**2. Analysis**

90 Continuum Mechanics – Progress in Fundamentals and Engineering Applications

0 10 20 30 40 50 60 70 80 90 100

**Time (ms)**

30th

40th

50th

60th

0 250 500 750 1000 1250 1500

**Frequency (Hz)**

Davidson and Smith (1969) first studied fluid-structure interactions using the TMM and verified their model with their own experimental data. Their data were used widely by subsequent researchers (Davidson & Samsury, 1972; Hatfield & Davidson, 1983) to verify analytical models which did not include viscosity. Hatfield *et al.* (1982) applied the component synthesis method in the frequency domain. In their method, fluid-structure interaction was included in terms of junction coupling. Their simulation predictions were validated with Davidson and Smith's (1969) experimental data. Bundy *et al.* [9] introduced

Fig. 2. Frequency spectrum of the pressure signal shown in Fig. 1 Pump speed: 1500 rpm; number of pumping elements: 10; fundamental pump rotational frequency: 25 Hz

Fig. 1. Pressure waveform measured at the outlet of a ten-vane power steering pump running at 1500 rpm. The periodic waveform is generated by the rotating elements of the

10th 20th


pumping mechanism.

10<sup>5</sup>

10<sup>2</sup>

10<sup>3</sup>

**Pressure Amplitude (Pa)**

10<sup>4</sup>


0

**Dynamic Pressure Amplitude (kPa)**

100

200

#### **2.1 Axial motion**

For a three-dimensional tubing system, fluidstructural coupling must be considered because tubing discontinuities, such as bends, cause unbalanced forces to act on both the tubing and fluid. Fig. 3 displays the coordinate system and state variables in a straight tube segment used in the following analysis.

Assuming axisymmetric, two-dimensional, laminar, viscous, compressible flow and negligible temperature variation (*i.e.*, constant fluid viscosity), the linearized NavierStokes equations reduce to (Chen, 2001):

$$\frac{\partial \operatorname{\boldsymbol{\mathcal{O}}} \boldsymbol{v}\_{z}}{\partial \boldsymbol{\upbeta}} = -\frac{1}{\rho\_{f}} \frac{\partial}{\partial \boldsymbol{\upbeta}} \frac{\boldsymbol{p}}{\boldsymbol{z}} + \nu \left[ \frac{\partial^{2} \boldsymbol{v}\_{z}}{\partial \boldsymbol{\upbeta}} + \frac{1}{r} \frac{\partial \operatorname{\boldsymbol{\upbeta}} \boldsymbol{v}\_{z}}{\partial \boldsymbol{r}} \right] \tag{1}$$

where *vz* , *vr* , and *p* denote the deviation of axial velocity, radial velocity, and pressure from the steady state, respectively.

Combining the continuity equation and equation of state for a liquid, gives:

$$\frac{1}{\beta} \frac{\partial}{\partial} \frac{p}{t} + \frac{\partial}{\partial} \frac{v\_r}{r} + \frac{v\_r}{r} + \frac{\partial}{\partial} \frac{v\_z}{z} = 0 \tag{2}$$

where is the fluid bulk modulus.

By averaging *vz* over the cross section, applying the boundary condition at the inner radius of the tubing, *u u <sup>f</sup> <sup>z</sup>* , and transforming to the Laplace domain, the following equation is obtained:

Noise and Vibration in Complex Hydraulic Tubing Systems 93

 () ( ) *<sup>z</sup> f f <sup>z</sup> f f <sup>f</sup> <sup>F</sup> A A sU A sU zs s*

The Poisson effect, longitudinal motion resulting in radial strain of the tubing or vice versa, was not included in previous work (Chen, 1992). The axial strain of the tubing, *<sup>z</sup>*

<sup>1</sup> *<sup>z</sup> <sup>z</sup> z r*

2 2 2

> 2 2 2

( )

*o i*

*e*

 *<sup>e</sup>* is an effective fluid bulk modulus that accounts for compliance of the tubing wall. When the Poisson effect is neglected, Equations (10) and (11) reduce to equations for

*o i pr r r*

 

<sup>2</sup> *<sup>i</sup> <sup>r</sup>*

1 2

*z i <sup>z</sup>*

 *ur f p z EA E r r*

For conservation of mass to hold, the axial change in volume of a fluid element results from pressure and expansion of the tubing. Radial expansion of the tubing is caused by pressure,

> *<sup>f</sup>* 2 1 *z*

The Bourdon effect occurs at bends where the fluid-filled tubing cross-section is ovalized. Bending of the tubing results in a change of cross-sectional area and thus fluid motion. The fluid pressure gradient in the bend produces a bending moment in the tubing, and the balancing bending moment in the tubing then displaces the fluid. For curved tubing, the

> 11 12 *<sup>z</sup> <sup>y</sup> <sup>u</sup> Ap Ah*

*<sup>f</sup> <sup>p</sup> z EA* 

 

1 1 2 2 1 1

(8)

are the elastic modulus and Poisson's ratio for the tubing

(10)

(11)

(12)

are the cylindrical coordinates.

(9)

(7)

, in a

Substituting Equation (3) into Equation (6), and rearranging Equation (6) yields:

 *u z E*

For thick-walled tubing, the radial and tangential stresses can be represented as:

 

 *u*

Bourdon coupling is described by Reissner *et al*. (1952) and Tentarelli (1990):

*z* 

and axial motion of the tubing results from Poisson coupling:

 

**2.1.2 Poisson effect** 

where

where

longitudinal motion of a bar.

**2.1.3 Bourdon coupling** 

cylindrical coordinates is written as:

is the stress, *E* and

Combining Equations (8) and (9), gives:

material, respectively, and subscripts *z* , *r* and

Fig. 3. Hydraulic line coordinate system and state variables: *u* is translational displacement of tubing; , angular displacement of tubing; *f,* force acting on the tubing; *h,* moment acting on the tubing; *<sup>f</sup> p* , fluid pressure; *uf* , fluid displacement; and subscripts *x*, *y*, and *z* the axes for the Cartesian coordinates (adapted from Chen, 1992).

$$\frac{\partial^{\gamma}P}{\partial \mathbf{\hat{c}}} = \frac{\rho\_f s^2}{\Omega(s)} \mathcal{U}\_f - \rho\_f s^2 \left[ \frac{1 + \Omega(s)}{\Omega(s)} \right] \mathcal{U}\_z \tag{3}$$

where <sup>1</sup> 2 ( ) <sup>1</sup> *<sup>i</sup> i ii J jr s s jr s J jr s* , *<sup>f</sup>* is the fluid density, and Jo and J1 are the zero and

firstorder Bessel functions of the first kind, respectively, and *s* denotes the Laplace transformation.

