pith. sign in

arxiv: 2507.22147 · v4 · submitted 2025-07-29 · 🧮 math.OC

Frequency-Domain Analysis of the Euler-Bernoulli and Timoshenko Beams with Attached Masses

Alexander Zuyev , Julia Kalosha This is my paper

Pith reviewed 2026-05-19 02:08 UTC · model grok-4.3

classification 🧮 math.OC
keywords Euler-Bernoulli beamTimoshenko beamtransfer functionBode plotvariational principlefrequency-domain analysislocal dampingcontrol actuator
0
0 comments X

The pith

Transfer functions are obtained for Timoshenko and Euler-Bernoulli beams with attached masses and local damping via variational principles.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper derives the equations of motion for a simply supported beam equipped with a spring-loaded control actuator and local damping by applying Hamilton's variational principle while incorporating the relevant interface conditions. It then constructs transfer functions for both the Timoshenko model, which includes shear deformation and rotary inertia, and the Euler-Bernoulli model, using measurements from a point sensor as the system output. Comparative Bode plots illustrate how the frequency responses change with different output signal choices and damping coefficient values. A sympathetic reader would care because these models underpin vibration control in flexible mechanical systems, where accurate frequency-domain descriptions guide actuator and sensor placement decisions.

Core claim

By incorporating interface conditions for the lumped control actuator and local damping directly into Hamilton's variational principle, the equations of motion are obtained in state-space form for the beam-rigid body system. Transfer functions are then derived for the Timoshenko and Euler-Bernoulli models with point-sensor outputs, and comparative Bode plots are generated to examine the effects of output signal selection and damping coefficients on the frequency responses of the two models.

What carries the argument

The transfer function computed from the state-space realization obtained via Hamilton's variational principle after embedding the lumped actuator and local damping interface conditions.

Load-bearing premise

The interface conditions involving the lumped control actuator and local damping can be incorporated directly into the variational formulation without additional modeling approximations beyond those already present in the Euler-Bernoulli and Timoshenko theories.

What would settle it

An experiment that records the frequency response of a physical simply supported beam fitted with a spring-loaded actuator and local damping at a point sensor location and checks whether the measured magnitude and phase curves match the Bode plots computed from the derived transfer functions.

Figures

Figures reproduced from arXiv: 2507.22147 by Alexander Zuyev, Julia Kalosha.

Figure 2
Figure 2. Figure 2: Comparative Bode plots. Outputs y2(t) (top) and y˜2(t) (bottom). (a) |H2(2iπν)|, Timoshenko model. (b) |H˜2(2iπν)|, Euler–Bernoulli model. In [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 4
Figure 4. Figure 4: Magnitude Bode plots for systems with different damping coefficients. Outputs y2(t) (top) and y˜2(t) (bottom). (a) |H2(2iπν)|, Timoshenko model. (b) |H˜2(2iπν)|, Euler–Bernoulli model. 6 Conclusion We conclude from the numeric investigations that both the Timoshenko and Euler–Bernoulli models show good agreement in terms of the localization of transfer functions poles for relatively low frequencies. As obs… view at source ↗
read the original abstract

This work focuses on the frequency-domain modeling of a control system with a flexible beam and a rigid body. A simply supported beam is equipped with a spring-loaded control actuator and possesses local damping effect. Using Hamilton's variational principle, the equations of motion are derived in the state space form taking into account interface conditions involving lumped control and local damping. The transfer functions are obtained for the Timoshenko and Euler--Bernoulli beam models with the output measurements provided by a point sensor. Comparative Bode plots are presented for the two beam models with different choices of output signals and damping coefficients.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 2 minor

Summary. The manuscript derives state-space equations and transfer functions for a simply supported Euler-Bernoulli beam and Timoshenko beam equipped with a spring-loaded control actuator and local damping. Hamilton's variational principle is used to obtain the equations while incorporating interface conditions at the actuator location; transfer functions are formed with point-sensor outputs, and comparative Bode plots are shown for the two beam theories under varying output choices and damping coefficients.

