**2. Mathematical model**

turylater, Mann *et al.* [14] discovered unstable regions in a stability lobe graph that existed underneath the stability boundary for the milling operation. They resemble islands in the fact that they are ovular areas contained within the stable regions, complicating the previ‐ ously thought simple stability lobe model. It was found that stability lobes taken from mo‐ dal parameters of the machine at rest (static) were not as accurate as the stability lobes produced from the dynamic modal properties. Zaghbani & Songmene [25] obtained these dynamic properties using operational modal analysis (OMA). OMA uses the autoregressive moving average method and least square complex exponential method to obtain these val‐ ues, producing a dynamic stability lobe that more accurately represents stable cutting condi‐ tions. These stability lobes have proven to be an invaluable asset to machinists and machine programmers. They provide a quick and easy reference to choose machining parameters

Tool wear is an often-overlooked factor that contributes to chatter. With the aid of more powerful computers this variable can now be included in simulations. The cutting tool is not indestructible and will change its shape while machining, and consequently affects the sta‐ bility of the system and stability lobes [7]. As the tool becomes worn, its limits of stability increase. Therefore, the axial depth of cut can be increased while maintaining the same spin‐ dle speed that would have previously created chatter. The rate of wear was incorporated in‐ to the stability lobe calculations for the tools so that it was now also a function of wear. To verify their calculations, the tools were ground to certain stages of wear and then tested ex‐ perimentally. They were found to be in strong agreement. Tool wear, however, is not some‐ thing that machine shops want increased. Chatter increases the rates of tool wear, shortening their lifespan, and increasing the amount of money the shops must spend on new tools. Li *et al.* [13] determined that the coherence function of two crossed accelerations in the bending vibration of the tool shank approaches unity at the onset of chatter. A thresh‐ old needs to be set [16] and then detected using simple mechanism to alert the operator to

In most of the previous stability prediction methods, a Frequency Response Function (FRF) is required to perform the calculations. FRF refers to how the machine's structure reacts to vibration. It is required to do an impact test to acquire the system's FRF [17]. In this case, an accelerometer is placed at the end of the top of the tool, and a hammer is used to strike the tool. The accelerometer will measure the displacement of the tool, telling the engineer how the machine reacts to vibration. This test gives crucial information about the machine, such as the damping of the structure and its natural frequencies. This method of obtaining infor‐ mation is impractical; because the FRF of the machine is always changing, it would require the impact test to be performed at all the different stages of machining. Also, having to do this interrupts the manufacturing processing and having machines sitting idle costs the com‐ pany money. An offline method of obtaining this information could greatly benefit machin‐ ing companies by eliminating the need for the impact test. Adetoro *et al.* [1] proposed that the machine, tool and work piece could be modelled using finite element analysis.A com‐ puter simulation would be able to predict the FRF during all phases of the machining proc‐ ess. As the part is machined and becomes thinner, its response to vibration changes

that should produce a chatter free cut [2].

70 Advances in Vibration Engineering and Structural Dynamics

change the machining conditions and avoid increased tool wear.

Computer Numeric Control (CNC) machines are quite often found in industries where a great deal of machining occurs. These machines are generally 3-, 4- or 5-axis, depending on the number of degrees of freedom the device has. Having the tool translate in the *X*, *Y* and Z direction accounts for the first three degrees-of-freedom (DOF). Rotation about the spindle axes account for any further DOF. The spindle contains the motors that rotate the tools and all the mechanisms that hold the tool in place. Figure 1 displays a sample spindle configura‐ tion and a typical tool/holder configuration is shown in Figure 2.

**Figure 1.** Typical Spindle Configuration.

**Figure 3.** Spinning Beam.

**Figure 4.** Degrees-of-Freedom (DOF) of the system

is given by

At an arbitrary cross section, located at *z* from *O*, *u* and *v* are lateral displacements of a point *P* in the *X* and *Y* directions, respectively. The cross section is allowed to rotate or twist about

where *i* and *j* are unit vectors in the *X* and *Y* directions, respectively. The velocity of point P

*rij* = -f + +f (uy vx ) ( ) (1)

Free Vibration Analysis of Spinning Spindles: A Calibrated Dynamic Stiffness Matrix Method

http://dx.doi.org/10.5772/51174

73

the *OZ* axis. The position vector *r* of the point *P* after deformation is given by

**Figure 2.** Typical Tool/Holder Configuration.

In this section, following the assumptions made by Banerjee & Su [5], discarding torsional vibrations, neglecting the rotary ineriaand shear deformation effects, the development of the governing differential equations of motion for coupled Bending-Bending (B-B) vibratins of a spinning beam is first briefly discussed. Then, based on the general procedure presented by Banerjee [4], the development of Dynamic Stiffness Matrix (DSM) formulation of the prob‐ lem is conciselypresented. Figure 3 shows the spinnning beam, represented by a cylinder in a right-handed rectangular Cartesian coordinates system. The beam length is *L*, mass per unit length is *m=ρA*, and the bending rigidities are *EI xx* and *EI yy*. See Figure 4 for the De‐ grees-of-Freedom (DOF) of the system.

Free Vibration Analysis of Spinning Spindles: A Calibrated Dynamic Stiffness Matrix Method http://dx.doi.org/10.5772/51174 73

**Figure 3.** Spinning Beam.

**Figure 1.** Typical Spindle Configuration.

72 Advances in Vibration Engineering and Structural Dynamics

**Figure 2.** Typical Tool/Holder Configuration.

grees-of-Freedom (DOF) of the system.

