CC BY
4Magazine of Civil Engineering. 2024. 17(4). Article No. 12810.
Magazine of Civil Engineering issn
2712-8172
journal homepage: http://engstroy.spbstu.ru/
Research article UDC 624.07
DOI: 10.34910/MCE.128.10
Providing free vibrations and stability of a multi-span beam under temperature changes by selecting the support system
I.V. Kudryavtsev ^ © , V.A. Ivanov, O.I. Rabetskaya, A.E. Mityaev
Siberian Federal University, Krasnoyarsk, Russian Federation M [email protected]
Keywords: Euler-Bernoulli beam theory, multi-span beam, periodic structures, temperature, axial compression, free vibration, structural design, buckling, transverse oscillation, topology optimization, finite element method, pipeline
Abstract. This study presents a newly developed method for reasonable selection of boundary conditions and number of pinned intermediate supports for a straight multi-span beam. This method might help to obtain the required values of the first frequency of free vibrations and the critical load from the action of axial force, resulting from the changing the beam temperature. The method is based on known concepts of beam vibration and stability theories and uses support coefficients as a criterion for selecting the appropriate support system for a multi-span beam. These coefficients are obtained by solving the corresponding differential equations of the dynamic behavior of the beam and are determined only by the support conditions. Comparative calculations of the straight pipeline using the developed and finite element methods for beam and shell models were carried out, which showed good convergence. Normalization of the values of the support coefficients allowed to combine both conditions, for the first natural vibration frequency and the first critical force, and express it as a single criterion for the selection of the support system. The selection of the support system is shown as three general methods of fixing multi-beam beams with a constant span length. This approach can be applied to any straight beams and support conditions for which support coefficient values are known. To this end, a general algorithm for selecting a support system with known support coefficients and requirements for their normalization is given. The results obtained can be used in the calculation and design of any multi-span beam structures to control the values of their free vibration frequencies and stability by selecting an appropriate support system during the engineering design process.
Funding: The research was financially supported by Russian Foundation for Basic Research, Krasnoyarsk Territory and Krasnoyarsk Regional Fund of Science, project No. 20-48-242922. Available online: https://www.sf-kras.ru/konkursy/reestr/6277-razrabotka-proektnykh-metodov-obespecheniya-trebuemykh-dinamicheskikh-parametrov-mnogoproletnykh-sterzhnevykh-konstruktsij-posredstvom-obosnovannogo-vybora-skhemy-rasstanovki-i-vidov-opor-s-uchetom-vliyaniya-temperatury.
Citation: Kudryavtsev, I.V., Ivanov, V.A., Rabetskaya, O.I., Mityaev, A.E. Providing free vibrations and stability of multi-span beam at temperature changes by supports selection. Magazine of Civil Engineering. 2024. 17(4). Article no.12810. DOI: 10.34910/MCE.128.10
1. Introduction
A large number of extended periodic structures are subject to forced vibrations and temperature changes. These include pipelines, oil lines, steam lines, railway rails, pull rods, cables, beams, etc. Serviceability conditions for such structures ensure the permissible values of the first natural frequency of vibration / and the first critical force Pcr1 or temperature ATcr1 at which the loss of stability occurs:
© Kudryavtsev, I.V., Ivanov, V.A., Rabetskaya, O.I., Mityaev, A.E., 2024. Published by Peter the Great St. Petersburg Polytechnic University.
f1 >[ f ], Pcr1 >[P] or ATcr1 > [AT]. (1)
In this paper, these multi-span structures are modeled on the basis of Euler-Bernoulli beam theory for cases of transverse free vibrations and stability loss. Theoretical foundation for calculating the vibrations and stability of multi-span beams have been described in many papers. This beam dynamic behavior is commonly calculated by partial differential equations with specified boundary conditions, which are determined by the supports of the beam [1-39]. As it is difficult to obtain an analytical solution to this problem, numerical calculation methods and specialized computer programs are used. They are mostly based on the finite element method [40-42]. At the same time, numerical methods allow to obtain only individual particular solutions. This complicates their use for quick qualitative assessment of structural decisions in order to meet serviceability conditions (1).
Recently, applied calculation methods and various reference books on dynamics were developed to help engineers conduct the necessary calculations without setting up and solving differential equations [43, 44]. However, almost all the literature on vibrations and stability is focused only on assessing the dynamic behavior of already existing structures with given supports without the possibility of their design calculation. Nevertheless, engineers mostly need to determine the required support system with boundary conditions and number of pinned intermediate supports for a given beam to ensure the serviceability conditions (1).
The literature on vibration protection [45-51] mainly considers the following vibration control methods: vibration isolation, additional damping, balancing, etc. In the optimal design of beam structures and topology optimization [52-76], the authors focus on making changes to the beam itself, wherein changing variables must be continuous and smooth, which is difficult for discrete changes in the number and type of supports. It is also true for beam stability calculations.
