Influence of structural flexibility on the nonlinear stiffness of hydraulic system

Abstract

Hydraulic system has been widely used in many mechatronic systems. Accurate identification of the hydraulic stiffness is critical to the design and control of such kind of system. It is widely recognized that the nonlinear hydraulic stiffness is influenced by many factors such as the compressibility of the fluid, the flexibility of the fluid supply circular tube, and the working status of the system. It is very difficult to accurately formulate the hydraulic stiffness due to the complex coupling effects. In this article, the concept of the volume modulus is first introduced to characterize the flexibility of the structure as a container. A hydraulic cylinder consisting of flexible circular tubes is used as an example to illustrate the relationship between the volume modulus and Young’s modulus of the circular tube. A novel formulation of the hydraulic stiffness is then proposed by taking into account the structural flexibility via the volume modulus of the circular tube. Finally, the influences of the circular tube parameters on the hydraulic stiffness are analyzed. Experiments are also carried out to verify the presented formulation. The proposed method can be used to design hydraulic system for achieving desired static and dynamic performances.

Keywords

Hydraulic system stiffness structural flexibility volume modulus influence

Introduction

Hydraulic system has been widely used in many applications such as machine tools,¹ metrology equipment,² motion and vibration simulators,^3,4 and vibration isolation platforms.⁵ The stiffness characteristics of hydraulic system are critical to the estimation of deformation, vibration frequencies, dynamic responses, and so on. A large amount of researches have been focused on the design, simulation, and control of hydraulic system^1–8 so as to achieve desired performances. It has been widely recognized that the static and dynamic performances of a hydraulic system are significantly affected by the nonlinear stiffness characteristics of the system.

The stiffness characteristics of a hydraulic system are determined by various factors,^9,10 among which the stiffness of each hydraulic cylinder is the most important one. Conventionally, the stiffness of a hydraulic cylinder is thought to be the result of deformation of pressurized fluid in the cylinder. Therefore, the stiffness is determined by the bulk modulus of fluid, the sectional area, and length at both sides of the piston.¹¹ Practice shows that the error of hydraulic stiffness identified using the traditional method is often unacceptable. Usually, the hydraulic cylinder is relatively large, and the influence of the cylinder flexibility on the hydraulic stiffness cannot be ignored. In some other engineering applications, relatively smaller cylinder and longer fluid supply circular tube are utilized, and the volume of the circular tube might be relatively large. The deformation of circular tube due to pressure fluctuation might be considerable and it results in decrease in stiffness of the hydraulic cylinder. Under this circumstance, the influence of the circular tube might be significant. The volume and volume modulus of circular tube and cylinder need to be taken into account in calculating the stiffness of the hydraulic cylinder. However, it is unclear about the relationships between the volume modulus of the circular tube and the related parameters of the circular tube.

In this article, the nonlinear stiffness of a hydraulic system when considering the flexibility of circular tube is investigated. The volume modulus of the circular tube is first derived to formulate the stiffness of a single hydraulic cylinder, and the influences of circular tube parameters on the stiffness of hydraulic cylinder are analyzed.

Volume modulus of a hydraulic system

The fluctuation of the fluid pressure will lead to deformation of the structure in the hydraulic system. In this section, the concept of volume modulus is introduced, which is defined as the ratio of infinitesimal pressure increment to the resulting relative increment in the capacity of the container. It will be found that the volume modulus is very helpful for formulating the hydraulic stiffness when the structural flexibility must be considered.

Considering an infinitely long circular tube under fluid pressure, as shown in Figure 1, $ξ = r_{o} / r_{i}$ denotes the radius ratio of the circular tube, and p denotes the fluid pressure. $σ_{r}$ , $σ_{c}$ , and $σ_{a}$ are the radial, the circumferential, and the axial stress, respectively, in the circular tube at position with radius r. According to Lame’s formula, the stress in three directions in the circular tube can be expressed as

σ_{r} = \frac{1 - r_{o}^{2} / r^{2}}{ξ^{2} - 1} \cdot p

(1)

σ_{c} = \frac{1 + r_{o}^{2} / r^{2}}{ξ^{2} - 1} \cdot p

(2)

σ_{a} = \frac{1}{ξ^{2} - 1} \cdot p

(3)

Figure 1.