In this section, following the assumptions made by Banerjee & Su [5], discarding torsional vibrations, neglecting the rotary ineriaand shear deformation effects, the development of the governing differential equations of motion for coupled Bending-Bending (B-B) vibratins of a spinning beam is first briefly discussed. Then, based on the general procedure presented by Banerjee [4], the development of Dynamic Stiffness Matrix (DSM) formulation of the prob‐ lem is conciselypresented. Figure 3 shows the spinnning beam, represented by a cylinder in a right-handed rectangular Cartesian coordinates system. The beam length is *L*, mass per unit length is *m=ρA*, and the bending rigidities are *EI xx* and *EI yy*. See Figure 4 for the De‐

**Figure 4.** Degrees-of-Freedom (DOF) of the system

At an arbitrary cross section, located at *z* from *O*, *u* and *v* are lateral displacements of a point *P* in the *X* and *Y* directions, respectively. The cross section is allowed to rotate or twist about the *OZ* axis. The position vector *r* of the point *P* after deformation is given by

$$\mathbf{r} = \left(\mathbf{u} - \phi \mathbf{y}\right)\mathbf{i} + \left(\mathbf{v} + \phi \mathbf{x}\right)\mathbf{j} \tag{1}$$

where *i* and *j* are unit vectors in the *X* and *Y* directions, respectively. The velocity of point P is given by

$$\mathbf{v} = \dot{\mathbf{r}} + \boldsymbol{\Omega} \times \mathbf{r}, \text{ where } \boldsymbol{\Omega} = \boldsymbol{\Omega}\mathbf{k} \tag{2}$$

(EIyyU'' '' - m(Ω<sup>2</sup> <sup>+</sup> <sup>ω</sup>2)U) <sup>+</sup> <sup>2</sup>mΩωV=0, (11)

Free Vibration Analysis of Spinning Spindles: A Calibrated Dynamic Stiffness Matrix Method

(EIxxV'' '' - m(Ω<sup>2</sup> <sup>+</sup> <sup>ω</sup>2)V) <sup>+</sup> <sup>2</sup>mΩωU=0. (12)

EIyy <sup>V</sup>=0, (13)

http://dx.doi.org/10.5772/51174

75

EIxx U=0. (14)

2(1 - η2)<sup>2</sup> W=0 (15)

<sup>ω</sup><sup>2</sup> (16)

2(1 - η2)<sup>2</sup> (17)

η2} (18)

Introducing ξ=z / L and D=d / dξ , which are non-dimensional length and the differential

2mΩωL<sup>4</sup>

2mΩωL<sup>4</sup>

2 λy

EIyy , and <sup>η</sup><sup>2</sup> <sup>=</sup> <sup>Ω</sup><sup>2</sup>

The solution of the differential equation is sought in the form W=erξ , and when substituted

<sup>4</sup> <sup>+</sup> <sup>λ</sup><sup>x</sup> 2 λy

<sup>2</sup> - <sup>λ</sup><sup>y</sup>

<sup>2</sup> - <sup>λ</sup><sup>y</sup>

From the above solutions of U and V, the corresponding bending rotation about *X* and *Y*

2)2(1 <sup>+</sup> <sup>η</sup>2)<sup>2</sup> <sup>+</sup> <sup>16</sup>λ<sup>x</sup>

2)2(1 <sup>+</sup> <sup>η</sup>2)<sup>2</sup> <sup>+</sup> <sup>16</sup>λ<sup>x</sup>

2 λy 2 η2} ,

2 λy 2

operator into equations (11) and (12) leads to

<sup>D</sup><sup>8</sup> - (λ<sup>x</sup>

written in terms of W , satisfied by both U and V, where

λx <sup>2</sup> <sup>=</sup> <sup>m</sup>ω<sup>2</sup> L4 EIxx , λ<sup>y</sup>

r <sup>8</sup> - (λ<sup>x</sup> <sup>2</sup> <sup>+</sup> <sup>λ</sup><sup>y</sup>

<sup>2</sup> {(λ<sup>x</sup> <sup>2</sup> <sup>+</sup> <sup>λ</sup><sup>y</sup>

axes, Θx and Θy , respectively, are given by

<sup>β</sup><sup>2</sup> <sup>=</sup> <sup>1</sup> <sup>2</sup> {(λ<sup>x</sup> <sup>2</sup> <sup>+</sup> <sup>λ</sup><sup>y</sup>

r1,3 = ± α, r2,4 = ± β, r5,7 = ± i α, r6,8 = ± i β,

2)(1 <sup>+</sup> <sup>η</sup>2) <sup>+</sup> (λ<sup>x</sup>

2)(1 <sup>+</sup> <sup>η</sup>2) - (λ<sup>x</sup>

into (15), leads to

<sup>α</sup><sup>2</sup> <sup>=</sup> <sup>1</sup>

where

and

<sup>2</sup> <sup>+</sup> <sup>λ</sup><sup>y</sup>

<sup>D</sup><sup>4</sup> - m(ω<sup>2</sup> <sup>+</sup> <sup>Ω</sup>2)L4

<sup>D</sup><sup>4</sup> - m(ω<sup>2</sup> <sup>+</sup> <sup>Ω</sup>2)L4

EIyy U +

EIxx V +

2)(1 <sup>+</sup> <sup>η</sup>2)D4 <sup>+</sup> <sup>λ</sup><sup>x</sup>

<sup>2</sup> <sup>=</sup> <sup>m</sup>ω<sup>2</sup> L4

2)(1 + η2)r

The above equations are combined to form the following 8th-order differential equation,

