**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.

**Figure 3.** The free body diagram of a small element *ds* constrained by the space-fixed point. The geometrical relations between *x*, *y*, and are

$$\frac{\partial \mathbf{x}(\mathbf{s},t)}{\partial \mathbf{s}} = \cos \theta(\mathbf{s},t) \tag{3}$$

Vibration Method in Stability Analysis of Planar Constrained Elastica 49

(8)

<sup>1</sup> ( ) *xe A xe F s F R Hs l* (9)

<sup>1</sup> ( ) *ye A ye F s Q R Hs l* (10)

<sup>1</sup> 1 1 ( ) sin *yd yd F s R Hs l t* (11)

*l* , and *l* represent the material

*l* to 1 1 *s l* , where

*l* to 1 1 *s l*

*yd F* . The step dashed curve in

<sup>2</sup> (,) 4 (,) *s t <sup>M</sup> s t*

The readers are reminded that the functions *x* , *y* , , *M* , *xF* , and *yF* in Equations (3)-(8) are dimensionless and are all written explicitly in terms of *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

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

<sup>1</sup> ( )*t* is a small number. This change of contact point is reflected in Equations (6) and (7). ( ) *R t <sup>x</sup>* and ( ) *R t <sup>y</sup>* are the *x-* and *y-*component forces exerted by the point constraint on the

We denote the static solutions of Equations (3)-(8) as ( ) *<sup>e</sup> x s* , ( ) *<sup>e</sup> y s* , ( ) *<sup>e</sup> s* , ( ) *M s <sup>e</sup>* , ( ) *xe F s* , and ( ) *ye F s* . It is assumed that these static solutions are known. In the case when contact occurs,

*H* is the Heaviside step function. During vibration, the function (,) *yF st* may be regarded as the superposition of ( ) *ye F s* in Equation (10) and a small harmonic perturbation, expressed

11 1 ( , ) ( ) ( ( ) ( )) *y ye ye F st F s R Hs l Hs l*

A variable with subscript "*d*" represents a small perturbation of its static counterpart with

Figure 4 is a graphical interpretation of Equation (11). The solid step lines in Figure 4(a) represent ( ) *ye F s* . After sliding occurs the cross-hatched area disappears and the contact

is represented by the first bracket on the right hand side of Equation (11). Figure 4(b) shows the superposition of reactive point force *Ryd* to the right of the new contact point. This action is represented by the second term in the second bracket on the right hand side of

point moves from 1 *s l* to 1 1 *s l* . This shift of the contact point from s= <sup>1</sup>

Equation (11). Finally, Figure 4(c) shows the superposition of ˆ

Figure 4(c) represents the final (,) *yF st* in Equation (11).

After defining a new variable as

*s*

coordinate of a point on the elastica. It is noted that *s*=0, 1

the relations between *xe F* , *ye F* and *AF* , *QA* are

mathematically as

subscript "*e*."

consequence, the contact point on the elastica may change from *s*= <sup>1</sup>

elastica during vibration. is the dimensionless Dirac delta function.

$$\frac{\partial \mathbf{y}(\mathbf{s},t)}{\partial \mathbf{s}} = \sin \theta(\mathbf{s},t) \tag{4}$$

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

$$\frac{\partial \mathcal{M}(\mathbf{s}, t)}{\partial \mathbf{s}} = F\_x(\mathbf{s}, t) \sin \theta(\mathbf{s}, t) - F\_y(\mathbf{s}, t) \cos \theta(\mathbf{s}, t) \tag{5}$$

$$\frac{\partial F\_{\mathbf{x}}(\mathbf{s},t)}{\partial \mathbf{s}} - R\_{\mathbf{x}}(t)\delta(s - l\_1 - \eta\_1(t)) = \frac{1}{4\pi^2} \frac{\partial^2 \mathbf{x}(\mathbf{s},t)}{\partial t^2} \tag{6}$$

$$\frac{\partial F\_y(s, t)}{\partial s} - R\_y(t)\delta(s - l\_1 - \eta\_1(t)) = \frac{1}{4\pi^2} \frac{\partial^2 y(s, t)}{\partial t^2} \tag{7}$$

(,) *xF st* and (,) *yF st* are the internal forces in the *x-* and *y*-directions. The moment-curvature relation of the Euler-Bernoulli beam model is

Vibration Method in Stability Analysis of Planar Constrained Elastica 49

$$\frac{\partial \Theta(\mathbf{s}, t)}{\partial \mathbf{s}} = 4\pi^2 M(\mathbf{s}, t) \tag{8}$$

The readers are reminded that the functions *x* , *y* , , *M* , *xF* , and *yF* in Equations (3)-(8) are dimensionless and are all written explicitly in terms of *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, 1 *l* , and *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*= <sup>1</sup> *l* to 1 1 *s l* , where <sup>1</sup> ( )*t* is a small number. This change of contact point is reflected in Equations (6) and (7). ( ) *R t <sup>x</sup>* and ( ) *R t <sup>y</sup>* are the *x-* and *y-*component forces exerted by the point constraint on the elastica during vibration. is the dimensionless Dirac delta function.

