant librement sur des appuis inégalement espacés, Ibid., 1857. (In Fr.).

4 Bress J.A.C. Cours de mécanique appliquée par Bresse Troisième partie. Calcul des moments de flexion dans une poutre à plusieurs travées solidaires, Paris: Gauthier Villars, 1865. (In Fr.).

Fig. 1. The tension-compression element-rod

Let us establish the equation of longitudinal vibration of an element-rod. The following relationships can be established [34]:

— N = £ EA

and

(1)

(2)

£ = U ,

where EA is the longitudinal stiffness, s is the axial strain, (...)/ denotes a derivative with respect to x. Substituting expression (2) into expression (1) and taking its derivative with respect to x:

1 A7"/ //

— N = u EA

In addition, we have the following relation:

(3)

,W д и

N +Я = р—Г = ри ot

or

r! 1 1

N' — + q— = ii. P P

(4)

(5)

where p is the density of the material, a dot over u denotes a partial derivative with respect to time. By differentiating Eq. (5) with respect to x, we obtain

3 1, д . 1. д .... — (N -) + — (q-) = —(u). ox p ox p ox

or (if p = const(x))

A tII 1 , / 1 (

N —vq — = (u) . P P

Substituting Eq. (2) into Eq. (7), we obtain

N" — + q' — = s. P P

Substituting Eq. (1) into Eq. (8), we obtain

1 ,,// /1 1 vy — N +q — = —N. p p EA

From Eq. (9), we have the following functional form in terms of the function N(x, t) :

(6)

(7)

(8)

(9)

и l

¡Up 2 EA

P

dxdt

(10)

For the longitudinal vibration of a rod, N(t,x) is assumed to be in the form:

x x x x IN I

N(x, t) = N (t)(1 - -) + N2 (t) - = [(1 - -) -] j n I = H(x)N(t),

e e e e L 2 J

(11)

x x

where H(x) = [(1--) —], N(t) = {N (t) N2 (t)}T , the superscript T denotes the transpose of the

l ^e

matrix, and le - the initial length of an element rod.

By substituting the expression (11) into each term of the functional form (10), we obtain

t2 le f 2 ^

t2 ^ 1 „Г 1

J Jl i±(N7)2 dxdt = J J| 1NT -(H')TH1N J012P J JJI2 P

t2 le f f

dxdt.

2 EA

dxdt =

ff[itfr — HTHN JL ^ EA

dxdt

t2 le f 1 t2 le i 1

ffl— Nq' dxdt = ff| NT — Hq

t, о Vp J t, о V p

dxdt .

(12)

(13)

(14)

1

From the expressions (12), (13) and (14), by setting Zflx = f — (H')TH1 dx and it is called the

" 0 P

le ^ le 1

inverse mass matrix (IMM), V= f — HTHdx - the flexibility matrix (FM), P = f — Hq'dx . Then,

J EA 0 p

" 1 -1" " к h '

le vax = — , v ea 3 6

1 i

(sym.) — (sym.) e

h _ 3 _

it is not difficult to obtain Z™ = —

p

The various types of axial element forces can be set for matrices Z™ and Va of size

, . . . r. Xx X . 7TX 2nX . «ffl,

(2 + n) x (2 + n) by using //(x) = [(l--) — sin— sin- ••• sin-] and

^e ^e ^e ^e ^e

{N{t)} = {Nx{t) N2(t) R,(t) /2,(0 ••• R„(t)}T, here n = 1,2,....

When n = 1, we have a pair of matrices Z" and V"x with size 3*3 as follows:

z;x = -p

1 -1 Г-л <nx\,i

— — —— cos(—)dx l l J l2 l

I

г Л Ttx

J~2cos(~T)dx

1

^e 2

• Л

Vax = , V1

1

EA

l l

e e

3 6 l

3

( sym.)

(• n 2 nx (sym.) I — Cos (-)dx

0 le le

Now, let's rewrite the functional form (10) as below:

L{N) = NTZaxN -^NTVaxN - NT P^dt.

С x л x

J(1 --)sln(—)dx

0 le le

I

r x лx I — sln(—)dx

0 le le

I,

r . 2 .ЛX

J sin (—)dx

(15)

The equation of free longitudinal vibration can be obtained based on the functional form (15): 130 | ISSN 2227-6858 www.dvfu.ru/vestnikis

VaxN + ZaxN = 0, (16)

where det(Zax) = 0, det(Vm) * 0.

Next, consider an element-rod under external transverse forces q(x) as shown in Fig. 2. The displacement v(x), the transverse force Q(x) and the bending moment M(x) represent the response of this element-rod due to applied load q(x).

and

Fig. 2. The bending element-rod:_____the initial state,......the deformed state

Let us establish the equation of bending vibration of an element-rod. The following relationships can be established [34]:

// -m

v// =-, (17)

EI '

Q = M', (18)

where EI is the bending stiffness, (...)" is the second derivative of the function.

