## 1. Introduction

The primary goal of the research in constrained elastica is to understand the behavior of a thin elastic strip under end thrust when it is subject to lateral constraints. It finds applications in a variety of practical problems, such as in compliant foil journal bearing, corrugated fiberboard, deep drilling, structural core sandwich panel, sheet forming, non-woven fabrics manufacturing, and stent deployment procedure. By assuming small deformation, Feodosyev (1977) included the problem of a buckled beam constrained by a pair of parallel walls as an exercise for a university strength and material course. Vaillette and Adams (1983) derived a critical axial compressive force an infinitely long constrained elastica can support. Adams and Benson (1986) studied the post-buckling behavior of an elastic plate in a rigid channel. Chateau and Nguyen (1991) considered the effect of dry friction on the buckling of a constrained elastica. Adan et al. (1994) showed that when a column with initial imperfection positioned at a distance from a plane wall is subject to compression, contact zones may develop leading to buckling mode transition. Domokos et al. (1997), Holms et al. (1999), and Chai (1998, 2002) investigated the planar buckling patterns of an elastica constrained inside a pair of parallel plane walls. It was observed that both point contact and line contact with the constraint walls are possible. Kuru et al. (2000) studied the buckling behavior of drilling pipes in directional wells. Roman and Pocheau (1999, 2002) used an elastica model to investigate the post-buckling response of bilaterally constrained thin plates subject to a prescribed height reduction. Chen and Li (2007) and Lu and Chen (2008) studied the deformation of a planar elastica inside a circular channel with clearance. Denoel and Detournay (2011) proposed an Eulerian formulation of the constrained elastica problem.

The emphasis of these studies was placed on the static deformations of the constrained elastica. Very often, multiple equilibria under a specified set of loading condition are possible. Since only stable equilibrium configurations can exist in practice, there is a need to determine the stability of each of these equilibria in order to predict the behavior of the constrained elastica as the external load varies. For an unconstrained elastica, vibration method is commonly used to determine its stability; see Perkins (1990), Patricio et al. (1998), Santillan, et al. (2006), and Chen and Lin (2008). This conventional method, however, becomes useless in the case of constrained elastica.

The difficulty of the conventional vibration method arises from the existence of unilateral constraints. A unilateral constraint is capable of exerting compressive force onto the structure, but not tension. Mathematically, this type of constraints can be represented by a set of inequality equations. This poses challenges in determining the critical states of the loaded structure. In order to overcome this difficulty, the conventional stability analysis needs modification. In this chapter, we introduce a vibration method which is capable of determining the stability of a constrained elastica once the equilibrium configuration is known. The key of solving the vibration problem in constrained elastica is to take into account the sliding between the elastica and the space-fixed unilateral constraint during vibration.

In this chapter, we consider the vibration of an elastica constrained by a space-fixed point constraint. This particular constrained elastica problem is used to demonstrate the vibration method which is suitable to analyze the stability of a structure under unilateral constraint. In Section 2, we describe the studied problem in detail. In Section 3, we describe the static load-deflection relation. In Section 4, we introduce the theoretical formulation of the vibration method. In Section 5 an imperfect system when the point constraint is not at the mid-span is analyzed. In Section 6, several conclusions are summarized.

## 2. Problem description

Figure 1 shows an inextensible elastic strip with the right end fully clamped at a point B. On the left hand side there is a straight channel with an opening at point A. The distance between points A and B is *L*. Part of the strip is allowed to slide without friction and clearance inside the channel. A longitudinal pushing force *xy*-coordinate system is fixed at point A. A point H fixed at position

The elastic strip is assumed to be straight and stress-free when

## 3. Load-deflection relation

The equilibrium equation at any point (*x*,*y*) of the buckled strip between points A and B, as shown in Figure 1, can be written as

*x,y*). *EI* is the flexural rigidity of the elastic strip. *s* is the length of the strip measured from point A. For convenience we introduce the following dimensionless parameters (with asterisks):

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 quantities described henceforth are dimensionless.

The length of the elastica being pushed in through the opening is*l* is the dimensionless length of the elastica between points A to B. Figure 2 shows the relation between the edge thrust *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 deformation will be described in detail in Section 4.

The symmetric deformation before contact occurs is called deformation (1), whose locus starts at (

At point (

The theoretical load-deflection curves shown in Figure 2 give us a mental picture how the elastica evolves as the pushing force

## 4. Vibration and stability analyses

### 4.1. Lagrangian and Eulerian descriptions

As mentioned above, the deformation patterns discussed in Section 3 may not necessarily be stable. If the deformation is unstable, then it can not be realized in practice. In order to study the vibration and stability properties of the elastica, we first derive the equations of motion of a small element *ds* supported by the point constraint, as shown in Figure 3.

The geometrical relations between *x*, *y*, and

The balance of moment and forces in the *x-* and *y*-directions results in

*x-* and *y*-directions. The moment-curvature relation of the Euler-Bernoulli beam model is

The readers are reminded that the functions*s* and *t* for clarity. These six equations can be called the Lagrangian version of the governing equations because a material element *ds* at location *s* is isolated as the free body. *s* may be called the Lagrangian coordinate of a point on the elastica. It is noted that *s*=0, *l* represent the material points at the left end, the contact point, and the right end, respectively, when the elastica is in equilibrium. During vibration, the elastica may “slide” on the point constraint. As a consequence, the contact point on the elastica may change from *s*=*x-* and *y-*component forces exerted by the point constraint on the elastica during vibration.