We denote the static solutions of Equations (3)-(8) as ( ) *<sup>e</sup> x s* , ( ) *<sup>e</sup> y s* , ( ) *<sup>e</sup> s* , ( ) *M s <sup>e</sup>* , ( ) *xe F s* , and ( ) *ye F s* . It is assumed that these static solutions are known. In the case when contact occurs, the relations between *xe F* , *ye F* and *AF* , *QA* are

$$F\_{\rm xc}(\mathbf{s}) = -F\_A + R\_{\rm xc}H(\mathbf{s} - l\_1) \tag{9}$$

$$F\_{ye}(\mathbf{s}) = \mathbf{Q}\_A + R\_{ye}H\left(\mathbf{s} - l\_1\right) \tag{10}$$

*H* is the Heaviside step function. During vibration, the function (,) *yF st* may be regarded as the superposition of ( ) *ye F s* in Equation (10) and a small harmonic perturbation, expressed mathematically as

$$F\_y(s, t) = F\_{ye}(s) + \left[ R\_{ye} \left( H(s - l\_1 - \eta\_1) - H(s - l\_1) \right) \right] +$$

$$\left[ F\_{yd}(s - \eta\_1) + R\_{yd} H \left( s - l\_1 - \eta\_1 \right) \right] \sin \alpha t \tag{11}$$

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 ( ) *ye F s* . After sliding occurs the cross-hatched area disappears and the contact point moves from 1 *s l* to 1 1 *s l* . This shift of the contact point from s= <sup>1</sup> *l* to 1 1 *s l* is represented by the first bracket on the right hand side of Equation (11). Figure 4(b) shows the superposition of reactive point force *Ryd* to the right of the new contact point. This action is represented by the second term in the second bracket on the right hand side of Equation (11). Finally, Figure 4(c) shows the superposition of ˆ *yd F* . The step dashed curve in Figure 4(c) represents the final (,) *yF st* in Equation (11).

After defining a new variable as

48 Advances in Computational Stability Analysis

**Figure 3.** The free body diagram of a small element *ds* constrained by the space-fixed point.

*s* 

*s* 

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

*x*

*y*

*s*

*x*

*y*

relation of the Euler-Bernoulli beam model is

(,) cos ( , ) *xst s t*

(,) sin ( , ) *yst s t*

(,) ( , )sin ( , ) ( , )cos ( , ) *x y Mst F st st F st st*

(,) *xF st* and (,) *yF st* are the internal forces in the *x-* and *y*-directions. The moment-curvature

*F st <sup>y</sup> s t Rt s l t s t*

(,) <sup>1</sup> (,) () () <sup>4</sup>

*F st xst Rt s l t s t*

(,) 1 (,) () () <sup>4</sup>

(3)

(4)

(5)

2

2

(6)

(7)

1 1 2 2

1 1 2 2

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

$$
\varepsilon = s - \eta\_{1'} \tag{12}
$$

Vibration Method in Stability Analysis of Planar Constrained Elastica 51

ˆ( , ) ( ) ( )sin *e d xt x x t* (15)

ˆ( , ) ( ) ( )sin *e d y ty y t* (16)

ˆ( , ) ( ) ( )sin *e d t t* (17)

<sup>ˆ</sup> ( , ) ( ) ( )sin *<sup>M</sup> e d tM M t* (18)

(19)

(20)

(21)

*t*

*t*

(23)

(24)

( ) sin *Rt R R t x xe xd* (25)

( ) sin *Rt R R t y ye yd* (26)

(22)

2

2

1 2 2

1 2 2

 defined in Equation (12) may be called the Eulerian coordinate of a point on the elastica. <sup>1</sup> *l* represents the point of the elastica passing through the point constraint at any instance during vibration. It can be a different material point at a different instant. The physical meaning of 1 *l* is like fixing a control window at the point constraint. Therefore,

ˆ(,) <sup>ˆ</sup> cos ( , ) *x t <sup>t</sup>*

ˆ(,) <sup>ˆ</sup> sin ( , ) *y t <sup>t</sup>* 

<sup>ˆ</sup> (,) ˆ ˆˆ ˆ ( , )sin ( , ) ( , )cos ( , ) *x y M t F t tF t t*

<sup>ˆ</sup> (,) <sup>1</sup> ˆ(,) ( ) <sup>4</sup>

*F t <sup>y</sup> <sup>t</sup> Rt l*

<sup>2</sup> ˆ(,) <sup>ˆ</sup> 4 (,) *<sup>t</sup> <sup>M</sup> <sup>t</sup>*

<sup>ˆ</sup> (,) 1 (,) <sup>ˆ</sup> ( ) <sup>4</sup>

*F t x t Rt l*

*x*

*y*

, the Lagrangian version of the governing equations (3)-(8) can now be

we call this type of description an Eulerian one.

*x*

*y*

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

By noting that 1

*s*

transformed into the Eulerian version as

<sup>ˆ</sup> (,) () *<sup>x</sup> xe FtF* <sup>1</sup> ( ) sin *xd xd F RH l t* (14)

Equation (11) can be rewritten as

$$\hat{F}\_y(\mathbf{c}, t) = F\_{ye}(\mathbf{c}) + \begin{bmatrix} F\_{yd}(\mathbf{c}) + R\_{yd}H(\mathbf{c} - l\_1) \end{bmatrix} \sin \alpha t \tag{13}$$

where 1 <sup>ˆ</sup> (,) ( ,) *y y FtF t* . Apparently, ˆ *yF* and *yF* are two different functions. It is noted that ( ) *ye F* is the static solution as obtained from the static analysis, except that the independent variable *s* is replaced by . Similarly, the other perturbed functions may be written as