This paper is aimed at a new approach for a reasonable selection of a support system for a straight multi-span beam that provides the required values of the first natural vibration frequency and first critical force taking into account the temperature.
To achieve this goal, the following tasks were solved;
1) to develop a design calculation approach for a reasonable selection of a support system for a straight multi-span beam that provides its required dynamic behavior (the first natural vibration frequency and the first critical force) taking into account the temperature;
2) to get a single criterion based on support coefficients for the design approach;
3) to propose a method, which makes it possible for an engineer to quickly and reasonably perform a multifactor assessment or design of multi-span beams to achieve the required dynamic behavior;
4) to justify the method by comparative calculations of a pipeline using different approaches.
2. Methods
The equation of free vibrations of a beam takes into account the action of the axial force P (Fig. 1a) and thus looks as follows [2, 3]:
EJ . d4y+P iz+m iy - 0 (2)
EJmm 4 +P 2 + m -.2 " 0 (2)
dx dx dt
As a function of the deflection y ( x, t ) for the bending mode of free vibrations at the first natural frequency, the following equation should be considered:
[Xft i
— I sin (rat). (3)
For an unambiguous solution of equation (2), setting four boundary conditions that reflect the beam restraint conditions on supports is necessary.
s s vrs s
A
run
'\y(x)
P(AT)
zx
rrrh
n
a)
b)
Figure 1. Effect of the beam temperature on its dynamic state: a) temperature design diagram;
b) frequency-temperature curve.
By substituting the deflection function (3) into equation (2) and temporarily taking P = 0 as well as the hinge restraint conditions, the solution for the first natural vibration frequency is calculated as follows:
f =
a
EJ„
2nl1
(4)
m
When the axial compressive force P acts on the beam, the first natural vibration frequency can be defined by Galefs formula [78-80]:
f1( P<>0 ) = f1( P=0 )
1 -
P
P
cr1
The axial force P is calculated through the temperature by equation [37]:
P = af -AT -ES.
(5)
(6)
The buckling force P for the first instability mode is calculated as follows [37]:
crl
Pcr1 =
n2EJ ■ „2 • l2
(7)
A typical curve of the dependence of the first natural vibration frequency on temperature is shown in Fig. 1b. By combining equations (4-7), the condition for the first natural vibration frequency of the beam is calculated, taking into account its temperature and the support restraints:
f1(Ar ) = ^ Î iE (n2 Jmin V-l 2 at -Ar ) > [ f ].
(8)
The beam buckling corresponds to the case of its zero natural vibration frequency (Fig. 1b). In equation (8), this corresponds to the equality to zero of the expression in brackets under the root, which makes it possible to use this dependence for assessing stability. For example, the first critical temperature of a beam is defined as follows:
ATcr1 =
n2 J ■ Jl mm
• l2-at •S
>[Ar ].
(9)
The obtained analytical dependence (8) ensures the fulfillment of all serviceability conditions (1).
2.1. Structural Design Approach
Let us consider the values of the support coefficients a and ^ (Table 1) for three common support systems (Fig. 2) for multi-span beams [43, 44].
l
®
N
2
N
"/7^7 II /7^7 "ffhl
®
N
/7^7 7/ X
"^77 // /7^7 /7^7
Figure 2. Patterns of beam support systems. Table 1. Original values of the support coefficients a and p.
Pattern no.
Coefficient
Number of intermediate supports, N
0
1
2
3
4
5
6
7
10
a
M a
M a
4.730
0.5 3.927 0.7
3.927 0.699 3.393 0.879
3.557 0.814 3.261 0.939
3.393 0.879 3.210 0.964
3.310 0.917 3.186 0.977
3.1416 1
3.260 0.939 3.173 0.983
3.230 0.954 3.164 0.988
3.210 0.964 3.159 0.99
3.196 0.971 3.156 0.992
3.186 0.977 3.153 0.994
3.180 0.978 3.151 0.996
y
y
x
8
9
If all intermediate supports (Fig. 2) are located equidistant from each other, the length of each span is calculated as follows:
l =
l
(10)
- N +1
The values of the support coefficients a and p in Table 1 make it possible to control the dynamic behavior of the beam by selecting their required values.
2.2. Normalization of Support Coefficients
The serviceability condition (8) includes two support coefficients a and p, but it is necessary to derive a single criterion for the selection method of the appropriate support system. For this purpose, we normalize the initial values of the support coefficients in Table 1 for all patterns (Fig. 2). To do this, it is necessary to divide the initial values of the coefficients (Table 1) by the support coefficients of free beam and take into account equation (10):
s2 x2
f
a =
a-(N + 1)Y ,_( N + O
, M _
V
n
y
(11)
V
M y
Normalized values of the support coefficients a' and p' for the patterns in Fig. 2 are shown in Table 2.
Table 2. Normalized values of the support coefficients a' andp'.
