**2. Theory**

Figure 1 shows the rotating tapered beam considered in the present paper. The beam could have step jumps in cross section and rotates at speed . The *X* -axis coincides with the centroidal axis of the beam, the *Y* -axis is parallel with the axis of rotation and the *Z* -axis lies in the plane of rotation. *L* is the length of the beam, *Lk* is the length of the segment *k* and *Ld* is the length of the last segment of the beam. The displacement in the *Y* direction is denoted as *w* and the section rotation is denoted as . Only displacements in the *X Y* plane are taken into account and the Coriolis effects are not considered.

The centrifugal force of a beam element at a distance *R x k k* from the axis of rotation can be expressed as

$$d\overline{F}\_k = \overline{\eta}^2 \left( \overline{R}\_k + \overline{\mathbf{x}}\_k \right) dm \tag{1}$$

where *dm* = ( ) *A x dx kk k* is its mass, with *ρ* the mass density of material, and ( ) *A x k k* , is the cross-sectional area at *<sup>k</sup> x* . Figure 2. The centrifugal force ( ) *N x k k* generated by is

$$d\overline{N}\_k(\overline{\mathbf{x}}\_k) = \overline{\eta}^2 \rho \left(\overline{R}\_k + \overline{\mathbf{x}}\_k\right) A\_k(\overline{\mathbf{x}}\_k) d\overline{\mathbf{x}}\_k \tag{2}$$

The finite element method was used by (Hodges & Rutkowski, 1981). (Vinod et al., 2007) presented a study about spectral finite element formulation for a rotating beam subjected to small duration impact. (Gunda & Ganguli, 2008) developed a new beam finite element whose basis functions were obtained by the exact solution of the governing static homogenous differential equation of a stiff string, which resulted from an approximation in the rotating beam equation. (Singh et al., 2007) used the Genetic Programming to create an approximate model of rotating beams. (Gunda et al., 2007) introduced a low degree of freedom model for dynamic analysis of rotating tapered beams based on a numerically efficient superelement, developed using a combination of polynomials and Fourier series as shape functions. (Kumar & Ganguli, 2009) looked for rotating beams whose eigenpair, frequency and mode-shape, is the same as that of uniform non rotating beams for a particular mode. An interesting paper (Ganesh & Ganguli, 2011) presented physics based basis function for vibration analysis of high speed rotating beams using the finite element method. The basis function gave rise to shape functions which depend on position of the element in the beam, material, geometric properties and rotational speed of the beam.

The present study tries to provide not only solutions for practical engineering situations but they also may be useful as benchmark for comparing other numerical models. The proposed differential quadrature method, offers a useful and accurate procedure for the solution of linear and non linear partial differential equations. It was used by Bellman in the 1970's. He used this method to calculate the natural frequencies of transverse vibration of a rotating cantilever beam. (Bellman & Casti, 1971). Other authors have used the differential quadrature method and recognized it as an effective technique for solving this kind of problems, (Bert &

Numerical results are obtained for the natural frequencies of transverse vibration and the mode shapes of rotating beams considering the elastic restraints, with non uniform variation of the cross-sectional area. Some of those cases have also been solved using the finite

Figure 1 shows the rotating tapered beam considered in the present paper. The beam could

centroidal axis of the beam, the *Y* -axis is parallel with the axis of rotation and the *Z* -axis lies in the plane of rotation. *L* is the length of the beam, *Lk* is the length of the segment *k* and *Ld* is the length of the last segment of the beam. The displacement in the *Y* direction is

The centrifugal force of a beam element at a distance *R x k k* from the axis of rotation can be

<sup>2</sup> ( ) *k kk dF R x dm* 

<sup>2</sup> () ( ) () *kk k k kk k dN x R x A x dx*

*A x dx kk k* is its mass, with *ρ* the mass density of material, and ( ) *A x k k* , is the

. The *X* -axis coincides with the

. Only displacements in the *X Y*

(1)

is

(2)

Malik, 1996; Shu & Chen, 1999; Choi et al., 2000; Liu & Wu, 2001; Shu, 2000).

element method, and the sets of results are in excellent agreement.

plane are taken into account and the Coriolis effects are not considered.

cross-sectional area at *<sup>k</sup> x* . Figure 2. The centrifugal force ( ) *N x k k* generated by

 

have step jumps in cross section and rotates at speed

denoted as *w* and the section rotation is denoted as

**2. Theory** 

expressed as

where *dm* = ( ) 