**Figure 4.** Schematic diagram to demonstrate the perturbation of (,) *yF st* , refer to Equation (11).

Vibration Method in Stability Analysis of Planar Constrained Elastica 51

$$\hat{F}\_{\mathbf{x}}(\mathbf{z},t) = F\_{\mathbf{x}\mathbf{c}}(\mathbf{z}) + \left[F\_{\mathbf{x}d}(\mathbf{z}) + R\_{\mathbf{x}d}H(\mathbf{c} - l\_1)\right] \sin\alpha t \tag{14}$$

$$
\hat{\mathbf{x}}(\varepsilon, t) = \mathbf{x}\_e(\varepsilon) + \mathbf{x}\_d(\varepsilon)\sin\alpha t \tag{15}
$$

$$
\hat{y}(\varepsilon, t) = y\_e(\varepsilon) + y\_d(\varepsilon)\sin\alpha t \tag{16}
$$

$$
\hat{\boldsymbol{\Theta}}(\varepsilon, t) = \boldsymbol{\Theta}\_{\varepsilon}(\varepsilon) + \boldsymbol{\Theta}\_{d}(\varepsilon)\sin\alpha t \tag{17}
$$

$$
\hat{M}(\varepsilon, t) = M\_{\varepsilon}(\varepsilon) + M\_{d}(\varepsilon)\sin\alpha t \tag{18}
$$

 defined in Equation (12) may be called the Eulerian coordinate of a point on the elastica. <sup>1</sup> *l* represents the point of the elastica passing through the point constraint at any instance during vibration. It can be a different material point at a different instant. The physical meaning of 1 *l* is like fixing a control window at the point constraint. Therefore, we call this type of description an Eulerian one.

50 Advances in Computational Stability Analysis

Equation (11) can be rewritten as

written as

(a)

(b)

(c)

where 1 <sup>ˆ</sup> (,) ( ,) *y y FtF t* . Apparently, ˆ

<sup>1</sup> *s* , (12)

*yF* and *yF* are two different functions. It is noted

<sup>ˆ</sup> (,) () *y ye FtF* <sup>1</sup> ( ) sin *yd yd F RH l t* (13)

that ( ) *ye F* is the static solution as obtained from the static analysis, except that the independent variable *s* is replaced by . Similarly, the other perturbed functions may be

**Figure 4.** Schematic diagram to demonstrate the perturbation of (,) *yF st* , refer to Equation (11).

By noting that 1 *s* , the Lagrangian version of the governing equations (3)-(8) can now be transformed into the Eulerian version as

$$\frac{\partial \hat{\mathbf{x}}(\varepsilon, t)}{\partial \varepsilon} = \cos \hat{\theta}(\varepsilon, t) \tag{19}$$

$$\frac{\partial \hat{y}(\varepsilon, t)}{\partial \varepsilon} = \sin \hat{\theta}(\varepsilon, t) \tag{20}$$

$$\frac{\partial \hat{M}(\varepsilon, t)}{\partial \varepsilon} = \hat{F}\_x(\varepsilon, t) \sin \hat{\theta}(\varepsilon, t) - \hat{F}\_y(\varepsilon, t) \cos \hat{\theta}(\varepsilon, t) \tag{21}$$

$$\frac{\partial \hat{\vec{F}}\_{\text{x}}(\varepsilon, t)}{\partial \varepsilon} - R\_{\text{x}}(t) \delta \left(\varepsilon - l\_{1}\right) = \frac{1}{4\pi^{2}} \frac{\hat{\sigma}^{2} \hat{\vec{\chi}}(\varepsilon, t)}{\hat{\sigma} t^{2}} \tag{22}$$

$$\frac{\partial \hat{F}\_y(\varepsilon, t)}{\partial \varepsilon} - R\_y(t)\delta\left(\varepsilon - l\_1\right) = \frac{1}{4\pi^2} \frac{\hat{\sigma}^2 \hat{y}(\varepsilon, t)}{\hat{\sigma}t^2} \tag{23}$$

$$\frac{\partial \hat{\Theta}(\varepsilon, t)}{\partial \varepsilon} = 4\pi^2 \hat{M}(\varepsilon, t) \tag{24}$$

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

$$R\_{\chi}(t) = R\_{\chi t} + R\_{\chi d} \sin \alpha t \tag{25}$$

$$R\_y(t) = R\_{ye} + R\_{yd} \sin \alpha t \tag{26}$$

$$
\mathfrak{n}\_1(t) = \mathfrak{n}\_{1d} \sin \alpha t \tag{27}
$$

Vibration Method in Stability Analysis of Planar Constrained Elastica 53

<sup>1</sup> ( ) *d d <sup>l</sup> <sup>x</sup>* (38)

() 0 *<sup>d</sup> <sup>l</sup> <sup>y</sup>* (39)

*<sup>M</sup>* (40)

0 0 ( ) sin *<sup>d</sup> t t* (41)

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

2 <sup>1</sup> () 4 () *d ed <sup>l</sup> <sup>l</sup>*

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

**Figure 5.** The boundary conditions at the opening A of the feeding channel. (a) In equilibrium position, the material point A' coincides with point A. When the elastica vibrates, the material point A' (b)

> 0 0 1 <sup>ˆ</sup> (,) (,) 0 *<sup>s</sup> st t*

Following the similar linearization procedure as at point B, we can linearize boundary

2

(42)

0 1 <sup>0</sup> <sup>0</sup> () 4 () ( ) *d ed <sup>M</sup> <sup>d</sup>* (43)

retreats in and (c) protrudes out of the channel.