Significance. If the interface conditions are correctly embedded, the work supplies analytical frequency-domain models that enable direct comparison of Euler-Bernoulli versus Timoshenko predictions for actuator-driven flexible structures. Such transfer functions are useful for subsequent controller synthesis in vibration-suppression applications.

major comments (1)
  1. [Derivation of equations of motion] The section deriving the equations of motion via Hamilton's principle: the incorporation of the point actuator and local damping is asserted to occur through interface conditions, yet the manuscript does not display the explicit jump conditions (discontinuity in shear force for Euler-Bernoulli or in shear and moment for Timoshenko) or the corresponding weak-form boundary terms that arise after integration by parts. Without these terms the resulting transfer functions omit or misrepresent the actuator contribution, which is load-bearing for the central claim that the Bode plots accurately reflect the closed-loop frequency response.
minor comments (2)
  1. [Numerical results] Figure captions for the Bode plots should explicitly state the numerical values of the damping coefficients and spring stiffness used in each curve.
  2. [Transfer-function derivation] Notation for the point-sensor output (e.g., displacement, velocity, or strain) should be introduced once and used consistently in the transfer-function definitions.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the thorough review and the recommendation for major revision. The comment on the derivation section is well-taken and highlights an opportunity to improve clarity. We address it point by point below and will revise the manuscript to incorporate the suggested details.

read point-by-point responses

  1. Referee: The section deriving the equations of motion via Hamilton's principle: the incorporation of the point actuator and local damping is asserted to occur through interface conditions, yet the manuscript does not display the explicit jump conditions (discontinuity in shear force for Euler-Bernoulli or in shear and moment for Timoshenko) or the corresponding weak-form boundary terms that arise after integration by parts. Without these terms the resulting transfer functions omit or misrepresent the actuator contribution, which is load-bearing for the central claim that the Bode plots accurately reflect the closed-loop frequency response.

    Authors: We agree that explicit display of the jump conditions and weak-form terms would strengthen the presentation. In our derivation, Hamilton's principle is applied to the total energy including the actuator spring potential and the local damping dissipation; the virtual work of the actuator force and damping force at the attachment point x = x_a naturally produces the interface conditions upon integration by parts. For the Euler-Bernoulli model this yields a jump in shear force equal to the actuator force (plus damping term), while the Timoshenko model produces corresponding jumps in both shear force and bending moment. To address the referee's concern directly, we will add a new subsection (or expanded paragraph) in the revised manuscript that explicitly states these jump conditions and shows the relevant boundary terms after integration by parts. This addition will confirm that the actuator contribution is correctly embedded in the state-space matrices and transfer functions used for the Bode plots. The numerical results themselves remain unchanged, as the interface conditions were already accounted for in the original formulation. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected in derivation chain

full rationale

The paper starts from Hamilton's variational principle as an external physical law, incorporates interface conditions for the actuator and damping into the formulation, derives the state-space equations, and obtains transfer functions for the Euler-Bernoulli and Timoshenko models. These steps constitute a direct modeling derivation without any reduction of predictions to fitted parameters from the same dataset, without self-definitional loops, and without load-bearing self-citations that substitute for independent justification. The comparative Bode plots follow as outputs of the derived models rather than inputs renamed as results. The derivation is therefore self-contained against standard external benchmarks in beam theory.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

The derivation rests on the applicability of Hamilton's variational principle to a distributed beam system with discrete interface conditions for the actuator and damper; no new physical constants or entities are introduced beyond the standard beam parameters.

free parameters (2)
  • damping coefficient
    Local damping effect is included as a modeling choice whose specific numerical value is varied in the Bode plots but not derived from first principles.
  • spring stiffness of actuator
    The control actuator is modeled as spring-loaded; its stiffness enters the interface conditions and is treated as a tunable parameter.
axioms (2)
  • domain assumption Hamilton's variational principle yields the correct equations of motion when interface conditions for lumped control and local damping are imposed.
    Invoked to derive the state-space form for both beam models.
  • standard math The simply supported boundary conditions and point-sensor output are compatible with the chosen beam theories.
    Used without further justification in the frequency-domain analysis.

pith-pipeline@v0.9.0 · 5624 in / 1410 out tokens · 29847 ms · 2026-05-19T02:08:20.133309+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

34 extracted references · 34 canonical work pages

  1. [1]