Applying Newton's second law to the tubing wall, yields:

$$\,^1F\_\tau + \frac{\partial \,^1F\_z}{\partial \,^2z} = \rho A s^2 \mathcal{U}\_z \tag{4}$$

where *F* is the friction force per unit length acting on the inner tubing wall, *A* is the crosssectional area of the tubing, and is the density of the tubing.

Applying Newton's second law to the fluid gives:

$$-A\_f \frac{\partial}{\partial z} \mathbf{F}\_\mathbf{r} = \rho\_f A\_f \mathbf{s}^2 \mathbf{U}\_f \tag{5}$$

where *Af* is the cross-sectional area of the fluid.

Combining Equations (4) and (5), gives:

$$\frac{\partial}{\partial z}\frac{F\_z}{z} - A\_f \frac{\partial}{\partial z}\frac{P}{z} = \rho A s^2 \mathcal{U}\_z + \rho\_f A\_f s^2 \mathcal{U}\_f \tag{6}$$

Substituting Equation (3) into Equation (6), and rearranging Equation (6) yields:

$$\frac{\partial \, F\_z}{\partial \, z} = \left\{ \rho A - \rho\_f A\_f \left| 1 + \frac{1}{\Omega(s)} \right| \right\} s^2 \mathcal{U}\_z + \rho\_f A\_f \left| 1 + \frac{1}{\Omega(s)} \right| s^2 \mathcal{U}\_f \tag{7}$$

#### **2.1.2 Poisson effect**

92 Continuum Mechanics – Progress in Fundamentals and Engineering Applications

*hx , <sup>x</sup>*

*x hy , <sup>y</sup>*

*fx , ux pf , uf hz , <sup>z</sup>*

2

*f*

, 

*z z*

Fig. 3. Hydraulic line coordinate system and state variables: *u* is translational displacement

on the tubing; *<sup>f</sup> p* , fluid pressure; *uf* , fluid displacement; and subscripts *x*, *y*, and *z* the axes

2 1 () () ( )

firstorder Bessel functions of the first kind, respectively, and *s* denotes the Laplace

*z* 2

is the density of the tubing.

*f ff f <sup>P</sup> A F AsU*

*<sup>z</sup> <sup>f</sup> <sup>z</sup> ff f <sup>F</sup> <sup>P</sup> A As U A s U*

is the friction force per unit length acting on the inner tubing wall, *A* is the cross-

*<sup>F</sup> F As U z*

 

> *z*

*z*

2

2 2

 

*P s s*

*zs s*

, angular displacement of tubing; *f,* force acting on the tubing; *h,* moment acting

*f f z*

*Us U*

*y*

*fy , uy*

for the Cartesian coordinates (adapted from Chen, 1992).

 

Applying Newton's second law to the tubing wall, yields:

of tubing;

where

sectional area of the tubing, and

*s*

transformation.

where *F*

<sup>1</sup> 2 ( ) <sup>1</sup> *<sup>i</sup> i ii*

*J jr s*

 

Applying Newton's second law to the fluid gives:

where *Af* is the cross-sectional area of the fluid.

Combining Equations (4) and (5), gives:

*jr s J jr s*

*fz , uz*

*ri*

*<sup>f</sup>* is the fluid density, and Jo and J1 are the zero and

(4)

(5)

(6)

*z* 

(3)

The Poisson effect, longitudinal motion resulting in radial strain of the tubing or vice versa, was not included in previous work (Chen, 1992). The axial strain of the tubing, *<sup>z</sup>* , in a cylindrical coordinates is written as:

$$
\varepsilon\_z = \frac{\partial \ln\_z}{\partial \, z} = \frac{1}{E} (\sigma\_z - \mu \sigma\_r - \mu \sigma\_\theta) \tag{8}
$$

where is the stress, *E* and are the elastic modulus and Poisson's ratio for the tubing material, respectively, and subscripts *z* , *r* and are the cylindrical coordinates.

For thick-walled tubing, the radial and tangential stresses can be represented as:

$$
\sigma\_r + \sigma\_\theta = \frac{2pr\_i^2}{r\_o^2 - r\_i^2} \tag{9}
$$

Combining Equations (8) and (9), gives:

$$\frac{\partial \ln\_z}{\partial z} = \frac{1}{EA} f\_z - \frac{2}{E} \frac{r\_i^2 \mu}{(r\_o^2 - r\_i^2)} p \tag{10}$$

For conservation of mass to hold, the axial change in volume of a fluid element results from pressure and expansion of the tubing. Radial expansion of the tubing is caused by pressure, and axial motion of the tubing results from Poisson coupling:

$$\frac{\partial \ln \mu\_f}{\partial \ z} = \frac{2}{EA} \frac{\mu}{\varepsilon} f\_z - \frac{1}{\mathcal{B}\_\varepsilon} p \tag{11}$$

where *<sup>e</sup>* is an effective fluid bulk modulus that accounts for compliance of the tubing wall. When the Poisson effect is neglected, Equations (10) and (11) reduce to equations for longitudinal motion of a bar.

#### **2.1.3 Bourdon coupling**

The Bourdon effect occurs at bends where the fluid-filled tubing cross-section is ovalized. Bending of the tubing results in a change of cross-sectional area and thus fluid motion. The fluid pressure gradient in the bend produces a bending moment in the tubing, and the balancing bending moment in the tubing then displaces the fluid. For curved tubing, the Bourdon coupling is described by Reissner *et al*. (1952) and Tentarelli (1990):

$$\frac{\partial \ln\_z}{\partial \; z} = A\_{11}p + A\_{12}h\_y \tag{12}$$

$$\frac{\partial \oint\_y \phi\_y}{\partial z} = A\_{21}p + A\_{22}h\_y \tag{13}$$

Noise and Vibration in Complex Hydraulic Tubing Systems 95

0 01 1

*x x y f f y x x f f y y*

*U U GA H H I Is z F F A As EI*

0 10 0

0 01 1

*y y f f x x y y f f x x*

*U U GA H H I Is z F F A As EI*

0 10 0

<sup>2</sup> 0 1 0 *z z z z H H Js*

00 0

00 0

0

, is used to include the Bourdon effect by replacing the flexural

 

Equations (15) (18) can be represented in the following form:

where *<sup>k</sup>*<sup>0</sup> *<sup>z</sup> S* is the substate vector at the inlet.

*EI* in Equation (14).

*z*

 

yields:

matrix *<sup>i</sup> T :* 

correction factor,

stiffness *EI* with

2

2

*z GJ*

0 01

; 1,4 *S ASk k kk*

where *Ak* is coefficient matrix, <sup>T</sup> *S PF U U* <sup>1</sup> *zfz* , <sup>T</sup> *S UHF* <sup>2</sup> *x yx y* , T *S UHF* <sup>3</sup> *y xy x* and <sup>T</sup> *S H* <sup>4</sup> *z z* .