condition (42) to the form

The condition of zero slope at opening A requires that

shown in Figures 5(b) and 5(c). We denote this small length of movement as

into Equations (19)-(24) and ignoring the higher-order terms, we arrive at the following linear equations for the six functions ( ) *<sup>d</sup> x* , ( ) *<sup>d</sup> y* , ( ) *<sup>d</sup>* , ( ) *Md* , ( ) *xd F* , and ( ) *yd F* :

$$\frac{d\mathbf{x}\_d(\varepsilon)}{d\varepsilon} = -\theta\_d(\varepsilon)\sin\theta\_\varepsilon(\varepsilon)\tag{28}$$

$$\frac{d\boldsymbol{y}\_d(\varepsilon)}{d\varepsilon} = \theta\_d(\varepsilon) \cos \theta\_\varepsilon(\varepsilon) \tag{29}$$

$$\frac{d\theta\_d(\varepsilon)}{d\varepsilon} = 4\pi^2 M\_d(\varepsilon) \tag{30}$$

$$\frac{dM\_d(\varepsilon)}{d\varepsilon} = \left[F\_{\varepsilon\varepsilon}(\varepsilon)\theta\_d(\varepsilon) - F\_{yd}(\varepsilon) - R\_{yd}H(\varepsilon - l\_1)\right] \cos\theta\_\varepsilon(\varepsilon)$$

$$+ \left[F\_{y\varepsilon}(\varepsilon)\Theta\_d(\varepsilon) + F\_{xd}(\varepsilon) + R\_{xd}H(\varepsilon - l\_1)\right] \sin\Theta\_\varepsilon(\varepsilon)\tag{31}$$

$$\frac{dF\_{\rm xd}(\varepsilon)}{d\varepsilon} = -\frac{1}{4\pi^2} \alpha^2 \left[ \chi\_d(\varepsilon) - \cos\theta\_\varepsilon(\varepsilon)\eta\_{1d} \right] \tag{32}$$

$$\frac{dF\_{yd}(\varepsilon)}{d\varepsilon} = -\frac{1}{4\pi^2} \alpha^2 \left[ \mathcal{Y}\_d(\varepsilon) - \sin \theta\_\varepsilon(\varepsilon) \mathfrak{n}\_{1d} \right] \tag{33}$$

#### **4.2. Boundary conditions**

The exact boundary conditions at the fixed end B are

$$\left.\infty(\mathbf{s},t)\right|\_{\mathbf{s}=l} = \hat{\mathbf{x}}(\mathbf{s},t)\Big|\_{\mathbf{c}=l-\eta\_1} = 1\tag{34}$$

$$\left. \left. y(s, t) \right|\_{s=l} = \hat{y}(\varepsilon, t) \right|\_{\varepsilon=l-\eta\_1} = 0 \tag{35}$$

$$\left. \Theta(s, t) \right|\_{s=l} = \hat{\Theta}(\varepsilon, t) \Big|\_{\varepsilon=l-\eta\_1} = 0 \tag{36}$$

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

$$\left. \left( \infty \right) \right|\_{\varepsilon = l - \eta\_1} + \left. \varkappa\_d \left( \varepsilon \right) \right|\_{\varepsilon = l - \eta\_1} \sin \alpha t = 1 \tag{37}$$

Both 1 ( ) *<sup>e</sup> <sup>l</sup> <sup>x</sup>* and 1 ( ) *<sup>d</sup> <sup>l</sup> <sup>x</sup>* in Equation (37) can be expanded as a Taylor series with respect to *l* . After ignoring the higher-order terms, Equation (37) can be linearized to

Vibration Method in Stability Analysis of Planar Constrained Elastica 53

$$\left.\pi\_d(\varepsilon)\right|\_{\varepsilon=l} = \eta\_{1d} \tag{38}$$

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

52 Advances in Computational Stability Analysis

*d*

**4.2. Boundary conditions** 

Both 1 ( ) *<sup>e</sup> <sup>l</sup> <sup>x</sup>* 

*xd*

*d*

*yd*

*dF*

*dF*

The exact boundary conditions at the fixed end B are

By using Equation (15), we can rewrite (34) into

( ) *<sup>d</sup> <sup>l</sup> <sup>x</sup>*

and 1

1 1 ( ) sin *<sup>d</sup> t t* (27)

(28)

(29)

(30)

into Equations (19)-(24) and ignoring the higher-order terms, we arrive at the following

( ) ( )sin ( ) *<sup>d</sup> d e*

( ) ( )cos ( ) *<sup>d</sup> d e*

<sup>2</sup> ( ) 4 () *<sup>d</sup>*

( ) () () () cos ( ) *<sup>d</sup> xe d yd yd e*

*<sup>d</sup> <sup>M</sup>*

2

2

*<sup>y</sup> <sup>d</sup>*

( ) <sup>1</sup> ( ) cos ( ) <sup>4</sup>

( ) <sup>1</sup> ( ) sin ( ) <sup>4</sup>

<sup>ˆ</sup> (,) (,) 0 *s l <sup>l</sup> st t*

These boundary conditions can be linearized as follows. Take Equation (34) as an example.