The total axial force at the cross section located at *R x k k* is

$$\overline{N}\_{k}(\overline{\mathbf{x}}\_{k}) = \overline{\eta}^{2} \rho \prod\_{\overline{\mathbf{x}}\_{k}}^{\iota\_{\overline{k}}} (\overline{\mathcal{R}}\_{k} + \overline{\mathbf{x}}\_{k}) A\_{k}(\overline{\mathbf{x}}\_{k}) d\overline{\mathbf{x}}\_{k} + \overline{F}\_{k+1} = \overline{\eta}^{2} \rho \left( \overline{\mathcal{R}}\_{k} \bigwedge\_{\overline{\mathbf{x}}\_{k}}^{\iota\_{\overline{k}}} A\_{k}(\overline{\mathbf{x}}\_{k}) d\overline{\mathbf{x}}\_{k} + \int\_{\overline{\mathbf{x}}\_{k}}^{\iota\_{\overline{k}}} A\_{k}(\overline{\mathbf{x}}\_{k}) \overline{\mathbf{x}}\_{k} d\overline{\mathbf{x}}\_{k} \right) + \overline{F}\_{k+1} \tag{3}$$

*Fk*1 is the outboard force at the end of the segment *k* , due to the adjacent segments *k*+1 to *d*.

Fig. 1. Rotating beam model

Fig. 2. Rotating beam segment *k* of length *Lk*

Finally, the tensile force can be written as

$$\overline{N}\_{k}(\overline{\mathbf{x}}\_{k}) = \overline{\eta}^{2} \rho \left( \overline{R}\_{k} \, V\_{k}(L\_{k}) + \Phi\_{k}(L\_{k}) - \overline{R}\_{k} \, V\_{k}(\overline{\mathbf{x}}\_{k}) - \Phi\_{k}(\overline{\mathbf{x}}\_{k}) \right) + \overline{F}\_{k+1} \tag{4}$$

with

Free Vibration Analysis of Centrifugally

*k k k k*

*k k*

*dx dx*

The term <sup>2</sup> () () *kk kk* 

 

and for the last segment *k*=*d*, at *d d x L* :

<sup>2</sup> (0) ;; ; (0) *kk k*

> ( ) ( ) (0) *k k*

1 <sup>1</sup> ; (0) *k*

*k*

*k A x*

( ) () ; (0)

*k*

*k*

*<sup>F</sup> <sup>N</sup> EA* 

*k*

*a x*

*kk k k kk xL I <sup>L</sup> xl r s*

*L LA r* ; ;

*<sup>A</sup>* ; ( ) ( ) (0)

*k*

*b x*

*N x N x*

Stiffened Non Uniform Timoshenko Beams 295

2 2

2 2

2

 

(13a,b) <sup>2</sup>

.

2

*k k k k*

*k k*

*Ix x* included in equation (13.b) was introduced by Banerjee, 2001.

*dx dx*

2 2

*kk kk kk kk*

 

<sup>1</sup> ( ) (0) 0 *WL W kk k* ; 1 ( ) (0) 0 *kk k L* (14a,b)

<sup>1</sup> ( ) (0) 0 *QL Q kk k* ; 1 ( ) (0) 0 *ML M kk k* (15a,b)

1 1 <sup>1</sup> (0) 0 0 *<sup>W</sup> Q KW* ; 1 1 <sup>1</sup> *M K* (0) (0) 0 (16a,b)

*d d* () 0 0 *<sup>d</sup> Wd QL K W* ; ( ) (0) 0 *ML K dd d d* (17a,b)

*k k*

*k k V x*

*k k*

*k*

;

*l A* ; 2

*k*

*Q x Q x EA* () ( ) (0)

*x*

*<sup>L</sup>* ; *k k k k kk k W x W x x x L*

;

( ) ( ) (0) *k k*

*k k*

*k k k k k <sup>L</sup> <sup>M</sup> x Mx*

*EI* ;

*l A*

*x*

;

 2 1 4 2 1 (0) (0) *<sup>A</sup> <sup>L</sup> EI* 

 

( ) ( ) ( ) () ()

() () () () () ()

Replacing equations (11) into equations (12), the differential equations of motion become:

() () ( ) () () ( ) ( )

*kk kk kk*

*dW x d x GA x x EI x*

*dI x d x <sup>E</sup> Ix x Ix x*

*kk kk k k kk kk k k k k k k k k*

*dN x dW x dW x dW x d x N x GA x dx dx dx dx dx*

*k k kk kk*

This term generates more realistic results especially for high rotational speeds, <sup>2</sup>

The conditions for displacements and forces between adjacent segments, *k* and *k*+1, are:

The boundary conditions of the beam at its ends are, for the first segment *k*=1, at <sup>1</sup> *x* 0 :

( ) ( ) () () ()

*dA x dW x <sup>G</sup> x Ax Wx*

 

*kk kk*

*k k*

Figure 3 shows the beam elastically restrained at both ends.