Pattern
Number of intermediate supports, N
no. i^uei nuiei IL 0 1 2 3 4 5 6 7 8 9 10
1 a' 1 4 9 16 25 36 49 64 81 100 121
M' 1 4 9 16 25 36 49 64 81 100 121
2 a' 2.267 6.250 11.54 18.66 27.75 38.77 51.80 66.82 83.83 102.8 124.0
M' 4 8.187 13.58 20.71 29.73 40.83 53.84 68.87 85.91 104.8 126.5
3 a' 1.563 4,666 9.697 16.71 25.72 36.72 49.70 64.71 81.75 100.7 121.7
M' 2.041 5.177 10.21 17.22 26.19 37.26 50.20 65.30 82.31 101.2 122.0
Now, the new support coefficients a' and p' can be compared as both are in the numerator and already take into account the number of intermediate supports.
2.3. Single Criterion for the Structural Design Approach
The analysis of the data from Table 2 shows that the new values of the support coefficients are close to each other for each support pattern, and at N > 2 the difference in their values is within 2-3 %.
Moreover, with the increase in N, this difference rapidly decreases because the initial support coefficients (Table 1) rapidly converge to the characteristic constants:
a - n, | -1 at N ^œ. (12)
This allows us to introduce a single criterion for the selection of the support system:
amin -a'-|'. (13)
If condition (13) is substituted in equation (8), then we get a square equation relative to the sought coefficient amin :
-a^in - n2EJmin + amin -12atATES + 4l4m [ f ]2 = 0. The solution of quadratic equation (14) with the required root sign is as follows:
a min CAT
4m
C2 +_
AT _2
f
l
n^EJ . min
(14)
(15)
where CAt is the temperature effect coefficient:
_at •AT •Sl
CAT =
2
2n2 J„
(16)
If the temperature effect is irrelevant, we assume AT = 0, and the condition (15) is simplified:
2l2 [ fi ]
m
a ■ = ^min
n \ EJ„
(17)
To ensure all operability conditions (1), it is necessary to select from Table 2 such pattern and number of intermediate supports for which values of support coefficients a' and y' are equal or more than the calculated value:
min
(a l') > amin = CAT +
2 ,4
C 2 + 4m [ f1 rl
CAT +
n2 EJm
(18)
The selected support system for a beam with the corresponding values of the coefficients a' and p,' ensure the simultaneous fulfillment of both serviceability conditions (1), which can be verified by checking the first vibration frequency and the first critical force or temperature using the new dependencies:
f1 (AT ) = ^
2l2 V P
cr1 2
Ef
m
n2Jmin -A•l2 •at -AT•S |>[f ];
V
i
2 t 4l4 [ f ]2 m ^
n J -
R ^min
a'2E
>[ p];
(19)
(20)
AT_i =
I'
cr1 ,2 o
l •at • S
2T 4l4 [ f ]2 ^
n2 J -
R ^min
m
a'2E
>[AT ].
(21)
If we take the condition [f ] - 0 in (19-21), the form of these equations is almost identical to their original version (4, 7, 9).
2.4. Algorithm for a Reasonable Selection of Support System for a Beam
The proposed design approach for normalizing the values of the support coefficients allows implementing the method for a reasonable selection of the support system for a beam to ensure the serviceability conditions (1). This method consists of the following steps:
1. Preparation of a set of possible patterns of beam support systems (as in Fig. 2).
2. Determination of the support coefficients a and p for each pattern from the selected set based on the
reference literature or by calculation.
3. Normalization of the values of the support coefficients a and p for each pattern so that they take close
values: a'« p.
4. Calculation of the minimum required value of the support coefficient amin (15).
5. Selection of a pattern from the obtained set so that the following condition is satisfied:
min (a, p')>amm.
Let us consider an example of calculating an extended structure by the developed method.
3. Results and Discussions
Consider a straight pipeline with a circular cross-section and the following characteristics: length l = 1.5 m, outer diameter D = 15 mm, wall thickness t = 1 mm, material: aluminum alloy E = 7.1 x 105 MPa, density p = 2,770 kg/m3, and CTE at = 2.3 * 10-5. Initially, the beam is rigidly fixed at both ends without intermediate supports. The objective is to select a support system that will provide the first natural vibration frequency [ / ] = 250 Hz at a temperature AT = 90 °C. We perform the calculation
by the developed method and verify the obtained results by the finite element method using the ANSYS software.
