A FINITE ELEMENT FORCE METHOD APPLIED TO FREE VIBRATION OF ROD SYSTEMS Текст научной статьи по специальности «Медицинские технологии»

XOK 624.04, 534.015

doi: 10.48612/dnitii/2024_50_34-46

A FINITE ELEMENT FORCE METHOD APPLIED TO FREE VIBRATION OF ROD SYSTEMS

V. V. Lalin H. H. Ngo A. M. Vavilova

Peter the Great St. Petersburg Polytechnic University, St. Petersburg

Abstract

In this study, the novel formulation of rod structures for dynamic analysis by the finite element force method using the element forces is proposed. The general equation of natural oscillation in which the unknowns of equation are nodal forces. The modified flexibility matrix is developed via the combination of consistent and lumped flexibility matrices with the specific ratio. The rate of convergence of frequencies using the modified flexibility matrix is considered. Numerical examples for the structural rods and frames are given to verify the effectiveness and practical applicability of the present study. It has been found that for a rod system when the use of modified flexibility formulation provides a good rate of convergence.

The Keywords

element forces, force method, finite element force method, flexibility matrices, inverse mass matrices

Date of receipt in edition

05.02.2024

Date of acceptance for printing

16.02.2024

Introduction

Rod systems are widely used in various scientific and engineering fields, including aerospace, architecture, medicine, and robotics. In free vibration analysis, the dynamic response of a rod system is expressed through two basic parameters such as the natural frequencies and mode shapes by using two well-known methods: the force method (FM) and displacement method (DM). Besides, the finite element method (FEM) is most commonly used in developing commercial software applications (Lira, Scad, etc.) for structural analysis and design. Obviously, the FEM is implemented in the form of the DM that it is called the finite element displacement method (FEDM) [1, 2]. On the other hand, the FEM is developed in the form of the FM that it is called the finite element force method [3, 4] or finite element flexibility method (FEFM) [5]. In addition, the mixed and hybrid finite element methods are presented in [6].

The early stage of development of the FM by using matrix calculations is found in [7 - 10]. Then, different formulations and variations of the FM have been proposed and used in engineering practice. Some of these methods are: the integrated force method (N. Patnaik) [11], the graph-theoretical force method (A. Kaveh) [12, 13], the generalized flexibility method (V. A. Meleshko) [14], the loop resultant method (V.V. Lalin) [15]. Furthermore, the

FM has been extended to handle various other problems in structural analysis, such as stability [16] and nonlinear [17, 18] analyses, as well as optimal design [19 - 22].

In this study, the FEFM is extended to the modified flexibility matrix for free vibration problems and it will be called the modified FEFM or MFEFM. In order to form the modified flexibility matrix, a suitable ratio between the consistent and lump flexibility matrices is determined. Three examples are analysed. The numerical results are compared to solutions of analytical models and Scad software to demonstrate the accuracy and the validity of the present formulation.

Formulations

The FEFM using the element forces for the equation of longitudinal and bending vibrations is described as follows. The rotational inertia of the cross-section of a rod is not considered.

The functional form of forces

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

z

M Û -I M

D

CÛ

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:

J_

EA

N=£,

(1)

and

s = u', (2)

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

.

EA

In addition, we have the following relation:

d2u

(3)

N +q = p

8t

,

or

*T> 1 1 "

,

(4)

(5)

c o

4J

(C

1_

■>

<U <U

2 2 i ®

2 Î sü

■ +J

< (U

§ S

zl5

xS

Z ¡

IH — -J (U

Sa

* ÍI > <

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

or

N"- + q'- = (u'y. P P

Substituting Eq. (2) into Eq. (7), we obtain Substituting Eq. (1) into Eq. (8), we obtain

.

p p EA

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

h K

(7)

(8)

(9)

(10)

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).

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:

vn =

M

,

and

where EI is the bending stiffness, (...)" is the second derivative of the function. By setting the second derivative of Eq. (11) with respect to x, we obtain

(11)

(12)

vIV =

-M EI

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

where p is the density.

Substituting Eq. (13) into Eq. (14), we obtain ,

or

M"+v = -q. P P

Let's set up the second derivative of Eq. (16) with respect to x, we obtain

(Zl M")"+(v"f=-q//. p p

Substituting Eq. (11) into Eq. (17), we obtain

.

p EI p

From Eq. (18), the functional form can be expressed in terms of the function M(x,t):

.