The four spring constants are denoted as: 1 1 *W Wd* ,,, *K KKK <sup>d</sup>* .

The expressions and parameters in dimensionless form are defined as follows:

(0) ; (0) *<sup>A</sup> <sup>L</sup> EI* 

*k k*

*k I x*

*k k*

*k*

*k <sup>R</sup> <sup>R</sup>*

*k*

*<sup>I</sup>* ; ( ) ( ) (0)

*v x*

*k*

*EA* ( ) () ; (0)

*k*

2 1 4 2 1

*dx dx*

$$V\_k(\overline{\mathbf{x}}\_k) = \bigcap\_{0}^{\overline{\mathbf{x}}\_k} A\_k(\overline{\mathbf{x}}\_k) \, d\overline{\mathbf{x}}\_k \; ; \; \Phi\_k(\overline{\mathbf{x}}\_k) = \int\_0^{\overline{\mathbf{x}}\_k} A\_k(\overline{\mathbf{x}}\_k) \, \overline{\mathbf{x}}\_k \, d\overline{\mathbf{x}}\_k \tag{5a,b}$$

The expressions for shear force and bending moment at an instant *t* in the rotating beam are

$$\sqrt{Q}\_{k}(\overline{\mathbf{x}}\_{k},t) = \overline{N}\_{k}(\overline{\mathbf{x}}\_{k})\frac{\partial \overline{w}\_{k}(\overline{\mathbf{x}}\_{k},t)}{\partial \overline{\mathbf{x}}\_{k}} + \kappa \,\mathrm{GA}\_{k}(\overline{\mathbf{x}}\_{k}) \left(\frac{\partial \overline{w}\_{k}(\overline{\mathbf{x}}\_{k},t)}{\partial \overline{\mathbf{x}}\_{k}} - \overline{\nu}\_{k}(\overline{\mathbf{x}}\_{k},t)\right) \tag{6}$$

$$i\overline{\mathcal{M}}\_k^\*(\overline{\mathbf{x}}\_{k'}t) = EI\_k(\overline{\mathbf{x}}\_k) \frac{\partial \overline{\nu}\_k(\overline{\mathbf{x}}\_{k'}t)}{\partial \overline{\mathbf{x}}\_k} \tag{7}$$

where ( ) *k k I x* is the second moment of area of the beam cross-section; *t* the time; ( ,) *w xt <sup>k</sup>* the transverse displacement; ( ,) *<sup>k</sup> x t* the section rotation; *E* the Young's modulus; the Poisson's ratio; *G E* / 2(1 ) the shear modulus and is the shear factor.

The governing differential equations of motion of a rotating Timoshenko beams (Banerjee, 2001) are:

$$\frac{\partial \overline{\mathcal{Q}}\_{k}^{\*}(\overline{\mathbf{x}}\_{k},t)}{\partial \overline{\mathbf{x}}\_{k}} = \rho A\_{k}(\overline{\mathbf{x}}\_{k}) \frac{\partial^{2} \overline{w}\_{k}(\overline{\mathbf{x}}\_{k},t)}{\partial t^{2}}$$

$$\rho \overline{\mathcal{Q}}\_{k}^{\*}(\overline{\mathbf{x}}\_{k},t) - N\_{k}(\overline{\mathbf{x}}\_{k}) \frac{\partial \overline{w}\_{k}(\overline{\mathbf{x}}\_{k},t)}{\partial \overline{\mathbf{x}}\_{k}} + \frac{\partial \overline{M}\_{k}^{\*}(\overline{\mathbf{x}}\_{k},t)}{\partial \overline{\mathbf{x}}\_{k}} + \rho I\_{k}(\overline{\mathbf{x}}\_{k}) \overline{\rho}^{2} \overline{\rho} \overline{\nu}\_{k}(\overline{\mathbf{x}}\_{k},t) = \rho I\_{k}(\overline{\mathbf{x}}\_{k}) \frac{\partial^{2} \overline{\nu}\_{k}(\overline{\mathbf{x}}\_{k},t)}{\partial t^{2}}$$

