Influence of parameters of PM controller on vibration performance of liquid hydrostatic guide-way system

Abstract

The vibration of liquid hydrostatic guide-way system during the working process makes a full impact on the machining precision of the numerically-controlled machine tool. The hydrostatic guide-way under the regulation of PM (Progressive Mengen) controller is taken as the research object. First, based on the force balance equation of hydrostatic guide-way, two kinds of vibration model of hydrostatic guide-way are established, respectively, and oil film stiffness equation and damping coefficient formula are derived. Second, the inherent frequency and amplitude of the guide-way system are derived. Third, the influence of PM controller parameters on the inherent frequency and amplitude of the guide-way system is analysed theoretically. The result indicates that the inherent frequency of the guide-way system changes with PM controller parameter and it makes a full impact on the active vibration amplitude of the hydrostatic guide-way system, but not the passive vibration amplitude. This article is provided a reference for the practical application of PM controller.

Keywords

Liquid hydrostatic guide-way system PM controller oil film stiffness damping coefficient inherent frequency vibration amplitude

Introduction

Liquid hydrostatic guide-way system is an important support part of lathe, its dynamic and static performance depends largely on the guide-way stiffness, oil film thickness, vibration frequency and oil chamber pressure and other parameters, and these parameters are closely related to PM controller.¹ So there has attracted many domestic and foreign experts and scholars to conduct in-depth study on PM controller.

Gao et al.^2,3 introduced the structure, working principle and characteristics of PM controller. In this article, the influence of PM controller parameters on the static stiffness and dynamic characteristics of liquid hydrostatic guide-way of computer numerical control (CNC) machine tool is studied in detail and determined the reasonable value range of PM controller parameters. The dynamic characteristic of open-type hydrostatic guide-way is analysed and compared under the linearization and nonlinear solution based on PM controller and provided the theoretical basis for PM controller selection.⁴

According to the hydrostatic bearing system in the project engineering, the finite element model of working turntable is established and obtained the curve of the dynamic characteristics. The influence of the number of oil chambers on the vibration frequency of hydrostatic bearing turntable is studied based on numerical simulation method. The results provide a theoretical basis for further study of the dynamic characteristics of the hydrostatic bearing system.⁵ The transmission process of the unbalanced force during the vibration of the hydrostatic vibration table is analysed, and the design of the air spring as the support and the vibration isolator is studied in combination with the actual situation.⁶

The parameters of the model measured with the machine table layout vibrator and suspension vibrator are compared with those of machine tool table, and the influence of the driving point position on the excitation frequency is analysed. The advantages and disadvantages of the two evaluation schemes of the dynamic performance of the machine tool are reviewed. The practical instruction is given according to the theory of the model analysis.⁷ The hydraulic system widely used in heavy machine tools is introduced; the modularization is adopted to accomplish the installation of the machine tool and reduce the cycle of the machine tool design and processing.⁸ The feed-forward motion control strategy is designed, and the zero-pole cancellation and zero-phase compensation of each axis are applied to make the dynamic response of each axis coordinate and improve the contour movement accuracy of each motion axis.⁹

Many domestic and foreign scholars are committed to the hydrostatic guide-way performance research and optimization analysis, but the vibration performance of hydrostatic guide-way in displacement under load is rarely reported. Therefore, the mathematical model of the hydrostatic guide-way is established, and the influence of PM controller parameters on the vibration performance of static pressure guide-way is studied.

Vibration model of hydrostatic guide-way

Active vibration model of closed-type liquid hydrostatic guide-way

Based on the PM controller, closed-type liquid hydrostatic guide-way system is shown in Figure 1, which is mainly composed of the oil cavity 1, oil cavity 2, guide-way, PM controller, pressure gauge, overflow valve, filter, check valve, hydraulic pump, fuel tank and other components.

Figure 1.

Closed-type hydrostatic slide system.

The shape of opposed oil cavity on guide-way is rectangular. Figure 2 shows the structure of rectangular oil chamber.