1 1 ( ) ( ) sin 1 *e d l l xx t* 

respect to *l* . After ignoring the higher-order terms, Equation (37) can be linearized to

*x*

*d*

2 1

2 1

*d ed*

*d ed*

1

1

1

<sup>1</sup>

<sup>1</sup> () () () sin ( ) *ye d xd xd <sup>e</sup> F F RH l* (31)

(32)

(33)

(,) (,) 1 <sup>ˆ</sup> *s l <sup>l</sup> xst x t* (34)

(,) (,) 0 <sup>ˆ</sup> *s l <sup>l</sup> yst y t* (35)

(36)

(37)

in Equation (37) can be expanded as a Taylor series with

linear equations for the six functions ( ) *<sup>d</sup> x* , ( ) *<sup>d</sup> y* , ( ) *<sup>d</sup>* , ( ) *Md* , ( ) *xd F* , and ( ) *yd F* :

*dx d*

> *dy d*

*d*

*dM F F RH l*

$$\left. \left. y\_d(\mathfrak{e}) \right|\_{\mathfrak{e}=l} = 0 \right. \tag{39}$$

$$\left. \Theta\_d(\varepsilon) \right|\_{\varepsilon=l} = 4\pi^2 M\_e(\varepsilon) \Big|\_{\varepsilon=l} \eta\_{1d} \tag{40}$$

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

$$
\mathfrak{n}\_0(t) = \mathfrak{n}\_{0d} \sin \alpha t \tag{41}
$$

**Figure 5.** The boundary conditions at the opening A of the feeding channel. (a) In equilibrium position, the material point A' coincides with point A. When the elastica vibrates, the material point A' (b) retreats in and (c) protrudes out of the channel.

The condition of zero slope at opening A requires that

$$\left. \Theta(s, t) \right|\_{s=\eta\_0} = \hat{\Theta}(\varepsilon, t) \Big|\_{\varepsilon=\eta\_0 - \eta\_1} = 0 \tag{42}$$

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

$$\left. \Theta\_d(\varepsilon) \right|\_{\varepsilon=0} = -4\pi^2 M\_\varepsilon(\varepsilon) \Big|\_{\varepsilon=0} (\eta\_{0d} - \eta\_{1d}) \tag{43}$$

Similarly, we can derive

$$\left. \propto\_{d} (\varepsilon) \right|\_{\varepsilon=0} = - (\eta\_{0d} - \eta\_{1d}) \tag{44}$$

$$\left.y\_d(\varepsilon)\right|\_{\varepsilon=0} = 0\tag{45}$$

Vibration Method in Stability Analysis of Planar Constrained Elastica 55

only when is equal to an eigenvalue of the system of equations. The unknowns to be found are the six functions ( ) *<sup>d</sup> x* , ( ) *<sup>d</sup> y* , ( ) *<sup>d</sup>* , ( ) *Md* , ( ) *xd F* , ( ) *yd F* , the amplitude of sliding at the point constraint 1*<sup>d</sup>* , and the two dynamic constraint reactions *Rxd* and *Ryd* . It is noted that in Equations (32)-(33) only appears in the form of <sup>2</sup> . Therefore, if the characteristic value <sup>2</sup> is positive, the corresponding mode is stable with natural frequency

A shooting method is used to solve for the characteristic value <sup>2</sup> . Since the linear vibration mode shape is independent of the amplitude, we can set 0 () 1 *Md* . After guessing six variables 0 ( ) *<sup>d</sup> <sup>x</sup>* , 0 ( ) *yd <sup>F</sup>* , *Rxd* , *Ryd* , 1*<sup>d</sup>* , and <sup>2</sup> , we can integrate the homogeneous equations (28)-(33) like an initial value problem all the way from 0 to *l* . The three boundary conditions (38)-(40) at *l* and the three constraint equations (50)-(51) and (53) at <sup>1</sup> *l* are used to check the accuracy of the guesses. If the guesses are not satisfactory, a new set of guesses is adopted. The stability of the deformations in Figure 2 is determined in this manner. It is noted that sometimes the term 0 ( ) *Md* of a mode shape happens to be zero. In such a case, the assumption 0 () 1 *Md* will yield no solution. When this situation occurs, a different variable is set as 1 in the shooting method; for instance, 0 ( ) *<sup>d</sup> <sup>x</sup>* =1.

Figure 6 shows the <sup>2</sup> of the first two modes as functions of the end force *AF* for deformation (2). The <sup>2</sup> of the first mode becomes negative when *AF* reaches 2.03268, at which the symmetry-breaking bifurcation occurs. It is noted that the <sup>2</sup> of the second mode

**Figure 6.** <sup>2</sup> of the first two modes as functions of the end force *AF* for deformation (2).

. On the other hand, the equilibrium configuration is unstable if <sup>2</sup> is negative.

$$\left.F\_{xd}(\varepsilon)\right|\_{\varepsilon=0} = 0\tag{46}$$

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

$$\left. \Theta\_d(\mathfrak{e}) \right|\_{\mathfrak{e}=0} = 4\pi^2 M\_\mathfrak{e}(\mathfrak{e}) \Big|\_{\mathfrak{e}=0} \left. \varkappa\_d(\mathfrak{e}) \right|\_{\mathfrak{e}=0} \tag{47}$$

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

$$\left.\infty(\mathbf{s},t)\right|\_{\mathbf{s}=l\_1+\eta\_1} = \hat{\mathbf{x}}(\mathbf{c},t)\Big|\_{\mathbf{c}=l\_1} = 0.5\tag{48}$$

$$\left.y(s,t)\right|\_{s=l\_1+\eta\_1} = \hat{y}(\varepsilon,t)\Big|\_{\varepsilon=l\_1} = \hbar\tag{49}$$

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

$$\left.\infty\_d(\varepsilon)\right|\_{\varepsilon=l\_1} = 0\tag{50}$$

$$\left.y\_d(\mathfrak{e})\right|\_{\mathfrak{e}=l\_1} = 0\tag{51}$$

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

$$R\_x \cos \hat{\theta}(\varepsilon, t) + R\_y \sin \hat{\theta}(\varepsilon, t) = 0 \quad \text{at} \quad \varepsilon = l\_1 \tag{52}$$

