**4. Vibration and stability analyses**

46 Advances in Computational Stability Analysis

deformations, respectively.

(1)

quantities described henceforth are dimensionless.

deformation will be described in detail in Section 4.

(5b)

(4a) (5a)

(4b)

(2)

 <sup>2</sup> <sup>4</sup> *AA A <sup>d</sup> M Qx Fy ds*

The method of static analysis can be found in Chen and Ro (2010). In this section we introduce several deformation patterns of the constrained elastica. All the physical

The length of the elastica being pushed in through the opening is *l l* 1 , where *l* is the dimensionless length of the elastica between points A to B. Figure 2 shows the relation between the edge thrust *AF* and the length increment *l* . The height of the point constraint *h* is 0.03. The dashed and solid curves in this load-deflection diagram represent unstable and stable configurations, respectively. The method used in determining the stability of the static

(2)

(4a)

**Figure 2.** Load-deflection curves for *h*=0.03. The solid and dashed curves represent stable and unstable

The symmetric deformation before contact occurs is called deformation (1), whose locus starts at ( *AF* , *l* ) = (1,0) and ends at (0.99668, 0.0022188). The slope of this load-deflection curve is slightly negative. After the middle point C of the elastica touches the point constraint H, the deformation pattern initially remains symmetric, called deformation (2). The load-deflection curve of deformation (2) starts at ( *AF* , *l* )=(0.99668,0.0022188) and ends at (3.97314, 0.0026985). It is noted that the lower part of this load-deflection curve up to

(2)

(3)

(5b) (4b)

(5a)
