Closed-form dynamic stiffness (DS) formulations are developed for exact dynamic analysis of complicated damped beam structures with internal hinges and generalized boundary conditions (BCs). The damping effects are considered by introducing both the strain rate dependent viscous and the velocity-dependent viscous damping factors. A range of theories are adopted for the beam element, that is, the classical and Rayleigh-Love theories for the axial vibration and the Euler–Bernoulli and Timoshenko theories for the bending vibration. Based on the governing differential equations considering the damping effect, the elemental DS matrices are derived by substituting the exact shape functions into the generalized displacement and force BCs. Then, a hinge joint for a beam element is simulated by implementing an end release. Next, assembling the elemental DS matrices and introducing generalized BCs leads to the final DS matrix used for dynamic analysis of beam structures. The complex transcendental frequency equation of the beam structures considered herein is solved by a global complex roots and poles finding algorithm with a self-adaptive mesh generator. The proposed method is verified against the finite element method. The results obtained by using different theories are also compared. This research provides an exact tool for dynamic analysis of complicated damped beam structures.
AbramowitzMStegunIA (1968) Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Washington, DC: US Government Printing Office, Vol. 55.
2.
AdhikariSBhattacharyaS (2012) Dynamic analysis of wind turbine towers on flexible foundations. Shock and Vibration19: 37–56.
3.
AdhikariSKarličićDLiuX (2021) Dynamic stiffness of nonlocal damped nano-beams on elastic foundation. European Journal of Mechanics - A: Solids86: 104144.
4.
AlatiNFaillaGSantiniA (2014) Complex modal analysis of rods with viscous damping devices. Journal of Sound and Vibration333(7): 2130–2163.
5.
AranyLBhattacharyaSAdhikariS, et al. (2015) An analytical model to predict the natural frequency of offshore wind turbines on three-spring flexible foundations using two different beam models. Soil Dynamics and Earthquake Engineering74: 40–45.
6.
AsakuraJSakuraiTTadanoH, et al. (2009) A numerical method for nonlinear eigenvalue problems using contour integrals. JSIAM Letters1: 52–55.
7.
AustinAPKravanjaPTrefethenLN (2014) Numerical algorithms based on analytic function values at roots of Unity. SIAM Journal on Numerical Analysis52(4): 1795–1821.
8.
BagleyRLTorvikPJ (1983) Fractional calculus-a different approach to the analysis of viscoelastically damped structures. AIAA Journal21(5): 741–748.
9.
BanerjeeJR (2021) Frequency dependent mass and stiffness matrices of bar and beam elements and their equivalency with the dynamic stiffness matrix. Computers & Structures254: 106616.
10.
BanerjeeJRAnanthapuvirajahA (2019) An exact dynamic stiffness matrix for a beam incorporating rayleigh–love and Timoshenko theories. International Journal of Mechanical Sciences150: 337–347.
11.
BanerjeeJRPapkovSLiuX, et al. (2015) Dynamic stiffness matrix of a rectangular plate for the general case. Journal of Sound and Vibration342: 177–199.
12.
BanerjeeJRPapkovSOVoTP, et al. (2023) Dynamic stiffness formulation for a micro beam using timoshenko–ehrenfest and modified couple stress theories with applications. Journal of Vibration and Control29(1–2): 428–439.
13.
BozyigitB (2021) Seismic response of pile supported frames using the combination of dynamic stiffness approach and galerkin’s method. Engineering Structures244: 112822.
14.
CalvettiDMorigiSReichelL, et al. (2001) An iterative method with error estimators. Journal of Computational and Applied Mathematics127: 93–119.
15.
CannizzaroFFioreICaddemiS, et al. (2024) Exact closed-form dynamic stiffness matrix of damaged frames comprising Timoshenko–Ehrenfest beams. Journal of Vibration and Control31(1).
16.
ChenPYSivanY (2017) Robust location of optical fiber modes via the argument principle method. Computer Physics Communications214: 105–116.
17.
DelvesLLynessJ (1967) A numerical method for locating the zeros of an analytic function. Mathematics of Computation21(100): 543–560.
18.
Di MatteoA (2023) Dynamic response of beams excited by moving oscillators: approximate analytical solutions for general boundary conditions. Computers & Structures280: 106989.
19.
DoyleJF (2021) Wave Propagation in Structures. Cham: Springer.
20.
DuigouLPotier-FerryMPotier-FerryM (2003) Iterative algorithms for non-linear eigenvalue problems. Application to vibrations of viscoelastic shells. Computer Methods in Applied Mechanics and Engineering192(11–12): 1323–1335.
21.
DziedziewiczSWareckaMLechR, et al. (2023) Self-adaptive mesh generator for global complex roots and poles finding algorithm. IEEE Transactions on Microwave Theory and Techniques71(7): 2854–2863.
22.
ElyasiPNeyaBNFiroozjaeeAR (2023) Free vibration of viscoelastic nonlocally damped tapered axially functionally graded beams using the state-space formulation. Engineering Structures288: 116183.
23.
