**Noise and Vibration in Complex Hydraulic Tubing Systems**

Chuan-Chiang Chen *Mechanical Engineering Department, California State Polytechnic University Pomona, USA* 

## **1. Introduction**

88 Continuum Mechanics – Progress in Fundamentals and Engineering Applications

Xiao, C., Mirshams, R., Whang, S., & Yin, W. (2001). Tensile behavior and fracture in nickel

Zhang, T., Zhu, Q., Huang, W., Xie, Z., & Xin, X. (2008). Stress field and failure probability

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Vol. 182, (2008), pp. (540-545).

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analysis for the single cell of planar solid oxide fuel cells. *Journal of Power Sources*,

In hydraulic systems, pumps are the major source of noise and vibration. It generates flow ripples which interact with other hydraulic components, such as transmission lines and valves to create harmonic pressure waves, *i.e.*, fluid-borne noise (FBN). Fig. 1 shows a typical oscillating pressure measured at the outlet of a ten-vane pump running at 1500 rpm. Fig. 2 gives the frequency spectrum for the pressure signal which contains harmonic components of the fundamental frequency, 25 Hz, which correlates with the pump operating speed. The largest peak is at 250 Hz, which corresponds to the shaft speed times the number of the pumping elements (10 vanes in this case). The FBN propagates along as well as interacts with the tubing and other components to result in airborne noise (ABN) and structure-borne noise (SBN, *i.e.*, structural vibration). These noises can become excessive, and lead to damage the tubing system and other components. Therefore, to study the pressure wave propagation in the hydraulic tubing system, it is important to take the fluid-structure interaction into account to further the understanding of noise transmission mechanism.

Fluid-structure interaction can be divided into three categories: junction coupling, Poisson coupling, and Bourdon coupling. Junction coupling occurs at discontinuities, such as bends and tees, where the pressure interacts with the structure to cause structural vibration. In unsteady flow, the pressure varies along the tube. Differences in pressure exert axial and transverse forces during power transmission at bends and other locations where the diametrical geometry changes. Moreover, the pressure is related to the longitudinal stresses in the pipe because of the radial contraction or expansion via Poisson coupling (Hatfield & Davidson, 1983). Furthermore, the cross-sectional shape of the line in a bend is not circular because of action by the bending forces. This effect, known as the Bourdon effect (Tentarelli, 1990), influences the structural modes at low frequencies.

Several approaches have been used (To & Kaladi, 1985; Everstine 1986; Nanayakkara & Perreia, 1986), such as the transfer matrix and finite element (FEM) methods, to model the fluid-structural coupling. In this study, the transfer matrix method (TMM) is used because of its simplicity. Even though FEM may offer better accuracy, it is more complicated and time-consuming than TMM.

Noise and Vibration in Complex Hydraulic Tubing Systems 91

structural damping which was neglected by other researchers in previous experimental and

Brown and Tentarelli (1988) arranged the 1414 transfer matrices for *n* segments and then assembled them into a global 14(n-1)14(n-1) sparse matrix. This approach was beneficial because, by solving the linear equations, the state variables at every point were obtained. Their algorithm also avoided round-off error at higher frequencies. Fluid friction was not considered in their analysis. Chen (1992), and Chen and Hastings (1992; 1994) considered both the fluid-structure interaction caused by discontinuities and the viscosity of the fluid in a distributed parameter, transfer matrix model of the transmission line in an automotive

Most researchers verified their models with a simplified experimental system; for example, L-tube or U-tube systems. Until now, the system model has not been verified in a complex tubing system. In this book, a transfer matrix system model incorporating the acoustic characteristics of termination is developed to predict the fluidborne noise in a complex three-dimensional tubing system. The results show good agreements between simulated

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

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

> 2 2

(2)

(1)

1 1

*z z z*

*v p v v t z rr r*

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

0 *rr z 1 p vv v β t rr z* 

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

*f*

 

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

theoretical investigations.

power steering system.

and experimental data.

segment used in the following analysis.

equations reduce to (Chen, 2001):

the steady state, respectively.

is the fluid bulk modulus.

where 

obtained:

**2. Analysis** 

**2.1 Axial motion** 

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 pumping mechanism.

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

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 structural damping which was neglected by other researchers in previous experimental and theoretical investigations.

Brown and Tentarelli (1988) arranged the 1414 transfer matrices for *n* segments and then assembled them into a global 14(n-1)14(n-1) sparse matrix. This approach was beneficial because, by solving the linear equations, the state variables at every point were obtained. Their algorithm also avoided round-off error at higher frequencies. Fluid friction was not considered in their analysis. Chen (1992), and Chen and Hastings (1992; 1994) considered both the fluid-structure interaction caused by discontinuities and the viscosity of the fluid in a distributed parameter, transfer matrix model of the transmission line in an automotive power steering system.

Most researchers verified their models with a simplified experimental system; for example, L-tube or U-tube systems. Until now, the system model has not been verified in a complex tubing system. In this book, a transfer matrix system model incorporating the acoustic characteristics of termination is developed to predict the fluidborne noise in a complex three-dimensional tubing system. The results show good agreements between simulated and experimental data.
