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TECHNICAL PAPERS

# Experiment and Theory on the Nonlinear Vibration of a Shallow Arch Under Harmonic Excitation at the End

[+] Author and Article Information
Jen-San Chen

Department of Mechanical Engineering, National Taiwan University, Taipei, Taiwan 10617jschen@ccms.ntu.edu.tw

Cheng-Han Yang

Department of Mechanical Engineering, National Taiwan University, Taipei, Taiwan 10617

J. Appl. Mech 74(6), 1061-1070 (Nov 07, 2005) (10 pages) doi:10.1115/1.2165231 History: Received May 21, 2005; Revised November 07, 2005

## Abstract

In this paper we study, both theoretically and experimentally, the nonlinear vibration of a shallow arch with one end attached to an electro-mechanical shaker. In the experiment we generate harmonic magnetic force on the central core of the shaker by controlling the electric current flowing into the shaker. The end motion of the arch is in general not harmonic, especially when the amplitude of lateral vibration is large. In the case when the excitation frequency is close to the $nth$ natural frequency of the arch, we found that geometrical imperfection is the key for the $nth$ mode to be excited. Analytical formula relating the amplitude of the steady state response and the geometrical imperfection can be derived via a multiple scale analysis. In the case when the excitation frequency is close to two times of the $nth$ natural frequency two stable steady state responses can exist simultaneously. As a consequence jump phenomenon is observed when the excitation frequency sweeps upward. The effect of geometrical imperfection on the steady state response is minimal in this case. The multiple scale analysis not only predicts the amplitudes and phases of both the stable and unstable solutions, but also predicts analytically the frequency at which jump phenomenon occurs.

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## Figures

Figure 1

Schematic diagram of the experimental setup

Figure 2

Theoretical model of the arch-shaker assembly

Figure 3

Power spectrums of the arch itself (upper graph) and the arch-shaker assembly (lower graph)

Figure 4

Solid line is the measured lateral displacement history at the middle point of the arch after an initial displacement at the end of the arch-shaker assembly. Dashed line is the calculated response based on the estimated damping.

Figure 10

Time history of the end motion of the two states A and B in Fig. 7

Figure 5

Steady state amplitude profiles when the assembly is excited at (a) ω1=38Hz, (b) ω2=78Hz, (c) ω3=138Hz, and (d) ω4=216Hz. Closed dots represent experimental measurements. Solid and dashed lines are the theoretical predictions including and excluding geometrical imperfections, respectively.

Figure 6

(a) Amplitude and (b) phase of α4 as functions of the frequency deviation Δγ. The solid lines are from multiple scale analysis. The closed dots are from numerically integrating the complete equations of motion.

Figure 7

Steady state amplitude of component α2. “•” represents the measurement during the sweeping-down process. “×” represents the measurement during the sweeping-up process. Solid (stable) and dashed (unstable) lines are multiple scale predictions.

Figure 8

Amplitude profiles corresponding to the two steady states A and B in Fig. 7

Figure 9

Time history of the two states A and B in Fig. 7 measured at location 11cm from the midpoint

## Errata

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