    A. M. A HMED and A. M. R IFAI , Euler–Bernoulli and Timoshenko beam theories analytical and numerical comprehensive revision, European Journal of Engineering and T echnology Research 6 no. 7 (2021), 20–32. https://doi.org/10.24018/ejeng.2021.6.7.2626

    work page doi:10.24018/ejeng.2021.6.7.2626 2021

  2. [2]

    T. E. A LBERTS , S. L. D ICKERSON , and W. J. B OOK , On the transfer function modeling of flexible structures with distributed damping, Robotics: Theory and Applications, ASME (1986), 23–30

    work page 1986

  3. [3]

    O. J. A LDRAIHEM , R. C. W ETHERHOLD , and T. S INGH , Distributed control of laminated beams: Timoshenko theory vs. Euler–Bernoulli theory, Journal of Intelligent Material Systems and Structures 8 no. 2 (1997), 149–

    work page 1997

  4. [4]

    https://doi.org/10.1177/1045389X9700800205

    work page doi:10.1177/1045389x9700800205

  5. [5]

    K. B ARTECKI , A general transfer function representation for a class of h yperbolic distributed parameter systems, International Journal of Applied Mathematics and Computer Science 23 no. 2 (2013), 291–307. https://doi.org/10.2478/amcs-2013-0022

    work page doi:10.2478/amcs-2013-0022 2013

  6. [6]

    B AUR, P

    U. B AUR, P. B ENNER , and L. F ENG , Model order reduction for linear and nonlinear systems: A system-theoretic perspective, Archives of Computational Methods in Engineering 21 (2014), 331–358. https://doi.org/10.1007/s11831-014-9111-2

    work page doi:10.1007/s11831-014-9111-2 2014

  7. [7]

    C URTAIN and K

    R. C URTAIN and K. M ORRIS , Transfer functions of distributed parameter systems: A tu torial, Automatica 45 no. 5 (2009), 1101–1116. https://doi.org/10.1016/j.automatica.2009.01.008

    work page doi:10.1016/j.automatica.2009.01.008 2009

  8. [8]

    D EMIR and H

    O. D EMIR and H. Ö ZBAY, On reduced order modeling of flexible structures from frequ ency response data, in 2014 European Control Conference (ECC) , IEEE, Strasbourg, France, June 2014, pp. 1133–1138. https://doi.org/10.1109/ECC.2014.6862456

    work page doi:10.1109/ecc.2014.6862456 2014

  9. [9]

    C. L. D YM and I. H. S HAMES , Solid Mechanics: A V ariational Approach, Augmented Editio n, Springer New Y ork, NY , 2013.https://doi.org/10.1007/978-1-4614-6034-3

    work page doi:10.1007/978-1-4614-6034-3 2013

  10. [10]

    E LISHAKOFF , Exact solution of Timoshenko–Ehrenfest equations, in Handbook on Timoshenko– Ehrenfest Beam and Uflyand–Mindlin Plate Theories , World Scientific Book, 2019, pp

    I. E LISHAKOFF , Exact solution of Timoshenko–Ehrenfest equations, in Handbook on Timoshenko– Ehrenfest Beam and Uflyand–Mindlin Plate Theories , World Scientific Book, 2019, pp. 107–138. https://doi.org/10.1142/9789813236523_0002

    work page doi:10.1142/9789813236523_0002 2019

  11. [11]

    J. M. G ERE and S. P. T IMOSHENKO , Mechanics of materials. 3rd Ed. , Chapman and Hall. Ltd., London. England, 1991

    work page 1991

  12. [12]

    G UIVER , H

    C. G UIVER , H. L OGEMANN , and M. R. O PMEER , Transfer functions of infinite-dimensional systems: positive realness and stabilization, Mathematics of Control, Signals, and Systems 29 no. 20 (2017), 1–61. https://doi.org/10.1007/s00498-017-0203-z

    work page doi:10.1007/s00498-017-0203-z 2017

  13. [13]

    J OVANOVIC and B

    M. J OVANOVIC and B. B AMIEH , A formula for frequency responses of distributed systems with one spatial variable, Systems Control Letters 55 no. 1 (2006), 27–37. https://doi.org/10.1016/j.sysconle.2005.04.014

    work page doi:10.1016/j.sysconle.2005.04.014 2006

  14. [14]

