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

Alexander Zuyev; Julia Kalosha

arxiv: 2507.22147 · v5 · pith:4HZIYZ7Unew · 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

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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: 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: 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.

Desk Editor's Note private letter to a colleague

Paper gives side-by-side transfer functions and Bode plots for Euler-Bernoulli versus Timoshenko beams with a spring-loaded actuator and point damping, but the handling of interface conditions at the attachment point needs explicit verification.

read the letter

The paper derives transfer functions for a simply supported beam under a spring-loaded actuator and local damping, then compares the frequency responses of the Euler-Bernoulli and Timoshenko models through Bode plots for different sensor locations and damping values. It does this by applying Hamilton's principle to obtain the state-space form while including the lumped elements via interface conditions. The main practical output is the direct visual comparison of how the two models behave in the frequency domain under the same control setup. That comparison is the useful part. Engineers who pick between these models for vibration control or positioning tasks can see the differences in resonance and roll-off without running their own derivations. The variational steps follow the usual energy formulation and produce the expected transfer-function structure for point outputs. The plots illustrate the effect of increasing damping on peak heights, which lines up with standard expectations. The soft spot sits at the interface conditions. A concentrated actuator force or damper on a beam produces a discontinuity in shear (or moment for Euler-Bernoulli), and this must appear either as a Dirac term in the strong form or as an adjusted boundary term when the weak form is written. Adding only the potential energy of the spring without modifying the test-function space or writing the jump explicitly can leave the actuator contribution incomplete. The abstract states that interface conditions are taken into account, but the letter of the weak formulation would need to be checked to confirm the jumps are present. The numerical generation of the Bode plots also lacks detail on discretization or modal truncation, so high-frequency accuracy between the two models is not yet verified. This work is aimed at control engineers and applied mathematicians who already work with distributed-parameter beam models and need concrete frequency-domain data for actuator placement or sensor choice. It is not a theoretical advance on beam equations themselves, but the explicit comparison adds usable information. A serious referee should see it, mainly to confirm the interface terms and the numerical procedure behind the plots.

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)

[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)

[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.
[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

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

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

Core claim

What carries the argument

Load-bearing premise

What would settle it

discussion (0)