Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Alexander Zuyev, Julia Kalosha

TL;DR

Problem: analyze the frequency-domain behavior of a damped flexible beam with an attached mass using both $Timoshenko$ and $Euler--Bernoulli$ theories. Approach: derive the two models via Hamilton's variational principle with a lumped mass and local damping, transform to the Laplace domain, and construct transfer functions $H_1(s)$ and $H_2(s)$ to compare sensor responses. Contributions: explicit frequency-domain representations for both models and a comparative study showing good low-frequency agreement and higher-frequency divergence due to shear effects, plus insights on damping impact near resonances. Significance: informs design and control of damped beam systems with attached masses and guides damping identification and sensor-actuator placement in engineering applications.

Abstract

This work is focused on the frequency-domain modeling of a simply supported flexible beam with an attached mass in the presence of dissipation. The considered system is equipped with a spring-loaded control actuator and possesses local damping effect. With the help of Hamilton's variational principle, the equations of motion are derived in the state space form with account of interface conditions involving lumped control and local damping. The transfer functions are obtained for Timoshenko and Euler--Bernoulli beam models with the output measurements provided by a sensor. Comparative Bode plots are presented for the two beam models with different choices of damping coefficients.

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

TL;DR

Problem: analyze the frequency-domain behavior of a damped flexible beam with an attached mass using both and theories. Approach: derive the two models via Hamilton's variational principle with a lumped mass and local damping, transform to the Laplace domain, and construct transfer functions and to compare sensor responses. Contributions: explicit frequency-domain representations for both models and a comparative study showing good low-frequency agreement and higher-frequency divergence due to shear effects, plus insights on damping impact near resonances. Significance: informs design and control of damped beam systems with attached masses and guides damping identification and sensor-actuator placement in engineering applications.

Abstract

This work is focused on the frequency-domain modeling of a simply supported flexible beam with an attached mass in the presence of dissipation. The considered system is equipped with a spring-loaded control actuator and possesses local damping effect. With the help of Hamilton's variational principle, the equations of motion are derived in the state space form with account of interface conditions involving lumped control and local damping. The transfer functions are obtained for Timoshenko and Euler--Bernoulli beam models with the output measurements provided by a sensor. Comparative Bode plots are presented for the two beam models with different choices of damping coefficients.

Paper Structure

This paper contains 6 sections, 2 theorems, 38 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Analytic Modeling of a Timoshenko Beam with a Rigid Body
  3. Transfer Function for the Damped Timoshenko Beam Model
  4. Damped Euler--Bernoulli Beam Model in the Frequency Domain
  5. Comparative Analysis of the Euler--Bernoulli and Timoshenko Beam Models in the Frequency Domain
  6. Conclusion

Key Result

Proposition 3.1

The transfer functions $H_1(s)$ and $H_2(s)$ of the control system TBeq1 -- TBic2 with outputs $y_1(t) = w(\ell_k,t)$ and $y_2(t) = \psi'(\ell_k,t)$, respectively, are given as follows: components $z_2(x,s)$, $z_4(x,s)$, $z_5(x,s)$ are defined in TBmxpz, and $M_{j4}^{-1}$, $j=\overline{1,4}$, are elements of the inverse matrix corresponding to TBmtrxM--TBmtrxMelm.

Figures (4)

  • Figure 1: Comparative Bode plots. Output $y_1(t)$.
  • Figure 2: Comparative Bode plots. Outputs $y_2(t)$ (top) and $\tilde{y}_2(t)$ (bottom).
  • Figure 3: Magnitude Bode plots for systems with different damping coefficients. Output $y_1(t)$.
  • Figure 4: Magnitude Bode plots for systems with different damping coefficients. Outputs $y_2(t)$ (top) and $\tilde{y}_2(t)$ (bottom).

Theorems & Definitions (2)

  • Proposition 3.1
  • Proposition 4.1