Assuming simple harmonic oscillation

$$
\overline{w}\_k \left( \overline{\mathbf{x}}\_{k'} t \right) = \overline{\mathcal{V}}\_k \left( \overline{\mathbf{x}}\_k \right) e^{i \alpha t} \quad ; \quad \overline{\varphi}\_k \left( \overline{\mathbf{x}}\_{k'} t \right) = \overline{\Psi}\_k \left( \overline{\mathbf{x}}\_k \right) e^{i \alpha t} \tag{9a,b}
$$

where is the circular frequency in radian per second. The bending moment and the shear force are expressed as

$$
\overline{Q}\_k^\*(\overline{\mathbf{x}}\_k, t) = \overline{Q}\_k(\overline{\mathbf{x}}\_k)e^{i\alpha t} \; ; \; \overline{M}\_k^\*(\overline{\mathbf{x}}\_k, t) = \overline{M}\_k(\overline{\mathbf{x}}\_k)e^{i\alpha t} \tag{10a, b}
$$

where

$$\overline{Q}\_{k}(\overline{\mathbf{x}}\_{k}) = \left(\overline{\mathbf{N}}\_{k}(\overline{\mathbf{x}}\_{k}) + \kappa \operatorname{GA}\_{k}(\overline{\mathbf{x}}\_{k})\right) \frac{d\overline{\mathcal{W}}\_{k}(\overline{\mathbf{x}}\_{k})}{d\overline{\mathbf{x}}\_{k}} - \kappa \operatorname{GA}\_{k}(\overline{\mathbf{x}}\_{k}) \overline{\Psi}\_{k}(\overline{\mathbf{x}}\_{k}) \\ ; \overline{M}\_{k}(\overline{\mathbf{x}}\_{k}) = EI\_{k}(\overline{\mathbf{x}}\_{k}) \frac{d\overline{\Psi}\_{k}(\overline{\mathbf{x}}\_{k})}{d\overline{\mathbf{x}}\_{k}} \tag{11a.b}$$

Substituting equations (9-10) into equations (8), the equations of motion for the free vibration of the segment *k* of the rotating beam result in:

$$\begin{aligned} -\frac{d\overline{Q}\_k(\overline{\mathbf{x}}\_k)}{d\overline{\mathbf{x}}\_k} &= \rho A\_k(\overline{\mathbf{x}}\_k) \, o^2 \overline{\mathcal{W}}\_k(\overline{\mathbf{x}}\_k) \\ -\overline{Q}\_k(\overline{\mathbf{x}}\_k) + \overline{N}\_k(\overline{\mathbf{x}}\_k) \frac{d\overline{\mathcal{W}}\_k(\overline{\mathbf{x}}\_k)}{d\overline{\mathbf{x}}\_k} - \frac{d\overline{\mathcal{M}}\_k(\overline{\mathbf{x}}\_k)}{d\overline{\mathbf{x}}\_k} - \rho I\_k(\overline{\mathbf{x}}\_k) \overline{\eta}^2 \, \overline{\Psi}\_k(\overline{\mathbf{x}}\_k) &= \rho I\_k(\overline{\mathbf{x}}\_k) \, o^2 \, \overline{\Psi}\_k(\overline{\mathbf{x}}\_k) \end{aligned} \tag{12a,b}$$

The expressions for shear force and bending moment at an instant *t* in the rotating beam are

\* ( ,) ( ,) ( ,) ( ) ( ) ( ,) *k k k k kk kk k k k k*

\* ( ,) ( ,) ( ) *k k*

where ( ) *k k I x* is the second moment of area of the beam cross-section; *t* the time; ( ,) *w xt <sup>k</sup>* the

The governing differential equations of motion of a rotating Timoshenko beams (Banerjee,

( ,) ( ,) ( )

*k k k k k k*

*Qxt wxt A x x t*

( ,) ( ,) ( ,) ( ,) ( ) ( ) ( ,) ( )

; ( ,) ( ) *i t*

*wxt Mxt x t Qxt Nx Ix xt Ix*

the shear modulus and

\* 2

; \*

> 

Substituting equations (9-10) into equations (8), the equations of motion for the free

*dQ x Ax Wx*

( ) () ()

() () () () () () () ()

*kk kk kk kk kk kk*

*dW x dM x Qx Nx Ix x Ix x*

2

*kk kk*

 

*kk kk kk kk kk*

*x t M x t EI x*

*wxt wxt Qxt Nx GA x x t*

*kk kk*

*k k*

*x x* 

0 () () *k x*

*k*

*x t* the section rotation; *E* the Young's modulus;

2 \* 2

*kk kk xt xe*

( ,) ( ) *i t Mkk kk xt Mxe*

*kk kk k k*

 

*x x t*

*x*

*kk kkk k x A x x dx* (5a,b)

 

(7)

is the shear factor.