The kinetic and potential energies of the beam (*T* and U) are given by:

$$T = \frac{1}{2} \int\_0^L \left| \mathbf{v} \right|^2 \, m d\mathbf{z} = \frac{1}{2} m \int\_0^L [\mathbf{u}^2 + \mathbf{v}^2 + 2\Omega(\mathbf{u}\mathbf{v} - \mathbf{u}\mathbf{v}) + \Omega^2(\mathbf{u}^2 + \mathbf{v}^2)] d\mathbf{z},\tag{3}$$

$$\mathbf{U} = \frac{1}{2} \mathbf{E} \mathbf{I}\_{\infty} \int\_{0}^{L} \mathbf{v} \,^{\top} \mathbf{dz} + \frac{1}{2} \mathbf{E} \mathbf{I}\_{\text{yy}} \int\_{0}^{L} \mathbf{u} \,^{\top} \mathbf{dz} \tag{4}$$

Using the Hamilton Principle in the usual notation state

$$
\left\| \int\_{t\_1}^{t\_2} (T - U)dt = 0, \right\| \tag{5}
$$

where t1 and t2 are the time intervals in the dynamic trajectory and δ is the variational oper‐ ator. Substituting the kinetic and potential energies in the Hamilton Principle, collecting like terms and integrating by parts,leads to the following set of equations.

$$\text{H}\text{H}\_{\text{yy}}\text{u}^{\prime\prime} - \text{m}\text{\text{\textdegree}}2^{\text{\textdegree}}\text{u} + \text{m}\text{\textdegree} - \text{2}\text{m}\text{\textdegree}\text{\textdegree} = \text{0},\tag{6}$$

$$-2\mathbf{m}\Omega\mathbf{u}\mathbf{\hat{}} - \mathbf{E}\mathbf{I}\_{\infty}\mathbf{v}\stackrel{\textstyle \mathbf{v}}{\dashv} - \mathbf{m}\mathbf{v} + \mathbf{m}\Omega^{2}\mathbf{v} = \mathbf{0}.\tag{7}$$

The resulting loads are then found to be in the following forms, written for Shear forces as

$$S\_x \equiv EI\_{\infty} \mu^{\prime \prime \prime} \quad \text{and} \quad S\_y \equiv EI\_{\text{yy}} \upsilon^{\prime} \,. \tag{8}$$

and for Bending Moments as

$$M\_x = EI\_{\infty} \boldsymbol{\upsilon}^{\prime} \quad \text{and} \quad M\_y = -EI\_{\text{yy}} \boldsymbol{\mu}^{\prime} \text{ .} \tag{9}$$

Assuming the simple harmonic motion, the form of

$$\mathbf{u}(\mathbf{z}, \ t) = \mathbf{U}(\mathbf{z}) \sin \omega t, \quad \text{and} \quad \mathbf{v}(\mathbf{z}, \ t) = \mathbf{V}(\mathbf{z}) \cos \omega t \tag{10}$$

where *ω* frequency of oscillation and U and V are the amplitudes of u and v . Substituting equations (10) into equations (6) and (7), they can be re-written as:

Free Vibration Analysis of Spinning Spindles: A Calibrated Dynamic Stiffness Matrix Method http://dx.doi.org/10.5772/51174 75

$$\left\{ \text{EI}\_{\text{yy}} \text{U}^{\text{\dots} \dots} \text{-m} \{ \text{\Omega}^2 + \omega^2 \} \text{U} \right\} + 2 \text{m} \Omega \omega \text{V} = 0,\tag{11}$$

$$\left\langle \mathrm{EI}\_{\infty} \mathrm{V}^{\prime \,\,\,\,\,\mathrm{m}} \left( \Omega^{2} + \omega^{2} \right) \mathrm{V} \right\rangle + 2 \mathrm{m} \Omega \omega \mathrm{U} = 0. \tag{12}$$

Introducing ξ=z / L and D=d / dξ , which are non-dimensional length and the differential operator into equations (11) and (12) leads to

$$\mathbf{U}\left[\mathbf{D}^4 \cdot \frac{\text{m}\{\omega^2 + \Omega^2\}\mathbf{L}^4}{\text{EI}\_{\text{YY}}}\right]\mathbf{U} + \frac{2\mathbf{m}\Omega\boldsymbol{\omega}\mathbf{L}^4}{\text{EI}\_{\text{YY}}}\mathbf{V} = \mathbf{0},\tag{13}$$

$$\mathbb{E}\left[\mathbf{D}^4 \text{ - } \frac{\text{m}\{\omega^2 + \Omega^2\}\mathbf{L}^4}{\text{EI}\_{\text{xx}}}\right] \mathbf{V} + \frac{2\mathbf{m}\Omega\omega\mathbf{L}^4}{\text{EI}\_{\text{xx}}} \mathbf{U} = \mathbf{0}.\tag{14}$$

The above equations are combined to form the following 8th-order differential equation,

$$\left[\mathbf{D}^{8} \cdot \left(\lambda\_{\mathbf{x}}^{2} + \lambda\_{\mathbf{y}}^{2}\right) \mathbf{(1} + \eta^{2}\right) \mathbf{D}^{4} + \lambda\_{\mathbf{x}}^{2} \lambda\_{\mathbf{y}}^{2} \mathbf{(1} \cdot \eta^{2}\right) \mathbf{W} = 0 \tag{15}$$

written in terms of W , satisfied by both U and V, where

$$
\lambda\_\chi^2 = \frac{m\omega^2 \mathcal{L}^4}{\mathcal{E}\mathcal{I}\_\infty}, \ \lambda\_\chi^2 = \frac{m\omega^2 \mathcal{L}^4}{\mathcal{E}\mathcal{I}\_\mathcal{Y}}, \text{ and } \eta^2 = \frac{\Omega^2}{\omega^2} \tag{16}
$$