3.1. Analytical Solution by the Developed Method
First, let us check the current dynamic parameters of the beam under the initial restraint conditions. According to Table 2, these correspond to the support coefficients a ' = 2.267 and p ' = 4. By dependences
(19-21) at AT = 0 °C and [ / ] = 0 Hz, we obtain the following initial dynamic characteristics of the beam:
/ = 39.76 Hz, ATcr = 18.79° C. (22)
Instability is sure to arise when heating the beam to Tcr1 = 18.79 °C, which prevents us from using
such initial restraint conditions. We cannot use initial restraint conditions at AT = 0 °C as, in this case, the
first natural vibration frequency is only /1 = 39.76 Hz. So it becomes necessary to select another support
system. For this, according to the proposed method, it is necessary to calculate the required support coefficient according to equation (15):
amin = 26.76. (23)
From Table 2, we select a support system with min (a ', p ' ) > 26.76; for example, Pattern 2 with N = 4, for which the support coefficients are equal:
a ' = 27.75; p ' = 29.73. (24)
Let us check the actual values of the first natural vibration frequency and the critical temperature of the beam using the equations (19) and (21):
/2 = 290.16 Hz; ATcr2 = 102.78°C, (25)
which fulfill the serviceability conditions of this task ([ /1 ] = 250 Hz at AT = 90 °C).
In Fig. 3, the two curves show the dependences of the first vibration frequency on temperature for both variants (22, 25).
N = 4
T, oC
Figure 3. Dependence of the first natural vibration frequency on temperature.
In Fig. 3, the curve f (N) for N = 4 clearly shows the rise of the first natural vibration frequency of the structure in the entire temperature range under consideration.
3.2. Numerical Solution of the Problem
Let us verify the obtained results using the finite element method (FEM) for the beam and shell models of the straight pipeline. The beam model has 1500 Beam189 finite elements, and the shell model has 14328 Shell281 finite elements. Fig. 4 shows some typical numerical calculation results for the shell model.
c)
d)
1 NODAL SOLUTION ANSYS R17.1
STEP-1 SOB =1 FACT=137.669 TJSUM (AVGJ RSYS=Q DMX =.11128 SMX -.11128 DEC 8 2020 18:21:20
0 .012364 ■ 24729 . 14 9 4 5 . )74167 .09691€ .037093 .061822 .086551 .11128
Figure 4. Results of the numerical calculation: a) first vibration mode at N = 0 and AT = 0°C; b) instability at Tcri, N = 0 and \f = 0 Hz; c) first vibration mode at N = 4 and AT = 90°C; d) instability
at Tcr2, N = 4 and \f = 0 Hz.
The numerical calculation results are summarized in Table 3.
Table 3. Comparison of the calculation results.
f1, Hz at Tcr1, C at f1, Hz at Tcr2, °C at Tcr2, °C at
AT = 0 °C f] = 0 Hz AT = 90°C [f = 0 Hz f] = 250 Hz
N = 0 N = 4
Developed method 39.76 18.79 290.16 139.62 102.78
FEM, Beam4 39.75 18.78 290.39 139.93 102.91 Deviation, %_0.00961_0.00786_0.0812_0.219_0.119
FEM, Shell281 Deviation, % 39.78 0.0532 18.77 0.0353 288.73 0.494 137.67 1.40 100.23 1.52
The comparison of the calculation results obtained by the proposed method with numerical solutions by FEM shows that the maximum difference is 1.52 %.
Comparison of the proposed method with the papers of other authors shows a complete coincidence for cases when the solution is based on the differential equation (2) and the formula (5), for example, in [78-83]. This approach is correct for the Euler-Bernoulli beam theory, which has some limitations when used. For example, there are some conditions of the beam size ratios [84, 85]. Formula (5) also has its limitations, mostly depending on the ratio [86]:
Pl2 FJ
(26)
In later works, other authors have developed refined solutions based on more complex dependencies [86-90]. However, the difference in results is less than 5 %, which is acceptable for beam theory, which usually implies a preliminary design calculation.
4. Conclusions
1. The paper developed a design calculation approach for a reasonable selection of a support system for a straight multi-span beam that provides its required dynamic behavior. A new method was developed to help engineers reasonably select the support system for straight multi-span beams, which ensures their serviceability under given requirements for the minimum first natural vibration frequency and the first critical force or temperature.
2. The proposed method has a simple analytical formulation, which also makes it possible for an engineer to quickly and reasonably perform a multifactor assessment or design of any multi-span beam to achieve the required dynamic behavior.
3. The method was justified by the comparative calculations of a pipeline by using the beam and shell models, which showed good convergence for all monitored parameters. Comparison of the method with the works of other authors also showed good convergence in all controlled parameters.
4. The proposed method is to be developed further for the plane multi-support beams consisting of straight and curved sections in order to ensure their dynamic, stressed, and deformed state.
References
1. Abramovich, H., Elishakoff, I. Application of the Krein's Method for Determination of Natural Frequencies of Periodically Supported Beam Based on Simplified Bresse-Timoshenko Equations. Acta Mechanica. 1987. 66(1-4). Pp. 39-59. DOI: 10.1007/BF01184284
2. Banakh, L.Y. Vibrations of Mechanical Systems with Regular Structure. Berlin: Springer, 2010. 262 p.
3. Benaroya, H., Nagurka, M., Han, S. Mechanical Vibration. London: CRC Press, 2017. 602 p.
4. Billing, J.R. Estimation of the natural frequencies of continuous multi-span bridges. Ontario: Ministry of transportation and communication, 1979. 20 p.