By setting the second derivative of Eq. (17) with respect to x, we obtain (if EI = const(x))

vIF = —. (19)

EI

In addition, we have the equation of motion for displacement v(x):

EIvir + pv = q. (20)

Substituting Eq. (19) into Eq. (20), we obtain

-.M"+pv = q, (21)

—M"+v = — q. (22)

P P

Let's set up the second derivative of Eq. (22) with respect to x, we obtain

(±M">i'+iy"f = -qu. (23)

P P

Substituting Eq. (17) into Eq. (23), we obtain

-M1V + —M + — q"= 0. (24)

P EI P

From Eq. (24), the functional form can be expressed in terms of the function M(x, t):

'2 'e (1 1 11 1

Z(M) = jn--(M"f---M2+-Mq" dxdt. (25)

tj 0 V2 P 2 EI P J

Assuming the function M (x, t) for the bending vibration of a rod as

M (x, t) = Q (t(x) + Mi (tP2 (x) + 02 (tP3 (x) + M2 (tp4 (x), (26)

-M "

or

where 3.(x) - the Hermite polynomials (i = 1,2,3,4 ).

The expression (26) can be represented in matrix form:

M(x,t) =(x) 32(x) 33(x) 34(x)]^^HF , (27)

where H = [3i(x) 32(x) 33(x) 34(x)] and F = | = {Qi(t) Mx(t) Q2(t) M2(t)]T.

[M,(t) J

By substituting the expression (27) into each term of the functional form (25), we have

0 v 2 P

P

JJ(1V")' W' = J} 11FT H">'H'F

12 h i i

dxd',

I-L^

2 EI

dxdt = \\{-FT—H

ij 0 v 2 EI

' 2 /e f \ ' 2 /e i I X

JJI- Mq" dxd' = JJ( FT - Hq

HFdx

dxdt.

dxd'.

t, 0

1

From the expressions (28), (29) and (30) by setting Z* = J — (H")TH"dx - the

(28)

(29)

(30) IMM,

P

Vbe =

n

f — HTHdx - the FM, and P = f — Hq''dx, we obtain

J ш J p

1

EI

rybe _ 1

Z 0 =

P

—2

T

6 -12 6

/2 /3 /e2

4 -6 2

I /2 К

12 -6

( sym.) 7 /e2

4

/e

Vbe = —

, V EI

13/e 11/2 9/e "13/e2

35 210 70 420

/3 13/e2 -/3

105 420 140

( sym.) 13/e 11/2

35 210 /3 105

The various types of bending element forces can be set for matrices Zh and Vh of size

(4 + n) x (4 + n) by using H = [3 (x) Э2 (x) Э3 (x) 34 (x) Ubxe (x) U\e (x)

F = {Qx{t) M1(t) 02(t) M2(t) Sx(t) S2(t) - ЗД}Г,п=1,2,...

Ub,;{x)-\ and

For n = 1, we have a pair of matrices Z\e and Vh with size 5*5 as follows:

rbe

32 6 ' 12 6 " j

w ~ T F тЭ— (U—) dx

Ke e e e 0

1 1

£ 35

11/2 - 13/;

210 70

т 3'(U1be)''dx

Zbe = - K 1 pK

^ к

(sym.)

К „ „

e e ъ £

i ? л ъ i K

12 — be )'dx Ъ Vbe = —K

H И т Эз (U1 ) dx I, V EI I

4 т 34'(U1be)''dx ъ

/; 13/. 105 420

13/, 35

(sym.)

420

-J3

140

- 11/,2 210

105

. 3Ubedx Ъ

т 32Ubedx I

т 3Ubedx I.

т Э4

3Ubedx ъ

т (U;e )''(U1be)' ' dx-ь 0 ъ

We can rewrite the functional form (25) as

- u:eu:edx-b

0

f 1 1 ^

L(M)=U-FTZbeF — FTVbeF + FTP dt. (31)

The equation of free bending vibration can be obtained based on the functional form (31):

VbeF + ZbeF = 0, (32)

where det(Zbe) = 0, det( Vbe) * 0 .

The equation of natural oscillation of a plane element is shown as below:

Vp,F + Zp,F = 0, (33)

where det(Zpl) = 0, det(Vpl) * 0, and for n = 0:

i- - 1 n 3 l n

k- 0 0 — 0 0 : ¿L 0 0 lie 0 0 n

K' l' b k3 6 -

K 12 2 6 2 „ - 12 2 6 2 - K 13l' lll'2 A 9le - 13l„2b

K -rK —r 0 —— r — r - K —' —e- 0 —

к 13 у f y f ' 12 у ъ

35 210 70 420

к 4 - fi 2 ъ к l3 131z _ l3 ъ

к - гг 0 — гг 2 гг ъ к J^ 0 i3^ -t ъ г

1 к l У l1 ' l ' Ъ тлЛ 1 к 105 420 140 ъ , Iy

- к 8 ее Ъ Vf =-к Ъ 1 2_ _У_

О к 1 ъ 0 ЕА к le А А Ъ here Гу -

р к 1 0 0 ъ к 7 0 0 ъ A

к ъ

к l ъ

к ъ

к (sym.) Ц- S ^ ryъ к ^

13/ - 11l2

к ' ' Г ' l2 У ъ

к е е ъ

к 4

У

ы

к 35 210 ъ

к ъ

к l3 ъ

2 ъ к е ъ

ТГУ ъ S 105 ы

The equation of natural oscillation of a space element is shown as below:

VspF + ZspF = 0, (34)

where det(Zsp) = 0, det(Vsp) * 0, and for n = 0.