Solving Equation (16) by employing boundary conditions at the inlet (*z* = 0) of each section

Relating the two end conditions for a given section *i*, yields the 1414 field transfer

A three-dimensional tubing system can be treated as a combination of short straight lines with different orientations resulting in coupling of the fluid pressure, and forces and moments in the tubing wall. Each section of tubing is modeled by a 1414 transfer matrix with state variable vectors. Details on the assembly of the 1414 matrix can be found in Chen [11]. Each bend is broken into three straight-line segments. For these segments, the

A transformation matrix [*R*] transfers the force and displacement from one section to another, couples structural vibration and fluid pressure waves at points of discontinuity, and transforms the coordinate system from one section to the next. Force and moment

*A L <sup>k</sup> k k z L <sup>z</sup> S eS*

[ ]

0 01

 

(19)

(20)

*i ii* <sup>1</sup>*S TS* (21)

  2

2

(18)

(16)

(17)

where

$$A\_{11} = \frac{1}{R\_v} \left(1 - \frac{b^2}{a^2}\right) \left| 1 - \frac{2R\_v(r\_o - r\_i)}{ab\sqrt{12\left(1 - \nu^2\right)}} \right| \frac{b^2 - a^2}{Ea\left(r\_o - r\_i\right)^2} \left[\sqrt{3\left(1 - \nu^2\right)} - \frac{R\_v(r\_o - r\_i)}{ab}\right],$$

$$\begin{aligned} A\_{22} &= \frac{a\sqrt{12\left(1-\nu^{2}\right)}}{2\pi E b^{2} R\_{v} \left(r\_{o} - \eta\_{i}\right)^{2}}, & A\_{12} &= \frac{-1}{\pi R\_{v} ab} \frac{b^{2} - a^{2}}{E a \left(r\_{o} - \eta\_{i}\right)^{2}} \bigg[\sqrt{3\left(1-\nu^{2}\right)} - \frac{R\_{v} \left(r\_{o} - \eta\_{i}\right)}{ab}\Big], \\\ A\_{22} &= a^{2} \quad \lceil \sqrt{1-\nu^{2}} \rceil \quad R\_{v} \left(r\_{o} - \eta\_{i}\right) \bigg[\ \qquad \qquad \qquad \qquad \ldots \quad \ldots \quad \qquad \qquad \qquad \qquad \qquad \ldots \quad \ \end{aligned}$$

 2 21 2 ( ) 3 1 *vo i o i b a Rr r <sup>A</sup> Ea r r ab* , *Rv* is the radius of curvature of the bend, and *a*

and *b* are the major and minor axes of the elliptical cross section, respectively.

Equations (12) and (13) reduce to the common flexural motion equations for *a=b* (*i.e*., circular cross-section). When there is no fluid pressure present, *A*22 can be approximated as a flexural stiffness with a correction factor to account for the ovalization effect. The effect produces a reduction in stiffness at bends in the transmission line.

Several straight short-length segments are used to model the bends and twists in the threedimensional tubing line. To account for the ovalization effect, a correction factor is used to adjust the flexural stiffness for the curved line. The correction factor ( ) for the flexural stiffness is formulated as Vigness (1943):

$$\kappa = \frac{1 + 12\left[4\left(r\_o - r\_i\right)R\_v\sqrt{\left(r\_o + r\_i\right)^2}\right]}{10 + 12\left[4\left(r\_o - r\_i\right)R\_v\sqrt{\left(r\_o + r\_i\right)^2}\right]}\tag{14}$$

The product of and a flexural stiffness can be shown to be a simplified form of *A*<sup>22</sup> (Reissner *et al*., 1956).

#### **2.2 Flexural and torsional motion**

Rearranging Equations (3), (7), (10) and (11), and considering the flexural motions in the *x-z* and *y-z* planes, and torsion about the *z*-axis in Laplace domain, four groups of linear, first order differential equations are obtained (Chen 2001):

$$
\frac{\partial}{\partial z} \begin{bmatrix} P \\ F\_z \\ \Omega\_f \\ \Omega\_z \end{bmatrix} = - \begin{bmatrix} 0 & 0 & \frac{-\rho\_f s^2}{\Omega(s)} & \rho\_f s^2 \left[ \frac{1+\Omega(s)}{\Omega(s)} \right] \\\\ 0 & 0 & -\rho\_f A\_f s^2 \left[ \frac{1+\Omega(s)}{\Omega(s)} \right] & -s^2 \left[ \rho A - \rho\_f A\_f \left[ \frac{1+\Omega(s)}{\Omega(s)} \right] \right] \\\\ \frac{1}{\beta\_e} & \frac{-2\mu}{EA} & 0 & 0 \\\\ \frac{2\mu r\_i^2}{E \left(r\_o^2 - r\_i^2\right)} & \frac{-1}{EA} & 0 & 0 \end{bmatrix} \begin{bmatrix} P \\ F\_z \\ \mathcal{U}\_f \\ \mathcal{U}\_z \end{bmatrix} \tag{15}
$$

<sup>11</sup> 2 2 <sup>2</sup>

12 1

Equations (12) and (13) reduce to the common flexural motion equations for *a=b* (*i.e*., circular cross-section). When there is no fluid pressure present, *A*22 can be approximated as a flexural stiffness with a correction factor to account for the ovalization effect. The effect

Several straight short-length segments are used to model the bends and twists in the threedimensional tubing line. To account for the ovalization effect, a correction factor is used to

> 

and a flexural stiffness can be shown to be a simplified form of *A*<sup>22</sup>

*r rR r r r rR r r*

1 12 [4 ] 10 12 [4 ] *oiv oi oiv oi*

 

Rearranging Equations (3), (7), (10) and (11), and considering the flexural motions in the *x-z* and *y-z* planes, and torsion about the *z*-axis in Laplace domain, four groups of linear, first

2

*f*

1 2 0 0

<sup>2</sup> <sup>1</sup> 0 0

1 () 0 0

2 2

*f f f f*

*A s sA A*

1 () 1 () 0 0

*z z f f z e z*

*F s s F z U U U EA U*

*P s s P*

2 2

2

*s*

( ) ( )

 

*f*

( ) ( )

*s s*

*s s*

*v o i*

*z* 

and *b* are the major and minor axes of the elliptical cross section, respectively.

where

*A*

 

*b a Rr r <sup>A</sup> Ea r r ab* 

stiffness is formulated as Vigness (1943):

**2.2 Flexural and torsional motion** 

2 2 2

*i o i*

*r Er r EA*

22 2 2 12 1

2 2

*o i*

*a*

21 2

The product of

(Reissner *et al*., 1956).