The solution of the differential equation is sought in the form W=erξ , and when substituted into (15), leads to

$$\mathbf{r}^8 \cdot \left(\lambda\_\mathbf{x}^2 + \lambda\_\mathbf{y}^2\right) \mathbf{(1} + \eta^2\right) \mathbf{r}^4 + \lambda\_\mathbf{x}^2 \lambda\_\mathbf{y}^2 \mathbf{(1} - \eta^2)^2 \tag{17}$$

where

**v**=**r**˙ + *Ω* ×**r**, *where Ω* =*Ω***k** (2)

*<sup>t</sup>* d-= *T U dt* ò (5)

'' '' 2 EI u m yy - +- = Ω u mu 2mΩv 0, && & (6)

xx - - -+ = 2mΩu EI v mv mΩ v 0. & && (7)

u(z, t)=U(z)sin*ωt*, and v(z, t)=V(z)cos*ωt* (10)

u''2dz (4)

, (8)

. (9)

The kinetic and potential energies of the beam (*T* and U) are given by:

0 0

74 Advances in Vibration Engineering and Structural Dynamics

U= <sup>1</sup>

Using the Hamilton Principle in the usual notation state

and for Bending Moments as

<sup>2</sup> EIxx *∫* 0 *L*

2 2 2 22 2

1 <sup>2</sup> EIyy *∫* 0 *L*

( ) 0,

where t1 and t2 are the time intervals in the dynamic trajectory and δ is the variational oper‐ ator. Substituting the kinetic and potential energies in the Hamilton Principle, collecting like

'' '' 2

, and *My* <sup>=</sup> <sup>−</sup>*EI*yy*<sup>u</sup>* ″

where *ω* frequency of oscillation and U and V are the amplitudes of u and v . Substituting

The resulting loads are then found to be in the following forms, written for Shear forces as

*Sx* <sup>=</sup>*EI*xx*<sup>u</sup>* ″″, and *Sy* <sup>=</sup> *EI*yy*<sup>v</sup>* ″

*L L T v mdz m* = ò = ò + + W - +W + & & && *dz* (3)

1 1 | | [u v 2 (uv uv) (u v )] , 2 2

v''2dz +

2 1

*t*

terms and integrating by parts,leads to the following set of equations.

*Mx* <sup>=</sup>*EI*xx*<sup>v</sup>* ″

equations (10) into equations (6) and (7), they can be re-written as:

Assuming the simple harmonic motion, the form of

$$\mathbf{r}\_{1,3} = \pm \sqrt{\alpha}, \; \mathbf{r}\_{2,4} = \pm \sqrt{\beta}, \; \mathbf{r}\_{5,7} = \pm i \sqrt{\alpha}, \; \mathbf{r}\_{6,8} = \pm i \sqrt{\beta}.$$

and

$$
\alpha^2 = \frac{1}{2} \left[ (\lambda\_\infty^2 + \lambda\_\chi^2)(1+\eta^2) + \sqrt{(\lambda\_\chi^2 - \lambda\_\chi^2)^2 (1+\eta^2)^2 + 16\lambda\_\chi^2 \lambda\_\chi^2 \eta^2} \right],
$$

$$
\beta^2 = \frac{1}{2} \left[ (\lambda\_\infty^2 + \lambda\_\chi^2)(1+\eta^2) - \sqrt{(\lambda\_\chi^2 - \lambda\_\chi^2)^2 (1+\eta^2)^2 + 16\lambda\_\chi^2 \lambda\_\chi^2 \eta^2} \right] \tag{18}
$$

From the above solutions of U and V, the corresponding bending rotation about *X* and *Y* axes, Θx and Θy , respectively, are given by

Θ x <sup>=</sup> dV dz = − 1L dV dξ = − 1L ( α B <sup>1</sup>cos α ξ − α B <sup>2</sup>sin α ξ + α B <sup>3</sup>cosh α ξ + α B <sup>4</sup>sinh α ξ + β B 5cos β ξ − β B 6sin β ξ + β B 7cosh β ξ + β B <sup>8</sup>sinh β ξ ) , Θ y <sup>=</sup> dU dz = 1L dU dξ = − 1L ( α A 1cos α ξ − α A 2sin α ξ + α A 3cosh α ξ + α A <sup>4</sup>sinh α ξ + β A 5cos β ξ − β A 6sin β ξ + β A <sup>7</sup>cosh β ξ + β A 8sinh β ξ ) . (19)

with

For Displacements:

For Forces we have

where

and

A 1 = k α B 1 , A 2 = k α B 2 , A 3 = k α B 3 , A 4 = k α B 4 ,

A 5 = k β B 5 , A 6 = k β B 6 , A 7 = k β B 7, A 8 = k β B 8

k α =

then introduced into the governing equations.

*A t z* =0:

*A t z* =L:

*A t z* =0:

*A t z* =L: *S x* = *S x* 2 , *S y* = *S y* 2 , *M x* = *M x* 2 = *M* y = *M*y2 .