5. Bottega, W.J. Engineering Vibrations. New York: CRC Press, 2006. 750 p.
6. Cai, C.W., Cheung, Y.K., Chan, H.C. Transverse vibration analysis of plane trusses by analytical method. Journal of Sound and Vibration. 1989. 133(1). Pp. 139-150. DOI: 10.1016/0022-460X(89)90988-7
7. Chonan, S., Sasaki, M. Vibration and stability of elastically supported multi-span beams under conservative and non-conservative loads. Journal of Sound and Vibration. 1985. 99(4). Pp. 545-556. DOI: 10.1016/0022-460X(85)90539-5
8. Chuang, C.H., Hou, G.J. Eigenvalue sensitivity analysis of planar frames with variable joint and support locations. AIAA Journal. 1992. 30(8). Pp. 2138-2147. DOI: 10.2514/3.11192
9. Clough, R.E. Dynamics of Structures. 3th ed. New York: McGraw-Hill College, 1995. 752 p.
10. Geradin, M., Rixen, D.J. Mechanical Vibrations. London: John Wiley & Sons, 2015. 617 p.
11. Hagedorn, P. Vibrations and waves in continuous mechanical systems. New Jersey: John Wiley. & Sons, 2007. 388 p.
12. Den Hartog, J.P. Mechanical Vibrations. New York: Dover Publications. 1985. 449 p.
13. Inman, D.J. Engineering Vibration. New Jersey: Pearson Education, 2014. 720 p.
14. Kelly, S.G. Advanced Vibration Analysis. Boca Raton: CRC/Taylor & Francis, 2007. 637 p.
15. Lee, H.P., Ng, T.Y. In-Plane Vibration of Planar Frame Structures. Journal of Sound and Vibration. 1994. 172(3). Pp. 420-427. DOI: 10.1006/jsvi.1994.1185
16. Lin, H., Chang, S.C. Free vibration analysis of multi-span beams with intermediate flexible constraints. Journal of Sound and Vibration. 2005. 281(1-2). Pp. 155-169. DOI: 10.1016/j.jsv.2004.01.010
17. Lin, H., Tsai, Y. Free vibration analysis of a uniform multi-span beam carrying multiple spring-mass systems. Journal of Sound and Vibration. 2007. 302(3). Pp. 442-456. DOI: 10.1016/j.jsv.2006.06.080
18. Lin, Y.K. Free Vibration of a Continuous Beam on Elastic Supports. International Journal of Mechanical Sciences. 1962. 4(5). Pp. 409-423. DOI: 10.1016/S0020-7403(62)80027-7
19. Liu, H., Nguyen, H., Xiang, Y. Vibration analysis of a multi-span continuous beam with cracks. Applied Mechanics and Materials. 2012. 256-259. Pp. 964-972. DOI: 10.4028/www.scientific.net/AMM.256-259.964
20. Luo, J., Zhu, S., Zhai, W. Exact closed-form solution for free vibration of Euler-Bernoulli and Timoshenko beams with intermediate elastic supports. International Journal of Mechanical Sciences. 2022. 213. Article no. 106842. DOI: 10.1016/j. ij mecsci .2021.106842
21. Maurizi, M.J., Bambill, D.V., Bellés, P.M., De Rosa, M.A., Grossi, R.O., Marcelo, A.C., Zannier, L. Free vibrations of Bernoulli-Euler beams with intermediate elastic support: A concise thematic recension. Journal of Sound and Vibration. 2005. 281(3-5). Pp. 1238-1239. DOI: 10.1016/j.jsv.2004.06.014
22. Mead, D.J. Free wave propagation in periodically supported, infinite beams. Journal of Sound and Vibration. 1970. 11(2). Pp. 181-197. DOI: 10.1016/S0022-460X(70)80062-1