2 *vo i*

*Eb R r r*

2

,

2

( ) 3 1 *vo i*

produces a reduction in stiffness at bends in the transmission line.

order differential equations are obtained (Chen 2001):

adjust the flexural stiffness for the curved line. The correction factor (

21 22 *<sup>y</sup> <sup>A</sup> <sup>y</sup> <sup>p</sup> A h*

2 2 2

(13)

*vo i vo i*

*b a Rr r <sup>A</sup> R ab Ea r r ab*

, *Rv* is the radius of curvature of the bend, and *a*

<sup>1</sup> ( ) 3 1 *vo i*

2

2

(14)

) for the flexural

(15)

,

,

2 2

<sup>1</sup> 2( ) ( ) 1 1 31

12 2

*v o i*

*<sup>b</sup> Rr r b a Rr r <sup>A</sup> R a Ea r r ab ab*

$$\frac{\partial}{\partial z} \begin{bmatrix} \mathcal{U}\_x \\ H\_y \\ F\_x \\ \Phi\_y \end{bmatrix} = - \begin{bmatrix} 0 & 0 & -1/\mathcal{G}A & -1 \\ 0 & 0 & 1 & -\left(\rho I + \rho\_f I\_f\right)s^2 \\ -\left(\rho A + \rho\_f A\_f\right)s^2 & 0 & 0 & 0 \\ 0 & -1/\mathcal{E}I & 0 & 0 \end{bmatrix} \begin{bmatrix} \mathcal{U}\_x \\ H\_y \\ F\_x \\ \Phi\_y \end{bmatrix} \tag{16}$$

$$\frac{\partial}{\partial z} \begin{bmatrix} \mathcal{U}\_y \\ H\_x \\ F\_y \\ \Phi\_x \end{bmatrix} = - \begin{bmatrix} 0 & 0 & -\mathbf{1}/\mathbf{GA} & \mathbf{1} \\ 0 & 0 & -\mathbf{1} & -\left(\rho I + \rho\_f I\_f\right)s^2 \\ -\left(\rho A + \rho\_f A\_f\right)s^2 & 0 & 0 & 0 \\ 0 & -\mathbf{1}/EI & 0 & 0 \end{bmatrix} \begin{bmatrix} \mathcal{U}\_y \\ H\_x \\ F\_y \\ \Phi\_x \end{bmatrix} \tag{17}$$

$$
\frac{\partial}{\partial z} \begin{bmatrix} H\_z \\ \Phi\_z \end{bmatrix} = - \begin{bmatrix} 0 & -\rho ls^2 \\ -1/G \end{bmatrix} \begin{bmatrix} H\_z \\ \Phi\_z \end{bmatrix} \tag{18}
$$

Equations (15) (18) can be represented in the following form:

$$\frac{\partial}{\partial z} \begin{bmatrix} \mathbf{S}\_k \end{bmatrix} = - \begin{bmatrix} A\_k \end{bmatrix} \begin{bmatrix} \mathbf{S}\_k \end{bmatrix}; k = 1, 4 \tag{19}$$

where *Ak* is coefficient matrix, <sup>T</sup> *S PF U U* <sup>1</sup> *zfz* , <sup>T</sup> *S UHF* <sup>2</sup> *x yx y* , T *S UHF* <sup>3</sup> *y xy x* and <sup>T</sup> *S H* <sup>4</sup> *z z* .

Solving Equation (16) by employing boundary conditions at the inlet (*z* = 0) of each section yields:

$$\left[\mathbf{S}\_{k}\right]\_{z=L} = e^{-\left[A\_{k}\right]L} \left[\mathbf{S}\_{k}\right]\_{z=0} \tag{20}$$

where *<sup>k</sup>*<sup>0</sup> *<sup>z</sup> S* is the substate vector at the inlet.

Relating the two end conditions for a given section *i*, yields the 1414 field transfer matrix *<sup>i</sup> T :* 

$$\left[\left[\mathbf{S}\right]\_{i+1} = \left[T\right]\_i \left[\mathbf{S}\right]\_i\right] \tag{21}$$

A three-dimensional tubing system can be treated as a combination of short straight lines with different orientations resulting in coupling of the fluid pressure, and forces and moments in the tubing wall. Each section of tubing is modeled by a 1414 transfer matrix with state variable vectors. Details on the assembly of the 1414 matrix can be found in Chen [11]. Each bend is broken into three straight-line segments. For these segments, the correction factor, , is used to include the Bourdon effect by replacing the flexural stiffness *EI* with *EI* in Equation (14).

A transformation matrix [*R*] transfers the force and displacement from one section to another, couples structural vibration and fluid pressure waves at points of discontinuity, and transforms the coordinate system from one section to the next. Force and moment

Noise and Vibration in Complex Hydraulic Tubing Systems 97

*S S TR I TR S*

 

 \* \* 1 1 21 21 1:14, 1:14 1:14, 1:14 <sup>2</sup> 14 14

14 ( 14) ( 14) 1 14 1

 \* \* <sup>1</sup> 21 1 21 2 32

*<sup>S</sup> TR I <sup>S</sup> TR*

To solve for the unknown state variables in Equation (31), MATLAB command "**\**", which solves the system of linear equations by Gaussian elimination, was used in the simulation.

Impedance characteristics of hydraulic components have an important effect on pressure pulsations in hydraulic circuits. These pressure oscillations lead to vibrations and are a source of noise. By using plane wave propagation theory, impedances can be estimated

Fig. 4 displays an acoustic impedance representation for the hydraulic circuit Five parameters can be used to define this system: the source impedance ( *Zs* ), source flow ripple ( *Qs* ), line impedance ( *Zc* ), line propagation constant ( ), and termination impedance ( *Zt* ).

using the two-microphone technique (ASTM E 1050-90, and ASTM C 384-108a).

*Z c <sup>s</sup> Z t*

*L*

*Pz Q z*

*z*

*<sup>a</sup> \* <sup>n</sup> n n ,n <sup>b</sup> d d n d d*

*a a*

1:14,1:14 1: ,1: 1

By rearranging the equations for all tubing sections, the global matrix is obtained:

*TR I S*

**2.4 Acoustic impedance of hydraulic system components** 

 ,1 \*

*TR I*

*n n*

0

*<sup>S</sup> <sup>S</sup> TR <sup>I</sup> <sup>S</sup>* 

*b*

Similarly, the equation of the last section of tubing can be written as:

1

1, *nn n nn* 1 1, *TR MR TR* and 1, *nn n n TR R T* .

where \*1

where \*

21 21 1 *TR TR MR* .

The source pressure ( *Ps* ) is derived from *Zs* .