Figure 2.

Rectangular oil chamber.

Initial state analysis

In engineering, the oil film of opposed oil cavity in closed-type hydrostatic guide-way is generally equal thickness of the state, so that the initial oil film thickness of the oil cavity 1 and oil cavity 2 are equal, where h₁₀ = h₂₀ = h₀. In the initial state, the guide-way is in equilibrium, the force balance equation is expressed as

m g + p_{2, 0} A_{e, 2} = p_{1, 0} A_{e, 1}

(1)

where p_1,0 and p_2,0 are the initial pressures of the oil cavity 1 and oil cavity 2, respectively. The guide-way’s quality is m. A_e,1 and A_e,2 represent the effective bearing area of oil cavities 1 and 2,¹⁰ it is represented as

A_{e, i} = (A_{i} - a_{i}) (B_{i} - b_{i})

(2)

where A and B are the length and width of the oil cavity, respectively; a, b are the length and width of the oil cavity sealing oil side, respectively; i = 1,2.

In the initial state, the flow rates q₁₀ and q₂₀ of the oil cavities 1, 2 are consistent with the PM controller flows q_PM10 and q_PM20, the flow equation is defined as

q_{i,,0} = q_{PM, i,,0}

(3)

where $q_{i,,0} = {\bar{B}}_{i} p_{i,,0} h_{0}^{3} / μ$ , $q_{PM, i,,0} = q_{i, 0, 0} [1 + ((p_{i,,0} / p_{s}) (K_{r, i} - 1))]$ .

Where the oil’s dynamic viscosity is μ; the oil film’s initial thickness is h₀; q_i00 and K_r are the initial flow and specific flow rate of the PM controller, respectively; the oil chamber pressure is p; the system pressure is p_s.

Working state analysis

In the working condition, the vibration load on the guide-way is f, the displacement is x, the pressure of the oil cavities 1 and 2 are p₁ and p₂. The external load, the gravity, the bearing capacity of the oil cavity and the inertia force on the guide-way make the guide-way keep balance, the force balance equation of the guide-way is established as

m g + f - p_{1} A_{e, 1} + p_{2} A_{e, 2} = m \overset{\cdot\cdot}{x}

(4)

where the guide-way’s vibration external load is f, x is displacement of guide-way, p₁ and p₂ are the pressures of the oil cavity 1 and oil cavity 2, respectively.

The hydrostatic guide-way produced oil film squeeze is affected on oil cavity, which is equivalent to the liquid damping.¹¹ The instantaneous flow rates of the oil cavity 1 and 2 are inconsistent with flow qPM of the PM controller, the flow of the oil cavity can be written as

q_{i} = q_{PM, i} + {(- 1)}^{i + 1} A_{b, i} \dot{x}

(5)

where $q_{i} = ({\bar{B}}_{i} p_{i} / μ) {[h_{0} + {(- 1)}^{i} x]}^{3}$ , $q_{PM, i} = q_{i, 0, 0} [1 + (p_{i} / p_{s}) (K_{r, i} - 1)]$ .

Mathematical model derivation

The equation (4) less equation (1), the results can be obtained as

f - Δ p_{1} A_{e, 1} + Δ p_{2} A_{e, 2} = m \overset{\cdot\cdot}{x}

(6)

where Δp₁ = p₁–p₁₀, Δp₂ = p₂–p₂₀.

The equation (5) less equation (3), the results can be obtained as

\begin{array}{l} \frac{{\bar{B}}_{i} p_{i}}{μ} {[h_{0} + {(- 1)}^{i} x]}^{3} - \frac{{\bar{B}}_{i} p_{i,,0} h_{0}^{3}}{μ} \\ = \frac{q_{i, 0, 0}}{p_{s}} (K_{r, i} - 1) Δ p_{i} + {(- 1)}^{i + 1} A_{b, i} \dot{x} \end{array}

(7)

where Δp_i = p_i–p_i,₀, Δq_PM,i = q_PM,i–q_PM,i,0.

