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Research Papers

# Equilibrium, Stability, and Dynamics of Rectangular Liquid-Filled Vessels

[+] Author and Article Information
Russell Trahan

Department of Aerospace Engineering, Texas A&M University, College Station, TX 77843rtrahan3@tamu.edu

Tamás Kalmár-Nagy

Department of Aerospace Engineering, Texas A&M University, College Station, TX 77843jcnd@kalmarnagy.com

This statement can be proved by considering the $h=0$, $w=2$ case. Equation 20 reduces to $tan3 θ−tan θ=0,$ which clearly has three real roots. By a continuity argument, Eq. 20 has three real roots in the region left of the double root curve.

$A>1$ would result in the liquid losing contact with the vessel.

J. Comput. Nonlinear Dynam 6(4), 041012 (May 20, 2011) (11 pages) doi:10.1115/1.4003915 History: Received May 28, 2010; Revised March 31, 2011; Published May 20, 2011; Online May 20, 2011

## Abstract

Here we focus on the stability and dynamic characteristics of a rectangular, liquid-filled vessel. The position vector of the center of gravity of the liquid volume is derived and used to express the equilibrium angles of the vessel. Analysis of the potential function determines the stability of these equilibria, and bifurcation diagrams are constructed to demonstrate the co-existence of several equilibrium configurations of the vessel. To validate the results, a vessel of rectangular cross section was built. The results of the experiments agree well with the theoretical predictions of stability. The dynamics of the unforced and forced systems with a threshold constraint is discussed in the context of the nonlinear Mathieu equation.

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## Figures

Figure 1

Vessel geometry

Figure 2

Liquid cross section geometry

Figure 3

Vessel cases

Figure 4

h−w−θ plot of θ≠0 equilibria

Figure 5

Number of solutions of expression 17

Figure 6

Number of real roots in Eq. 20

Figure 7

Number of solutions in Eq. 20 with constraint 21

Figure 8

Total number of equilibria for the vessel in terms of h and w. Solid lines demarcate regions with different numbers of equilibria. The shaded region refers to the validity domain of case 2.

Figure 9

Coexistence of equilibria for w=2, h=0.5

Figure 10

Bifurcation diagrams

Figure 11

Unfolding of the bifurcation diagrams near the cusp. Shaded region refers to case 2 equilibria existing.

Figure 12

Test bucket design and apparatus

Figure 13

Experimental results

Figure 14

Plots of p(θ) for w=2 and various h values

Figure 15

Unforced phase plot of the equation of motion for w=2.0

Figure 16

Example trajectories for h=0.55, w=2, ω=0.5, A=0.5, ϕ=0

Figure 17

Lifespan of trajectories for various initial conditions, amplitudes, and frequencies for h=0.55 and w=2. The color scale denotes the lifespan in time units.

Figure 18

Zooming in on the lifespan plot for ω=0.75, A=0.75, ϕ=0. The three initial conditions for (d) are marked on (c).

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