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Research Papers

Complex Modal Analysis of a Nonmodally Damped Continuous Beam

[+] Author and Article Information
Xing Xing

Dynamics and Vibrations Research Lab,
Department of Mechanical Engineering,
Michigan State University,
East Lansing, MI 48823
e-mail: xingxing@msu.edu

Brian F. Feeny

Department of Mechanical Engineering,
Michigan State University,
East Lansing, MI 48823
e-mail: feeny@egr.msu.edu

Contributed by the Technical Committee on Vibration and Sound of ASME for publication in the JOURNAL OF VIBRATION AND ACOUSTICS. Manuscript received July 15, 2014; final manuscript received February 17, 2015; published online March 13, 2015. Assoc. Editor: Walter Lacarbonara.

J. Vib. Acoust 137(4), 041006 (Aug 01, 2015) (9 pages) Paper No: VIB-14-1260; doi: 10.1115/1.4029899 History: Received July 15, 2014; Revised February 17, 2015; Online March 13, 2015

This work represents an investigation of the complex modes of continuous vibration systems with nonmodal damping. As an example, a cantilevered beam with damping at the free end is studied. Assumed modes are applied to discretize the eigenvalue problem in state-variable form and then to obtain estimates of the true complex normal modes and frequencies. The finite element method (FEM) is also used to get the mass, stiffness, and damping matrices and further to solve a state-variable eigenvalue problem. A comparison between the complex modes and eigenvalues obtained from the assumed-mode analysis and the finite element analysis shows that the methods produce consistent results. The convergence behavior when using different assumed mode functions is investigated. The assumed-mode method is then used to study the effects of the end-damping coefficient on the estimated normal modes and modal damping. Most modes remain underdamped regardless of the end-damping coefficient. There is an optimal end-damping coefficient for vibration decay, which correlates with the maximum modal nonsynchronicity.

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Figures

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Fig. 1

Cantilevered beam with damper at the free end

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Fig. 5

Variation of the eigenvalues with increasing number n of assumed modes (uniform-beam modes) for the real mode pair

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Fig. 6

Variation of the eigenvalues with increasing number n of assumed modes defined as modified Duncan polynomials

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Fig. 7

Real (solid lines) and imaginary (dashed lines) parts of the displacement modal vectors generated from the FEM. In the top left, two real modes are grouped as a “real mode pair” (c = 50 kg/s).

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Fig. 2

Real (solid lines) and imaginary (dashed lines) parts of the sampled displacement modal functions generated from the assumed-mode method. In the top left, two real modes are grouped as a “real mode pair” (c = 50 kg/s).

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Fig. 3

Real and imaginary parts parameterized in x and plotted against each other in the complex plane (assumed-mode method shown with solid lines, and FEM shown with dots). The top left shows two modes of the real mode pair plotted against each other (c = 50 kg/s).

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Fig. 4

Variation of the eigenvalues with increasing number n of assumed modes (uniform-beam modes)

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Fig. 9

Variation of the nonmodal frequencies with increasing damping coefficient

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Fig. 10

Variation of the eigenvalues with increasing damping coefficient

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Fig. 11

Variation of the damping ratio with increasing damping coefficient

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Fig. 12

Variation of the nonsynchronicity index with increasing damping coefficient

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Fig. 8

Mode shapes from the assumed-mode method with varying damping coefficients. (When c = 1 kg/s, mode 1 is complex, so real modes 1* and 2* are real part and imaginary part of mode 1, respectively.) (a) Real parts and (b) imaginary parts except for the cases of the real mode pair (c = 50 and 1000 kg/s), in which the second real modes are plotted.

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