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

$$
\left[\boldsymbol{R}\_{y\epsilon}\boldsymbol{\Theta}\_{d}(\mathbf{c}) + \boldsymbol{R}\_{xd}\right] \cos\Theta\_{\epsilon}(\mathbf{c}) + \left[-\boldsymbol{R}\_{\rm xe}\boldsymbol{\Theta}\_{d}(\mathbf{c}) + \boldsymbol{R}\_{yd}\right] \sin\Theta\_{\epsilon}(\mathbf{c}) = \mathbf{0} \quad \text{at} \quad \mathbf{c} = l\_{1} \tag{53}
$$

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 is equal to an eigenvalue of the system of equations. The unknowns to be found are the six functions ( ) *<sup>d</sup> x* , ( ) *<sup>d</sup> y* , ( ) *<sup>d</sup>* , ( ) *Md* , ( ) *xd F* , ( ) *yd F* , the amplitude of sliding at the point constraint 1*<sup>d</sup>* , and the two dynamic constraint reactions *Rxd* and *Ryd* . It is noted that in Equations (32)-(33) only appears in the form of <sup>2</sup> . Therefore, if the characteristic value <sup>2</sup> is positive, the corresponding mode is stable with natural frequency . On the other hand, the equilibrium configuration is unstable if <sup>2</sup> is negative.

54 Advances in Computational Stability Analysis

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

constraint. Mathematically, this condition can be written as

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

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

2

When contact occurs, it is required that the elastica always passes through the point

1 1 1 ( , ) ( , ) 0.5 <sup>ˆ</sup> *s l <sup>l</sup> xst x t*

1 1 1

1

1

We also require that the dynamic reactive force must be always normal to the elastica at the

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

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

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

0 1 <sup>0</sup> () ( ) *<sup>d</sup> d d <sup>x</sup>* (44)

<sup>0</sup> () 0 *<sup>d</sup> <sup>y</sup>* (45)

<sup>0</sup> () 0 *xd <sup>F</sup>* (46)

(48)

() 0 *<sup>d</sup> <sup>l</sup> <sup>x</sup>* (50)

() 0 *<sup>d</sup> <sup>l</sup> <sup>y</sup>* (51)

ˆ ˆ cos ( , ) sin ( , ) 0 *R tR t x y* at 1 *<sup>l</sup>* (52)

( ) cos ( ) ( ) sin ( ) 0 *RR RR ye d xd e xe d yd e* at 1 *<sup>l</sup>* (53)

(,) (,) <sup>ˆ</sup> *s l <sup>l</sup> <sup>y</sup> st y t h* (49)

0 0 <sup>0</sup> () 4 () () *d ed M x* (47)

Similarly, we can derive

**4.3. Constraint equations** 

point constraint, or mathematically,

linearized to

**4.4. Solution method** 

A shooting method is used to solve for the characteristic value <sup>2</sup> . Since the linear vibration mode shape is independent of the amplitude, we can set 0 () 1 *Md* . After guessing six variables 0 ( ) *<sup>d</sup> <sup>x</sup>* , 0 ( ) *yd <sup>F</sup>* , *Rxd* , *Ryd* , 1*<sup>d</sup>* , and <sup>2</sup> , we can integrate the homogeneous equations (28)-(33) like an initial value problem all the way from 0 to *l* . The three boundary conditions (38)-(40) at *l* and the three constraint equations (50)-(51) and (53) at <sup>1</sup> *l* are used to check the accuracy of the guesses. If the guesses are not satisfactory, a new set of guesses is adopted. The stability of the deformations in Figure 2 is determined in this manner. It is noted that sometimes the term 0 ( ) *Md* of a mode shape happens to be zero. In such a case, the assumption 0 () 1 *Md* will yield no solution. When this situation occurs, a different variable is set as 1 in the shooting method; for instance, 0 ( ) *<sup>d</sup> <sup>x</sup>* =1.

Figure 6 shows the <sup>2</sup> of the first two modes as functions of the end force *AF* for deformation (2). The <sup>2</sup> of the first mode becomes negative when *AF* reaches 2.03268, at which the symmetry-breaking bifurcation occurs. It is noted that the <sup>2</sup> of the second mode

**Figure 6.** <sup>2</sup> of the first two modes as functions of the end force *AF* for deformation (2).

becomes negative when *AF* reaches 3.97314. This happens to be the point at which deformation (3) begins to appear.

Vibration Method in Stability Analysis of Planar Constrained Elastica 57

mode shapes do not necessarily satisfy the exact boundary conditions at points A and B.

Figures 8(a) and 8(b) show the lowest <sup>2</sup> , i.e., <sup>2</sup> <sup>1</sup> , as a function of the length increment *<sup>l</sup>* for deformations (4a) and (5a), respectively. <sup>2</sup> 1 of deformation (4a) is always positive. On the other hand, <sup>2</sup> 1 of deformation (5a) becomes negative when *l* reaches 0.0031711. This is the point at which the load-deflection locus of deformation (5a) reaches its top in Figure 2. The mode shapes of deformations (4a) and (5a) are also depicted in the graphs. Similarly, the <sup>2</sup> 1 of deformations (4b) and (5b) are plotted in Figures 9(a) and 9(b). Since <sup>2</sup> 1 is always

**Figure 8.** <sup>2</sup> 1 as a function of the length increment *l* for deformations (a) (4a) and (b) (5a). The <sup>2</sup> 1 of

deformation (5a) becomes negative when *l* reaches 0.0031711.