23. Miles, J. Vibration of beams on many supports. ASCE Journal of Engineering Mechanics. 1956. 82. Pp. 1-9.
24. Mills-Curran, W.C., Lust, R.V., Schmit, L.A. Approximations method for space frame synthesis. AIAA Journal. 1983. 21(11). Pp. 1571-1580. DOI: 10.2514/3.60154
25. Naguleswaran, S. Transverse vibration of an Euler-Bernoulli uniform beam on up to five resilient supports including ends. Journal of Sound and Vibration. 2003. 261(2). Pp. 372-384. DOI: 10.1016/S0022-460X(02)01238-5
26. Sen Gupta, G. Natural flexural waves and the normal modes of periodically-supported beams and plates. Journal of Sound and Vibration. 1970. 13(1). Pp. 89-101. DOI: 10.1016/S0022-460X(70)80082-7
27. Wang, D., Friswell, M.I., Lei, Y. Maximizing the natural frequency of a beam with an intermediate elastic support. Journal of Sound and Vibration. 2006. 291(3-5). Pp. 1229-1238. DOI: 10.1016/j.jsv.2005.06.028
28. Wang, R. Vibration of multi-span Timoshenko beams to a moving force. Journal of Sound and Vibration. 1997. 207(5). Pp. 731742. DOI: 10.1006/jsvi.1997.1188
29. West, H.H., Mafi, M. Eigenvalues for Beam-Columns on Elastic Supports. Journal of Structural Engineering (United States). 1984. 110 (6). Pp. 1305-1320. DOI: 10.1061/(ASCE)0733-9445(1984)110:6(1305)
30. Zhao, Z., Wen, S., Li, F., Zhang, C. Free vibration analysis of multi-span Timoshenko beams using the assumed mode method. Archive of Applied Mechanics. 2018. 88(7). Pp. 1213-1228. DOI: 10.1007/s00419-018-1368-8
31. Zhu, L., Elishakoff, I., Lin, Y.K. Free and forced vibrations of periodic multispan beams. Shock and Vibration. 1994. 1(3). Pp. 217232. DOI: 10.3233/SAV-1994-1302
32. Bayo, E., Loureiro, A. An efficient and direct method for buckling analysis of steel frame structures. Journal of Constructional Steel Research. 2001. 57(12). Pp. 1321-1336. DOI: 10.1016/S0143-974X(01)00040-2
33. Kobelev, V. Closed-form solution for optimization of buckling column. Mechanics Based Design of Structures and Machines. 2021. 51(10). Pp. 5596-5635. DOI: 10.1080/15397734.2021.2008259
34. Liu, Z., Hu, H., Huang, C. Derivative of Buckling Load with Respect to Support Locations. Journal of Engineering Mechanics. 2000. 126(6). Pp. 559-564. DOI: 10.1061/(ASCE)0733-9399(2000)126:6(559)
35. §akar, G., Ozturk, H., Sabuncu, M. Dynamic stability of multi-span frames subjected to periodic loading. Journal of Constructional Steel Research. 2012. 70. Pp. 65-70. DOI: 10.1016/j.jcsr.2011.10.009
36. Thomsen, J.J. Vibrations and Stability. Advanced Theory, Analysis, and Tools. 2nd ed. London: Springer, 2003. 420 p. DOI: 10.1007/978-3-662-10793-5
37. Timoshenko, S.P., Gere, J.M. Theory of Elastic Stability, 2nd ed. New York: Dover Publications. 2009. 560 p.
38. Trahair, N.S. Buckling analysis design of steel frames. Journal of Constructional Steel Research. 2009. 65(7). Pp. 1459-1463. DOI: 10.1016/j.jcsr.2009.03.012
39. West, H.H., Mafi, M. Eigenvalues for Beam-Columns on Elastic Supports. Journal of Structural Engineering (United States). 1984. 110(6). Pp. 1305-1320. DOI: 10.1061/(ASCE)0733-9445(1984)110:6(1305)
40. Shrikhande, M. Finite Element Method and Computational Structural Dynamic. New Delhi: PHI Learning Private Limited, 2008. 234 p.
41. Moaveni, S. Finite Element Analysis: Theory and Application with ANSYS. 4th ed. New York: Pearson Education, 2015. 929 p.
42. Stolarski, T. Engineering Analysis with ANSYS Software. London: Butterworth-Heinemann, 2018. 553 p.
43. Blevins, R.D. Formulas for Dynamics, Acoustics and Vibration. London: John Wiley & Sons, 2016. 464 p.
44. Birger, I.A., Panovko, Y.G. Strength, stability, fluctuations. Moscow: Mashinostroenie, 1968. 567 p.
45. Baz, A.M. Active and Passive Vibration Damping. New York: John Wiley & Sons. 2019. 737 p.
46. Inman, D.J. Vibration with Control. Berlin, 2017. 426 p.
47. Barez, F. Dynamic Systems Vibration and Control. San Jose: San Jose State University, 2018. 283 p.
48. Genta, G. Vibration Dynamics and Control. New York: Springer New York, 2009. DOI: 10.1007/978-0-387-79580-5
49. Nashif A.D., Jones D.I. Vibration Damping. New York: John Wiley & Sons, 1985. 480 p.
50. Thorby, D. Structural Dynamics and Vibration in Practice. Boston: Butterworth-Heinemann, 2008. 419 p. DOI: 10.1016/B978-0-7506-8002-8.X0001 -6
51. Yao, P.-F. Modeling and Control in Vibrational and Structural Dynamics: A Differential Geometrical Approach. Boca Raton: CRC Press, 2011. 417 p. DOI: 10.1201/b11042
52. Albarracin, C.M., Zannier, L., Grossi, R.O. Some observations in the dynamics of beams with intermediate supports. Journal of Sound and Vibration. 2004. 271(1-2). Pp. 475-480. DOI: 10.1016/S0022-460X(03)00631-X
53. Andreassen, E., Ferrari, F., Sigmund, O., Diaz, A. R. Frequency response as a surrogate eigenvalue problem in topology optimization. International Journal for Numerical Methods in Engineering. 2018. 113(8). Pp. 1214-1229. DOI: 10.1002/nme.5563
54. Aydin, E., Dutkiewicz, M., Ozturk, B., Sonmez, M. Optimization of elastic spring supports for cantilever beams. Structural and Multidisciplinary Optimization. 2020. 62(1). Pp. 55-81. DOI: 10.1007/s00158-019-02469-3
55. Bojczuk, D., Rebosz-Kurdek, A. Optimal design of bar structures with their supports in problems of stability and free vibrations. Journal of Theoretical and Applied Mechanics (Poland). 2014. 52(2). Pp. 533-546.