*z=0*

Fig. 4. Acoustic representation of a hydraulic circuit

*Q s*

*Ps*

14 ( 14) ( 14) 1 14 1

0

(29)

(30)

0

*a*

(31)

0

0

0

*b a*

1

1 1, <sup>1</sup>

*S TR S I S*

*<sup>n</sup> <sup>c</sup> <sup>d</sup> <sup>n</sup> n n <sup>n</sup>*

*Z*

*b*

0

equilibrium, conservation of mass flow, and structural continuity are considered when deriving the transformation matrix (Chen, 1992). Finally, the relationship between one end of the system and the other is obtained by multiplying [*R*] and [*T*] for each line section:

$$\begin{bmatrix} \mathbf{S} \end{bmatrix}\_{n+1} = \begin{bmatrix} \mathbf{R} \end{bmatrix}\_{n} \begin{bmatrix} T \end{bmatrix}\_{n} \cdot \cdots \cdot \begin{bmatrix} \mathbf{R} \end{bmatrix}\_{1} \begin{bmatrix} T \end{bmatrix}\_{1} \begin{bmatrix} \mathbf{S} \end{bmatrix}\_{1} \tag{22}$$

#### **2.3 Implementation of the matrix partitioning algorithm**

The transfer matrix method solves the equations of motion step by step and determines the unknown variables (translational displacement, angular displacement, force and moment) simultaneously in the solution process. Because of the transfer matrix chain multiplication, as shown in Equation (22), numerical errors occur and build up as the multiplicative process progresses. In this study, matrix partitioning was applied to the system of equations to eliminate the long chain of matrix multiplication.

In most tubing systems, the boundary conditions at each end are defined because the tubing is attached to the pump outlet and the rotary valve inlet. Therefore, to reduce numerical error, matrix partitioning originally developed by (Clark, 1956) was used. With known boundary conditions at the ends, the state variables are re-arranged as follows:

$$\begin{bmatrix} \mathbf{S} \end{bmatrix}\_1^\* = \begin{Bmatrix} \mathbf{S}\_1^{\ a} \\ \mathbf{S}\_1^{\ b} \end{Bmatrix} \tag{23}$$

$$\begin{bmatrix} \mathbf{S} \end{bmatrix}\_{n+1}^{\*} = \begin{Bmatrix} \mathbf{S}\_{n+1} \\ \mathbf{S}\_{n+1} \end{Bmatrix} \tag{24}$$

where 1 *<sup>a</sup> S* and 1 *<sup>a</sup> Sn* are the known state variables, and 1 *<sup>b</sup> S* and 1 *<sup>b</sup> Sn* are the unknown variables.

By using the matrices *M* <sup>1</sup> *R* and *MR <sup>n</sup>*<sup>1</sup> , the following equations are obtained:

$$\begin{bmatrix} \mathbf{S} \end{bmatrix}\_1 \stackrel{\*}{=} \begin{bmatrix} MR \end{bmatrix}\_1 \begin{bmatrix} \mathbf{S} \end{bmatrix}\_1 \tag{25}$$

$$\left[\left[\mathbf{S}\right]\_{n+1}^{\*} = \left[MR\right]\_{n+1}\left[\mathbf{S}\right]\_{n+1} \tag{26}$$

$$\left[\left[S\right]\_1\right]\_1 = \left[MR\right]\_1^{-1} \left[S\right]\_1^\* \tag{27}$$

The relationship between one end and the other for the first element is:

$$\begin{bmatrix} \mathbf{S} \end{bmatrix}\_2 = \begin{bmatrix} T\mathbf{R} \end{bmatrix}\_{21} \begin{bmatrix} \mathbf{S} \end{bmatrix}\_1 \tag{28}$$

where 21 1 1 *TR R T* .

Combining Equations (27) and (28) and arranging the unknown variables on the left side, yields:

$$
\begin{bmatrix}
\begin{array}{c}
\text{TR}^\*\_{21} & - & I \\
\text{1:14, } b+1:14 & 1:14
\end{array}
\end{bmatrix}
\begin{bmatrix}
\text{S}\_1^{\;b} \\
\text{S}\_2
\end{bmatrix} = -\begin{bmatrix}
\text{TR}
\end{bmatrix}^\*\_{21} \begin{bmatrix}
\text{S}\_1^{\;a} \\
\text{0}
\end{bmatrix}
\tag{29}
$$

where \*1 21 21 1 *TR TR MR* .

96 Continuum Mechanics – Progress in Fundamentals and Engineering Applications

equilibrium, conservation of mass flow, and structural continuity are considered when deriving the transformation matrix (Chen, 1992). Finally, the relationship between one end of the system and the other is obtained by multiplying [*R*] and [*T*] for each line section:

The transfer matrix method solves the equations of motion step by step and determines the unknown variables (translational displacement, angular displacement, force and moment) simultaneously in the solution process. Because of the transfer matrix chain multiplication, as shown in Equation (22), numerical errors occur and build up as the multiplicative process progresses. In this study, matrix partitioning was applied to the system of equations to

In most tubing systems, the boundary conditions at each end are defined because the tubing is attached to the pump outlet and the rotary valve inlet. Therefore, to reduce numerical error, matrix partitioning originally developed by (Clark, 1956) was used. With known

> \* <sup>1</sup> 1

 <sup>1</sup> 1

\*

\*

1 \*

Combining Equations (27) and (28) and arranging the unknown variables on the left side,

*a \* <sup>n</sup> n b n*

*S*

1

1

 

*S* 

*S*

*S* 

*a b S*

boundary conditions at the ends, the state variables are re-arranged as follows:

*S*

*<sup>a</sup> Sn* are the known state variables, and 1

The relationship between one end and the other for the first element is:

By using the matrices *M* <sup>1</sup> *R* and *MR <sup>n</sup>*<sup>1</sup> , the following equations are obtained:

**2.3 Implementation of the matrix partitioning algorithm** 

eliminate the long chain of matrix multiplication.

where 1

variables.

yields:

*<sup>a</sup> S* and 1

where 21 1 1 *TR R T* .

*n nn* 1 1 1 1 *S RT RTS* (22)

(23)

(24)

*<sup>b</sup> Sn* are the unknown

*<sup>b</sup> S* and 1

1 1 1 *S MR S* (25)

*n nn* 1 1 1 *S MR S* (26)

1 1 1 *S MR S* (27)

*S TR S* 2 21 1 (28)

Similarly, the equation of the last section of tubing can be written as:

$$
\begin{bmatrix}
\text{TR}^\*\_{n+1,n} & \begin{matrix} \text{0} \\ \text{I}^\*\_{n+1,n} & -I \end{matrix} \\
\begin{bmatrix}
\text{1:14,1:14} & \text{1:d,1:d} \\
\text{14:(d+14)} & \text{(d+14):1}
\end{bmatrix}
\end{bmatrix}
\begin{bmatrix}
\text{S}\_n \\
\text{S}\_{n+1} \\
\text{(d+14):1}
\end{bmatrix} = 
\begin{bmatrix}
\text{S}\_{n+1}{}^a \\
\text{0} \\
\text{14:1}
\end{bmatrix}
\tag{30}
$$

where \* 1, *nn n nn* 1 1, *TR MR TR* and 1, *nn n n TR R T* . By rearranging the equations for all tubing sections, the global matrix is obtained:

 \* \* <sup>1</sup> 21 1 21 2 32 ,1 \* 1 1, <sup>1</sup> 0 0 0 0 *a b n n <sup>n</sup> <sup>c</sup> <sup>d</sup> <sup>n</sup> n n <sup>n</sup> <sup>S</sup> TR I <sup>S</sup> TR TR I S TR I S TR S I S* (31)

To solve for the unknown state variables in Equation (31), MATLAB command "**\**", which solves the system of linear equations by Gaussian elimination, was used in the simulation.