This may become obvious when the amplitude of vibration is increased dramatically.

The first two mode shapes when *AF* =1.5 are depicted in Figure 7. The solid and dashed curves represent the static and the vibrating mode shapes of the constrained elastica, respectively. The first mode (a) is asymmetric and the second mode (b) is symmetric. To examine whether sliding at the contact point occurs we examine the values of 0*<sup>d</sup>* and <sup>1</sup>*<sup>d</sup>* of each mode. For the asymmetric mode we found that the ratio 0*<sup>d</sup>* / <sup>1</sup>*<sup>d</sup>* is approximately 1:400. This means that during vibration the protruding and retreating of the elastica at the channel opening A is negligible compared to the sliding at the contact point. In other words, the elastica length within the domain of interest is almost constant during vibration. This can also be observed from Figure 7(a). For the symmetric mode, we found that the ratio 0*<sup>d</sup>* / <sup>1</sup>*<sup>d</sup>* is approximately 2:1. Therefore, sliding at the contact point still occurs. From Figure 7(b) we can observe that the lengths of the elastica on both sides of the point constraint increase (or decrease during the other half of the period) the same amount. Since the elastica is inextensible, the protruding 0*<sup>d</sup>* at the channel opening has to be twice the amount of the sliding 1*<sup>d</sup>* at the contact point. It is noted that a vibration analysis of a constrained structure will cause an erroneous result if sliding at the constraint is neglected.

**Figure 7.** The first two mode shapes of the constrained elastica when *AF* =1.5.

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.

56 Advances in Computational Stability Analysis

deformation (3) begins to appear.

constraint is neglected.

becomes negative when *AF* reaches 3.97314. This happens to be the point at which

The first two mode shapes when *AF* =1.5 are depicted in Figure 7. The solid and dashed curves represent the static and the vibrating mode shapes of the constrained elastica, respectively. The first mode (a) is asymmetric and the second mode (b) is symmetric. To examine whether sliding at the contact point occurs we examine the values of 0*<sup>d</sup>* and <sup>1</sup>*<sup>d</sup>* of each mode. For the asymmetric mode we found that the ratio 0*<sup>d</sup>* / <sup>1</sup>*<sup>d</sup>* is approximately 1:400. This means that during vibration the protruding and retreating of the elastica at the channel opening A is negligible compared to the sliding at the contact point. In other words, the elastica length within the domain of interest is almost constant during vibration. This can also be observed from Figure 7(a). For the symmetric mode, we found that the ratio 0*<sup>d</sup>* / <sup>1</sup>*<sup>d</sup>* is approximately 2:1. Therefore, sliding at the contact point still occurs. From Figure 7(b) we can observe that the lengths of the elastica on both sides of the point constraint increase (or decrease during the other half of the period) the same amount. Since the elastica is inextensible, the protruding 0*<sup>d</sup>* at the channel opening has to be twice the amount of the sliding 1*<sup>d</sup>* at the contact point. It is noted that a vibration analysis of a constrained structure will cause an erroneous result if sliding at the

**Figure 7.** The first two mode shapes of the constrained elastica when *AF* =1.5.

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 Figures 8(a) and 8(b) show the lowest <sup>2</sup> , i.e., <sup>2</sup> <sup>1</sup> , as a function of the length increment *<sup>l</sup>* for deformations (4a) and (5a), respectively. <sup>2</sup> 1 of deformation (4a) is always positive. On the other hand, <sup>2</sup> 1 of deformation (5a) becomes negative when *l* reaches 0.0031711. This is the point at which the load-deflection locus of deformation (5a) reaches its top in Figure 2. The mode shapes of deformations (4a) and (5a) are also depicted in the graphs. Similarly, the <sup>2</sup> 1 of deformations (4b) and (5b) are plotted in Figures 9(a) and 9(b). Since <sup>2</sup> 1 is always

**Figure 8.** <sup>2</sup> 1 as a function of the length increment *l* for deformations (a) (4a) and (b) (5a). The <sup>2</sup> 1 of deformation (5a) becomes negative when *l* reaches 0.0031711.

Vibration Method in Stability Analysis of Planar Constrained Elastica 59

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 *H* to the left of the ideal center (open circle). If the point constraint is on the right, *H* is considered to be

**Figure 10.** The point constraint H (black dot) is at a distance *H* to the left of the ideal center (open

In Figure 11 we describe the change of the load-deflection relation when *H* is increased from 0 (ideal case) to 0.01 and 0.05. The height *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 *H* =0.01, on the other hand, is stable before point ( *AF* , *<sup>l</sup>* )=(1.974, 0.00552). Figure 12 shows the <sup>2</sup> 1 along the locus of deformation 6(a) when *H* =0.01. It is shown that <sup>2</sup> becomes negative when *<sup>l</sup>* =0.00552. For a larger *H* =0.05,

Inspecting Figure 11 reveals something unusual about the degeneration of the symmetrybreaking bifurcation due to the offset of the point constraint. For the ideal case with *H* =0, part of upper branch (deformations 4(a) and (5a)) is stable until it reaches a maximum. On the other hand, the lower branch (deformations 4(b) and (5b)) is always unstable. When *H* increases from 0 to 0.01, part of the lower branch is stable until it reaches a maximum at *l* =0.00552. On the other hand, the upper branch is always unstable. It is not clear how the

circle). If H is on the right, *H* is considered to be negative.

deformation 6(b) is stable throughout the range in Figure 11.

load-deflection curves evolve as *H* varies.

negative.