Oil film is generally range in 30 μm, the amplitude of the guide-way system is controlled below 10 μm. If the amplitude is too large, the movement precision of the guide-way system is greatly reduced and the stability is deteriorated, the hydrostatic support guide-way system needs to be readjusted and corrected. Therefore, the vibration of the guide-way system is basically a small amplitude vibration; the linearization error of the guide-way system is smaller.

The equation (7) is a nonlinear differential equation, and the result is very difficult to solve. The equation (7) is shown in Taylor series and ignores the higher order term.¹² The linearization equation is expressed as

\begin{array}{l} \frac{{\bar{B}}_{i} h_{0}^{3}}{μ} Δ p_{i} + \frac{3 {\bar{B}}_{i} p_{i,,0} h_{0}^{2}}{μ} {(- 1)}^{i} x \\ = \frac{q_{i, 0, 0} (K_{r, i} - 1)}{p_{s}} Δ p_{i} + {(- 1)}^{i + 1} A_{b, i} \dot{x} \end{array}

(8)

Taking equation (8) into equation (6), the mathematical model of active vibration in closed-type hydrostatic guide-way is obtained as follows

m \overset{\cdot\cdot}{x} + c \dot{x} + k x = f

(9)

where

\begin{array}{l} c = \sum_{i = 1}^{2} \frac{A_{b, i} A_{e, i}}{\frac{{\bar{B}}_{i} h_{0}^{3}}{μ} - \frac{q_{i, 0, 0}}{p_{s}} (K_{r, i} - 1)}, \\ k = \sum_{i = 1}^{2} \frac{3 {\bar{B}}_{i} p_{i,,0} A_{e, i} h_{0}^{2}}{{\bar{B}}_{i} h_{0}^{3} - μ \frac{q_{i, 0, 0}}{p_{s}} (K_{r, i} - 1)} \end{array}

Active vibration model of open-type hydrostatic guide-way

The open-type hydrostatic guide-way is supported by a single oil cavity, so that the vibration analysis process is similar to a closed-type guide-way, and the data of the oil cavity 2 are set to zero, shown in Figure 3.

Figure 3.

Open-type hydrostatic slide system.

Referring to the vibration analysis process of the closed-type guide-way, the mathematical model of the active vibration on open-type hydrostatic guide-way is presented as

m \overset{\cdot\cdot}{x} + c \dot{x} + k x = f

(10)

where

c = \frac{A_{b, 1} A_{e, 1}}{\frac{{\bar{B}}_{1} h_{0}^{3}}{μ} - \frac{q_{1, 0, 0}}{p_{s}} (K_{r, 1} - 1)}, k = \frac{3 {\bar{B}}_{1} p_{1, 0} A_{e, 1} h_{0}^{2}}{{\bar{B}}_{1} h_{0}^{3} - μ \frac{q_{1, 0, 0}}{p_{s}} (K_{r, 1} - 1)}

Based on equations (9) and (10), the active vibration model of hydrostatic guide-way is oscillation. Closed-type oil cavity is equivalent to two pairs of oil cavities in parallel.

Passive vibration model of closed-type liquid hydrostatic guide -way

Initial state analysis

The initial state of the passive vibration on closed-type hydrostatic guide-way system is the same as the active vibration, as written in equations (1)–(3).