FaillaG (2016) An exact generalised function approach to frequency response analysis of beams and plane frames with the inclusion of viscoelastic damping. Journal of Sound and Vibration360: 171–202.
24.
GartiaAKChakravertyS (2024) Free vibration of bi-directional functionally graded nanobeams resting on winkler–pasternak foundations. Journal of Vibration Engineering & Technologies12: 1–17.
25.
GasdiaFMarshallRA (2021) A new longwave mode propagator for the earth–ionosphere waveguide. IEEE Transactions on Antennas and Propagation69(12): 8675–8688.
26.
GuoAZLCaoZZhangZ, et al. (2019) Frequency domain-based analysis of floor vibrations using the dynamic stiffness matrix method. Journal of Vibration and Control25(4): 763–776.
27.
HanFDanDChengW (2018a) An exact solution for dynamic analysis of a complex double-beam system. Composite Structures193: 295–305.
28.
HanFDanDChengW (2018b) Extension of dynamic stiffness method to complicated damped structures. Computers & Structures208: 143–150.
29.
HanFDanDChengW, et al. (2020) A novel analysis method for damping characteristic of a type of double-beam systems with viscoelastic layer. Applied Mathematical Modelling80: 911–928.
30.
HračovSNáprstekJ (2017) Approximate complex eigensolution of proportionally damped linear systems supplemented with a passive damper. Procedia Engineering199: 1677–1682.
31.
HullA (1994) A closed form solution of a longitudinal bar with a viscous boundary condition. Journal of Sound and Vibration169(1): 19–28.
32.
ImakuraAMorikuniKTakayasuA (2020) Verified partial eigenvalue computations using contour integrals for hermitian generalized eigenproblems. Journal of Computational and Applied Mathematics369: 112543.
33.
JafariAEzzatiMAtaiAA (2019) Static and free vibration analysis of timoshenko beam based on combined peridynamic-classical theory besides fem formulation. Computers & Structures213: 72–81.
34.
JasinskiMDziedziewiczSJozwickaM, et al. (2019) An improvement of global complex roots and poles finding algorithm for propagation and radiation problems. In: 2019 13th European Conference on Antennas and Propagation (EuCAP), Krakow, Poland, 31 March–5 April 2019, 1–5.
35.
KimJChangSP (2001) Dynamic stiffness matrix of an inclined cable. Engineering Structures23(12): 1614–1621.
36.
KoloušekV (1941) Anwendung des Gesetzes der virtuellen Verschiebungen und des Reziprozitätssatzes in der Stabwerksdynamik. Ingenieur-Archiv12(6): 363–370.
37.
KowalczykP (2018) Global complex roots and poles finding algorithm based on phase analysis for propagation and radiation problems. IEEE Transactions on Antennas and Propagation66(12): 7198–7205.
38.
KravanjaPBarelMV (1999) A derivative-free algorithm for computing zeros of analytic functions. Computing63: 69–91.
39.
KravanjaPBarelMVRagosO, et al. (2000) Zeal: a mathematical software package for computing zeros of analytic functions. Computer Physics Communications124: 212–232.
40.
LaoudBBenyoucefSBachiriA, et al. (2024) Impact of material distribution on the dynamic response of a bidirectional fg beam under general boundary conditions and supported by nonlinear substrate. Acta Mechanica235: 1–17.
41.
LeeU (2009) Spectral Element Method in Structural Dynamics. Hoboken, NJ: John Wiley & Sons.
42.
LeungAY (2012) Dynamic Stiffness and Substructures. Berlin: Springer Science & Business Media.
43.
LiHYinXWuW (2018) Dynamic stiffness formulation for in-plane and bending vibrations of plates with two opposite edges simply supported. Journal of Vibration and Control24(9): 1652–1669.
44.
LimJYouCDayyaniI (2020) Multi-objective topology optimization and structural analysis of periodic spaceframe structures. Materials & Design190: 108552.
45.
LingMChenSLiQ, et al. (2018) Dynamic stiffness matrix for free vibration analysis of flexure hinges based on non-uniform timoshenko beam. Journal of Sound and Vibration437: 40–52.
46.
LingMSongDZhangX, et al. (2022) Analysis and design of spatial compliant mechanisms using a 3-D dynamic stiffness model. Mechanism and Machine Theory168: 104581.
47.
LingMZhouHChenL (2023) Dynamic stiffness matrix with Timoshenko beam theory and linear frequency solution for use in compliant mechanisms. Journal of Mechanisms and Robotics15(6): 061002.
48.
LiuXLiuXZhouW (2020) An analytical spectral stiffness method for buckling of rectangular plates on winkler foundation subject to general boundary conditions. Applied Mathematical Modelling86: 36–53.
49.
LiuXZhaoYZhouW, et al. (2022) Dynamic stiffness method for exact longitudinal free vibration of rods and trusses using simple and advanced theories. Applied Mathematical Modelling104: 401–420.
50.
LiuXZhaoYLuT, et al. (2023) Closed-form dynamic stiffness formulation for modal and dynamic response analysis of pile group foundations. Computers and Geotechnics159: 105481.
51.