2 λ y2 η

*U* = *U* 1 , *V* = *V* 1 , Θ x = Θ x 1 , Θ y = Θ y 1

*U* = *U* 2 , *V* = *V* 2 , Θ x = Θx2 , Θ y = Θy2

*S x* = *S x* 1 , *S y* = *S y* 1 , *M x* = *M x* 1 = *M* y = *M* y 1 ,

**δ** = *U* 1 *V* 1 Θ x 1 Θ y 1 *U* 2 *V* 2 Θ x 2 Θ y 2 *T* ,

**R** = *R* 1 *R* 2 *R* 3 *R* 4 *R* 5 *R* 6 *R* 7 *R* 8 *T*

Substituting the boundary conditions into the governing equations we find

2 λ y2 η

Free Vibration Analysis of Spinning Spindles: A Calibrated Dynamic Stiffness Matrix Method

. (21)

http://dx.doi.org/10.5772/51174

77

**δ**=**B R** (24)

, (25)

(22)

(23)

β 2 − λ y2 ( 1 + η 2 )

To obtain the dynamic stiffness matrix (DSM) of the system the boundary conditions are

α 2 − λ y2 ( 1 + η 2 ) , and k β =

By doing similar substitutions we find

Sx =( EIyy <sup>L</sup><sup>3</sup> ) ( −α <sup>α</sup>A1cos <sup>α</sup> ξ + α α A 2sin α ξ + α α A <sup>3</sup>cosh α ξ + α α A <sup>4</sup>sinh α ξ − β β A 5cos β ξ + β β A <sup>6</sup>sin β ξ + β β A 7cosh β ξ + β β A <sup>8</sup>sinh β ξ ) S y = ( EIxx L3 ) ( − α α B <sup>1</sup>cos α ξ + α α B <sup>2</sup>sin α ξ + α α B 3cosh α ξ + α α B <sup>4</sup>sinh α ξ − β β B <sup>5</sup>cos β ξ + β β B <sup>6</sup>sin β ξ + β β B <sup>7</sup>cosh β ξ + β β B 8sinh β ξ ) M x = ( EIyy L2 ) ( −αB <sup>1</sup>sin α ξ −αB <sup>2</sup>cos α ξ + αB 3sinh α ξ + αB <sup>4</sup>cosh α ξ −βB <sup>5</sup>sin β ξ + βB 6cos β ξ + βB 7sinh β ξ + βB 8cosh β ξ ) M y = ( EIxx L2 ) ( −αA <sup>1</sup>sin α ξ −αA 2cos α ξ + αA <sup>3</sup>sinh α ξ + αA 4cosh α ξ − βA 5sin β ξ + βA 6cos β ξ + βA 7sinh β ξ +βA <sup>8</sup>cosh β ξ ) , (20)

where

Free Vibration Analysis of Spinning Spindles: A Calibrated Dynamic Stiffness Matrix Method http://dx.doi.org/10.5772/51174 77

$$\begin{aligned} \mathbf{A}\_{1} &= \mathbf{k}\_{\alpha} \mathbf{B}\_{1\prime} & \mathbf{A}\_{2} &= \mathbf{k}\_{\alpha} \mathbf{B}\_{2\prime} & \mathbf{A}\_{3} &= \mathbf{k}\_{\alpha} \mathbf{B}\_{3\prime} & \mathbf{A}\_{4} &= \mathbf{k}\_{\alpha} \mathbf{B}\_{4\prime} \\ \mathbf{A}\_{5} &= \mathbf{k}\_{\beta} \mathbf{B}\_{5\prime} & \mathbf{A}\_{6} &= \mathbf{k}\_{\beta} \mathbf{B}\_{6\prime} & \mathbf{A}\_{7} &= \mathbf{k}\_{\beta} \mathbf{B}\_{7\prime} & \mathbf{A}\_{8} &= \mathbf{k}\_{\beta} \mathbf{B}\_{8} \end{aligned}$$

with

<sup>Θ</sup><sup>x</sup> <sup>=</sup> dV

76 Advances in Vibration Engineering and Structural Dynamics

− 1

<sup>Θ</sup><sup>y</sup> <sup>=</sup> dU dz <sup>=</sup> <sup>1</sup> L dU dξ =

− 1

By doing similar substitutions we find

Sx =( EIyy <sup>L</sup><sup>3</sup> ) (

Sy =( EIxx <sup>L</sup><sup>3</sup> ) (

Mx =( EIyy

My =( EIxx

where

<sup>L</sup><sup>2</sup> ) (

dz <sup>=</sup> <sup>−</sup> <sup>1</sup> L dV dξ =

<sup>L</sup> ( <sup>α</sup>B1cos αξ− <sup>α</sup>B2sin αξ<sup>+</sup>

<sup>L</sup> ( <sup>α</sup>A1cos αξ− <sup>α</sup>A2sin αξ<sup>+</sup>

+β βA8sinh βξ

−α αB1cos αξ + α αB2sin

β βB5cos βξ + β βB6sin βξ+ β βB7cosh βξ + β βB8sinh βξ

−αA1sin αξ−αA2cos αξ+ αA3sinh αξ + αA4cosh αξ−

<sup>L</sup><sup>2</sup> ) ( −αB1sin αξ−αB2cos αξ<sup>+</sup>

+βA8cosh βξ

βB8sinh βξ

βB6sin βξ + βB7cosh βξ+

αB3cosh αξ + αB4sinh αξ + βB5cos βξ−

αA3cosh αξ + αA4sinh αξ + βA5cos βξ−

βA6sin βξ + βA7cosh βξ + βA8sinh βξ

−α αA1cos αξ + α αA2sin αξ+ α αA3cosh αξ + α αA4sinh αξ−

β βA5cos βξ + β βA6sin βξ + β βA7cosh βξ

αξ + α αB3cosh αξ + α αB4sinh αξ−

αB3sinh αξ + αB4cosh αξ−βB5sin βξ+ βB6cos βξ + βB7sinh βξ + βB8cosh βξ

βA5sin βξ + βA6cos βξ + βA7sinh βξ

),

(19)

(20)

).