Numerical results of structural rods

Example 1: Consider a rod with different boundary conditions.

Initial data: Young's modulus E = 2.1*1011 (N/m2), the length of a rod L = 1 (m), the square cross-section with a square side 0.1 (m) and the mass per unit length of a rod p = 7830 (kg/m3).

The exact solutions evaluated using the differential equation of vibration of a rod. The n-th natural frequency of longitudinal vibration of a rod with fixed-free ends is given by

(2n + V)n E , here n = 0,1,2,...; the n-th natural frequency of bending vibration of a rod with

2 L Vp

а, л

\EI

fixed-fixed ends is given by œ = I-, here ta = 4.7300, ta = 7.8532, ta = 10.9956, ta =

v L

^pA

7T

14.1372, ta = 17.2788, K=~ (2n + -), ( n > 5 ).

The numerical results of the first five natural frequencies are performed for rods from 1 to 20 finite elements (FE) using the developed program with the help of Matlab software. The normal, italic and bold values are obtained from the first type (n = 0), second type (n = 1) and third type (n = 2) of element forces, respectively.

In figures 3 and 4, numerical results of the first two natural frequencies using different types of element forces are compared with exact solutions.

pl

2

Fig. 3. The first (left) and second (right) natural frequencies of axial vibration

The error between numerical and exact solutions can be calculated by

\Numerical - Exact\

S =■

Exact

(35)

The errors of numerical results are generated by three types of element forces and given in tables 1 and 2.

Table 1

The errors of the longitudinal natural vibration frequencies

Si 2FE 6FE 10FE 14FE 18FE 20FE n

0.0259 0.0029 0.0010 0.0005 0.0003 0.0003 0

1 ü.üüü6 Ü.ÜÜÜÜ Ü.ÜÜÜÜ Ü.ÜÜÜÜ Ü.ÜÜÜÜ Ü.ÜÜÜÜ 1

0.0003 0.0000 0.0000 0.0000 0.0000 0.0000 2

0.1946 0.0259 0.0093 0.0047 0.0029 0.0023 0

2 0.01б2 ü.üüü6 Ü.ÜÜÜ2 Ü.ÜÜÜ1 Ü.ÜÜÜÜ Ü.ÜÜÜÜ 1

0.0016 0.0003 0.0001 0.0001 0.0000 0.0000 2

- 0.0719 0.0259 0.0132 0.0080 0.0064 0

3 - Ü.ÜÜ25 ü.üüü6 Ü.ÜÜÜ2 Ü.ÜÜÜ1 Ü.ÜÜÜ1 1

- 0.0008 0.0003 0.0002 0.0001 0.0001 2

- 0.1365 0.0508 0.0259 0.0156 0.0126 0

4 - Ü.ÜÜ73 Ü.ÜÜ15 ü.üüü6 Ü.ÜÜÜ3 Ü.ÜÜÜ2 1

- 0.0012 0.0006 0.0003 0.0002 0.0002 2

- 0.1946 0.0837 0.0428 0.0259 0.0209 0

5 - 0.01б2 Ü.ÜÜ32 Ü.ÜÜ12 ü.üüü6 Ü.ÜÜÜ4 1

- 0.0016 0.0009 0.0005 0.0003 0.0003 2

From table 1 and figure 3, we can draw the following facts:

- The rate of convergence of frequencies of the first type (n = 0) is slower as compared to

the second (n = 1) and third (n = 2) types (e.g., values £1(„=0) = 0.0259 3(«=2) = 0.0003 by using 2 FE).

S n=1) = 0.0006,

- The rate of convergence of the third {n = 2) type is much fast as compared to the first

(n = 0) and second (n = 1) types for most frequencies (e.g., values

2{n=2)

= 0.0016.

S2{n=1) = 0.0162 .

S2{n=0) = 0 1946 by using 2 FE).

Fig. 4. The first (left) and second (right) natural frequencies of bending vibration

The errors of the natural frequencies of bending vibration

Si 2FE 3FE 4FE 5FE 6FE 7FE 8FE 9FE n

0.0162 0.0041 0.0013 0.0005 0.0003 0.0001 0.0001 0.0001 0

1 0.0008 0.0001 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 1

0.0001 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 2

0.3292 0.0200 0.0092 0.0040 0.0020 0.0011 0.0006 0.0004 0

2 0.0012 0.0017 0.0003 0.0001 0.0000 0.0000 0.0000 0.0000 1

0.0001 0.0001 0.0001 0.0000 0.0000 0.0000 0.0000 0.0000 2

- 0.2101 0.0214 0.0138 0.0072 0.0040 0.0024 0.0015 0

3 - 0.0013 0.0024 0.0005 0.0002 0.0001 0.0000 0.0000 1

- 0.0001 0.0002 0.0001 0.0000 0.0000 0.0000 0.0000 2

- 0.4548 0.1689 0.0218 0.0175 0.0103 0.0062 0.0040 0

4 - 0.0192 0.0013 0.0029 0.0008 0.0003 0.0002 0.0001 1

- 0.0005 0.0001 0.0002 0.0001 0.0001 0.0000 0.0000 2

- - 0.2942 0.1485 0.0219 0.0203 0.0131 0.0085 0

5 - - 0.0129 0.0014 0.0033 0.0010 0.0005 0.0002 1

- - 0.0003 0.0001 0.0002 0.0001 0.0001 0.0000 2

Table 2

The following facts can be drawn in table 2 and figure 4:

- The rate of convergence of frequencies of the first type (n = 0) is much slower as compared to the second (n = 1) and third (n = 2) types for most cases from 2 to 9 FE (e.g., values

»=0) = 0.3292 :

„=,) = 0.0012 :

s2( n=2) = 0.0001 by using 2 FE).

