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Research Papers

Modeling Dynamic Response of Hydraulic Fluid Within Tapered Transmission Lines

[+] Author and Article Information
Jeremy W. ven der Buhs

Department of Mechanical Engineering,
University of Saskatchewan,
Saskatoon, SK S7N 5A9, Canada
e-mail: jeremy.venderbuhs@usask.ca

Travis K. Wiens

Department of Mechanical Engineering,
University of Saskatchewan,
Saskatoon, SK S7N 5A9, Canada
e-mail: t.wiens@usask.ca

1Corresponding author.

Contributed by the Dynamic Systems Division of ASME for publication in the JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT,AND CONTROL. Manuscript received June 28, 2017; final manuscript received June 21, 2018; published online July 23, 2018. Assoc. Editor: Heikki Handroos.

J. Dyn. Sys., Meas., Control 140(12), 121008 (Jul 23, 2018) (13 pages) Paper No: DS-17-1329; doi: 10.1115/1.4040667 History: Received June 28, 2017; Revised June 21, 2018

This paper examines modeling of the laminar dynamic fluid responses within hydraulic transmission lines that have a tapered shape between the inlet and the outlet. There are excellent models available for fast simulation of pressure and flow dynamics within uniform lines; however, the established models for tapered lines either cannot be implemented in the time domain, are complex to implement, or have long simulation times. The enhanced transmission line method (TLM) structure is applied in this paper since it can be computed quickly in the time domain and has shown to accurately model the effects of frequency-dependent friction. This paper presents a method of optimizing the TLM weighting functions, minimizing the error between the TLM transmission matrix terms and a numerical ordinary differential equation (ODE) solution calculated using a boundary value solver. Optimizations have shown that using the TLM to model tapered lines can provide a fair approximation when compared in the frequency domain. Two-dimensional (2D) interpolation of a look-up table is possible allowing for quick selection of the optimized parameters. Further investigation into the effects of pipe wall elasticity and its inclusion into the TLM is also performed. Also, an experiment was performed to validate high frequency harmonic peaks present in the frequency response, which yielded acceptable results when compared to the theory, and the proposed tapered TLM. This model can be used in numerous applications where line dynamic effects must be accounted for, especially with digital hydraulic switched inertance converters where high frequencies are present.

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References

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ven der Buhs, J. , and Wiens, T. , 2017, “ Transmission Line Models,” Linköping, Sweden, accessed July 3, 2018 https://github.com/tkw954/usask_tlm
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Figures

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Fig. 1

Block diagram of the TLM [1]

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Fig. 2

Schematic of a rigidly walled tapered transmission line. For one-dimensional flow, θ is assumed to be small.

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Fig. 3

Transmission matrix frequency response for β=0.001 and λ=0.75

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Fig. 4

Transmission matrix frequency response for β=0.1 and λ=0.9

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Fig. 5

Error analysis for tapered TLM. The black lines show the region of acceptable error, as defined by ε<0.5.

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Fig. 6

Schematic of a tapered transmission line with elastic wall effects. Note that constant wall thickness is maintained throughout the length of the pipe.

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Fig. 7

Normalized wave speed as a function of axial position within a tapered elastic pipe for different pipe stiffness values. Shows trend as pipe becomes more rigid, the wave speed slope tends to 0. Fluid properties and pipe dimensions are given in Table 1.

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Fig. 8

Scaled wave speed as a function of axial position within a tapered elastic pipe for different pipe stiffness values. Shows as pipe becomes more rigid, the wave speed function also tends toward linearity. Fluid properties and pipe dimensions are given in Table 1.

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Fig. 9

Computer-aided design drawing of the experimental apparatus 3D printed out of acrylonitrile butadiene styrene. Note: there is a break in the drawing.

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Fig. 10

A MLS digital sequence implements in the time domain. There are 20 samples shown at a sampling frequency of 10 kHz.

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Fig. 11

Comparison between the observed experimental result, the numerical ODE solution, and the proposed tapered TLM. Note the notch at 3.13 kHz, this is confidently believed to be the result of line vibration, a phenomenon extensively studied and quantified by D'Souza and Oldenburger [14].

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Fig. 12

matlabsimulink model of the tapered TLM

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Fig. 13

Simulated results for β=0.001 and λ=0.75

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Fig. 14

Simulated results for β=0.1 and λ=0.9

Tables

Errata

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