{Ik2 p 2 ei p j

The element forces

For the longitudinal vibration of a rod, is assumed to be in the form:

,

e e

(13)

(14)

(15)

(16)

(17)

(18)

(19)

(20)

where H(x) = [(1 - —) —], {N(t)} - {N^t) N2 (0}r> the superscript T denotes the trans-^i ^i

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

By substituting the expression (20) into each term of the functional form (10), we obtain:

BNT>'fH'N

,

(21)

{J

z

M

O

-I

M

D CD

c o

4J

(C

1_

■>

<u

<U

I ®

Si a

* *

■ +J

< <U

§ t

zj5

z I

IH —

-J <U

* I

> <

Veil 1 . %'ril^T 1

(Nfyxdt = J —HTHNjbcdt,

(23)

From the expressions (21), (22) and (23), by setting Z"x = J— (Ilf II dx and it is called the inverse

mass matrix (IMM), V'a

= f— J FA

1 -r,

H1 Hdx — the consistent flexibility matrix (CFM), P= [—Hq'dx.

EA [p

1

Then, it is not difficult to obtain Zax =

Ph

1 -1 / 2 f

j/ax _ 'e

(sym.) 1 ' c ~ 6EA (sym.) 2

The element forces allows its consistent flexibility matrix to be reduced to the diagonal matrix, thus V™ = is called the lumped flexibility matrix (LFM).

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

(24)

The equation of longitudinal vibration can be obtained based on the functional form (24): ,

where det(Zav) = 0, det(Fav) * 0.

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

,

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

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

where II = [ (x) 32 (jc) 33 (x) 34 (x)] and

p = mo Ar,(o>r={0(o Ml(O Q2(t) M2(t)}T.

By substituting the expression (27) into each term of the functional form (19), we have

hi.

r, o

2 p

№dt = \ J

r, 0

—FT —{H")TH"F 2 p

\

,

7

(25)

(26)

(27)

(28)

(29)

h i. / i '>( i \ JJ dxdt = \J FT-Hq" Vbcdt.

t,a\P / r, o V Z7 7

(30)

From the expressions (28), (29) and (30) by setting Z Vhce = j^jHTHdx — the CFM, and P = |—Hq"dx. Here

i-(H")TH"dx - the IMM,

ÎP

and FLbe =

/

/3

/

It

2EI 2AEI 2EI 24EI

} is the LFM.

We can rewrite the functional form (19) as

.

(31)

2 2

i, -

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

VbeF + ZbeF = 0,(32)

where det(/'" ) = 0, dct(Fi?) * 0.

Using the element forces of proposed formulation that forces, moments (Nix, Qiy, Miz, N2x, Q2y, M2z) at two ends of the rod which are called the nodal forces as shown in Fig. 3.

Fig. 3. Finite element (FE) under tension, compression and bending forces

The equation of natural oscillation of a rod taking into account the axial and bending vibrations is shown as below:

.

(33)

z

M Û -I M

D CD

c o

4J

(C

1_

>

<u

(U

2 2 O 13

£ a

* Î

sJi

■ +J

< (U

8 8

zj5

z |

IH — -J (U

=S JB

> SI

> <

where R ~[Ni Qi M.f

z = —

ple

0 12

ñ

(sym. )

-1 0 0

0 -12 6

~¡r ~e

0 -s 2

1 0 0

12 -6

ñ le 4

,V =

0 0 te 6 EA 0 0

13 le 0 -13iB2

35 El 210 El 70 EI 420EI

I3 0 13!; -I2 Le

105£7 42 0B/ 140£/

le 3EA 0 0

(sym. ) 13ie 35 EI -ml 210 EI

In free vibration, it is assumed that the element forces are the longitudinal and bending harmonics in the time:

Z-o/V

R= 0. (34)

where CO and R are the frequency and force mode shape, respectively

We investigate the rate of convergence of frequencies using the modified flexibility matrix of a rod which is created by the combination of consistent and lumped flexibility matrices in a specific ratio, as shown below:

r A ^ 3 T '

EA v 3

V =

le_ A3 a. El ^ 35

(sym. )

ai

nié

210 EI

le s al i P'/. \ El 105 24

'e

1 6EA 0

0

le sal i .'"' I

EA 3 2 '

a.

a-,

9le 70 El 13 l2e 420El

0

EI 35 2

Ia EI

-13 li

a,--

1 420El -I3

a-, ——

1 140 El 0

-nil a i--

1 210 EI

, a± , ^

105 24 ■

here a, =-,/?,= —, a, =—,/?,= —. 1 5 1 5 2 9 2 9

Examples of natural vibrations for structural rods and frames of this study using the developed program of the FEFM and MFEFM with the help of Matlab software are given below.

Numerical results of structural rods

Example 1: Consider a rod with different boundary conditions.

Initial data: Young's modulus E = 2.1 x 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 nth natural

n7T E

frequency of longitudinal vibration of a rod with fixed-fixed ends is given by (On

-, here n = 1, 2,

3,...; the nth natural frequency of bending vibration of a rod with pinned-fixed ends 0)n =

'O

L

,

PA

here = 3.9266, X2 = 7.0686, A3 = 10.2102, A4 = 13.3518, A = —{An +1), (n >4).