Dimensions and stress in a pressurized circular tube.

$E_{h}$ and $μ_{h}$ denote Young’s modulus and Poisson’s ratio, respectively, of the circular tube. Notice that there might be differences between Young’s modulus of a circular tube along different directions. In this article, the circular tube is regarded as isotropic one so as to simplify the formulation and illustrate the method conveniently. If the difference along different directions must be considered, the results can also be derived in a similar way.

Considering that the shearing stress along three directions is equal to zero in such kind of circular tube, the strains in three directions at the point with radius r in the body of the circular tube can be expressed as

ε_{r} = (\frac{1 - 2 μ}{ξ^{2} - 1} - \frac{1 + μ}{ξ^{2} - 1} \cdot \frac{r_{o}^{2}}{r^{2}}) \cdot \frac{p}{E_{h}}

(4)

ε_{c} = (\frac{1 - 2 μ}{ξ^{2} - 1} + \frac{1 + μ}{ξ^{2} - 1} \cdot \frac{r_{o}^{2}}{r^{2}}) \cdot \frac{p}{E_{h}}

(5)

ε_{a} = \frac{1 - 2 μ}{ξ^{2} - 1} \cdot \frac{p}{E_{h}}

(6)

Suppose that the increment in pressure is $Δ p$ , and the increments in length and inner radius of the circular tube are $Δ l$ and $Δ r$ , respectively. The increment in volume of the circular tube can be approximated as $Δ V_{h} = 2 π r_{i} l Δ r + π r_{i}^{2} Δ l$ . The variables $Δ l$ and $Δ r$ can be calculated using axial strain $Δ ε_{a}$ and tangential strain $Δ ε_{c}$ .

The increments of strains in tangential and axial directions are

Δ ε_{c} = (\frac{1 - 2 μ}{ξ^{2} - 1} + \frac{1 + μ}{ξ^{2} - 1} \cdot \frac{r_{o}^{2}}{r^{2}}) \cdot \frac{Δ p}{E_{h}}

(7)

Δ ε_{a} = \frac{1 - 2 μ}{ξ^{2} - 1} \cdot \frac{Δ p}{E_{h}}

(8)

The variables $Δ l$ and $Δ r$ can be calculated using $Δ ε_{a}$ and $Δ ε_{c}$ as follows

Δ l = \int_{0}^{l} Δ ε_{a} d l = \frac{1 - 2 μ}{ξ^{2} - 1} \cdot l \cdot \frac{Δ p}{E_{h}}

(9)

Δ r = r_{i} {Δ ε_{c} |}_{r = r_{i}} = (1 + μ + \frac{2 - μ}{ξ^{2} - 1}) \cdot r_{i} \cdot \frac{Δ p}{E_{h}}

(10)

The increment in volume of the circular tube can thus be derived as

Δ V_{h} = \frac{2 (1 + μ) ξ^{2} + 3 - 6 μ}{ξ^{2} - 1} \cdot π r_{i}^{2} l \cdot \frac{Δ p}{E_{h}}

(11)

Considering that the nominal volume of the circular tube is $V_{h} = π r_{i}^{2} l$ , the volume modulus of the circular tube $K_{h}$ is defined as

K_{h} = Δ p \frac{V_{h}}{Δ V_{h}} = \frac{ξ^{2} - 1}{2 (1 + μ) ξ^{2} + 3 - 6 μ} E_{h}

(12)

$η$ , the ratio of the volume modulus to Young’s modulus of the circular tube, is defined as follows

η = \frac{K_{h}}{E_{h}} = \frac{ξ^{2} - 1}{2 (1 + μ) ξ^{2} + 3 - 6 μ}

(13)

The relationship between $ξ$ and $η$ under different Poisson’s ratios $μ$ is shown in Figure 2. The results indicate the following:

The value of $η$ depends on both the radius ratio $ξ$ and Poisson’s ratio $μ$ and is mainly influenced by radius ratio $ξ$ . Notice that $η \equiv 2 / 9$ for $ξ = \sqrt{3}$ .