    K ALOSHA and A

    J. K ALOSHA and A. Z UYEV , Asymptotic stabilization of a flexible beam with attached m ass, Ukrainian Mathematical Journal 73 (2022), 1537–1550. https://doi.org/10.1007/s11253-022-02012-6

    work page doi:10.1007/s11253-022-02012-6 2022

  15. [15]

    K ALOSHA , A

    J. K ALOSHA , A. Z UYEV , and P. B ENNER , On the eigenvalue distribution for a beam with attached mas ses, in Stabilization of Distributed Parameter Systems: Design Me thods and Applications (G. S KLYAR and A. Z UYEV , eds.), SEMA SIMAI Springer Series 2, Springer International Publishing, Cham, 2021, pp. 43–56 . https://doi.org/10.1007/978-3-030-61742-4_3

    work page doi:10.1007/978-3-030-61742-4_3 2021

  16. [16]

    D. S. K ARACHALIOS , I. V. G OSEA , and A. C. A NTOULAS , The Loewner framework for system identification and reduction, in Model Order Reduction: V olume 1: System- and Data-Driven Me thods and Algorithms (P. BENNER , S. G RIVET -TALOCIA , A. Q UARTERONI , G. R OZZA , W. S CHILDERS , and L. M. S ILVEIRA , eds.), de Gruyter, 2021, pp. 181–228. https://doi....

    work page doi:10.1515/9783110498967-006 2021

  17. [17]

    M. A. K OÇ, Finite element and numerical vibration analysis of a Timos henko and Euler–Bernoulli beams traversed by a moving high-speed train, Journal of the Brazilian Society of Mechanical Sciences and Engineering 43 no. 165 (2021). https://doi.org/10.1007/s40430-021-02835-7

    work page doi:10.1007/s40430-021-02835-7 2021

  18. [18]

    L ENCI , F

    S. L ENCI , F. C LEMENTI , and G. R EGA , Comparing nonlinear free vibrations of timoshenko beams with mechanical or geometric curvature definition, Procedia IUTAM 20 (2017), 34–41. https://doi.org/10.1016/j.piutam.2017.03.006

    work page doi:10.1016/j.piutam.2017.03.006 2017

  19. [19]

    L ENZ and H

    K. L ENZ and H. Ö ZBAY, Analysis and robust control techniques for an ideal flexibl e beam, in Multidisciplinary Engineering Systems: Design and Optimi zation T echniques and their Applica- tion (C. L EONDES , ed.), Control and Dynamic Systems 57, Academic Press, 1993, pp. 369–421. https://doi.org/10.1016/B978-0-12-012757-3.50015-4 . 9 A PREPRINT - J ULY 2025

    work page doi:10.1016/b978-0-12-012757-3.50015-4 1993

  20. [20]

    J. L EW, C. H YDE , and M. W ADE , Damage detection of flexible beams using transfer function correlation, in Proceedings of the ASME 1999 Design Engineering T echnical C onferences, International Design Engineering T echnical Conferences and Computers and Information in Eng ineering Conference V olume 7A: 17th Biennial Conference on Mechanical Vibration a...

    work page doi:10.1115/detc99/vib-8373 1999

  21. [21]

    L IU and Q

    Z. L IU and Q. Z HANG , Stability and regularity of solution to the Ttimoshenko be am equation with local Kelvin–V oigt damping, SIAM Journal on Control and Optimization 56 no. 6 (2018), 3919–3947. https://doi.org/10.1137/17M1146506

    work page doi:10.1137/17m1146506 2018

  22. [22]

    G. G. G. L UESCHEN , L. A. B ERGMAN , and D. M. M CFARLAND , Green’s functions for uniform Timoshenko beams, Journal of sound and vibration 194 no. 1 (1996), 93–102. https://doi.org/10.1006/jsvi.1996.0346

    work page doi:10.1006/jsvi.1996.0346 1996

  23. [23]

    M ALTI , M

    R. M ALTI , M. R. R APAI ´C, and V. T URKULOV , A unified framework for exponential stability analysis of irrational transfer functions in the parametric space, Annual Reviews in Control 57 (2024), 100935. https://doi.org/10.1016/j.arcontrol.2024.100935

    work page doi:10.1016/j.arcontrol.2024.100935 2024

  24. [24]