- The rate of convergence of frequencies of the third type (n = 2) is fast as compared to the

second (n = 1) and first (n = 0) types (e.g., values 5M =2) = 0.0005

SA( n=i) = 0.0192

n=0) = 0.4548 by using 3 FE).

Numerical results of structural frames

Example 2. The planar frame is given, see Fig. 5.

Initial data: The elastic modulus E = 2*108 (kN/m2), the cross-sectional area A = 0.0144 (m2), the moment of inertia I = L728*10"5 (m4), the mass per unit length of a rod p = 2750 (kg/m3). Let's consider three options for the finite element mesh: 3, 6 and 9 finite elements.

Fig. 5. The planar frame, 3 rods

Table 3 shows the natural frequencies (Hz) taking into account the axial and bending vibrations, the error of results is enclosed in brackets.

The first five natural frequencies of the planar frame

Table 3

3FE 6FE 9FE Exact

Ш1 n = 0 n = 1 n = 2 n = 0 n = 1 n = 2 n = 0 n = 1 n = 2

150.49 150.19 150.18 1 150.21 150.17 150.17 150.18 150.17 150.17 145.65

1 (0.033) (0.031) (0.031) (0.031) (0.031) (0.031) (0.031) (0.031) (0.031)

682.32 5 78.5 7 578.35 581.84 578.37 578.32 579.02 5 78.30 578.30 553.92

2 (0.231) (0.045) (0.044) (0.050) (0.044) (0.044) (0.045) (0.044) (0.044)

1505.7 968.60 966.66 978.31 967.22 966.68 969.67 966.66 966.62 898.52

3 (0.675) (0.078) (0.076) (0.088) (0.076) (0.076) (0.079) (0.076) (0.076)

1803.9 991.66 991.50 1006.7 992.05 991.59 995.10 991.53 991.50 909.85

4 (0.982) (0.090) (0.090) (0.106) (0.090) (0.090) (0.093) (0.090) (0.090)

1923.7 1 727.9 1629.6 с 1695.9 1628.4 1627.9 1641.7 1627.8 1627.7 1554.4

5 (0.237) (0.112) (0.048) (0.091) (0.048) (0.047) (0.056) (0.047) (0.047)

Note: The exact solutions are performed by SCAD with 30 FE.

Example 3. Consider the space frame as shown in Fig. 6.

Initial data: The elastic modulus E = 3*107 (kN/m2), the cross-sectional area A = 0.01 (m2), the coefficient of cross-sectional shape k = 5/6, Poisson's ratio v = 0.2, the moment of inertia I = 8.33*10-6 (m4), the mass per unit length of a rod p = 24517 (N/m3).

Let's consider three options for the finite element mesh: 8, 16 and 24 finite elements.

Fig. 6. The space frame, 8 rods

Table 4 shows the natural frequencies (Hz) taking into account the axial, bending and torsional vibrations.

Table 4

The first five natural frequencies of the space frame

8FE 16FE 24FE Exact

Wi n = 0 n = 1 n = 2 n = 0 n = 1 n = 2 n = 0 n = 1 n = 2

40.064 39.935 39.923 39.972 39.963 39.963 39.967 39.965 39.964 39.419

1 (0.016) (0.013) (0.013) (0.014) (0.014) (0.014) (0.014) (0.014) (0.014)

50.932 50.837 50.821 50.852 50.841 50.840 50.843 50.840 50.840 50.191

2 (0.014) (0.013) (0.013) (0.013) (0.013) (0.013) (0.013) (0.013) (0.013)

88.196 87.726 87.634 87.777 87.718 87.714 87.724 8 7.709 87.708 86.526

3 (0.019) (0.014) (0.013) (0.014) (0.014) (0.014) (0.014) (0.014) (0.014)

185.47 164.28 163.33 л 165.99 165.40 165.34 165.42 165.30 165.29 163.84

4 (0.132) (0.003) (0.003) (0.013) (0.009) (0.009) (0.010) (0.009) (0.009)

225.36 200.53 199.28 с 193.87 193.32 193.25 192.96 192.80 192.79 189.81

5 (0.187) (0.056) (0.050) (0.021) (0.018) (0.018) (0.017) (0.016) (0.016)

Note: The exact solutions are performed by SCAD with 80

'E.

From tables 3 and 4, we can see that when using 3FE for the planar frame and 8FE for the space frame, the frequencies obtained by the third type (n = 2) of element forces, which show high

accuracy even with high-order frequencies (e.g., values

«кn=2) = 0.048 :

n=D = 0.112.

ö5( n=0) = 0.237 in table 3; values ö5( n=2) = 0.050.

Ss( n=i) = 0.056,

«5( n=0) = 0.187 in table 4).