 

*kk kk*

2 2

   

*d x M x EI x*

; ( ) () () *k k*

(9a,b)

(10a,b)

(6)

the

(8a,b)

2

*k*

(12a,b)

*dx* (11a,b)

0 () () *k x V x A x dx kk kk k* ;

transverse displacement; ( ,) *<sup>k</sup>*

Assuming simple harmonic oscillation

Poisson's ratio; *G E* / 2(1 )

2001) are:

where 

where

\* 2

( ,) ( ) *i t wxt Wxe kk kk*

( ,) ( ) *i t Qxt Qx e kk kk*

*k*

*k k*

*k kk kk*

*dx*

*k k*

*dx dx*

*dx* 

 is the circular frequency in radian per second. The bending moment and the shear force are expressed as

 ( ) () () () () () *k k kk kk kk kk kk*

*dW x Q x N x GA x GA x x*

vibration of the segment *k* of the rotating beam result in:

\*

*k*

*k k*

Replacing equations (11) into equations (12), the differential equations of motion become:

2 2 2 2 2 () () ( ) () () ( ) ( ) ( ) ( ) () () () *kk kk k k kk kk k k k k k k k k k k k k kk kk kk k k dN x dW x dW x dW x d x N x GA x dx dx dx dx dx dA x dW x <sup>G</sup> x Ax Wx dx dx* (13a,b) <sup>2</sup> 2 2 2 ( ) ( ) ( ) () () () () () () () () *k k k k k k kk kk k k kk kk kk kk kk kk k k dW x d x GA x x EI x dx dx dI x d x <sup>E</sup> Ix x Ix x dx dx* 

The term <sup>2</sup> () () *kk kk Ix x* included in equation (13.b) was introduced by Banerjee, 2001. This term generates more realistic results especially for high rotational speeds, <sup>2</sup> .

The conditions for displacements and forces between adjacent segments, *k* and *k*+1, are:

$$
\overline{\mathcal{V}}\_k(L\_k) - \overline{\mathcal{V}}\_{k+1}(0) = 0 \; ; \quad \overline{\Psi}\_k(L\_k) - \overline{\Psi}\_{k+1}(0) = 0 \tag{14a,b}
$$

$$
\overline{Q}\_k(L\_k) - \overline{Q}\_{k+1}(0) = 0 \; ; \quad \overline{M}\_k(L\_k) - \overline{M}\_{k+1}(0) = 0 \tag{15a,b}
$$

Figure 3 shows the beam elastically restrained at both ends.

The boundary conditions of the beam at its ends are, for the first segment *k*=1, at <sup>1</sup> *x* 0 :

$$
\overline{Q}\_1(0) - \overline{K}\_{\,\,\,\nu 1} \cdot \overline{W}\_1(0) = 0 \\
\vdots \,\,\, \overline{M}\_1(0) - \overline{K}\_{\,\,\,\,\nu 1} \cdot \overline{\Psi}\_1(0) = 0 \tag{16a,b}
$$

and for the last segment *k*=*d*, at *d d x L* :

$$
\overline{Q}\_d(L\_d) - \overline{K}\_{\mathbb{W}d} \quad \overline{W}\_d(0) = 0 \\
\vdots \quad \overline{M}\_d(L\_d) - \overline{K}\_{\mathbb{W}d} \ \overline{\Psi}\_d(0) = 0 \tag{17a,b}
$$

The four spring constants are denoted as: 1 1 *W Wd* ,,, *K KKK <sup>d</sup>* . The expressions and parameters in dimensionless form are defined as follows:

2 1 4 2 1 (0) ; (0) *<sup>A</sup> <sup>L</sup> EI* 2 1 4 2 1 (0) (0) *<sup>A</sup> <sup>L</sup> EI* ; <sup>2</sup> (0) ;; ; (0) *kk k kk k k kk xL I <sup>L</sup> xl r s L LA r* ; ; *k k <sup>R</sup> <sup>R</sup> <sup>L</sup>* ; *k k k k kk k W x W x x x L* ; ( ) ( ) (0) *k k k k A x a x <sup>A</sup>* ; ( ) ( ) (0) *k k k k I x b x <sup>I</sup>* ; ( ) ( ) (0) *k k k k k V x v x l A* ; 2 ( ) ( ) (0) *k k k k k x x l A* ; 1 <sup>1</sup> ; (0) *k k k <sup>F</sup> <sup>N</sup> EA* ( ) () ; (0) *k k k k N x N x EA* ( ) () ; (0) *k k k k Q x Q x EA* () ( ) (0) *k k k k k <sup>L</sup> <sup>M</sup> x Mx EI* ;

Free Vibration Analysis of Centrifugally

in dimensionless form as follows:

the following adimensional equations:

or

where

*k*

;

*A A*

 1 1 0; *<sup>d</sup> d Wd d d*

*K l <sup>M</sup>*

that is equal to zero in the present study.

*<sup>l</sup> QKW*

Stiffened Non Uniform Timoshenko Beams 297

The equations (14), which satisfy continuity of displacement and rotation, can be expressed

and the equations (15) of compatibility of the bending moment and the shear force, result in

1

( ) ( ) ( ) ( ) <sup>0</sup> *kk k k*

*a dW x a*

*dx*

;

 

> 

*k*

*dW x Nx ax ax x dx*

 

1 1 1 1 1

*k k x x d x d x b x b x l dx l dx*

 

The boundary conditions at the end closest to the axis of rotation, segment 1, *x*=0, are:

*Q K lW* 1 11 1 (0) *<sup>W</sup>* 0 0 ; 1 1 1 1

*M Kl* 1 11 1 (0) 0 0 ; <sup>1</sup>

and at the other end of the rotating beam, segment *d* , *x*=1, they are:

Gauss - Lobato expression, (Shu, 2000). (See Fig. A.1 in Appendix A)

*k k k k k*

( ) () () () () 2(1 ) 2(1 )

 

*k k k k k k*

*dW x Nx ax ax x dx*

*k k*

; 1

1 1 (1) (0) 0

 *kk k k Q Q* 

*k I I*

.

(1) 1 0 *d d d d d*

**3. Differential Quadrature Method, DQM** 

1 1 (1) (0) 0 *kk k k lW l W* ; 1 (1) (0) 0 *k k* (23a,b)

 

1 (1) (0) 0 *<sup>k</sup> <sup>k</sup> k k*

*k k M M l l*

( ) ( ) ( ) ( ) ( ) 0; 2(1 ) 2(1 )

(24a,b)

1 1 1 1 1 0

1 11 1 0

> 1 11 1 0 (0) 0 0

*x d x <sup>b</sup> K l dx* 

1

*<sup>a</sup> dW x <sup>a</sup> K l N W*

2(1 ) 2(1 )

*d x K l <sup>b</sup> dx*

*dx*

where (1) *Nd* is an outboard force at the end of the beam, farthest from the axis of rotation,

In order to obtain the DQM analog equations from the governing equations of the rotating beam, the beam segment domain is discretized in a grid of *i* points, using the Chebyshev –

*N K lW*

(0) ( ) 0 (0) (0) 0 0 2(1 ) 2(1 ) *<sup>W</sup> x*

 

(1) ( ) <sup>1</sup> (1) (1) 1 0

*d x*

*d d d Wd d d d d*

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

 

 

 

> 

1

1

*x*

0

(25a,b)

(26a,b)

*x*

Fig. 3. Elastic restraints of the rotating beam

In each segment *k* of the beam*, x* varies between 0 and 1.