For a circular tube with thickness smaller than its inner radius, that is, radius ratio $ξ \leq 2$ , and the volume modulus $K_{P}$ is usually less than one-quarter of Young’s modulus of the circular tube. For a circular tube with radius ratio smaller than 1.25, the value of $η$ is smaller than 0.1.

It will be illustrated in the following section that the volume modulus of the circular tube has significant influence on the hydraulic stiffness.

Figure 2.

The relationship between $ξ$ and $η$ under different Poisson’s ratios $μ$ .

Stiffness considering the flexibility of the circular tube

Figure 3 is the sketch map of a hydraulic cylinder. The meaning and value of all parameters are listed in Table 1. The effective sectional areas at two sides of the piston are $A_{1} = π d_{1}^{2} / 4$ and $A_{2} = π (d_{1}^{2} - d_{2}^{2}) / 4$ . $V_{C} = A_{1} s$ denotes the nominal volume of the cylinder, and x denotes the position of the piston.

Figure 3.

The sketch map of the hydraulic limb.

Considering the hydraulic system, as shown in Figure 3, $V_{h 1}$ and $V_{h 2}$ denote the volumes of the circular tubes at the left and right side of the hydraulic cylinder, respectively, and $s_{1} = V_{h 1} / A_{1}$ and $s_{2} = V_{h 2} / A_{2}$ denote the equivalent length converted to the hydraulic cylinder. The bulk modulus of the fluid is $K_{F} = - Δ pV / Δ V$ , where V is the volume of the fluid, $Δ p$ and $Δ V$ are the variations of the pressure P and the volume V, respectively. On the assumption that there are no deformations for the hydraulic cylinder and the circular tubes during pressure fluctuation, and neither leaks nor pressure loss is considered, the conventional model of hydraulic stiffness can be easily obtained as follows

k = \frac{K_{F} A_{1}}{s_{1} + x} + \frac{K_{F} A_{2}}{s_{2} + s - x}

(14)

In fact, the volumes or, namely, the capacity of the circular tubes will also change due to the pressure fluctuation. Therefore, the deformation of the circular tube must be taken into account in calculating hydraulic stiffness.

$δ_{1} = - δ_{2} = δ$ denotes the displacement of the piston, as shown in Figure 3. The variation of volumes of fluid in the cylinder in both sides of the piston can be expressed as

Δ V_{i} = - A_{i} δ_{i} + Δ V_{hi} (i = 1, 2)

(15)

According to the definition of bulk modulus of the fluid, one can obtain that

K_{F} Δ V_{i} = - Δ p_{i} (V_{hi} + A_{i} u_{i}) (i = 1, 2)

(16)

where $u_{1} = x$ , $u_{2} = s - x$ . Considering that $K_{hi} = Δ p_{i} (V_{hi} / Δ V_{hi}) (i = 1, 2)$ , it yields

Δ p_{i} A_{i} = \frac{K_{F} A_{i} δ}{(1 + K_{F} / K_{hi}) s_{i} + u_{i}} (i = 1, 2)

(17)

The stiffness coefficient of the hydraulic system can be written as

k_{h} = \frac{F}{δ} = \frac{Δ p_{1} A_{1} - Δ p_{2} A_{2}}{δ}

(18)

Substituting equation (17) into equation (18), the hydraulic stiffness with considering the structural flexibility can be solved as

k_{h} = \frac{A_{1}}{(s_{1} + x) / K_{F} + s_{1} / K_{h 1}} + \frac{A_{2}}{(s_{2} + s - x) / K_{F} + s_{2} / K_{h 2}}

(19)

The proposed formulation indicates that the stiffness of a hydraulic system is the result of comprehensive effect of fluid compressibility and structural flexibility. It is a nonlinear function with respect to the piston position. For a specific hydraulic system, when the fluid pressure is high, the variation of the pressure and the piston position are often considerable; therefore, nonlinearity of hydraulic stiffness is usually significant. In low-to-middle pressure range, the variation of the pressure and the piston position are usually very small; therefore, the nonlinearity of hydraulic stiffness is not too remarkable.