#### **2.4 Acoustic impedance of hydraulic system components**

Impedance characteristics of hydraulic components have an important effect on pressure pulsations in hydraulic circuits. These pressure oscillations lead to vibrations and are a source of noise. By using plane wave propagation theory, impedances can be estimated using the two-microphone technique (ASTM E 1050-90, and ASTM C 384-108a).

Fig. 4 displays an acoustic impedance representation for the hydraulic circuit Five parameters can be used to define this system: the source impedance ( *Zs* ), source flow ripple ( *Qs* ), line impedance ( *Zc* ), line propagation constant ( ), and termination impedance ( *Zt* ). The source pressure ( *Ps* ) is derived from *Zs* .

Fig. 4. Acoustic representation of a hydraulic circuit

The pressure ( *Pz* ) and volumetric flow velocity ( *Qz* ) at any harmonic frequency at a distance *z* along the line are:

$$P\_z = P\_i e^{-\Gamma z} + P\_r e^{\Gamma z} \tag{32}$$

Noise and Vibration in Complex Hydraulic Tubing Systems 99

The measuring pressure signals before the valve, the termination impedance can be readily determined by Equation (37) and (40). Fig. 5 displays the estimated impedance of the rotary valve in power steering system at various opening positions. The data show that modeling this valve as a pure resistance is not appropriate as a strong reactive component of the

0 400 800 1200 1600

0 400 800 1200 1600

**Frequency (Hz)**

**Frequency (Hz)**

and 1J denote the zero- and first-order Bessel functions of the first kind, respectively.

is the angular frequency; *ir* is the inner radius of

is the density and kinematic viscosity of the fluid, respectively; and 0J

Fully turned wheel Half turned wheel

Steering wheel at neutral position

where *c* is the sound speed;

transmission line;

impedance is apparent.

109


Fig. 5. Amplitude and phase angle of the valve impedance


0

**Phase angle (deg)** 

100

200

1010

**Amplitude (Pa-s/m3)** 

1011

1012

$$Q\_z = \frac{1}{Z\_c} \left( P\_i e^{-\Gamma z} - P\_r e^{\Gamma z} \right) \tag{33}$$

where *Pi* and *Pr* are the complex incident and reflected pressures, respectively.

The termination reflection coefficient, *Cr* , is defined by the ratio of the reflected pressure to the incident pressure:

$$\mathbf{C}\_r = \frac{P\_r}{P\_i} \tag{34}$$

The pressures at locations 1*z* and 2 *z* are:

$$P\_{z1} = P\_i e^{-\Gamma \overline{z}\_1} + P\_r e^{\Gamma \overline{z}\_1} \tag{35}$$

$$P\_{z2} = P\_i e^{-\Gamma z\_2} + P\_r e^{\Gamma z\_2} \tag{36}$$

Solving Equations (35) and (36) for *Pi* and *Pr* , and substituting into Equation (34), the reflection coefficient is obtained:

$$R = \frac{-\left(P\_{z1}e^{-\Gamma z\_2} - P\_{z2}e^{-\Gamma z\_1}\right)}{P\_{z1}e^{\Gamma z\_2} - P\_{z2}e^{\Gamma z\_1}}\tag{37}$$

If the impedance of the termination is *Zt* , applying Equations (32) and (33) at the boundary *z* 0 (*i.e., z L* ) gives:

$$\left(\left(P\_i + P\_r\right)\_{z^\circ = 0} = \left(P\_t\right)\_{z^\circ = 0}\right) \tag{38}$$

$$\left[\left(P\_i - P\_r\right)\middle|Z\_c\right]\_{z^\circ = 0} = \left(P\_t \middle|Z\_t\right)\_{z^\circ = 0} \tag{39}$$

By rearranging the above two equations and combining with Equation (34), the termination impedance can be represented in terms of *Zc* and *Cr* :

$$Z\_t = \left(\frac{1 + \mathcal{C}\_r}{1 - \mathcal{C}\_r}\right) Z\_c \tag{40}$$

*Zc* is the characteristic impedance in the tubing given by Chen and Hastings (1992):

$$Z\_c = \frac{\rho\_f}{\pi} \frac{c}{r\_i^2} \left[1 - \frac{2}{j r\_i \sqrt{j \alpha / \nu}} \frac{\mathbf{J}\_1(j r\_i \sqrt{j \alpha / \nu})}{\mathbf{J}\_0(j r\_i \sqrt{j \alpha / \nu})}\right]^{-\frac{1}{f\_2}}\tag{41}$$

The pressure ( *Pz* ) and volumetric flow velocity ( *Qz* ) at any harmonic frequency at a

*z z P Pe Pe zi r*

The termination reflection coefficient, *Cr* , is defined by the ratio of the reflected pressure to

*<sup>r</sup> <sup>r</sup> i <sup>P</sup> <sup>C</sup>*

*z z P Pe Pe zi r*

*z z P Pe Pe zi r*

Solving Equations (35) and (36) for *Pi* and *Pr* , and substituting into Equation (34), the

*z z z z z z*

*Pe Pe*

If the impedance of the termination is *Zt* , applying Equations (32) and (33) at the boundary

By rearranging the above two equations and combining with Equation (34), the termination

1 1 *<sup>r</sup> t c r*

*Zc* is the characteristic impedance in the tubing given by Chen and Hastings (1992):

*f i*

*ii i <sup>c</sup> jr j <sup>Z</sup> r jr j jr j*

2

 *<sup>C</sup> Z Z C* 

1

0 <sup>2</sup> J( ) <sup>1</sup> J( )

*Pe Pe*

 2 1 2 1 1 2 1 2

*z z*

 

1 1

2 2

where *Pi* and *Pr* are the complex incident and reflected pressures, respectively.

1

2

*R*

impedance can be represented in terms of *Zc* and *Cr* :

*c*

 <sup>1</sup> *z z z ir c Q Pe Pe <sup>Z</sup>*

(32)

(33)

*<sup>P</sup>* (34)

(35)

(36)

(37)

'0 '0 *ir t z z PP P* (38)

1 2

 

 

(40)

(41)

' 0 ' 0 *i r c tt <sup>z</sup> <sup>z</sup> P P Z PZ* (39)

distance *z* along the line are:

the incident pressure:

The pressures at locations 1*z* and 2 *z* are:

reflection coefficient is obtained:

*z* 0 (*i.e., z L* ) gives:

where *c* is the sound speed; is the angular frequency; *ir* is the inner radius of transmission line; is the density and kinematic viscosity of the fluid, respectively; and 0J and 1J denote the zero- and first-order Bessel functions of the first kind, respectively.