The axial force, the shear force and the bending moment in the adimensional form become:

$$N\_k(\mathbf{x}) = \eta^2 \frac{l\_k^2}{s\_1^2} \left( R\_k \upsilon\_k(\mathbf{1}) + \phi\_k(\mathbf{1}) - R\_k \upsilon\_k(\mathbf{x}) - \phi\_k(\mathbf{x}) \right) + N\_{k+1}; \text{ with } s\_1 = \frac{L}{r\_1}; \text{ } r\_1^2 = \frac{I\_1(0)}{A\_1(0)} \tag{18}$$

$$Q\_k(\mathbf{x}) = \left(N\_k(\mathbf{x}) + \frac{\kappa}{2(1+\nu)}a\_k(\mathbf{x})\right)\frac{d\mathcal{W}\_k(\mathbf{x})}{d\mathbf{x}} - \frac{\kappa}{2(1+\nu)}a\_k(\mathbf{x})\mathcal{W}\_k(\mathbf{x})\tag{19}$$

$$M\_k(\mathbf{x}) = b\_k(\mathbf{x}) \frac{d\Psi\_k(\mathbf{x})}{d\mathbf{x}} \tag{20}$$

And the equations of motion in dimensionless form are:

 2 2 2 2 2 1 1 22 2 2 2 1 2 2 ( ) ( ) () () ( ) ( ) ( ) 2(1 ) ( ) ( ) () () () 2(1 ) *k k k k k k k k k k k k k kk k dW x s s dW x dW x d x axR x N x a x dx l dx l dx dx <sup>s</sup> da x dW x x a xW x l dx dx* (21) 2 2 2 2 2 1 1 1 2 22 2 2 ( ) ( ) ( ) ( ) ( ) () () 2(1 ) () () () () *k k k k k k k k k k kk kk dW x s s d x dx db x s sa x x bx dx l dx l dx dx bx x bx x* (22)

The equations (14), which satisfy continuity of displacement and rotation, can be expressed in dimensionless form as follows:

$$\Psi\_k \mathcal{W}\_k(\mathbf{1}) - l\_{k+1} \mathcal{W}\_{k+1}(\mathbf{0}) = 0 \; ; \; \Psi\_k(\mathbf{1}) - \Psi\_{k+1}(\mathbf{0}) = 0 \tag{23a,b}$$

and the equations (15) of compatibility of the bending moment and the shear force, result in the following adimensional equations:

$$a\_k \, \mathbb{Q}\_k(1) - a\_{k+1} \, \mathbb{Q}\_{k+1}(0) = 0 \\ \vdots \\ \frac{\beta\_k}{l\_k} \, M\_k(1) - \frac{\beta\_{k+1}}{l\_{k+1}} M\_{k+1}(0) = 0$$

or

296 Mechanical Engineering

1(0) *<sup>j</sup> <sup>j</sup> <sup>L</sup> K K*

K <sup>w</sup> <sup>d</sup>

L= Li

The axial force, the shear force and the bending moment in the adimensional form become:

 

( ) () () () () () 2(1 ) 2(1 ) *k*

( ) () () *<sup>k</sup>*

*dx*

*dx*

2 2 2 2

1 2

( ) ( ) ( ) ( ) ( ) () () 2(1 )

() () () ()

*kk kk*

*bx x bx x*

( ) ( ) () () () 2(1 )

( ) ( ) () () ( ) ( ) ( ) 2(1 )

*k k*

*dW x s s dW x dW x d x axR x N x a x*

*k k*

 

*l dx dx*

2 2 1 1 1 2 22 2 2

*k k k k*

22 2 2

*k k k k*

2 2 2

*k k*

*k k k k*

*dx l dx l dx dx*

*dW x s s d x dx db x*

*dx l dx l dx dx*

*<sup>s</sup> da x dW x x a xW x*

*k k k k k dW x Qx Nx ax ax x*

*k k d x Mx bx*

; with 1

 

> 

*k kk*

1 *L*

*<sup>r</sup>* ; <sup>2</sup> <sup>1</sup> 1

*r*

1 (0) (0) *I*

*<sup>A</sup>* (18)

(19)

(21)

(22)

*s*

(20)

*EI* ; with *j* =1 or *j*=*d.*

K <sup>d</sup>

and

K w

Fig. 3. Elastic restraints of the rotating beam

2 2

1

*s*

In each segment *k* of the beam*, x* varies between 0 and 1.