**Figure 9.** <sup>2</sup> 1 as a function of the length increment *l* for deformations (a) (4b) and (b) (5b). The <sup>2</sup> 1 of deformations (4b) and (5b) is always negative.

negative along the loci of deformations (4b) and (5b), we conclude that deformations (4b) and (5b) are unstable. From these stability analyses, it is concluded that after the symmetrybreaking bifurcation of deformation (2), the elastica branches to deformation 4(a) and continue to deform along locus (5a). After the *l* reaches 0.0031711, the elastica will jump to a remote self-contact configuration beyond the range of Figure 2.

### **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 *H* to the left of the ideal center (open circle). If the point constraint is on the right, *H* is considered to be negative.

58 Advances in Computational Stability Analysis

deformations (4b) and (5b) is always negative.

**5. Analysis of an imperfect system** 

a remote self-contact configuration beyond the range of Figure 2.

**Figure 9.** <sup>2</sup> 1 as a function of the length increment *l* for deformations (a) (4b) and (b) (5b). The <sup>2</sup> 1 of

negative along the loci of deformations (4b) and (5b), we conclude that deformations (4b) and (5b) are unstable. From these stability analyses, it is concluded that after the symmetrybreaking bifurcation of deformation (2), the elastica branches to deformation 4(a) and continue to deform along locus (5a). After the *l* reaches 0.0031711, the elastica will jump to

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

**Figure 10.** The point constraint H (black dot) is at a distance *H* to the left of the ideal center (open circle). If H is on the right, *H* is considered to be negative.

In Figure 11 we describe the change of the load-deflection relation when *H* is increased from 0 (ideal case) to 0.01 and 0.05. The height *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 *H* =0.01, on the other hand, is stable before point ( *AF* , *<sup>l</sup>* )=(1.974, 0.00552). Figure 12 shows the <sup>2</sup> 1 along the locus of deformation 6(a) when *H* =0.01. It is shown that <sup>2</sup> becomes negative when *<sup>l</sup>* =0.00552. For a larger *H* =0.05, deformation 6(b) is stable throughout the range in Figure 11.

Inspecting Figure 11 reveals something unusual about the degeneration of the symmetrybreaking bifurcation due to the offset of the point constraint. For the ideal case with *H* =0, part of upper branch (deformations 4(a) and (5a)) is stable until it reaches a maximum. On the other hand, the lower branch (deformations 4(b) and (5b)) is always unstable. When *H* increases from 0 to 0.01, part of the lower branch is stable until it reaches a maximum at *l* =0.00552. On the other hand, the upper branch is always unstable. It is not clear how the load-deflection curves evolve as *H* varies.

Vibration Method in Stability Analysis of Planar Constrained Elastica 61

In order to answer this question, we plot the load-deflection curves when *H* varies with smaller increment. Figure 13(a) shows the load-deflection curves when *H* =0, <sup>5</sup> 5 10 , and <sup>4</sup> 2 10 , respectively. For the case when *H* =0, part of the upper branch is stable, while the

**Figure 13.** Load-deflection curves for (a) *<sup>H</sup>* =0, <sup>5</sup> 5 10 , and <sup>4</sup> 2 10 ; (b) *H* =0, <sup>5</sup> 5 10 , and <sup>4</sup> 2 10 , respectively. The solid and dashed curves represent stable and unstable deformations,

respectively.

**Figure 11.** Load-deflection curves for *h*=0.03 and *H* =0 (ideal case), 0.01, and 0.05, respectively. The solid and dashed curves represent stable and unstable deformations, respectively.

**Figure 12.** <sup>2</sup> of the first mode as functions of *l* for deformation (6) in Figure 11.

In order to answer this question, we plot the load-deflection curves when *H* varies with smaller increment. Figure 13(a) shows the load-deflection curves when *H* =0, <sup>5</sup> 5 10 , and <sup>4</sup> 2 10 , respectively. For the case when *H* =0, part of the upper branch is stable, while the

60 Advances in Computational Stability Analysis

**Figure 11.** Load-deflection curves for *h*=0.03 and *H* =0 (ideal case), 0.01, and 0.05, respectively. The

solid and dashed curves represent stable and unstable deformations, respectively.

**Figure 12.** <sup>2</sup> of the first mode as functions of *l* for deformation (6) in Figure 11.

**Figure 13.** Load-deflection curves for (a) *<sup>H</sup>* =0, <sup>5</sup> 5 10 , and <sup>4</sup> 2 10 ; (b) *H* =0, <sup>5</sup> 5 10 , and <sup>4</sup> 2 10 , respectively. The solid and dashed curves represent stable and unstable deformations, respectively.

whole lower branch is unstable, as in the inset of Figure 2. When *H* increases by a small amount, the load-deflection curves degenerates into two branches veering away from each other. When *H* <sup>=</sup> <sup>5</sup> 5 10 , the stable range on the upper branch shrinks. When *H* continues to increase to <sup>4</sup> 2 10 , the stable range on the upper branch disappears altogether. For the lower branch, there exists a limit point. The locus with positive slope before the limit point is stable.

Vibration Method in Stability Analysis of Planar Constrained Elastica 63

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Figure 13(b) shows another scenario when *H* varies from 0 to <sup>5</sup> 5 10 , and <sup>4</sup> 2 10 . It is observed that for a negative *H* , the sharp corner degenerates into two smooth curves crossing each other. The one emerging from the lower part has a stable range which ends at the peak of the curve. The other curve emerging from the top is unstable all the way.