Conclusions

Present formulations provide frequencies of structural rods and frames with the use of flexibility matrix and inverse mass matrix using the mode shapes of a fixed-fixed rod. From above results, we concluded that increasing in parameters n = 1 and n = 2 correspond to first and second modes, results the decrease in the rate of convergence of frequencies as shown in tables 1, 2, 3 and 4. The variation in frequency is sudden decrement between parameters n = 0 and n = 1 (e.g., values 3(n=0) = 0.0259 , £1(B=1) = 0.0006 in table 1 or values ¿4(B=0) = 0.132, S4(n=1) = 0.003 in table 4), i.e.,

calculations using the element forces in the form of the finite element force method give fairly accurate results even with a coarse mesh.

APPENDIX

12

F '

12 12 Г

1

21(1 + v)

0 0 /, 0 0 0 0

0 6 2 0 - 12 2 T"r' 0 0 0

- 6 2 7Г '' 0 0 0 - 12 2 Tr' 0 - 6 . If ry

0 0 0 0 0 - 1 2 21,(1+ v)Г 0

4 2 — r 1 0 0 0 6 2 Fr' 0 2 2 — r 1

4 2 — r 1 z 0 - 6 2 ITr 0 0 0

1 0 0 0 0

12 2 irr 0 0 0

(sym.)

12 Ï Г

6 2 F ''

1

21 (1+ v)Г

ZS' = -

:

ea :

13/, 35

13/, 35

X

3 "

(sym.)

0 0

- 11/.2 210

0 105

11/,2 210

105

9/, 70

13/2 420

13/ 35

9/, 70

0

- 13/2 420

0 0

13/ 35

X 6 "

3 "

0 ъ ъ , ъ

- 13/2 ъ

13/2 420

140

11// 210

105

420 0

0

- /

140 0

- 11// 210

0 0

here r2 =.

A

2 I

r,2 = ^

A

105 ы

ВКЛАД АВТОРОВ I CONTRIBUTION OF THE AUTHORS

Х.Х. Нго, В.В. Лалин, Т.К.Ч. Ле - анализ и интерпретация результатов; подготовка и редактирование текста. Все авторы прочитали и одобрили окончательный вариант рукописи.

Н.Н. Ngo, V.V. Lalin, T.Q.T. Le - analysis and interpretation of results; draft manuscript preparation. All authors reviewed the results and approved the final version of the manuscript.

КОНФЛИКТ ИНТЕРЕСОВ | CONFLICT OF INTEREST Авторы заявляют об отсутствии конфликта интересов. The authors declare no conflict of interest.

СПИСОК ИСТОЧНИКОВ

1. Kinney J.K. Indeterminate structure analysis. Reading, 1st ed. Massachusetts, U.S.A.: Addison-wesley publising company, Inc, 1962. 651 p.

2. Розин Л.А. Стержневые системы как системы конечных элементов. Ленинград: ЛГУ, 1975. 237 с.

3. Розин Л.А. О методе сил в строительной механике // Метод конечных элементов и строительная механика (Труды ЛПИ), Ленинградский политехнический институт. 1976. C. 5-15.

4. Argyris J.H., Kelsey S. The matrix force method of structural analysis and some new applications. Aeronautical Research Council Reports & Memoranda. 1956. 42 p. URL: https://reports.aerade.cran-field.ac.uk/handle/1826.2/3602 (дата обращения: 06.03.2024).

5. Argyris J.H., Kelsey S. Initial Strains in the Matrix Force Method of Structural Analysis // The Aeronautical Journal. 1960. Vol. 64. Iss. 596. P. 493-495. https://doi.org/10.1017/S0368393100073326

6. Gallagher R.H. A Correlation Study of Methods of Matrix Structural Analysis: Report to the 14th Meeting, Structures and Materials Panel. Advisory Group for Aeronautical Research and Development, NATO, Paris, France. 1964. 343 p. URL: https://www.sto.nato.int/publications/AGARD (дата обращения: 06.03.2024).

7. Felippa C.A. A historical outline of matrix structural analysis: a play in three acts // Computers & Structures. 2001. № 14(79). P. 1313-1324. https://doi.org/10.1016/S0045-7949(01)00025-6

8. Gallagher R.H., Dhalla A.K. Direct Flexibility Finite Element Elastoplastic Analysis. Berlin, Germany: IAS-MiRT, 1971. P. 443-462. // NC State University Libraries. URL: http://www.lib.ncsu.edu/resol-ver/1840.20/29198 (дата обращения: 06.03.2024).