Working state analysis

In working condition, the displacement of the oil cavity is y, the guide-way’s displacement is x, the pressure of the oil cavities 1 and 2 are p₁ and p₂, respectively. The external load, gravity, oil bearing capacity and inertial force are balanced with each other, and the mechanical equilibrium equation of the guide-way is established as

m g - p_{1} A_{e, 1} + p_{2} A_{e, 2} = m \overset{\cdot\cdot}{x}

(11)

The squeezing effect of the oil film is formed by the extrusion of the hydrostatic guide-way to the oil cavity, which is equivalent to liquid damping, which caused the instantaneous flow q₁and q₂ of the oil cavities 1 and 2 to be inconsistent with the PM controller flow obtained as

q_{i} = q_{PM, i} + {(- 1)}^{i + 1} A_{b, i} {(x - y)}^{'}

(12)

where $q_{i} = ({\bar{B}}_{i} p_{i} / μ) {[h_{0} + {(- 1)}^{i} (x - y)]}^{3}$ , $q_{PM, i} = q_{i, 0, 0} [1 + ((p_{i} / p_{s}) (K_{r, i} - 1))]$ .

Mathematical model derivation

Equation (11) minus equation (1), can be obtained as

- Δ p_{1} A_{e, 1} + Δ p_{2} A_{e, 2} = m \overset{\cdot\cdot}{x}

(13)

Equation (12) minus equation (3), the results can be obtained as

\begin{array}{l} \frac{{\bar{B}}_{i} p_{i}}{μ} {[h_{0} + {(- 1)}^{i} (x - y)]}^{3} - \frac{{\bar{B}}_{i} p_{i,,0} h_{0}^{3}}{μ} \\ = \frac{q_{i, 0, 0} (K_{r, i} - 1)}{p_{s}} Δ p_{i} + {(- 1)}^{i + 1} A_{b, i} {(x - y)}^{'} \end{array}

(14)

Equation (14) is a nonlinear differential equation, which is linearized to be obtained as

\begin{array}{l} \frac{{\bar{B}}_{i} h_{0}^{3}}{μ} Δ p_{i} + \frac{3 {\bar{B}}_{i} p_{i,,0} h_{0}^{2}}{μ} {(- 1)}^{i} (x - y) \\ = \frac{q_{i, 0, 0} (K_{r, i} - 1)}{p_{s}} Δ p_{i} + {(- 1)}^{i + 1} A_{b, i} {(x - y)}^{'} \end{array}

(15)

Taking equation (15) into equation (13), the mathematical model of passive vibration of closed-type hydrostatic guide-way is obtained as

m \overset{\cdot\cdot}{x} + c \dot{x} + k x = c \dot{y} + k y

(16)

where

\begin{array}{l} c = \sum_{i = 1}^{2} \frac{A_{b, i} A_{e, i}}{\frac{{\bar{B}}_{i} h_{0}^{3}}{μ} - \frac{q_{i, 0, 0}}{p_{s}} (K_{r, i} - 1)}, \\ k = \sum_{i = 1}^{2} \frac{3 {\bar{B}}_{i} p_{i,,0} A_{e, i} h_{0}^{2}}{{\bar{B}}_{i} h_{0}^{3} - μ \frac{q_{i, 0, 0}}{p_{s}} (K_{r, i} - 1)} \end{array}

Passive vibration model of open-type hydrostatic guide-way

According to the vibration analysis process of the closed-type guide-way, the active vibration mathematical model of the open-type hydrostatic guide-way is written as

m \overset{\cdot\cdot}{x} + c \dot{x} + k x = c \dot{y} + k y

(17)

where

c = \frac{A_{b, 1} A_{e, 1}}{\frac{{\bar{B}}_{1} h_{0}^{3}}{μ} - \frac{q_{1, 0, 0}}{p_{s}} (K_{r, 1} - 1)}, k = \frac{3 {\bar{B}}_{1} p_{1, 0} A_{e, 1} h_{0}^{2}}{{\bar{B}}_{1} h_{0}^{3} - μ \frac{q_{1, 0, 0}}{p_{s}} (K_{r, 1} - 1)}

Based on equations (16) and (17), the passive vibration model of the hydrostatic guide-way is composed of the oscillation and the first-order differential. The closed-type oil cavities are equivalent to two opposed oil cavities in parallel.