We denote the static solutions of Equations (3)-(8) as

*H* is the Heaviside step function. During vibration, the function

A variable with subscript “*d*” represents a small perturbation of its static counterpart with subscript “*e*.”

Figure 4 is a graphical interpretation of Equation (11). The solid step lines in Figure 4(a) represent

After defining a new variable

Equation (11) can be rewritten as

where*s* is replaced by

By noting that

By substituting Equations (13)-(18), together with the relations

into Equations (19)-(24) and ignoring the higher-order terms, we arrive at the following linear equations for the six functions

### 4.2. Boundary conditions

The exact boundary conditions at the fixed end B are

These boundary conditions can be linearized as follows. Take Equation (34) as an example. By using Equation (15), we can rewrite (34) into

Both

Similarly, the boundary conditions (35)-(36) can be linearized to

The boundary condition at the left end A is more complicated. We denote the material point on the strip right at the opening A of the channel as point A’ when the elastica is in equilibrium, as shown in Figure 5(a). Since the strip is under a constant pushing force at the left end, A’ will retreat into and protrude out of the channel when the elastica vibrates, as shown in Figures 5(b) and 5(c). We denote this small length of movement as

The condition of zero slope at opening A requires that

Following the similar linearization procedure as at point B, we can linearize boundary condition (42) to the form

Similarly, we can derive

Finally, Equations (43) and (44) may be combined as

The three equations (45)-(47) are the linearized boundary conditions at point A.

### 4.3. Constraint equations

When contact occurs, it is required that the elastica always passes through the point constraint. Mathematically, this condition can be written as

After using Equations (15)-(16), Equations (48)-(49) can be rewritten as

We also require that the dynamic reactive force must be always normal to the elastica at the point constraint, or mathematically,

After using Equations (17), (25)-(26) and neglecting higher-order terms, Equation (52) can be linearized to

Equations (50), (51), and (53) are the three constraint equations.

### 4.4. Solution method

In summary, the six linearized differential equations (28)-(33), six boundary conditions (38)-(40), (45)-(47), and three constraint equations (50)-(51) and (53) admit nontrivial solutions only when

A shooting method is used to solve for the characteristic value

Figure 6 shows the

The first two mode shapes when

It is noted that the geometric conditions (50)-(51) at the point constraint are exact. Therefore, the vibrating elastica always passes through the point constraint no matter how large the vibration amplitude is. On the other hand, the boundary conditions at points A and B used in the calculation have been linearized from the exact boundary conditions. Therefore, the mode shapes do not necessarily satisfy the exact boundary conditions at points A and B. This may become obvious when the amplitude of vibration is increased dramatically.

Figures 8(a) and 8(b) show the lowest

## 5. Analysis of an imperfect system

The point constraint H in Figure 1 is at the middle between the two ends A and B. In practice, it is very difficult to place the point constraint accurately at the center. Instead, it is almost inevitable that the point constraint may be off the center somewhat. Figure 10 shows the configuration when the point constraint H (black dot) is at a distance

In Figure 11 we describe the change of the load-deflection relation when *h* remains to be 0.03. Focus is placed on how the offset affects the symmetry-breaking bifurcation when deformation (2) branches into asymmetric deformations 4(a) and 4(b) in Figure 2. It is observed that the sharp corner at the bifurcation point degenerates into two smooth curves, called deformations 6(a) and 6(b) in Figure 11. Both deformations 6(a) and 6(b) are asymmetric. The tops of deformation 6(a) and 6(b) are to the left and right, respectively, of the point constraint H. Deformation 6(b) is always unstable. Deformation 6(a) for

Inspecting Figure 11 reveals something unusual about the degeneration of the symmetry-breaking bifurcation due to the offset of the point constraint. For the ideal case with

In order to answer this question, we plot the load-deflection curves when

Figure 13(b) shows another scenario when

## 6. Conclusions

In this chapter we introduce a vibration method which is suitable to analyze the stability of a constrained elastica. A planar elastica constrained by a space-fixed point constraint is used to demonstrate the method. Generally speaking, static analysis allows one to find all the possible equilibrium configurations of a constrained elastica. In order to predict how the elastica behaves in reality, the stability of these equilibrium configurations needs to be determined. The key of the vibration method is to take into account the sliding between the elastica and the unilateral constraint during vibration. In order to accomplish this, Eulerian coordinates are defined to specify the positions of the material points on the elastica. After transforming the governing equations and the boundary conditions from the Lagrangian description to the Eulerian one, the natural frequencies and the vibration mode shapes of the constrained elastica can be calculated. The vibration method is applied to an elastica constrained by a point constraint in this chapter. The same principles can be extended to other similar problems as well, for instance; multiple point constraints (Chen et al., 2010) and plane constraints (Ro et al., 2010).