( ) (1) (1) ( ) ( ) *<sup>k</sup> k kk k kk k k <sup>l</sup> N x Rv Rv x x N*

And the equations of motion in dimensionless form are:

2 1 1

*k k k k*

2

2

*k*

*s sa x x bx*

2 1

K

1(0) *Wj Wj <sup>L</sup> K K EA* ;

$$a\_k \left[ \left( N\_k(\mathbf{x}) + \frac{\kappa}{2(1+\nu)} a\_k(\mathbf{x}) \right) \frac{d\mathcal{W}\_k(\mathbf{x})}{d\mathbf{x}} - \frac{\kappa}{2(1+\nu)} a\_k(\mathbf{x}) \mathcal{V}\_k(\mathbf{x}) \right]\_{\mathbf{x}=1} -$$

$$-a\_{k+1} \left[ \left( N\_{k+1}(\mathbf{x}) + \frac{\kappa}{2(1+\nu)} a\_{k+1}(\mathbf{x}) \right) \frac{d\mathcal{W}\_{k+1}(\mathbf{x})}{d\mathbf{x}} - \frac{\kappa}{2(1+\nu)} a\_{k+1}(\mathbf{x}) \mathcal{V}\_{k+1}(\mathbf{x}) \right] \bigg|\_{\mathbf{x}=0} = 0; \qquad \text{(24a.b)}$$

$$\frac{\rho\_k}{l\_k} b\_k(\mathbf{x}) \frac{d\mathcal{V}\_k(\mathbf{x})}{d\mathbf{x}} \bigg|\_{\mathbf{x}=1} - \frac{\rho\_{k+1}}{l\_{k+1}} b\_{k+1}(\mathbf{x}) \frac{d\mathcal{V}\_{k+1}(\mathbf{x})}{d\mathbf{x}} \bigg|\_{\mathbf{x}=0} = 0$$

where <sup>1</sup> 0 0 *k k A A* ; <sup>1</sup> 0 0 *k k I I* .

The boundary conditions at the end closest to the axis of rotation, segment 1, *x*=0, are:

$$\begin{aligned} \left(Q\_1(0) - K\_{\mathrm{VI}1} l\_1 \mathcal{W}\_1(0) = 0 \right) \left(N\_1(0) + \frac{\kappa a\_1(0)}{2(1+\nu)} \right) \frac{d \mathcal{W}\_1(\mathbf{x})}{d\mathbf{x}} \bigg|\_{\mathbf{x}=0} - \frac{\kappa a\_1(0) \mathcal{W}\_1(0)}{2(1+\nu)} - K\_{\mathrm{VI}1} l\_1 \mathcal{W}\_1(0) = 0 \\ \left(M\_1(0) - K\_{\mathrm{VI}1} l\_1 \mathcal{W}\_1(0) = 0 \right) \left. \frac{d \mathcal{W}\_1(\mathbf{x})}{d\mathbf{x}} \right|\_{\mathbf{x}=0} - K\_{\mathrm{VI}1} l\_1 \mathcal{W}\_1(0) = 0 \end{aligned} \tag{25a.b}$$

and at the other end of the rotating beam, segment *d* , *x*=1, they are:

$$\begin{split} \, \, ^{\prime Q}\_{d} \mathbf{Q}\_{d} (\mathbf{1}) - \, ^{\prime}\_{\text{V} \, \text{d}} \frac{l\_{d}}{a\_{d}} \mathcal{W}\_{d} (\mathbf{1}) = 0; \, \Big[ \, \, \, ^{\prime}\_{d} (\mathbf{1}) + \, \frac{\kappa \, ^{\prime}\_{d} (\mathbf{1})}{2(\mathbf{1} + \nu)} \Big] \frac{d \mathcal{W}\_{d} (\mathbf{x})}{d \mathbf{x}} \Big|\_{\mathbf{x} = 1} - \frac{\kappa \, ^{\prime}\_{d} (\mathbf{1})}{2(\mathbf{1} + \nu)} \Psi\_{d} (\mathbf{1}) - \frac{\mathcal{K}\_{\text{Wd}} l\_{d}}{a\_{d}} \mathcal{W}\_{d} (\mathbf{1}) = 0 \\ \, \, \, \, \, \mathcal{W}\_{d} (\mathbf{1}) - \frac{\mathcal{K}\_{\text{Wd}} l\_{d}}{\mathcal{P}\_{d}} \Psi\_{d} (\mathbf{1}) = 0 \; \, \, \, b\_{d} (\mathbf{1}) \frac{d \mathcal{W}\_{d} (\mathbf{x})}{d \mathbf{x}} \Big|\_{\mathbf{x} = 1} - \frac{\mathcal{K}\_{\text{Wd}} l\_{d}}{\mathcal{P}\_{d}} \, \, \Psi\_{d} (\mathbf{1}) = 0 \end{split} \tag{26a.b}$$

where (1) *Nd* is an outboard force at the end of the beam, farthest from the axis of rotation, that is equal to zero in the present study.