Vibration analysis of static pressure guide-way under simple harmonic load

Active vibration of hydrostatic guide-way under simple harmonic load

When the linear system input is simple harmonic motion, the output will be simple harmonic motion, and the output and input frequency is equal.^13,14

Active vibration load of liquid hydrostatic guide-way is f = a₁sinωt, the displacement is x =B₁sin(ωt-ψ₁), the first and second derivatives of the displacement for the guide-way are deduced as

{\begin{cases} \dot{x} = B_{1} ω \cos (ω t - ψ_{1}) \\ \overset{\cdot\cdot}{x} = - B_{1} ω^{2} \sin (ω t - ψ_{1}) \end{cases}

(18)

where the input external load’s vibration frequency is ω.

Taking equation (18) into equation (9), we can get

B_{1} \sqrt{{(k - m ω^{2})}^{2} + {(c ω)}^{2}} \sin (ω t - ψ_{1} + δ_{1}) = a_{1} \sin ω t

(19)

where $δ_{1} = \arctan [c ω / (k - m ω^{2})]$ .

According to equation (16), amplitude B₁ and phase difference Ψ₁ are written as

{\begin{cases} B_{1} = \frac{a_{1}}{\sqrt{{(k - m ω^{2})}^{2} + {(c ω)}^{2}}} \\ ψ_{1} = δ_{1} \end{cases}

(20)

Therefore, the active vibration of the hydrostatic guide-way under the simple harmonic period load is written as

x = a_{1} K_{1} \sin (ω t - δ_{1})

(21)

where K₁ is the amplitude magnification factor, $K_{1} = 1 / \sqrt{{(k - m ω^{2})}^{2} + {(c ω)}^{2}}$ .

Passive vibration of hydrostatic guide-way under simple harmonic load

During hydrostatic guide-way passive vibration, the oil cavity’s displacement is y, where y = a₂sinωt. The displacement of guide-way is x, where x = B₂sin(ωt-ψ₂), the first and second derivatives of the displacement for the guide-way are deduced as

{\begin{cases} \dot{y} = a_{2} ω \cos ω t \\ \dot{x} = B_{2} ω \cos (ω t - ψ_{2}) \\ \overset{\cdot\cdot}{x} = - B_{2} ω^{2} \sin (ω t - ψ_{2}) \end{cases}

(22)

where ω is the vibration frequency of load-oil chamber displacement y.

Taking equation (22) into equation (16), we can get

\begin{array}{l} B_{2} \sqrt{{(k - m ω^{2})}^{2} + {(c ω)}^{2}} \sin (ω t - ψ_{2} + δ_{2}) \\ = a_{2} \sqrt{k^{2} + {(c ω)}^{2}} \sin (ω t + δ_{3}) \end{array}

(23)

where $δ_{2} = \arctan [c ω / (k - m ω^{2})]$ , $δ_{3} = \arctan (c ω / k)$ .

According to equation (23), amplitude B₂ and phase difference Ψ₂ are written as

{\begin{cases} B_{2} = a_{2} \sqrt{\frac{k^{2} + {(c ω)}^{2}}{{(k - m ω^{2})}^{2} + {(c ω)}^{2}}} \\ ψ_{2} = δ_{2} - δ_{3} \end{cases}

(24)

Therefore, the passive vibration of the hydrostatic guide-way under the simple harmonic period load is written as

x = a_{2} K_{2} \sin (ω t - ψ_{2})

(25)

where K₂ is the amplitude magnification factor, $K_{2} = \sqrt{k^{2} + {(c ω)}^{2} / ({(k - m ω^{2})}^{2} + {(c ω)}^{2})}$ .

Influence of PM controller parameters on guide-way vibration

In the vibration engineering analysis, the parameters of the PM controller are adjusted to realize the inherent frequency of the system. The inherent frequency of the system is far greater than the actual load frequency, thus reducing the vibration amplitude and improving the vibration performance of the guide-way system.