The measuring pressure signals before the valve, the termination impedance can be readily determined by Equation (37) and (40). Fig. 5 displays the estimated impedance of the rotary valve in power steering system at various opening positions. The data show that modeling this valve as a pure resistance is not appropriate as a strong reactive component of the impedance is apparent.

Fig. 5. Amplitude and phase angle of the valve impedance

Noise and Vibration in Complex Hydraulic Tubing Systems 101

Four piezoelectric pressure transducers are used to measure the dynamic pressure in this system. The first pressure transducer (P1) is placed at the outlet of the pump to measure the source of pressure disturbance. The fourth one (P4) is located at the inlet of the rotary valve so that the amplitude ratio of outlet pressure to inlet pressure (P4/P1) can be measured and compared to the model prediction. The pressure signals are connected to the Kistler Charge amplifier and then to a HP3566A 8-channel analyzer. Data are saved in a computer and

Because the focus of this study is to investigate the fluidborne noise propagation in the tubing system and interaction with the tubing structure, the pressure frequency response of the tubing transmission line is investigated. To correlate the transfer matrix model with better accuracy, the sound speed in steel tubing and damping factor are experimentally estimated (Chen, 2001). The sound speed was optimized to be 1374 m/s. The frequencydependent damping in the system was estimated based on the Half-Power method. Figs. 7 and 8 display the frequency response for the outlet pressure of the pressure side transmission line (P4) with different valve opening due to the steering wheel positions for an all steel tubing system. The model prediction and experimental data match very well.

0 250 500 750 1000

Simulation Experiment

**Frequency (Hz)**

Fig. 7. Pressure response of an all steel tubing system with a fully turned steering wheel.

retrieved later for further analysis.

Good agreement was obtained.




**Pressure amplitude ratio, P4/P1 (dB)**

0

5

10

## **3. Experimental results**

An automotive hydraulic power steering tubing system was tested in this research. Detailed layout of this three-dimensional tubing transmission line was provided by the manufacturer. Since this study addresses pump induced noise, a system with a pump source was set up to verify the transfer matrix model for the tubing system. Fig. 6 illustrates the system layout. This includes the power steering pump, hydraulic transmission lines, a rack and pinion unit, steering wheel and column, and rotary valve. The steering pump is driven by an electric motor through a belt. A variable speed, AC controller is used to control the electric motor and vary the speed of pump. In this setup, a water-cooling system using a coil heat exchanger is used. Water from the building supply circulates through the coil heat exchanger connected to the return line and then flows into a drain.

Fig. 6. Experimental setup for an automotive hydraulic power steering tubing system

An automotive hydraulic power steering tubing system was tested in this research. Detailed layout of this three-dimensional tubing transmission line was provided by the manufacturer. Since this study addresses pump induced noise, a system with a pump source was set up to verify the transfer matrix model for the tubing system. Fig. 6 illustrates the system layout. This includes the power steering pump, hydraulic transmission lines, a rack and pinion unit, steering wheel and column, and rotary valve. The steering pump is driven by an electric motor through a belt. A variable speed, AC controller is used to control the electric motor and vary the speed of pump. In this setup, a water-cooling system using a coil heat exchanger is used. Water from the building supply circulates through the coil heat

> GP-2000 Variable speed AC controller

Sink

Load cell

Rotary valve

Steering column Steering wheel

Cooling system

Water in

Power steering pump driven by a belt

Return line

exchanger connected to the return line and then flows into a drain.

2 HP Electric motor

Fig. 6. Experimental setup for an automotive hydraulic power steering tubing system

P1

A1

Rack and pinion unit

P3 P4

**3. Experimental results** 

Pressure line

Stain-gage pressure transducer

Coaxial hose and tuning cable

HP Optical encoder

A2

P2

Four piezoelectric pressure transducers are used to measure the dynamic pressure in this system. The first pressure transducer (P1) is placed at the outlet of the pump to measure the source of pressure disturbance. The fourth one (P4) is located at the inlet of the rotary valve so that the amplitude ratio of outlet pressure to inlet pressure (P4/P1) can be measured and compared to the model prediction. The pressure signals are connected to the Kistler Charge amplifier and then to a HP3566A 8-channel analyzer. Data are saved in a computer and retrieved later for further analysis.

Because the focus of this study is to investigate the fluidborne noise propagation in the tubing system and interaction with the tubing structure, the pressure frequency response of the tubing transmission line is investigated. To correlate the transfer matrix model with better accuracy, the sound speed in steel tubing and damping factor are experimentally estimated (Chen, 2001). The sound speed was optimized to be 1374 m/s. The frequencydependent damping in the system was estimated based on the Half-Power method. Figs. 7 and 8 display the frequency response for the outlet pressure of the pressure side transmission line (P4) with different valve opening due to the steering wheel positions for an all steel tubing system. The model prediction and experimental data match very well. Good agreement was obtained.

Fig. 7. Pressure response of an all steel tubing system with a fully turned steering wheel.

Noise and Vibration in Complex Hydraulic Tubing Systems 103

Bundy, D.D.; Wiggert, D.C. & Hatfield F.J. (1991). The Influence of Structural Damping on

Chen C.-C. (1992). A Theoretical Analysis of Noise Reduction in Automotive Power

Chen C.-C. & Hastings M.C. (1992). Noise reduction in Power Steering Transmission Lines.

Chen C.-C. & Hastings M.C. (1994). Half-wavelength Tuning Cable for Passive Noise

Chen C.-C. (2001). An Investigation of Noise and Vibration in an Automotive Power

Clark RA. Torsional Wave Propagation in Hollow Cylindrical Bars. Journal of Acoustical

Davidson L.C. & Smith, J.E (1969). Liquid-Structure Coupling in Curved Pipes. The Shock

Davidson L.C. & Samsury D.R. (1972). Liquid-Structure Coupling in Curved Pipes – II. The

Everstine G.C. (1986). Dynamic Analysis of Fluid-Filled Piping Systems Using Finite

Hatfield F.J. & Davidson L.C. (1983). Experimental Validation of the Component Synthesis

Hatfield, F.J. , Wiggert D.C. & Otwell R.S. (1982). Fluid Structure Interaction in Piping by

Nanayakkara S. & Perreia, N.D. (1986). Wave Propagation and Attenuation in Piping

Reissner E., Clark R.A. & Gilroy R.I. (1952). Stresses and Deformations of Torsional Shells of

Tentarelli S.C. (1990). Propagation of Noise and Vibration in Complex Hydraulic

To C.W.S. & Kaladi V. (1985). Vibration of Piping Systems Containing a Moving Medium.