    M EI and B

    C. M EI and B. R. M ACE, Wave reflection and transmission in Timoshenko beams and wa ve anal- ysis of Timoshenko beam structures, Journal of Vibration and Acoustics 127 no. 4 (2005), 382–394. https://doi.org/10.1115/1.1924647

    work page doi:10.1115/1.1924647 2005

  25. [25]

    P ARK , Transfer function methods to measure dynamic mechanical p roperties of complex structures, Journal of Sound and Vibration 288 no

    J. P ARK , Transfer function methods to measure dynamic mechanical p roperties of complex structures, Journal of Sound and Vibration 288 no. 1–2 (2005), 57–79. https://doi.org/10.1016/j.jsv.2004.12.019

    work page doi:10.1016/j.jsv.2004.12.019 2005

  26. [26]

    R UGE and C

    P. R UGE and C. B IRK , A comparison of infinite Timoshenko and Euler–Bernoulli be am models on Winkler foundation in the frequency- and time-domain, Journal of Sound and Vibration 304 no. 3–5 (2007), 932–947. https://doi.org/10.1016/j.jsv.2007.04.001

    work page doi:10.1016/j.jsv.2007.04.001 2007

  27. [27]

    S. P. T IMOSHENKO , LXVI. on the correction for shear of the differential equat ion for transverse vibrations of prismatic bars, The London, Edinburgh, and Dublin Philosophical Magazine a nd Journal of Science 41 no. 245 (1921), 744–746. https://doi.org/10.1080/14786442108636264

    work page doi:10.1080/14786442108636264 1921

  28. [28]

    S. P. T IMOSHENKO , D. H. Y OUNG , and W. W EAVER , Vibration Problems in Engineering, 4th Edition: Wiley, Chichester, 1974. https://doi.org/10.1017/S0001924000035041

    work page doi:10.1017/s0001924000035041 1974

  29. [29]

    R. W. T RAILL -NASH and A. R. C OLLAR , The effects of shear flexibility and rotatory inertia on the bending vibrations of beams, The Quarterly Journal of Mechanics and Applied Mathematics 6 no. 2 (1953), 186–222. https://doi.org/10.1093/qjmam/6.2.186

    work page doi:10.1093/qjmam/6.2.186 1953

  30. [30]

    C. M. W ANG , Timoshenko beam-bending solutions in terms of Euler–Bern oulli solutions, Journal of Engineering Mechanics 121 no. 6 (1995), 763–765. https://doi.org/10.1061/(ASCE)0733-9399(1995)121:6( 763)

    work page doi:10.1061/(asce)0733-9399(1995)121:6( 1995

  31. [31]

    Z HANG , D

    X. Z HANG , D. T HOMPSON , and X. S HENG , Differences between Euler–Bernoulli and Timoshenko beam formulations for calculating the effects of moving loads on a periodically supported beam, Journal of Sound and Vibration 481 (2020), 115432. https://doi.org/10.1016/j.jsv.2020.115432

    work page doi:10.1016/j.jsv.2020.115432 2020

  32. [32]

    Z HANG and B

    Y. Z HANG and B. Y ANG , Medium-frequency vibration analysis of Timoshenko beam s tructures, International Journal of Structural Stability and Dynamic s 20 no. 13 (2020), 2041009. https://doi.org/10.1142/S0219455420410096

    work page doi:10.1142/s0219455420410096 2020

  33. [33]

    Z UYEV and O

    A. Z UYEV and O. S AWODNY , Stabilization and observability of a rotating Timoshenko beam model, Mathematical Problems in Engineering no. 1 (2007), 1–19. https://doi.org/10.1155/2007/57238

    work page doi:10.1155/2007/57238 2007

  34. [34]

    Z UYEV and I

    A. Z UYEV and I. V. G OSEA , Approximating a flexible beam model in the Loewner frame- work, in 2023 European Control Conference (ECC) , IEEE, Bucharest, Romania, 2023, pp. 1–7. https://doi.org/10.23919/ECC57647.2023.10178203. 10

    work page doi:10.23919/ecc57647.2023.10178203 2023