56. Dems, K., Turant, J. Sensitivity Analysis and Optimal Design of Elastic Hinges and Supports in Beam and Frame Structures. Mechanics of Structures and Machines. 1997. 25(4). Pp. 417-443. DOI: 10.1080/08905459708905297
57. Garstecki, A., Mrôz, Z. Optimal Design of Supports of Elastic Structures Subjected to Loads and Initial Distortions. Mechanics of Structures and Machines. 1987. 15(1). Pp. 47-68. DOI: 10.1080/08905458708905108
58. Grandhi, R.V. Structural optimization with frequency constraints. Proceedings of 4th Symposium on Multidisciplinary Analysis and Optimization. Cleveland, 1992. AIAA-92-4813. DOI: 10.2514/6.1992-4813
59. Kennedy, G.J., Fu, Y. Topology Optimization with Natural Frequency Constraints using a Quadratic Approximation of a Spectral Aggregate. Proceedings of AIAA Science and Technology Forum and Exposition. San Diego: American Institute of Aeronautics and Astronautics, 2022. DOI: 10.2514/6.2022-2244
60. Kesson, B., Olhoff, N. Minimum stiffness of optimally located supports for maximum value of beam eigenfrequencies. Journal of Sound and Vibration. 1988. 120(3). Pp. 457-463. DOI: 10.1016/S0022-460X(88)80218-9
61. Kobelev, V. Some exact analytical solutions in structural optimization. Mechanics based design of structures and machines. 2017. 45 (1). Pp. 43-61. DOI: 10.1080/15397734.2016.1143374
62. Leader, M.K., Chin, T.W., Kennedy, G.J. High-Resolution Topology Optimization with Stress and Natural Frequency Constraints. AIAA Journal. 2019. 57(8). Pp. 3562-3578. DOI: 10.2514/1.J057777
63. Mrôz, Z., Rozvany, G.I.N. Optimal design of structures with variable support conditions. Journal of Optimization Theory and Applications. 1975. 15 (1). Pp. 85-101. DOI: 10.1007/BF00933023
64. Olhoff, N. Optimal design with respect to structural eigenvalues. Proceedings of XVth Int. IUTAM Cong. Theoretical and Applied Mechanics. Amsterdam: North-Holland, 1980. Pp. 133-149.
65. Olhoff, N., Akesson, B. Minimum stiffness of optimally located supports for maximum value of column buckling loads. Structural Optimization. 1991. 3(3). Pp. 163-175. DOI: 10.1007/BF01743073
66. Rao, S.S. Engineering Optimization: Theory and Practice. New York: John Wiley & Sons, 2009. 848 p. DOI: 10.1002/9780470549124
67. Rozvany, G.I.N., Mroz, Z. Column Design: Optimization of Support Conditions and Segmentation. Journal of Structural Mechanics. 1977. 5(3). Pp. 279-290. DOI: 10.1080/03601217708907315
68. Sergeyev O.A., Kiselev V.G., Sergeyeva S.A. Optimal design of 3d frame structures taking into account the stress and multiple natural frequency constraints. Magazine of Civil Engineering. 2016. 61(1). Pp. 74-81. DOI: 10.5862/MCE.61.7
69. Son, J.H., Kwak, B.M. Optimization of Boundary Conditions for Maximum Fundamental Frequency of Vibrating Structures. AIAA Journal. 1993. 31(12). Pp. 2351-2357. DOI: 10.2514/3.11935
70. Studzinski, R., Pozorski, Z., Btaszczuk, J. Optimal Support System of Sandwich P|anels. Journal of Engineering Mechanics. 2015. 141(3). DOI: 10.1061/(ASCE)EM.1943-7889.0000857
71. Vanderplaats, G.N., Salajegheh, E. An efficient approximation technique for frequency constraints in frame optimization. International Journal for Numerical Methods in Engineering. 1988. 26(5). Pp. 1057-1069. DOI: 10.1002/nme.1620260505
72. Wang, B.P., Chen, J.L. Application of genetic algorithm for the support location optimization of beams. Computers and Structures. 1996. 58(4). Pp. 797-800. DOI: 10.1016/0045-7949(95)00184-I
73. Wang, C.Y. Minimum stiffness of an internal elastic support to maximize the fundamental frequency of a vibrating beam. Journal of Sound and Vibration. 2003. 259(1). Pp. 229-232. DOI: 10.1006/jsvi.2002.5100