LiuXTangDLiuX (2025) An enhanced argument principle algorithm for exact complex transcendental eigenvalue analysis of damped structures. Journal of Sound and Vibration595: 118751.
52.
Mesa LedesmaFLValerioGRodríguez BerralR, et al. (2021) Simulation-assisted efficient computation of the dispersion diagram of periodic structures: a comprehensive overview with applications to filters, leaky-wave antennas and metasurfaces. IEEE Antennas and Propagation Magazine63(5): 33–45.
53.
MeylanMHGrossL (2003) A parallel algorithm to find the zeros of a complex analytic function. ANZIAM Journal44: E236–E254.
54.
MoitaJAraújoAMartinsP, et al. (2011) A finite element model for the analysis of viscoelastic sandwich structures. Computers & Structures89(21): 1874–1881.
55.
MukherjeeASarkarSBanerjeeA (2021) Nonlinear eigenvalue analysis for spectral element method. Computers & Structures242: 106367.
56.
MulMLameckiAGómez-GarcíaR, et al. (2021) Inverse nonlinear eigenvalue problem framework for the synthesis of coupled-resonator filters with nonresonant nodes and arbitrary frequency-variant reactive couplings. IEEE Transactions on Microwave Theory and Techniques69(12): 5203–5216.
57.
MullerDE (1956) A method for solving algebraic equations using an automatic computer. Mathematics of Computation10(56): 208–215.
58.
NakatsukasaYSèteOTrefethenLN (2018) The AAA algorithm for rational approximation. SIAM Journal on Scientific Computing40(3): A1494–A1522.
59.
OrtaAHKersemansMVan Den AbeeleK (2022) A comparative study for calculating dispersion curves in viscoelastic multi-layered plates. Composite Structures294: 115779.
60.
PinkertJR (1976) An exact method for finding the roots of a complex polynomial. ACM Transactions on Mathematical Software2: 351–363.
61.
PressWHTeukolskySAVetterlingWT, et al. (1996) Numerical Recipes in Fortran 90: The Art of Scientific Computing. 2nd edition. Cambridge: Cambridge University Press.
62.
ProinovPDIvanovSI (2019) Convergence analysis of Sakurai–Torii–Sugiura iterative method for simultaneous approximation of polynomial zeros. Journal of Computational and Applied Mathematics357: 56–70.
SakuraiTSugiuraH (2003) A projection method for generalized eigenvalue problems using numerical integration. Journal of Computational and Applied Mathematics159(1): 119–128.
65.
SorrentinoSMarchesielloSPiomboB (2003) A new analytical technique for vibration analysis of non-proportionally damped beams. Journal of Sound and Vibration265(4): 765–782.
66.
SorrentinoSFasanaAMarchesielloS (2007) Analysis of non-homogeneous Timoshenko beams with generalized damping distributions. Journal of Sound and Vibration304(3–5): 779–792.
67.
TaimaMSEl-SayedTAFarghalySH (2021) Free vibration analysis of multistepped nonlocal Bernoulli–Euler beams using dynamic stiffness matrix method. Journal of Vibration and Control27(7-8): 774–789.
68.
ThinhTINguyenMC (2016) Dynamic stiffness method for free vibration of composite cylindrical shells containing fluid. Applied Mathematical Modelling40(21): 9286–9301.
69.
WeiZLiHYinX, et al. (2022) Vibration transmission from a machine with three degree of freedoms to beam structures by dynamic stiffness method. Shock and Vibration2022(1): 1956518.
70.
WilliamsFWWittrickWH (1970) An automatic computational procedure for calculating natural frequencies of skeletal structures. International Journal of Mechanical Sciences12(9): 781–791.
71.
WittrickWHWilliamsFW (1971) A general algorithm for computing natural frequencies of elastic structures. Journal of Mechanics and applied mathematics, XXIV(September 1970).
72.
WuWYinXLiH, et al. (2018) Power flow analysis of built-up plate structures using the dynamic stiffness method. Journal of Vibration and Control24(13): 2815–2831.
73.
XuBDanDZouY, et al. (2019) Research on characteristic function for cable inverse analysis based on dynamic stiffness theory and its application. Engineering Structures194: 384–395.
74.
YauLBen-IsraelA (1998) The newton and Halley methods for complex roots. The American Mathematical Monthly105(9): 806–818.
75.
YinXWuWZhongK, et al. (2018) Dynamic stiffness formulation for the vibrations of stiffened plate structures with consideration of in-plane deformation. Journal of Vibration and Control24(20): 4825–4838.
76.
YingXKatzIN (1988) A reliable argument principle algorithm to find the number of zeros of an analytic function in a bounded domain. Numerische Mathematik53: 143–163.
77.
ZhouHLingMYinY, et al. (2023) Exact vibration solution for three versions of timoshenko beam theory: a unified dynamic stiffness matrix method. Journal of Vibration and Control30(21–22): 4931–4945.
78.
ZhouHLingMYinY (2024) Perspectives on the generalized modeling of six beam theories: a unified dynamic stiffness matrix. Thin-Walled Structures200: 111863.
