# Linear modal analysis of L-shaped beam structures

Fotios Georgiades, Jerzy Warminski, Matthew P. Cartmell

Research output: Contribution to journalArticle

13 Citations (Scopus)

### Abstract

In this article a theoretical linear modal analysis of Euler–Bernoulli L-shaped beam structures is performed by solving two sets of coupled partial differential equations of motion. The first set, with two equations, corresponds to in-plane bending motions whilst the second set with four equations corresponds to out-of-plane motions with bending and torsion. The case is also shown of a single cantilever beam taking into account rotary inertia terms. At first for the case of examination of the results for the L-shaped beam structure, an individual modal analysis is presented for four selected beams which will be used for modelling an L-shaped beam structure; in order to investigate the influence of rotary inertia terms and shear effects. Then, a theoretical and numerical modal analysis is performed for four models of the L-shaped beam structure consisting of two sets of beams, in order to examine the effect of the orientation of the secondary beam (oriented in two ways) and also shear effects. The comparison of theoretical and finite element simulations shows a good agreement for both in-plane and out-of-plane motions, which validates the theoretical analysis. This work is essential to make progress with new investigations into the nonlinear equations for the L-shaped beam structures within Nonlinear Normal Mode theory.
Original language English 312-332 21 Mechanical Systems and Signal Processing 38 2 https://doi.org/10.1016/j.ymssp.2012.12.006 Published - 2013

### Fingerprint

Modal analysis
Cantilever beams
Nonlinear equations
Torsional stress
Partial differential equations
Equations of motion

### Keywords

• L-shaped beam structure
• elastic continua dynamics
• Euler–Bernoulli beams
• autoparametric structure

### Cite this

Georgiades, Fotios ; Warminski, Jerzy ; Cartmell, Matthew P. / Linear modal analysis of L-shaped beam structures. In: Mechanical Systems and Signal Processing. 2013 ; Vol. 38, No. 2. pp. 312-332.
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abstract = "In this article a theoretical linear modal analysis of Euler–Bernoulli L-shaped beam structures is performed by solving two sets of coupled partial differential equations of motion. The first set, with two equations, corresponds to in-plane bending motions whilst the second set with four equations corresponds to out-of-plane motions with bending and torsion. The case is also shown of a single cantilever beam taking into account rotary inertia terms. At first for the case of examination of the results for the L-shaped beam structure, an individual modal analysis is presented for four selected beams which will be used for modelling an L-shaped beam structure; in order to investigate the influence of rotary inertia terms and shear effects. Then, a theoretical and numerical modal analysis is performed for four models of the L-shaped beam structure consisting of two sets of beams, in order to examine the effect of the orientation of the secondary beam (oriented in two ways) and also shear effects. The comparison of theoretical and finite element simulations shows a good agreement for both in-plane and out-of-plane motions, which validates the theoretical analysis. This work is essential to make progress with new investigations into the nonlinear equations for the L-shaped beam structures within Nonlinear Normal Mode theory.",
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Linear modal analysis of L-shaped beam structures. / Georgiades, Fotios; Warminski, Jerzy; Cartmell, Matthew P.

In: Mechanical Systems and Signal Processing, Vol. 38, No. 2, 2013, p. 312-332.

Research output: Contribution to journalArticle

TY - JOUR

T1 - Linear modal analysis of L-shaped beam structures

AU - Warminski, Jerzy

AU - Cartmell, Matthew P.

PY - 2013

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N2 - In this article a theoretical linear modal analysis of Euler–Bernoulli L-shaped beam structures is performed by solving two sets of coupled partial differential equations of motion. The first set, with two equations, corresponds to in-plane bending motions whilst the second set with four equations corresponds to out-of-plane motions with bending and torsion. The case is also shown of a single cantilever beam taking into account rotary inertia terms. At first for the case of examination of the results for the L-shaped beam structure, an individual modal analysis is presented for four selected beams which will be used for modelling an L-shaped beam structure; in order to investigate the influence of rotary inertia terms and shear effects. Then, a theoretical and numerical modal analysis is performed for four models of the L-shaped beam structure consisting of two sets of beams, in order to examine the effect of the orientation of the secondary beam (oriented in two ways) and also shear effects. The comparison of theoretical and finite element simulations shows a good agreement for both in-plane and out-of-plane motions, which validates the theoretical analysis. This work is essential to make progress with new investigations into the nonlinear equations for the L-shaped beam structures within Nonlinear Normal Mode theory.

AB - In this article a theoretical linear modal analysis of Euler–Bernoulli L-shaped beam structures is performed by solving two sets of coupled partial differential equations of motion. The first set, with two equations, corresponds to in-plane bending motions whilst the second set with four equations corresponds to out-of-plane motions with bending and torsion. The case is also shown of a single cantilever beam taking into account rotary inertia terms. At first for the case of examination of the results for the L-shaped beam structure, an individual modal analysis is presented for four selected beams which will be used for modelling an L-shaped beam structure; in order to investigate the influence of rotary inertia terms and shear effects. Then, a theoretical and numerical modal analysis is performed for four models of the L-shaped beam structure consisting of two sets of beams, in order to examine the effect of the orientation of the secondary beam (oriented in two ways) and also shear effects. The comparison of theoretical and finite element simulations shows a good agreement for both in-plane and out-of-plane motions, which validates the theoretical analysis. This work is essential to make progress with new investigations into the nonlinear equations for the L-shaped beam structures within Nonlinear Normal Mode theory.

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KW - elastic continua dynamics

KW - Euler–Bernoulli beams

KW - autoparametric structure

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