On the other hand, the flexibility of the circular tubes has significant influence on the hydraulic stiffness. Even though a hydraulic system works in low-to-middle pressure range, if the volume of the hydraulic cylinder is small and the volume of the fluid supply hose is relatively large, there could be big error between results calculated using traditional formulation and the experiments. Specifically, the hydraulic stiffness is affected by parameters of the circular tube such as the length, the inner and outer radius, and Young’s modulus.

Influences of parameters on the hydraulic stiffness

In this section, the influences of parameters of the circular tube on the hydraulic stiffness are investigated. The concerned parameters are Young’s modulus $E_{h}$ , the radius ratio $ξ$ , the nominal volume ratio $ν = (V_{h 1} + V_{h 2}) / (A_{1} s)$ , and the length ratio $λ = l_{1} / l_{2}$ of the circular tube.

The relationship between the stiffness $k_{limb}$ and Young’s modulus $E_{h}$ is shown in Figure 4. The results show that the stiffness of hydraulic limb increases significantly as Young’s modulus increases. It means that the stiffness of hydraulic limb can be increased by selecting material with higher Young’s modulus.

Figure 4.

Stiffness $k_{limb}$ versus Young’s modulus $E_{h}$ .

The relationship between the stiffness $k_{limb}$ and the radius ratio $ξ$ of the circular tube is shown in Figure 5. The radius ratio $ξ$ of the circular tube is modified by changing the outer radius of the circular tube, while keeping the inner radius unchanged. The results show that the stiffness of hydraulic limb increases as the radius ratio increases. Besides, the incremental amplitude of the stiffness decreases as the radius ratio increases. This phenomenon coincides with the relationship between the volume modulus and the radius ratio, as shown in Figure 2.

Figure 5.

Stiffness $k_{limb}$ versus radius ratio $ξ$ .

The relationship between the stiffness $k_{limb}$ and the nominal volume ratio $ν = (V_{h 1} + V_{h 2}) / (A_{1} s)$ is shown in Figure 6. The nominal volume ratio $ν$ is modified by changing the length $l_{1}$ of the circular tube. Notice that $l_{2}$ changes simultaneously in this case according to $l_{2} = l_{0} + l_{1} + b_{1} + b_{2}$ . The results show that the stiffness of hydraulic limb increases as the nominal volume ratio decreases.

Figure 6.

Stiffness $k_{limb}$ versus nominal volume ratio $ν$ .

It can be seen that for most cases, the stiffness of hydraulic limb decreases as the position of piston x increases. This is probably due to the fact that the length of circular tube $l_{2}$ is much longer than $l_{1}$ . Suppose that $l_{1}$ and $l_{2}$ can be modified by moving the valve in Figure 3, thus the length ratio $λ = l_{1} / l_{2}$ of the circular tube can be modified to investigate its influence on the stiffness $k_{limb}$ . Notice that the total length $l_{sum} = l_{1} + l_{2}$ is kept unchanged in this case for comparison. The results in Figure 7 show that as the length ratio $λ$ increases, the stiffness for piston position near $x = 0$ decreases, whereas the stiffness for piston position near $x = s$ increases.

Figure 7.

Stiffness $k_{limb}$ versus length ratio $λ$ of the circular tube.

According to analysis above, it can be concluded that

Higher hydraulic stiffness can be obtained by increasing Young’s modulus or the radius ratio or by decreasing the nominal volume ratio.

Modification of the length ratio of the circular tube can modulate the hydraulic stiffness at different positions of the piston.