Element Techniques. Journal of Pressure Vessel Technology, 1986, Vol. 10, pp. 57-

Method for Prediction Vibration of Liquid-Filled Piping. The Shock and Vibration

Component Synthesis. ASME Journal of Fluid Engineers, 1982, Vol. 104, pp. 318-

Systems. Journal of Vibration, Acoustics, Stress, and Reliability in Design, 1986,

an Elliptical Cross Section with Applications to the Problems of Bending of Curved Tubes and the Bourdon Gage. Transaction of ASME, Journal of Applied Mechanics,

Tubing System. Ph.D. Dissertation, 1990, Lehigh University, Bethlehem,

Transaction of ASME, Journal of Pressure Vessel Technology, 1985. Vol. 107, pp.

Noise (K. W. Wang et al., eds.), 1994, ASME DE-75, pp. 355-361

Society of America, 1956; 28 (6), pp. 1163-1165

Bulletin, 1983, Vol. 53, No. 2, pp. 1-10

and Vibration Bulletin, 1969, Vol. 40, No. 4, pp. 197-207

Shock and Vibration Bulletin, 1972, Vol. 43. No. 1, pp. 123-135

113, pp. 424-579

Columbus, Ohio

1, pp. 67-72

Ohio

61

325

Vol. 108, pp. 441-446

1952, pp.37-48

Pennsylvania

344-349

Internal Pressure during a Transient Flow, ASME Journal of Fluid Engineers, Vol.

Steering Transmission Lines. M.S. Thesis, 1992, The Ohio State University,

Proceedings of the International Congress on Noise Control Engineering, 1992, Vol.

Control in Automotive Power Steering Systems. Active Control of Vibration and

Steering System. Ph.D. Dissertation, 2001, The Ohio State University, Columbus,

Fig. 8. Pressure response of an all steel tubing system with a steering wheel at neutral position

#### **4. Conclusions**

A distributed-parameter transfer-matrix model is developed to predict the fluidborne noise in a complex tubing system. This study provides a systematic approach to predict the pump-induced fluidborne noise by incorporating the experimentally determined acoustic characteristics of valve termination. The developed model was supported by experimental measurement with good agreements. Inclusion of Poisson and Bourdon effects in the model provide better predictions. Furthermore, the transfer matrix-partitioning algorithm presented here not only can reduce truncation error but also be more efficient in comparison with the matrix chain multiplication. It is also noted that the damping of the tubing system needs to be included to better predict the peak amplitude. The mathematical model presented can be applied to the analysis of noise in other hydraulic systems, such as those used in air conditioners and power plants. However, to fully characterize the noise propagation/transmission in the tubing system, SBN (not presented here, but can also be predicted by the developed model) should also be investigated because of fluid-structure interaction.

#### **5. References**

Brown F.T. & Tentarelli S.C. (1988). Analysis of Noise and Vibration in Complex Tubing System with Fluid-Filled Interactions. 43rd National Conference on Fluid Power, 1988, pp. 139-149

0 250 500 750 1000

Simulation Experiment

**Frequency (Hz)**

A distributed-parameter transfer-matrix model is developed to predict the fluidborne noise in a complex tubing system. This study provides a systematic approach to predict the pump-induced fluidborne noise by incorporating the experimentally determined acoustic characteristics of valve termination. The developed model was supported by experimental measurement with good agreements. Inclusion of Poisson and Bourdon effects in the model provide better predictions. Furthermore, the transfer matrix-partitioning algorithm presented here not only can reduce truncation error but also be more efficient in comparison with the matrix chain multiplication. It is also noted that the damping of the tubing system needs to be included to better predict the peak amplitude. The mathematical model presented can be applied to the analysis of noise in other hydraulic systems, such as those used in air conditioners and power plants. However, to fully characterize the noise propagation/transmission in the tubing system, SBN (not presented here, but can also be predicted by the developed model) should also be investigated because of fluid-structure

Brown F.T. & Tentarelli S.C. (1988). Analysis of Noise and Vibration in Complex Tubing

System with Fluid-Filled Interactions. 43rd National Conference on Fluid Power,

Fig. 8. Pressure response of an all steel tubing system with a steering wheel at neutral


position

interaction.

**5. References** 

1988, pp. 139-149

**4. Conclusions** 



**Pressure amplitude ratio, P4/P1 (dB)**

0

5

10


**Analysis Precision Machining Process** 

Machining is the process of removing the material in the form of chips by means of wedge shaped tool[1]. The need to manufacture high precision items and to machine difficult-to-cut materials led to the development of the newer machining processes. The dimensional tolerance achieved by precision machining technology is on the order of 0.01 μm and the surface roughness is on the order of 1 nm. The dimensions of the parts or elements of the parts produced may be as small as 1 μm, and the resolution and the repeatability of the machine used must be of the order of 0.01 μm (10 nm). The accuracy targets for ultraprecision component cannot be achieved by a simple extension of conventional machining processes and techniques. They are called precision machining processes, notwithstanding that the definition of conventional and traditional changes with time. Unlike conventional machining processes, precision machining processes are not based on the removing the metal in the form of chips using a wedge shaped tool. There are a variety of ways by which the material may be removed in precision machining processes. Some of them are abrasion

by abrasive particles, impact of water, thermal action, chemical action and so on.

world wide convinced precision materials removal theory is not built up until now.

There are two basic approaches to the analysis of metal cutting process, namely, the analytical and the numerical method. As the complexity associate with the precision machining process, which involve high strains, strain rates, size effects and temperature, various simplifications and idealizations are necessary and therefore important machining features such as the strain

When metal is removed by machining there is substantial increase in the specific energy required with decrease in chip size. It is generally believed this is due to the fact that all metals contain defects (grain boundaries, missing and impurity atoms, etc.), and when the size of the material removed decreases, the probability of encountering a stress-reducing defect decreases. Since the shear stress and strain in metal cutting is unusually high, discontinuous microcracks usually form on the metal-cutting shear plane. If the material being cut is very brittle, or the compressive stress on the shear plane is relatively low, microcracks grow into gross cracks giving rise to discontinuous chip formation[2]. When discontinuous microcracks form on the shear plane they weld and reform as strain proceeds, thus joining the transport of dislocations in accounting for the total slip of the shear plane. In the presence of a contaminant, the rewelding of microcracks decreases, resulting in decrease in the cutting force required for chip formation. Owing to the complexity of elastic-plastic deformation at nanometer scale, the

**1 Introduction** 

**Using Finite Element Method** 

*School of Mechanical Engineering, Tianjin University,* 

Xuesong Han

*P.R. China* 

Vigness I. (1943). Elastic Properties of Curved Tubes. Transaction of ASME, 1943, Vol.65, pp.105-117 **6** 