The numerical results of the first five natural frequencies are performed for rods from 2 to 16 finite elements. The normal, italic and bold values are obtained from Scad, the FEFM and MFEFM, respectively.

The error between numerical and exact solutions can be calculated by

S„ =

\Numerical - Exact\

Exact

(З5)

The error of numerical results generated by Scad, the FEFM and MFEFM, is given in tables 1 and 2.

Table 1

The errors of the longitudinal natural vibration frequencies

«n 2FE 4FE SFE 16FE

0.108 0.0З5 0.01б 0.011 Scad

l 0.103 0.026 0.007 0.002 FEFM

0.033 G.GG6 G.GGl G.GGG MFEFM

- 0.108 0.0З5 0.01б Scad

2 - 0.103 0.026 0.006 FEFM

- G.G33 G.GG6 G.GGl MFEFM

- 0.22З 0.0бб 0.024 Scad

З - 0.195 0.058 0.015 FEFM

- G.lGS G.Gl5 G.GG3 MFEFM

- - 0.109 0.0З5 Scad

4 - - 0.102 0.026 FEFM

- - G.G33 G.GG6 MFEFM

- - 0.1б1 0.049 Scad

5 - - 0.153 0.040 FEFM

- - G.G62 G.GlG MFEFM

From table 1, we can draw the following facts:

• The rate of convergence of frequencies of Scad is slower as compared to the FEFM and MFEFM (e. g., values 61(Scad) = 0.016, 61(FEFM) = 0.007, 61(MFEFM) = 0.001 by using 8 FE).

• The rate of convergence of the MFEFM is much fast as compared to the FEFM and Scad for most frequencies (e. g., values S5(MFEFM) = 0.062,S5(FEFM) = 0.153,S5(Scad) = 0.161 by using 8 FE).

Table 2

The errors of the natural frequencies of bending vibration

«n 2FE 4FE SFE ^FE

0.08З 0.04З 0.041 0.041 Scad

l 0.008 0.000 0.000 0.000 FEFM

G.GG4 G.GGG G.GGG G.GGG MFEFM

и z

H Û -I H

D CD

с о

4J

ra

n

>

<D (U

Sí Í3

о 13

î SÜ

■ -M

< (U

8 Ï

zl5

z ¡^

IH — -J <U

=S JB

Л > <

- 0.113 0.089 0.086 Scad

2 - 0.006 0.000 0.000 FEFM

- 0.004 0.000 0.000 MFEFM

- 0.263 0.153 0.142 Scad

3 - 0.023 0.002 0.000 FEFM

- 0.008 0.001 0.000 MFEFM

- - 0.232 0.204 Scad

4 - - 0.005 0.000 FEFM

- - 0.003 0.000 MFEFM

- - 0.321 0.268 Scad

5 - - 0.011 0.001 FEFM

- - 0.007 0.001 MFEFM

The following facts can be drawn in table 2:

• The rate of convergence of frequencies of Scad is much slower as compared to the FEFM and MFEFM for most cases from 2 to 16 FE (e. g., values Si(scad) = 0-083, S^fefm) = 0.008, S^mfefm) = 0.004 by using 2 FE).

• The rate ofconvergence offrequencies ofthe MFEFM is fast as compared to the FEFM, but in some cases both the FEFM and MFEFM have similar rate of convergence (e. g., values ^(fefm) = ^s(MFEFM) = 0.001 by using 16 FE).

Numerical results of structural frames

Example 2. The planar frame is given, see Fig. 4.

Initial data: The elastic modulus E=2 x 108 (kN/m2), the cross-sectional area A = 0.0144 (m2), the moment of inertia I = 1.728 x 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.

it

L

IJL

Fig. 4. 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.

Table 3

The first five natural frequencies of the planar frame

«n 3FE 6FE 9FE Exact

FEFM MFEFM FEFM MFEFM FEFM MFEFM

1 150.49 (0.0332) 148.37 (0.0187) 150.21 (0.0313) 149.71 (0.0279) 150.18 (0.0311) 149.97 (0.0297) 145.65

2 682.32 (0.2318) 624.65 (0.1277) 581.84 (0.0504) 575.25 (0.0385) 579.02 (0.0453) 576.31 (0.0404) 553.92

3 1505.7 (0.6758) 1096.4 (0.2202) 978.31 (0.0888) 951.95 (0.0595) 969.67 (0.0792) 958.86 (0.0672) 898.52

4 1803.9 (0.9826) 1605.1 (0.7641) 1006.7 (0.1064) 991.25 (0.0895) 995.10 (0.0937) 989.15 (0.0872) 909.85

5 1923.7 (0.2376) 1692.1 (0.0886) 1695.9 (0.0910) 1665.6 (0.0715) 1641.7 (0.0562) 1630.0 (0.0486) 1554.4

Note: The exact solutions are performed by Scad with 30 FE; FEFM and MFEFM — the results using the finite element force method and the modified finite element force method, respectively.