Experiments and discussion

As shown in Figure 8, stiffness of the hydraulic limb as specified in Appendix 1 is measured on a WDW-100 Electronic Testing Machine. Compressive force is applied on two sides of the hydraulic limb, and both force and displacements are measured to calculate the stiffness. The measuring accuracy of the force is ±0.5%, and the measuring accuracy of the displacement is ±0.01 mm.

Figure 8.

Experimental setup.

In order to avoid unnecessary disturbances and simplify the test, the chamber with rod is connected to a soft container and is full of non-pressurized hydraulic oil, which means the stiffness of this chamber is nearly zero. The chamber without rod is filled with pressurized hydraulic oil. This chamber is also connected with the circular tube by an on–off valve. If the valve is closed, the testing results stand for stiffness of the hydraulic limb without the circular tube. Or else, if the valve is open, the circular tube and the chamber without the rod serve as an entirety and the stiffness will decrease significantly. Besides, the stiffness is identified with fluid pressure near $p = 13.5 MPa$ .

Figure 9 shows comparison between simulation and experiment of stiffness for the hydraulic limb without the circular tube. In this circumstance, the proposed method is identical with the traditional one because there is no circular tube in the chamber without rod. It can be seen that the maximal relative error between the theoretical results and the experimental ones is 3.4%. Notice that the error depends mainly on the bulk modulus of the fluid, which is identified as $K_{F} = 1.56 \times 10^{9} MPa$ .

Figure 9.

Stiffness of the hydraulic limb without the circular tube.

Figure 10 shows comparison between simulation and experiment of stiffness for the hydraulic limb with the circular tube. In this experimental setup, the circular tube is made of wire-braided hydraulic hose. The volume modulus of the circular tube is identified as $K_{P} = 810 MPa$ . The nominal volume ratio of the circular tube can also be figured out to be $ν = 0.47$ . The maximal relative error between the theoretical results using the proposed method and the experimental ones is only 3.8%. However, the maximal and minimal relative errors between the theoretical results using the traditional method and the experimental ones are 123% and 63.8%, respectively. It can be seen that the stiffness calculated using the traditional method is much higher than that using the proposed method. Notice that the error of the traditional method depends on the volume modulus and the nominal volume ratio of the circular tube. If the material of the circular tube is steel, or the length of the circular tube is shorter, then the error of the traditional method will reduce.

Figure 10.

Stiffness of the hydraulic limb with the circular tube.

Conclusion

Structural flexibility is taken into account to formulate the nonlinear hydraulic stiffness of a hydraulic system. Theoretical analysis shows that larger Young’s modulus, larger radius ratio, and smaller volume ratio of the circular tube yield higher hydraulic stiffness. Besides, the nonlinear stiffness of a hydraulic limb varies with the changing of piston position and is significantly affected by the length ratios of the circular tubes at two sides of the cylinder. Experimental results have verified the presented method. The proposed method can be used to design hydraulic system for achieving desired static and dynamic performances.

Footnotes

Appendix 1

Table 1.

Parameters of a hydraulic system.

Parameter	Value	Unit	Meaning
$d_{0}$	0.060	m	Outer diameter of the cylinder
$d_{1}$	0.040	m	Inner diameter of the cylinder
$d_{2}$	0.020	m	Diameter of the rod
s	0.188	m	Effective moving range of the piston
$r_{o}$	0.0095	m	Outer radius of the circular tube
$r_{i}$	0.004	m	Inner radius of the circular tube
$l_{1}$	2.000	m	Length of the left circular tube
$l_{2}$	0	m	Length of the right circular tube
$E_{h}$	2.16 × 10⁹	Pa	Young’s modulus of the circular tube
$μ_{h}$	0.36	–	Poisson’s ratio of the circular tube
$K_{F}$	1.56 × 10⁹	Pa	Bulk modulus of the fluid

Academic Editor: Mark J Jackson

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was partially supported by the National Natural Science Foundation of China (nos 51435006, 51235005, 51405174, and 51421062).

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