74. Wang, D. Optimal Design of an Intermediate Support for a Beam with Elastically Restrained Boundaries. Journal of Vibration and Acoustics. Transactions of the ASME. 2011. 133(3). Article no. 031014. DOI: 10.1115/1.4003204
75. Won, K., Park, Y Optimal support positions for a structure to maximize its fundamental natural frequency. Journal of Sound and Vibration. 1998. 213(5). Pp. 801-812. DOI: 10.1006/jsvi.1997.1493
76. Zhu, J., Zhang, W. Maximization of structural natural frequency with optimal support layout. Structural and Multidisciplinary Optimization. 2006. 31(6). Pp. 462-469. DOI: 10.1007/s00158-005-0593-2
77. Amba-Rao, C.L. Effect of End Conditions on the Lateral Frequencies of Uniform Straight Columns. The Journal of the Acoustical Society of America. 1967. 42. Pp. 900-901. DOI: 10.1121/1.1910667
78. Galef, A.E. Bending Frequencies of Compressed Beams. Journal of the Acoustical Society of America. 1968. 44(2). P. 643. DOI: 10.1121/1.1911144
79. Bokaian, A. Natural frequencies of beams under compressive axial loads. Journal of Sound and Vibration. 1988. 126(1). Pp. 4965. DOI: 10.1016/0022-460X(88)90397-5
80. Bokaian, A. Author's reply. Journal of Sound and Vibration. 1989. 131(2). P. 351. DOI: 10.1016/0022-460X(89)90499-9
81. Bokaian, A. Natural frequencies of beams under tensile axial loads. Journal of Sound and Vibration. 1990. 142(3). Pp. 481-498. DOI: 10.1016/0022-460X(90)90663-K
82. Pilkington, D.F, Carr, J.B. Vibration of Beams Subjected to End and Axially Distributed Loading. Journal of Mechanical Engineering Science. 1970. 12(1). Pp. 70-72. DOI: 10.1243/JMES_JOUR_1970_012_012_02
83. Saiidi, M., Douglas, B., Feng, S. Prestress Force Effect on Vibration Frequency of Concrete Bridges. Journal of Structural Engineering (United States). 1994. 120(7). Pp. 2233-2241. DOI: 10.1061/(ASCE)0733-9445(1994)120:7(2233)
84. Kudryavtsev, I.V., Brungardt, M.V., Kudryavtseva, Y.M., Kolotov, A.V., Rabetskaya, O.I. Boundary revision of a beam model for a thin-walled waveguide at bending vibration. Paper presented at the Journal of Physics: Conference Series. 2021. 1889(2). Article no. 022109. DOI: 10.1088/1742-6596/1889/2/022109
85. Mityaev, A.E., Kudryavtsev, I.V., Khomutov, M.P., Brungardt, M.V., Rabetskaya, O.I. Mutual influence of the supports stiffness and the first natural frequency at bending vibrations of a spring-hinged beam. IOP Conference Series: Materials Science and Engineering. 2021. 1155. Article no. 012101. DOI: 10.1088/1757-899X/1155/1/012101
86. Seshu, P., Dhuri, K.D. Corrected Formulas for Natural Frequencies of Cantilever Beams under Uniform Axial Tension. AIAA Journal. 2007. 45(6). Pp. 1435-1438. DOI: 10.2514/1.28937
87. Han, S.M., Benaroya, H., Wei, T. Dynamics of transversely vibrating beams using four engineering theories. Journal of Sound and Vibration. 1999. 225(5). Pp. 935-988. DOI: 10.1006/jsvi.1999.2257
88. Stephen, N.G. Beam vibration under compressive axial load-upper and lower bound approximation. Journal of Sound and Vibration. 1989. 131(2). Pp. 345-350. DOI: 10.1016/0022-460X(89)90498-7
89. Valle, J., Fernández, D., Madrenas, J. Closed-form equation for natural frequencies of beams under full range of axial loads modeled with a spring-mass system. International Journal of Mechanical Sciences. 2019. 153-154, Pp. 380-390. DOI: 10.1016/j.ijmecsci.2019.02.014
90. Williams, F.W., Banerjee, J.R. Flexural vibration of axially loaded beams with linear or parabolic taper. Journal of Sound and Vibration. 1985. 99(1). Pp. 121-138. DOI: 10.1016/0022-460X(85)90449-3
Information about the authors:
Ilya Kudryavtsev,
ORCID: https://orcid.org/0000-0001-7933-3710
E-mail: [email protected]
Viktor Ivanov,
E-mail: [email protected]
Olga Rabetskaya,
E-mail: olga [email protected]
Alexander Mityaev,
E-mail: [email protected]
Received: 19.09.2021. Approved after reviewing: 09.04.2024. Accepted: 10.05.2024.