Based on equations (21) and (25), the mathematical relationship between the PM controller parameter Kr, the inherent frequency of the guide-way system ω_n, and the amplitude amplification coefficient K are written as

{\begin{cases} ω_{n} = \sqrt{\frac{k}{m}} \\ K_{a c t i v e} = \frac{1}{\sqrt{{(k - m ω^{2})}^{2} + {(c ω)}^{2}}} \\ K_{p a s s i v e} = \sqrt{\frac{k^{2} + {(c ω)}^{2}}{{(k - m ω^{2})}^{2} + {(c ω)}^{2}}} \end{cases}

(26)

where ω_n is the inherent frequency of the guide-way system

\begin{array}{l} c_{c l o s e d} = \sum_{i = 1}^{2} \frac{μ A_{b, i} A_{e, i}}{{\bar{B}}_{i} h_{0}^{3}} [1 + \frac{p_{i,,0}}{p_{s}} (K_{r, i} - 1)], \\ c_{o p e n e d} = \frac{μ A_{b, 1} A_{e, 1}}{{\bar{B}}_{1} h_{0}^{3}} [1 + \frac{p_{1, 0}}{p_{s}} (K_{r, 1} - 1)], \\ k_{c l o s e d} = \sum_{i = 1}^{2} \frac{3 p_{i,,0} A_{e, i}}{h_{0}} [1 + \frac{p_{i,,0}}{p_{s}} (K_{r, i} - 1)], \\ k_{o p e n e d} = \frac{3 p_{1, 0} A_{e, 1}}{h_{0}} [1 + \frac{p_{1, 0}}{p_{s}} (K_{r, 1} - 1)] \end{array}

From equation (26), the inherent frequency of hydrostatic guide-way increases with increasing amplitude amplification coefficient. The simplified model of the amplitude amplification coefficient K in different frequency ranges and the variation of the parameter K are shown in Table 1.

Table 1.

Amplitude magnification coefficient of hydrostatic guide-way system and changing trends.

Amplitude magnification factor K		Load frequency range
Amplitude magnification factor K		(0, ω₁)	(ω₁, ω₂)	(ω₂, ω₃)	(ω₃, +∞)
Active vibration	Simplify the model	1/(k-mω²)	1/cω	1/cω	1/(mω²-k)
	The changing trend of ω	Monotonically increasing	Monotonically decreasing	Monotonically decreasing	Monotonically decreasing
	The changing trend of K_r	Monotonically decreasing	Monotonically decreasing	Monotonically decreasing	Monotonically increasing
Passive vibration	Simplify the model	k/(k-mω²)	k/cω	1/cω	cω/(mω²-k)
	The changing trend of ω	Monotonically increasing	Monotonically decreasing	Basically unchanged	Monotonically decreasing
	The changing trend of K_r	Monotonically decreasing	Basically unchanged	Basically unchanged	Monotonically increasing

where ω₁, ω₂, and ω₃ are the demarcation points of the inherent frequency of the rail system, respectively.

In which, $ω_{1} = (- c + \sqrt{c^{2} + 4 m k}) / 2 m$ , $ω_{2} = k / c$ , $ω_{3} = (c + \sqrt{c^{2} + 4 m k}) / 2 m$ .

In actual production, the Internet frequency of hydrostatic guide-way system can be adjusted away from the load frequency by changing parameters Kr. In different load frequency range, changing parameter Kr, the amplitude magnification coefficient of the guide-way can be regulated, and the good vibration performance of the guide-way system is obtained.

Calculation examples

Initial parameters of hydrostatic guide-way system

The influence of the PM controller parameters on the vibration performance of the open-type hydrostatic guide-way is analysed, shown in Figure 3.

The initial parameters of the guide-way and the PM controller are shown in Tables 2 and 3. The dimensions of the rectangular oil cavity are shown in Table 4.

Table 2.

Basic parameters of hydrostatic guide-way.

Guide-way quality, m/t	Oil film thickness, h₀/μm	Oil cavity pressure, p_r/MPa
6	30	0.615
Oil type	Oil working temperature /°C	Dynamic viscosity μ/Pa s
VG15	10	0.04032

Table 3.