9. Domaszewski М., Borkowski A. On automatic selection of redundancies // Computers & Structures. 1979. Vol. 10. Iss. 4. P. 577-582. https://doi.org/10.1016/0045-7949(79)90002-6

10. Mukhopadhyay M. Free vibration of a free-free beam with rotary inertia affect - a flexibility matrix approach // Journal of Sound and Vibration. 1988. Vol. 125. № 3. P. 565-569. https://doi.org/10.1016/0022-460X(88)90262-3

11. Zhang D.W., Wei F.S. Structural Eigenderivative Analysis Using Practical and Simplified Dynamic Flexibility Method // American Institute of Aeronautics and Astronautics. 1999. Vol. 37. № 7. P. 865-873. https://doi.org/10.2514/2.7535

12. Rashed F. A coupled BEM-flexibility force method for bending analysis of internally supported plates // International Journal for Numerical Methods in Engineering. 2002. Vol. 54. Issue 10. P. 1431-1457. https://doi.org/10.1002/nme.472

13. Chen J.T., Chung I.L. Computation of dynamic stiffness and flexibility for arbitrarily shaped two-dimensional membranes // Structural Engineering and Mechanics. 2002. Vol. 13. P. 437-453. https://doi.org/10.12989/sem.2002.13.4.437

14. Stutz L.T., Castello D.A., Rochinha F.A. A flexibility-based continuum damage identification approach // Journal of Sound and Vibration. 2005. Vol. 279. Iss. 3-5. P. 641-667. https://doi.org/10.1016/j .jsv.2003. 11.043

15. Fried I., Coleman M. Improvable bounds on the largest eigenvalue of a completely positive finite element flexibility matrix // Journal of Sound and Vibration. 2005. Vol. 283. Iss. 1-2. P. 487-494. https://doi.org/10.1016/jjsv.2004.06.024

16. Stutz L.T., Tenenbaum R.A., Correa R.A.P. The Differential Evolution method applied to continuum damage identification via flexibility matrix // Journal of Sound and Vibration. 2015. Vol. 345. P. 86-102. https://doi.org/10.1016/jjsv.2015.01.049

17. Armando Lanzi, Enrique Luco J. Caughey Damping Series in Terms of Products of the Flexibility Matrix // Journal of Engineering Mechanics. 2017. Vol. 143. № 9. https://doi.org/10.1061/(ASCE)EM.1943-7889.0001306

18. Soyer E., Topcu A. Sparse self-stress matrices for the finite element force method // International Journal for Numerical Methods in Engineering. 2001. Vol. 50. Iss. 9. P. 2175-2194. https://doi.org/10.1002/nme.119

19. Sedaghati R., Suleman A., Tabarrok B. Structural Optimization with Frequency Constraints Using the Finite Element Force Method // AIAA Journal. 2002. № 2(40). P. 382-388. https://doi.org/10.2514/2.1657

20. Лалин В.В., Лалина И.И., Нго Х.Х., Вавилова A.M. Формулировка элементных сил для анализа устойчивости стержневых систем в форме метода конечных элементов в усилиях // Инновации и инвестиции. 2024. № 2. P. 270-275.

21. Patnaik N., Yadagiri S. Frequency analysis of structures by integrated force method // Journal of Sound and Vibration. 1982. Vol. 83. P. 93-109. https://doi.org/10.1016/S0022-460X(82)80078-3

22. Patnaik N., Hopkins A., Halford R. Integrated force method solution to indeterminate structural mechanics problems. Washington DC: National Aeronautics and Space Administration, 2004. 180 p. URL: https://ntrs.nasa.gov/api/citations/20040045162/downloads/20040045162.pdf (дата обращения: 06.03.2024).

23. Singh A., Yang F., Sedaghati R. Design optimization of stiffened panels using finite element integrated force method // Engineering Structures. 2018. Vol. 159. P. 99-109. https://doi.org/10.1016/j.eng-struct.2017.12.040

24. Wang Y., Senatore G. Extended integrated force method for the analysis of prestress-stable statically and kinematically indeterminate structures // International Journal of Solids and Structures. 2020. Vol. 202. P. 798-815. https://doi.org/10.1016/j.ijsolstr.2020.05.029

25. Kaveh A., Aalizadeh Arvanaq R. Free vibration of symmetric planar frames via the force method and canonical forms // International Journal for Numerical Methods in Biomedical Engineering. 2011. № 6(27). P. 936-961. https://doi.org/10.1002/cnm.1344

26. Kaveh A., Massoudi M.S., Massoudi M.J. Efficient finite element analysis using graph-theoretical force method; rectangular plane stress and plane strain serendipity family elements // Periodica Polytechnica Civil Engineering. 2014. Vol. 58. P. 3-22. https://doi.org/10.3311/PPci.7405

27. Lalin V.V., Ngo H.H. The Loop Resultant Method for Static Structural Analysis // International Journal for Computational Civil and Structural Engineering. 2022. № 1(18). P. 72-81. https://doi.org/10.22337/2587-9618-2021-18-1-72-81

28. Meleshko V.A., Rutman Y.L. Generalized Flexibility Method by the Example of Plane Elastoplastic Problem // Procedia Structural Integrity. 2017. Vol. 6. P. 140-145. https://doi.org/10.1016/j.prostr.2017.11.022

29. Peng Y., Liu Y. Advances in the Base Force Element Method. Springer Nature Singapore Pte Ltd., 2019. 470 p.

30. El-Saved M.E.M., Maijadi D., Sandgren E. Force method formulations based on hamilton's principle // Computers & Structures. 1991. Vol. 38. Iss. 3. P. 301-316. https://doi.org/10.1016/0045-7949(91)-90108-X

31. Kangwai R.D, Guest S.D. Symmetry-adapted equilibrium matrices // International Journal of Solids and Structures. 2000. Vol. 37. Iss. 11. P. 1525-1548. https://doi.org/10.1016/S0020-7683(98)00318-7

32. Treibergs A., Cherkaev A., Krtolica P. Compatibility conditions for discrete planar structures // International Journal of Solids and Structures. 2020. Vol. 184. P. 248-278. https://doi.org/10.1016/j .ijsolstr. 2019.06.008

33. Rao S.S. Vibration of Continuous Systems. John Wiley & Sons, Inc., 2019. 792 p.

34. Kabe A.M., Sako B.H. Structural Dynamics Fundamentals and Advanced Applications. Academic Press, Elsevier Inc., 2020. 960 p.

