Mechanical vibrations and flow fluctuation give rise to complex interactive vibration mechanisms in hydraulic pumps. The working conditions for a hydraulic pump are therefore required to be improved in the design stage or as early as possible. Considering the structural features, parameters, and operating environment of a hydraulic plunger pump, the vibration modes for two-degree-of-freedom system were established by using vibration theory and hydraulic technology. Afterwards, the analytical form of the natural frequency and the numerical solution of the steady-state response were deduced for a hydraulic plunger pump. Then, a method for the vibration analysis of a hydraulic pump was proposed. Finally, the dynamic responses of a hydraulic plunger pump are obtained through numerical simulation.

Since there have been increasingly higher requirements imposed on the engineering quality, the accuracy, and reliability of products, it is an urgent task to study and solve various vibration problems existing in industrial machinery. Due to increasing system complexity, rapid operations, and improved accuracy of mechanical devices, vibration is a serious issue. Therefore, both the static strength effect and the dynamic force effect must be taken into account while designing mechanical devices [

Many studies have been performed to analyse the dynamic performance of hydraulic components and systems. To prevent hydraulic systems from breaking down, analysis of their vibration is required. An effective method of reducing vibration and noise needs to be developed: its aim is to improve the performance of hydraulic devices and thus reduce the vibration and noise in hydraulic systems. Vibration and noise arise from the interaction between solids and the fluid here. Fluid-structure interaction (FSI) not only reflects the essence of vibration noise in hydraulic system but also is the

Vibration analysis: hydraulic component and system.

A hydraulic pump is the main vibration and noise source in a hydraulic system, and its working state determines the safe operation of hydraulic components therein. Therefore, it can be seen that mechanical vibrations and flow fluctuations not only affect the engineering quality but also reduce the lifespan of hydraulic components and systems, generate noise pollution, and even cause damage. Furthermore, they may give rise to accidents. Studies relating the vibration of hydraulic pump mainly focus on the analysis of test data and the reduction in vibration and noise [

Influenced by the design, structure, and operating environment of a hydraulic plunger pump as well as the inherent characteristic curve, flow pulsation is bound to be generated. Flow and pressure pulsation are two of the leading reasons for noise and vibration being generated by a hydraulic pump. This paper gave full considerations to the vibration problems caused by flow and pressure pulsation in a hydraulic plunger pump and converted the simplified formula into an excitation describing the vibration system of a plunger pump. In addition, a vibration mode for the two degrees of freedom in a hydraulic plunger pump was made available for practical calculation and was established to estimate the vibration of a hydraulic plunger pump under specific working conditions. Meanwhile, the model reveals the basic mechanisms of vibration and noise in a hydraulic plunger pump. By conducting the dynamic analysis using the proposed approach, a better understanding of hydraulic plunger pumps can be obtained.

The vibration model for the two degrees of freedom of a hydraulic plunger pump was developed based on the data and conditions regarding the structural features, parameters, variables, constraints, operational states, and flow pulsation. The model is shown in Figure

Vibration model for the two degrees of freedom of a hydraulic plunger pump.

According to the proposed vibration model, the positive directions of the acceleration and excitation were determined, which were in accordance with the positive direction of the coordinate axes. The deduced vibration differential equation is

One of the most influential factors in the vibration model of hydraulic plunger pump is the value of input vibration excitation; because flow pulsation is the source of pressure pulsation, flow pulsation has to be analysed to study pressure pulsation. Flow pulsation refers to the instantaneous flow variation when a hydraulic pump is running. When a hydraulic pump keeps operating continuously, the ever-changing sealing volume is expected to be generated in a majority of hydraulic pumps. Meanwhile, the instantaneous flow changes repeatedly, and some instantaneous, nonconstant flow may generate flow pulsation. The instantaneous actual flow of a hydraulic plunger pump may be given as follows.

