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Research Papers

Optimal Lumped Mass Matrices by Minimization of Modal Errors for Beam Elements

[+] Author and Article Information
Zhanxuan Zuo

Key Lab of Structures Dynamic and Control,
Ministry of Education,
School of Civil Engineering,
Harbin Institute of Technology,
Heilongjiang,
Harbin 150090, China

Shuang Li

Assistant Professor
Key Lab of Structures Dynamic and Control,
Ministry of Education,
School of Civil Engineering,
Harbin Institute of Technology,
Heilongjiang,
Harbin 150090, China
e-mail: shuangli@hit.edu.cn

Changhai Zhai

Professor
Key Lab of Structures Dynamic and Control,
Ministry of Education,
School of Civil Engineering,
Harbin Institute of Technology,
Heilongjiang,
Harbin 150090, China
e-mail: zch-hit@hit.edu.cn

Lili Xie

Professor
Key Lab of Structures Dynamic and Control,
Ministry of Education,
School of Civil Engineering,
Harbin Institute of Technology,
Heilongjiang,
Harbin 150090, China

1Corresponding authors.

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF VIBRATION AND ACOUSTICS. Manuscript received March 26, 2013; final manuscript received November 22, 2013; published online January 16, 2014. Assoc. Editor: Dr. Corina Sandu.

J. Vib. Acoust 136(2), 021015 (Jan 16, 2014) (7 pages) Paper No: VIB-13-1093; doi: 10.1115/1.4026247 History: Received March 26, 2013; Revised November 22, 2013

This paper clarifies the reason why error occurs in the mass lumping procedure and presents a new approach to construct lumped mass matrices for Euler–Bernoulli beam elements, which contain both translational and rotational degrees of freedom. Lumped mass matrices provide the proper translational inertia but change the rotational inertia compared with the continuous mass representation. Therefore, the optimal lumped mass matrices are expressed through the adoption of a variable rotational inertia parameter to counterbalance the decreased or increased rotational inertia. The goal of this study is to propose lumped mass matrices to minimize the modal error for beam elements. The accuracy of the new mass matrices is validated by a number of numerical tests.

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References

Figures

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Fig. 1

(a) Continuous mass beam element, and (b) lumped mass beam element

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Fig. 2

Rigid body rotation of the beam elements: (a) continuous mass beam element, and (b) lumped mass beam element

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Fig. 3

Flexural rotation of the beam elements: (a) continuous mass beam element, and (b) lumped mass beam element

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Fig. 4

The first mode shapes: (a) cantilever beam, and (b) simply supported beam

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Fig. 5

First natural frequency error of a one element model: (a) rigid body rotation dominated beam, and (b) flexural rotation dominated beam

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Fig. 6

First three mode shapes of the rigid body rotation dominated beam [12]

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Fig. 7

First three natural frequency errors of a 10-element model

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Fig. 8

Error curves for different element numbers per wavelength for the rigid body rotation dominated beam

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Fig. 9

Frequency error versus the rotational inertia coefficient of different N for the flexural rotation dominated beam

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Fig. 10

Error curves for different element numbers per wavelength for the flexural rotation dominated beam

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Fig. 11

Frequency error versus the rotational inertia coefficient of different N for the rigid body rotation dominated beam

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Fig. 12

Change trend of the optimal rotational inertia coefficient versus the mode number

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Fig. 13

Frequency error versus the rotational inertia coefficient for N = 10

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