Initial design parameters of the PM controller.

K _r	p_s/MPa	q_0,0/(L/min)	Initial q₀/(L/min)
3	10	0.059	0.066

Table 4.

Dimensional parameters of the rectangular oil cavity.

B/m	b/m	A/m	a/m
0.2	0.05	0.7	0.05

Adjust the inherent frequency of the guide-way system

Based on equation (26), according to the design parameters as shown in Tables 2 and 3, the inherent frequency of the guide-way system is calculated by changing the PM controller parameter K_r, as shown in Table 5.

Table 5.

Inherent frequency of open-type hydrostatic guide-way system.

K_r	4	5	6	7	8	9	10	11	12	13
Inherent frequency, ω_n/Hz	1088	1116	1143	1170	1196	1221	1246	1271	1295	1318

From the equation (26) and Table 5, it can be seen that increasing the PM controller parameter Kr and the oil film stiffness k of the hydrostatic guide-way, the inherent frequency of the guide-way system increases in turn, but the variation range is not large.

Analysis of vibration amplitude of guide-way system

Using the design parameters in Tables 2 and 3, base on equation (26), changing the parameters K_r of the PM controller, the active amplitude amplification factor of the guide-way system is calculated by MATLAB software, as shown in Figures 4 and 5.

Figure 4.

Active vibration amplitude magnification coefficient of different K_r guide system.

Figure 5.

Passive vibration amplitude magnification coefficient of different K_r slide system.

The boundary frequency of the hydrostatic guide-way is calculated according to the equation (26), which ω₁ = ω₂ = 1.8315 Hz, and ω₃ = 9.4920×10⁵ Hz. Working load frequency of the hydrostatic guide-way is generally 0–100 Hz, it can see that the hydrostatic guide-way system operating load frequency is generally in the range (ω₂, ω₃).

From Table 1, the active vibration amplitude magnification coefficient of hydrostatic guide-way decreases rapidly with the increase of the external load in the range of (0–10 Hz), which corresponding to the frequency range of (ω₂, ω₃)in Table 1; and it is decreases in the range of (10–100 Hz), the magnitude of the reduction is not large. With the PM controller parameter Kr increased in turn lower, but the amplitude is not reduced, which was corresponding to Table 1(ω₂, ω₃) frequency range.

Passive vibration amplitude coefficient of the hydrostatic guide-way in (10–100 Hz), the frequency range of change in the scope of the basic frequency (ω₂, ω₃) is very small, can be neglected; Figure 5 appeared in a step-shape curve, but the ordinate MATLAB software is basically unchanged, also shows that the passive vibration amplitude amplification the coefficient in (ω₂, ω₃) is not affected by the loading frequency and parameters of K_r in the range of influence.

Conclusion

Based on the analysis of the influence of PM controller parameters on the vibration performance of hydrostatic guide-way, the following conclusions are obtained: First, the active and passive vibration models of the hydrostatic guide-way are the combination of the oscillation and the first-order differential equation. Second, the closed-type oil cavities are equivalent to two pairs of oil cavities in parallel, the sum of the two oil chambers is the oil film stiffness and damping coefficient. Third, with the increase of the parameter K_r of the PM controller, the active vibration amplitude magnification coefficient of the hydrostatic guide-way system decreases and passive vibration amplitude magnification coefficient is not affected. With the increase of the external load frequency of guide-way system, the active vibration amplitude magnification coefficient of the guide-way decreases while the passive vibration amplitude magnification coefficient is not affected.

Footnotes

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

This research was supported by Hebei province natural science fund of China (grant nos.: E2016203324 and E51705445), the Yanshan University Dr. Fund of China (grant no.: B815) and the Yanshan University young teachers independent research project plan B of China (grant no.: 13LGB003). The mentioned supports are gratefully acknowledged.

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