When

When ^{2}) of the plunger;

Then

Converting the instantaneous actual flow into a function of time

Flow pulsation inevitably gives rise to pressure pulsation, which indicates that the flow and pressure output from the hydraulic plunger pump change with time. Therefore, the flow and pressure output by hydraulic plunger pump are not necessarily stable. The change in the volume of a hydraulic plunger pump always results in fluctuations in output pressure and fluid flow, leading to the generation of noise and vibration. According to the fundamental principles of fluid dynamics, the change of the flow in the closed cavity between the plunger and cylinder block of a hydraulic plunger pump inevitably leads to a pressure change. With respect to a compressible flow, its instantaneous pressure can be represented by the following formulas.

When

When

The flow and pressure pulsation in a hydraulic plunger pump are a nonharmonic periodic function, which can be represented by a harmonic Fourier series. Thereby, the dynamic response problem corresponding to the harmonic excitation of its Fourier series can be solved. The periodic excitation function in an interval of

The function

Therefore, the quantity of pressure pulsation induced by the quantity of flow pulsation can be expressed as

Pressure pulsation

From Figure

The excitation of the vibration system for the two-degree-of-freedom system of a hydraulic plunger pump caused by flow and pressure pulsation is given by

As seen from (

To study the inherent characteristics of a hydraulic plunger pump, the deduced characteristic equation, or frequency equation, of the two-degree-of-freedom system of a hydraulic plunger pump may be given as follows:

Equation (

Since

Therefore, the constants in pairs such as

There are two natural frequencies in this system, and correspondingly there are two natural vibration modes. The lower frequency

With the improvement and development of computer software, hardware, and technology, almost all engineering problems can be simulated quantitatively with high precision. Numerical simulation involves the solution of a mathematical problem whose exact solution is hard to find in practical engineering. The numerical simulation applied to mechanical vibration problems must discrete the time history for a dynamic response within the time domain so as to discretize the differential equation of motion and numerical equations at different moments. Meanwhile, the speed and acceleration at a given time are described by the combination of the displacements at adjacent time steps. As a result, the differential equation of motion of the system is converted into algebraic equations at discrete time steps. Then, the values corresponding to a series of discrete times are obtained by numerical integration of the differential equation of motion of the coupling system. There are many commonly used approaches for finding the dynamic response of such systems, including (1) central difference, (2) Houbolt, (3) Wilson-

The mass of the cylinder block of a certain hydraulic plunger pump was

According to the known conditions, here we have

Figure

Natural mode of vibration of the two-degree-of-freedom system.

Newmark-

The response of the two-degree-of-freedom system with time

As discovered during calculation, owing to the unstable initial running stage of the hydraulic plunger pump and the superposition of transient and steady-state vibrations, the system vibrated irregularly at large amplitude. However, the transient vibration gradually weakened and finally vanished after a period of time, and the system vibration reached a steady state. The amplitude of steady-state vibration of the plunger pump was acceptable in its smooth running stage. Meanwhile, compared with the retainer plate, seven plungers, and sliding shoes, the amplitude of the cylinder block of the plunger pump is much greater. The numerical calculation results can provide quantitative theoretical support for the accurate design of a hydraulic plunger pump.

The mechanical vibration analysis and the dynamic design are key points during the engineering design of mechanical products and are crucial for producing products with the required dynamic characteristics. While conducting vibration analysis and dynamic design on a hydraulic plunger pump using the proposed approach, the actual design level and dynamic characteristics of the hydraulic plunger pump are expected to be improved. This can also prevent malfunctions and accidents caused by breakdowns. Based on using vibration theory and hydraulic technology, this research has developed a vibration model for the dynamic analysis of a two-degree-of-freedom hydraulic plunger pump system. Meanwhile, the inherent characteristics and dynamic response of the hydraulic plunger pump were studied and were used to estimate the vibration of a hydraulic plunger pump under regulated working conditions. In addition, the basic mechanisms of noise and vibration in a hydraulic plunger pump were revealed.

The authors declare that there is no conflict of interests regarding the publication of this paper.

This research was supported by China Natural Science Foundation Project (Grant no. 51135003), the National Key Development Programme for Fundamental Research (973 Programme, Grant no. 2014CB046303), and Australian Research Council (ARC DP150102751).