)

)

)

),

$$\mathbf{k}\_{\alpha} = \frac{2\lambda\_{\mathbf{y}}^{2}\eta}{\alpha^{2} - \lambda\_{\mathbf{y}}^{2}(1+\eta^{2})^{\prime}} \text{ and } \mathbf{k}\_{\beta} = \frac{2\lambda\_{\mathbf{y}}^{2}\eta}{\beta^{2} - \lambda\_{\mathbf{y}}^{2}(1+\eta^{2})}. \tag{21}$$

To obtain the dynamic stiffness matrix (DSM) of the system the boundary conditions are then introduced into the governing equations.

For Displacements:

$$\begin{aligned} Atz = 0: \qquad \mathcal{U} = \mathcal{U}\_{1'} \; V = V\_{1'} \; \Theta\_{\mathbf{x}} = \Theta\_{\mathbf{x}1'} \; \Theta\_{\mathbf{y}} = \Theta\_{\mathbf{y}1} \\ Atz = \mathcal{L}: \qquad \mathcal{U} = \mathcal{U}\_{2'} \; V = V\_{2'} \; \Theta\_{\mathbf{x}} = \Theta\_{\mathbf{x}2'} \; \Theta\_{\mathbf{y}} = \Theta\_{\mathbf{y}2} \end{aligned} \tag{22}$$

For Forces we have

$$\begin{aligned} Atz &= 0: & \quad \mathcal{S}\_{\mathbf{x}} = \mathcal{S}\_{\mathbf{x}\cdot\mathbf{1}'} \; \mathcal{S}\_{\mathbf{y}} = \mathcal{S}\_{\mathbf{y}\cdot\mathbf{1}} \; \mathcal{M}\_{\mathbf{x}} = \mathcal{M}\_{\mathbf{x}\cdot\mathbf{1}} = \mathcal{M}\_{\mathbf{y}} = \mathcal{M}\_{\mathbf{y}\cdot\mathbf{1}} \\ Atz &= \mathbf{L}: & \quad \mathcal{S}\_{\mathbf{x}} = \mathcal{S}\_{\mathbf{x}\cdot\mathbf{2}'} \; \mathcal{S}\_{\mathbf{y}} = \mathcal{S}\_{\mathbf{y}\cdot\mathbf{2}} \; \mathcal{M}\_{\mathbf{x}} = \mathcal{M}\_{\mathbf{x}\cdot\mathbf{2}} = \mathcal{M}\_{\mathbf{y}} = \mathcal{M}\_{\mathbf{y}\cdot\mathbf{2}} \end{aligned} \tag{23}$$

Substituting the boundary conditions into the governing equations we find

$$
\boxed{\mathsf{\delta}=\mathsf{B}\,\mathsf{R}}\tag{24}
$$

where

$$\begin{aligned} \mathsf{S} &= \left[ \mathcal{U}\_1 V\_1 \Theta\_{\ge 1} \Theta\_{\ge 1} \mathcal{U}\_2 V\_2 \Theta\_{\ge 2} \Theta\_{\ge 2} \mathbf{I}^T, \\ \mathbf{R} &= \left[ \mathcal{R}\_1 \mathcal{R}\_2 \mathcal{R}\_3 \mathcal{R}\_4 \mathcal{R}\_5 \mathcal{R}\_6 \mathcal{R}\_7 \mathcal{R}\_8 \right]^T, \end{aligned} \tag{25}$$

and

$$\mathbf{B} = \begin{bmatrix} 0 & k\_a & 0 & k\_a & 0 & k\_b & 0 & k\_b \\ 0 & 1 & 0 & 1 & 0 & 1 & 0 & 1 \\ -\tau\_a & 0 & -\tau\_a & 0 & -\tau\_b & 0 & -\tau\_b & 0 \\ \chi\_a & 0 & \chi\_a & 0 & \chi\_b & 0 & \chi\_b & 0 \\ k\_a S\_a & k\_a C\_a & k\_a S\_{h\_a} & k\_a C\_{h\_a} & k\_b S\_{\rho} & k\_{\rho} C\_{\rho} & k\_{\rho} S\_{h\_b} & k\_{\rho} C\_{h\_b} & k\_{\rho} C\_{h\_b} \\ & S\_a & C\_a & S\_{h\_a} & C\_{h\_a} & S\_{\rho} & C\_{\rho} & S\_{h\_b} & C\_{h\_b} \\ -\tau\_a C\_a & \tau\_a S\_a & -\tau\_a C\_{h\_a} & -\tau\_a S\_{h\_a} & -\tau\_b C\_{\rho} & \tau\_{\rho} S\_{\rho} & -\tau\_b C\_{\rho} & -\tau\_{\rho} S\_{\lambda\_b} & -\tau\_{\rho} S\_{\lambda\_b} \\ \chi\_a C\_a & -\chi\_a S\_a & \chi\_a C\_{a\_a} & \chi\_a S\_{a\_b} & \chi\_b C\_{\rho} & -\chi\_{\rho} S\_{\rho} & \chi\_b C\_{\rho} & \chi\_{\rho} S\_{\lambda\_b} & \chi\_{\rho} S\_{\lambda\_b} \end{bmatrix} \tag{26}$$

with,

$$
\begin{aligned}
\boldsymbol{\tau}\_{\alpha} &= \frac{\sqrt{\alpha}}{L}, \; \boldsymbol{\tau}\_{\beta} = -\frac{\sqrt{\beta}}{L}, \; \omega\_{\alpha} = k\_{\alpha} \boldsymbol{\tau}\_{\alpha}, \; \omega\_{\beta} = k\_{\beta} \boldsymbol{\tau}\_{\beta}, \\
\boldsymbol{S}\_{\alpha} &= \mathrm{sinc}\sqrt{\alpha}, \; \quad \boldsymbol{C}\_{\alpha} = \mathrm{cos}\sqrt{\alpha}, \; \quad \boldsymbol{S}\_{h\_{\alpha}} = \mathrm{sinh}\sqrt{\alpha}, \; \quad \boldsymbol{C}\_{h\_{\alpha}} = \mathrm{cosh}\sqrt{\alpha}, \\
\boldsymbol{S}\_{\beta} &= \mathrm{sinc}\sqrt{\beta}, \; \quad \boldsymbol{C}\_{\beta} = \mathrm{cos}\sqrt{\beta}, \; \quad \boldsymbol{S}\_{h\_{\beta}} = \mathrm{sinh}\sqrt{\beta}, \; \quad \boldsymbol{C}\_{h\_{\beta}} = \mathrm{cosh}\sqrt{\beta}.
\end{aligned}
\tag{27}
$$