Example 3. Consider the space frame as shown in Fig. 5.

Initial data: The elastic modulus E = 3x107 (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 = 8.33x10-6 (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.

z

M

O

-I

M

D CD

Fig. 5. The space frame 8 rods

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

c o

4J

(C

1_

■>

<u

<U

2 5 O 13

Si a * *

■ +J

< <U

§ t

zl5

z ¡^

IH — -J (U

SI > <

Table 4

The first five natural frequencies of the planar frame

^n 8FE 16FE 24FE Exact

FEFM MFEFM FEFM MFEFM FEFM MFEFM

1 40.064 (0.0164) 39.725 (0.0078) 39.972 (0.0140) 39.895 (0.0121) 39.967 (0.0139) 39.933 (0.0130) 39.419

2 50.932 (0.0148) 49.125 (0.0212) 50.852 (0.0132) 50.396 (0.0041) 50.843 (0.0130) 50.641 (0.0090) 50.191

3 88.196 (0.0193) 86.240 (0.0033) 87.777 (0.0145) 87.349 (0.0095) 87.724 (0.0138) 87.540 (0.0117) 86.526

4 185.47 (0.1320) 172.85 (0.0550) 165.99 (0.0131) 164.20 (0.0022) 165.42 (0.0096) 164.71 (0.0053) 163.84

5 225.36 (0.1873) 208.01 (0.0959) 193.87 (0.0214) 191.96 (0.0113) 192.96 (0.0166) 192.25 (0.0129) 189.81

Note: The exact solutions are performed by Scad with 80 FE.

From tables 3 and 4, we can conclude that calculations using the element forces in the form of the MFEFM give fairly accurate results, even with a coarse mesh.

Conclusions

The numerical examples of structural rods with different boundary conditions show all values obtained from the FEFM, MFEFM and Scad that converge to the exact values. The results dealt with the element forces by using a modified flexibility matrix which show the MFEFM is far more efficient than the FEFM and Scad in terms of convergence speed. Illustrated examples such as the planar and space frames were considered in order to assess the performance of the proposed formulation.

The FEFM solution can be improved using the combined technique of the consistent and lumped flexibility matrices in appropriate ratios. In the MFEFM, this technique is used for longitudinal and bending vibrations, which is not necessarily in the same ratio but must satisfy the following conditions: a + ^ = 1 and

a2 + & = 1

References

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4. Kaveh, A., Massoudi, M. S. Recent Advances in the Finite Element Force Method, In: B.H.V. Topping and P. Ivanyi (eds) "Computational Methods for Engineering Technology". Saxe-Coburg Publications, Stirlingshire, UK, Chapter 12, 2014, pp. 305 - 324.

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z

M Û -I M

D CD

c o

4J

(C

1_

■>

<U (U

2 2

2 Î

sJi

■ +J

< (U

§ 8

ZÎ5

xS z i

IH — -J (U

Sa

* il > <

СИЛОВОЙ МЕТОД КОНЕЧНЫХ ЭЛЕМЕНТОВ, ПРИМЕНЯЕМЫЙ К СВОБОДНОЙ ВИБРАЦИИ СТЕРЖНЕВЫХ СИСТЕМ

В. В. Лалин Х. Х. Нго A. M. Вавилова

Санкт-Петербургский политехнический университет Петра Великого, г. Санкт-Петербург

Аннотация

Ключевые слова

элементарные силы, силовой метод, силовой метод конечных элементов, матрицы гибкости, обратные матрицы масс

В этом исследовании предлагается новая формулировка стержневых конструкций для динамического анализа методом конечных элементов с использованием элементных сил. Общее уравнение собственных колебаний, в котором неизвестными уравнения являются узловые силы. Модифицированная матрица гибкости разрабатывается путем сочетания согласованной и сосредоточенной матриц гибкости с определенным соотношением. Рассматривается скорость сходимости частот с использованием модифицированной матрицы гибкости. Численные примеры для конструктивных стержней и рам приведены для проверки

Дата поступления в редакцию

Дата принятия к печати

05.02.2024

16.02.2024

эффективности и практической применимости настоящего исследования. Было обнаружено, что для стержневой системы при использовании модифицированной формулы гибкости обеспечивается хорошая скорость сходимости.

Ссылка для цитирования:

V. V. Lalin, H. H. Ngo, A. M. Vavilova. A finite element force method applied to free vibration of rod systems. — Системные технологии. — 2024. — № 1 (50). — С. 34 - 46.