REFERENCES

1. Kinney J.K. Indeterminate structure analysis. Reading, 1st ed. Massachusetts, U.S.A., Addison-wesley publising company, Inc, 1962. 651 p.

2. Rozin L.A. Rod systems as finite element systems. Leningrad, Leningrad State University, 1975. 237 p. (In Russ.).

3. Rozin L.A. On the method force in structural mechanics. Finite element method and structural mechanics, Leningrad Polytechnic Institute, Proceedings of LPI, 1976, pp. 5-15. (In Russ.)

4. Argyris J.H., Kelsey S. The matrix force method of structural analysis and some new applications. Aeronautical Research Council Reports & Memoranda, 1956, 42 p. URL: https://reports.aerade.cran-field.ac.uk/handle/1826.2/3602 (accessed: March 6, 2024).

5. Argyris J.H., Kelsey S. Initial Strains in the Matrix Force Method of Structural Analysis. The Aeronautical Journal, 1960, vol. 64, issue 596, pp. 493-495. https://doi.org/10.1017/S0368393100073326

6. Gallagher R.H. A Correlation Study of Methods of Matrix Structural Analysis: Report to the 14th Meeting, Structures and Materials Panel. Advisory Group for Aeronautical Research and Development, NATO, Paris, France, 1964, 343 p. URL: https://www.sto.nato.int/publications/AGARD (accessed: March 6, 2024).

7. Felippa C.A. A historical outline of matrix structural analysis: a play in three acts. Computers & Structures, 2001, no. 14 (79), pp. 1313-1324. https://doi.org/10.1016/S0045-7949(01)00025-6

8. Gallagher R.H., Dhalla A.K. Direct Flexibility Finite Element Elastoplastic Analysis. IAS-MiRT, Berlin, Germany, 1971, pp. 443-462. NC State University Libraries. URL: http://www.lib.ncsu.edu/resol-ver/1840.20/29198 (accessed: March 6, 2024).

9. Domaszewski M., Borkowski A. On automatic selection of redundancies. Computers & Structures, 1979, vol. 10, iss. 4, pp. 577-582. https://doi.org/10.1016/0045-7949(79)90002-6

10. Mukhopadhyay M. Free vibration of a free-free beam with rotary inertia affect - a flexibility matrix approach. Journal of Sound and Vibration, 1988, vol. 125, no. 3, pp. 565-569. https://doi.org/10.1016/0022-460X(88)90262-3

11. Zhang D.W., Wei F.S. Structural Eigenderivative Analysis Using Practical and Simplified Dynamic Flexibility Method. American Institute of Aeronautics and Astronautics, 1999, vol. 37, no. 7, pp. 865-873. https://doi.org/10.2514Z2.7535

12. Rashed F. A coupled BEM-flexibility force method for bending analysis of internally supported plates. International Journal for Numerical Methods in Engineering, 2002, vol. 54, iss. 10, pp. 1431-1457. https://doi.org/10.1002/nme.472

13. Chen J.T., Chung I.L. Computation of dynamic stiffness and flexibility for arbitrarily shaped two-dimensional membranes. Structural Engineering and Mechanics, 2002, vol. 13, pp. 437-453. https://doi.org/10.12989/sem.2002.13.4.437

14. Stutz L.T., Castello D.A., Rochinha F.A. A flexibility-based continuum damage identification approach. Journal of Sound and Vibration, 2005, vol. 279, iss. 3-5, pp. 641-667. https://doi.org/10.1016/j .jsv.2003. 11.043

15. Fried I., Coleman M. Improvable bounds on the largest eigenvalue of a completely positive finite element flexibility matrix. Journal of Sound and Vibration, 2005, vol. 283, iss. 1-2, pp. 487-494. https://doi.org/10.1016/jjsv.2004.06.024

16. Stutz L.T., Tenenbaum R.A., Correa R.A.P. The Differential Evolution method applied to continuum damage identification via flexibility matrix. Journal of Sound and Vibration, 2015, vol. 345, pp. 86-102. https://doi.org/10.1016/jjsv.2015.01.049

17. Armando Lanzi, Enrique Luco J. Caughey Damping Series in Terms of Products ofthe Flexibility Matrix. Journal of Engineering Mechanics, 2017, vol. 143, no. 9. https://doi.org/10.1061/(ASCE)EM.1943-7889.0001306

18. Soyer E., Topcu A. Sparse self-stress matrices for the finite element force method. International Journal for Numerical Methods in Engineering, 2001, vol. 50, iss. 9, pp. 2175-2194. https://doi.org/10.1002/nme.119

19. Sedaghati R., Suleman A., Tabarrok B. Structural Optimization with Frequency Constraints Using the Finite Element Force Method. AIAA Journal, 2002, no. 2 (40), pp. 382-388. https://doi.org/10.2514/2.1657

20. Lalin V.V., Lalina I.I., Ngo H.H., Vavilova A.M. An element forces formulation of stability analysis using the finite element force method for rod systems. Innovation and Investment, 2024, no. 2, pp. 270-275.