Substituting similarly for the force equation

$$\boxed{\mathbf{F} = \mathbf{A} \,\, \mathbf{R}}\tag{28}$$

and

For that we find

as shown in Figure 5.

*ζα* =*kαα α*

*ζβ* =*kββ β*

and assuming *KB = A*, finally leads to

*E I yy*

*E I yy*

*<sup>L</sup>* <sup>3</sup> , *ηα* =*α α*

*<sup>L</sup>* <sup>3</sup> , *ηβ* =*β β*

**Figure 5.** Simplified Spindle sections, with bearings modeled as simply-supported BC.

*E I xx <sup>L</sup>* <sup>3</sup> , *γα* =*α*

*E I xx <sup>L</sup>* <sup>3</sup> , *γβ* =*β* *E I xx*

*E I xx*

The frequency-dependent dynamic stiffness matrix (DSM) of the spinning beam, **K(ω)** ,can be derived by eliminating **R** . The force amplitude is related to the displacement vector by

Once the correctness and accuracy of the DSM code was established, a real machine spindle was then modeled, where the non-uniform spindle was idealized as a piecewise uniform (stepped) beam. Each step was modeled as a single continuous element and the above steps in the DSM formulation are repeated several times to determine the stiffness matrix for each element of the spindle. The element Dynamic Stiffness matrices are then assembled and the boundary conditions are applied. The system is simplified to contain 12 elements (13 nodes),

*<sup>L</sup>* <sup>2</sup> , *λα* =*kαα*

Free Vibration Analysis of Spinning Spindles: A Calibrated Dynamic Stiffness Matrix Method

*<sup>L</sup>* <sup>2</sup> , *λβ* =*kββ*

*E I yy <sup>L</sup>* <sup>2</sup> ,

**F**=**Kδ** (32)

**F**=**K**(**B R**)=**A R** (33)

**K**(**ω**)=**AB−<sup>1</sup>** (34)

(31)

79

http://dx.doi.org/10.5772/51174

*E I yy <sup>L</sup>* <sup>2</sup> .

where

$$\mathbf{F} = \begin{bmatrix} \mathbf{S}\_{\times1} \mathbf{S}\_{y1} \mathbf{M}\_{\times1} \mathbf{M}\_{\times1} \mathbf{S}\_{\times2} \mathbf{S}\_{y2} \mathbf{M}\_{\times2} \mathbf{M}\_{\times2} \mathbf{M}\_{\times2} \end{bmatrix}^{T} \tag{29}$$

$$\mathbf{A} = \begin{bmatrix} -\zeta\_a & 0 & \zeta\_a & 0 & -\zeta\_\beta & 0 & \zeta\_\beta & 0\\ -\varepsilon\_a & 0 & \varepsilon\_a & 0 & -\varepsilon\_\beta & 0 & \varepsilon\_\beta & 0\\ 0 & -\gamma\_a & 0 & \gamma\_a & 0 & -\gamma\_\beta & 0 & \gamma\_\beta\\ 0 & \lambda\_a & 0 & -\lambda\_a & 0 & \lambda\_\beta & 0 & -\lambda\_\eta\\ \zeta\_a C\_a & -\zeta\_a S\_a & -\zeta\_a C\_{a\_a} & -\zeta\_a S\_{a\_a} & \zeta\_p C\_\rho & -\xi\_p S\_\rho & -\xi\_p C\_{\rho\_b} & -\xi\_p S\_{A\_b}\\ \varepsilon\_a C\_a & -\varepsilon\_a S\_a & -\varepsilon\_a C\_{a\_a} & -\varepsilon\_a S\_{a\_a} & \varepsilon\_p C\_\rho & -\varepsilon\_p S\_\rho & -\varepsilon\_p C\_{\rho\_b} & -\varepsilon\_p S\_{\rho\_b}\\ \gamma\_a S\_a & \gamma\_a C\_a & -\gamma\_a S\_{a\_a} & -\gamma\_a C\_{a\_a} & \gamma\_\beta S\_\beta & \gamma\_\beta C\_\rho & -\gamma\_\beta S\_{\rho\_b} & -\gamma\_\beta C\_{\rho\_b}\\ -\lambda\_a S\_a & -\lambda\_a C\_a & \lambda\_a S\_{a\_a} & \lambda\_a C\_{a\_a} & -\lambda\_p S\_\beta & -\lambda\_p C\_\rho & -\lambda\_p C\_{\rho\_b} & \lambda\_p S\_{\rho\_b} & \lambda\_p C\_{\rho\_b} \end{bmatrix} \tag{30}$$

and

0 0 00

é ê ê

78 Advances in Vibration Engineering and Structural Dynamics

= ê ê ê ê

**B**

with, 

where

**A**

ê

*τα* <sup>=</sup> *<sup>α</sup>*

a a bb a a bb aa a a a a bb b b b b

<sup>ê</sup> -t -t -t -t <sup>ê</sup> c c cc ê

a a b b

*<sup>L</sup>* , *τβ* <sup>=</sup> <sup>−</sup> *<sup>β</sup>*

**F**= *Sx*1*Sy*1Mx1My1*Sx*2*Sy*2Mx2My2

0 0 00 0 0 00

= z -z -z -z z -z -z -z

a a bb a a bb


*C S C SC SC S C S C SC SC S S C S CSC S C*

é ù ê ú

a a aa a a b b bb b b a a aa a a b b bb b b aa a a a a bb b b b b


e -e -e -e e -e -e -e g g -g -g g g -g -g

*h*

a

*S CS C*

aa a a a a


0 00 0 0 00 0

a a bb a ab b


> *h h h h h h h h h h h h*

a a b b a a b b a a b b

*<sup>h</sup> h h S CS C* <sup>a</sup> bb b b b b b b

*S<sup>α</sup>* =sin *α*, *C<sup>α</sup>* =cos *α*, *Sh <sup>α</sup>*

*S<sup>β</sup>* =sin *β*, *C<sup>β</sup>* =cos *β*, *Sh <sup>β</sup>*

Substituting similarly for the force equation

0 1 0 1 01 0 1

*k S kC k S kC kS kC kS kC S C S C SCS C C S C S CS C S C S C S C SC S*

a a aa a a b b bb b b a a aa a a b b bb b b

ê-t t -t -t -t t -t -t

c -c c c c -c c c êë

0 00 0 0 00 0

*k kk k*

a ab b

*h h h h h h h h h h h h h h h h*

*<sup>L</sup>* , *ωα* =*kατα*, *ωβ* =*kβτβ*,

=sinh *α*, *Ch <sup>α</sup>*

=sinh *β*, *Ch <sup>β</sup>*

=cosh *α*,

**F**=**A R** (28)

*<sup>T</sup>* , (29)

=cosh *β*.

a a b b a a b b a a b b a a b b

ù ú ú ú ú ú ú ú ú ú ú ú úû

(26)

(27)

(30)

$$\begin{split} \mathsf{V}\_{\alpha} &= k\_{\alpha} \alpha \mathsf{V} \overline{\alpha} \frac{\operatorname{E} I\_{yy}}{\operatorname{L}^{-3}}, \ \eta\_{\alpha} = \alpha \mathsf{V} \overline{\alpha} \frac{\operatorname{E} I\_{xx}}{\operatorname{L}^{-3}}, \ \mathcal{V}\_{\alpha} = \alpha \frac{\operatorname{E} I\_{xx}}{\operatorname{L}^{-2}}, \ \lambda\_{\alpha} = k\_{\alpha} \alpha \frac{\operatorname{E} I\_{yy}}{\operatorname{L}^{-2}}, \\ \ \mathsf{V}\_{\beta} &= k\_{\beta} \beta \mathsf{V} \overline{\beta} \frac{\operatorname{E} I\_{yy}}{\operatorname{L}^{-3}}, \ \eta\_{\beta} = \beta \mathsf{V} \overline{\beta} \frac{\operatorname{E} I\_{xx}}{\operatorname{L}^{-3}}, \ \mathcal{V}\_{\beta} = \beta \frac{\operatorname{E} I\_{xx}}{\operatorname{L}^{-2}}, \ \lambda\_{\beta} = k\_{\beta} \beta \frac{\operatorname{E} I\_{yy}}{\operatorname{L}^{-2}}. \end{split} \tag{31}$$

The frequency-dependent dynamic stiffness matrix (DSM) of the spinning beam, **K(ω)** ,can be derived by eliminating **R** . The force amplitude is related to the displacement vector by

$$\boxed{\text{F}=\text{K}\\$}\tag{32}$$

For that we find

$$\mathbf{F} = \mathbf{K}(\mathbf{B} \mid \mathbf{R}) = \mathbf{A} \mid \mathbf{R} \tag{33}$$

and assuming *KB = A*, finally leads to

$$\boxed{\mathbf{K}(\omega) = \mathbf{A}\mathbf{B}^{-1}\,}\tag{34}$$

Once the correctness and accuracy of the DSM code was established, a real machine spindle was then modeled, where the non-uniform spindle was idealized as a piecewise uniform (stepped) beam. Each step was modeled as a single continuous element and the above steps in the DSM formulation are repeated several times to determine the stiffness matrix for each element of the spindle. The element Dynamic Stiffness matrices are then assembled and the boundary conditions are applied. The system is simplified to contain 12 elements (13 nodes), as shown in Figure 5.

**Figure 5.** Simplified Spindle sections, with bearings modeled as simply-supported BC.

It is assumed that the entire system is made from the same material and the properties of tooling steel were used for all section. It was also assumed that the system is simply sup‐ ported at the locations of the bearings. The simply supported boundary conditions were then modified and replaced by linear spring elements (Figure 6); the spring stiffness values were varied in an attempt to achieve a fundamental frequency equivalent to the spindle sys‐ tem's natural frequency reported by the manufacturer.

critical spindle speed is 2.3×10<sup>6</sup>

RPM.

**Figure 7.** System Natural Frequency vs. Bearing Equivalent Spring Constant (in log scale).

**Figure 8.** Spindle Natural Frequency vs. Spindle RPM.

the spindle, i.e., 3 *.*5×10<sup>4</sup>

RPM which is well above the operating rotational speed of

http://dx.doi.org/10.5772/51174

81

Free Vibration Analysis of Spinning Spindles: A Calibrated Dynamic Stiffness Matrix Method

**Figure 6.** Spindle model, with bearings modeled as linear spring elements(modified BC).