21. Patnaik N., Yadagiri S. Frequency analysis of structures by integrated force method. Journal of Sound and Vibration, 1982, vol. 83, pp. 93-109. https://doi.org/10.1016/S0022-460X(82)80078-3

22. Patnaik N., Hopkins A., Halford R. Integrated force method solution to indeterminate structural mechanics problems. National Aeronautics and Space Administration. Washington DC 20546-0001, 2004, 180 p. URL: https://ntrs.nasa.gov/api/citations/20040045162/downloads/20040045162.pdf (accessed: March 6, 2024).

23. Singh A., Yang F., Sedaghati R. Design optimization of stiffened panels using finite element integrated force method. Engineering Structures, 2018, vol. 159, pp. 99-109. https://doi.org/10.1016/j.eng-struct.2017.12.040

24. Wang Y., Senatore G. Extended integrated force method for the analysis of prestress-stable statically and kinematicallyindeterminate structures. International Journal of Solids and Structures, 2020, vol. 202, pp. 798-815. https://doi.org/10.1016/j.ijsolstr.2020.05.029

25. Kaveh A., Aalizadeh Arvanaq R. Free vibration of symmetric planar frames via the force method and canonical forms. International Journal for Numerical Methods in Biomedical Engineering, 2011, no. 6(27), pp. 936-961. https://doi.org/10.1002/cnm.1344

26. Kaveh A., Massoudi M.S., Massoudi M.J. Efficient finite element analysis using graph-theoretical force method; rectangular plane stress and plane strain serendipity family elements. Periodica Polytechnica Civil Engineering, 2014, vol. 58, pp. 3-22. https://doi.org/10.3311/PPci.7405

27. Lalin V.V., Ngo H.H. The Loop Resultant Method for Static Structural Analysis. International Journal for Computational Civil and Structural Engineering, 2022, no. 1(18), pp. 72-81. https://doi.org/10.22337/2587-9618-2021-18-1-72-81

28. Meleshko V.A., Rutman Y.L. Generalized Flexibility Method by the Example of Plane Elastoplastic Problem. Procedia Structural Integrity, 2017, vol. 6, pp. 140-145. https://doi.org/10.1016/j .prostr.2017.11.022

29. Peng Y., Liu Y. Advances in the Base Force Element Method. Springer Nature Singapore Pte Ltd., 2019, 470 p.

30. El-Saved M.E.M., Marjadi D., Sandgren E. Force method formulations based on hamilton's principle. Computers & Structures, 1991, vol. 38, iss. 3, pp. 301-316. https://doi.org/10.1016/0045-7949(91)-90108-X

31. Kangwai R.D, Guest S.D. Symmetry-adapted equilibrium matrices. International Journal of Solids and Structures, 2000, vol. 37, iss. 11, pp. 1525-1548. https://doi.org/10.1016/S0020-7683(98)00318-7

32. Treibergs A., Cherkaev A., Krtolica P. Compatibility conditions for discrete planar structures. International Journal of Solids and Structures, 2020, vol. 184, pp. 248-278. https://doi.org/10.1016/j.ijsol-str.2019.06.008

33. Rao S.S. Vibration of Continuous Systems. John Wiley & Sons, Inc., 2019. 792 p.

34. Kabe A.M., Sako B.H. Structural Dynamics Fundamentals and Advanced Applications. Academic Press, Elsevier Inc., 2020. 960 p.

ИНФОРМАЦИЯ ОБ АВТОРАХ | INFORMATION ABOUT THE AUTHORS

Нго Хыу Хиеу - аспирант, Санкт-Петербургский политехнический университет Петра Великого, Санкт-Петербург, Российская Федерация, [email protected], https://orcid.org/0000-0002-7882-7062

Ngo Huu Hieu, Postgraduate Student, Peter the Great St. Petersburg Polytechnic University, Saint Petersburg, Russian Federation, [email protected], https://orcid.org/0000-0002-7882-7062

Лалин Владимир Владимирович - доктор технических наук, профессор, Санкт-Петербургский политехнический университет Петра Великого, Санкт-Петербург, Российская Федерация, [email protected], https://orcid.org/0000-0003-3850-424X

Lalin Vladimir Vladimirovich, Doctor of Engineering Sciences, Professor, Peter the Great St. Petersburg Polytechnic University, Saint Petersburg, Russian Federation, [email protected], https://orcid.org/0000-0003-3850-424X

Ле Ты Куанг Чунг - кандидат технических наук, Вьетнамский институт атомной энергии, Министерство науки и технологий, Ханой, Вьетнам, [email protected], https://orcid.org/0000-0002-6547-4632

Le Tu Quang Trung, PhD in Sci. Tech., Vietnam Atomic Energy Institute, Ministry of Science and Technology (Hanoi, Vietnam), [email protected], https://orcid.org/0000-0002-6547-4632

Статья поступила в редакцию / Received: 07.03.2024. Доработана после рецензирования / Revised: 20.05.2024. Принята к публикации / Accepted: 10.06.2024.