Vibration Analysis For Hydraulic Pump 4i283m

Mechanism and Machine Theory

Mechanism and Machine Theory 41 (2006) 487–504

www.elsevier.com/locate/mechmt

Dynamic vibration analysis of a swash-plate type water hydraulic motor H.X. Chen, Patrick S.K. Chua, G.H. Lim

*

School of Mechanical and Production Engineering, Nanyang Technological University, 50 Nanyang Avenue, Singapore 639798, Singapore Received 13 October 2004; received in revised form 3 August 2005; accepted 7 September 2005 Available online 3 November 2005

Abstract This paper presents the study of the dynamic analysis of a swash-plate water hydraulic motor in a modern water hydraulic system. A swash-plate mechanism is modeled as a system with three masses and 14 degrees of freedom (DOF). In order to evaluate the applicability of the dynamic model, the numerical simulation analysis of the dynamic response of the model due to pressure pulsation is presented and compared with experimental results. A series of the dynamic vibration characteristics of the water hydraulic piston motor are studied by the numerical simulation. It is effective for the model to simulate the vibration signal of the casing in the hydraulic motor. The waveform and frequency of the simulated signal is similar to the experimental signal. The simulated signals in other directions show that the vibration signals in all the directions mainly consist of the hydraulic pump and hydraulic motor rotational frequencies. 2005 Elsevier Ltd. All rights reserved. Keywords: Water hydraulic motor; Dynamic response; Numerical simulation; Vibration analysis; Swash plate; Pressure

1. Introduction With the increasing environmental impact of the operating oil-based hydraulic system and the concern raised by environmentally conscious organizations, a most exciting area of development in the fluid power industry over the past few years has been in the area of water hydraulics, which involves using tap water as a viable alternative to oil in fluid power transmission. Water hydraulic systems have been used in the farming, forestry, food, pharmaceutical and paper industries [1–3]. The axial piston motor is commonly applied to provide high torque and performance for water hydraulic system. Axial piston swash-plate type hydraulic motor is comprised of a discrete number of pistons that reciprocate in a sinusoidal fashion for the purposes of torque output. The basic force within axial piston pump is analyzed and summarized. The instantaneous torque exerted on the shaft is computed and its resonant

*

Corresponding author. Fax: +65 6791 1859. E-mail address: [email protected] (G.H. Lim).

0094-114X/$ - see front matter 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.mechmachtheory.2005.09.002

488

H.X. Chen et al. / Mechanism and Machine Theory 41 (2006) 487–504

Nomenclature A piston cross-section area CHZ damping coefficient of the interface between piston and cylinder port CSh damping coefficient of the interface between shaft and valve cover and outer shell CSw damping coefficient of the interface between piston and swash plate CVa damping coefficient in the interface between valve cover (valve port plate) and cylinder block fp fundamental frequency of the hydraulic pump fm(=5f0 = 1/Tm) fundamental frequency of the water hydraulic motor f0 fundamental frequency of the motor piston FCz exciting force on the cylinder block in Z-axis Finertia inertia-related force FPz exciting force on the piston in Z-axis ICx moment of inertia of the cylinder block around XC-axis ICy moment of inertia of the cylinder block around YC-axis IFx moment of inertia of the outer shell and swash plate around XF-axis IFy moment of inertia of the outer shell and swash plate around YF-axis IPx moment of inertia of the piston and retaining ring around XP-axis IPy moment of inertia of the piston and retaining ring around YP-axis KSw stiffness of the interface between piston and swash plate KBo stiffness of the bolt between outer shell and valve cover (port flange) KSh stiffness of the motor shaft bearing KHZ stiffness of the interface between piston and cylinder port KP spring stiffness KVa stiffness of the interface between valve port plate and cylinder block lx equivalent center coordinates of the exerting force FCz in the direction of YC-axis in the cylinder block cross-section ly equivalent center coordinates of the exerting force FCz in the direction of XC-axis in the cylinder block cross-section LPF distance between the point OP and the point OF, referred to Fig. 5 LF1, LF2 lengths, refer to Fig. 5 LC1, LC2 lengths, refer to Fig. 5 Lb, LSh lengths, refer to Fig. 5 MC cylinder masses MP piston and retaining ring masses MF masses of the outer shell and the swash plate n number of piston p inlet pulsation pressure in the piston r radius of the spring port circle R radius of the cylinder port circle, refer to Fig. 5 R0 radius of the accelerometer position of the outer shell RV radius of force point circle between cylinder and valve cover (valve port plate), refer to Fig. 5 TCx torque on the cylinder around XC-axis TCy torque on the cylinder around YC-axis Ti the instantaneous torque of one piston T0 period of the piston Tm period of the water hydraulic motor Tp period of the pump TPx the torque on the piston and retaining ring around axis XP TPy the torque on the cylinder block around axis YP

H.X. Chen et al. / Mechanism and Machine Theory 41 (2006) 487–504

TP d0 x0 / h #

489

the piston torque angle of swash plate hydraulic motor angular velocity circular motion around axis X circular motion around axis Y anger between the radius and +YF-axis

frequencies of the pump that occur at even-multiple of the ‘‘piston-’’ frequency has been discussed by Manring [4]. The mathematical equations describing swash-plate model are derived from the general hydraulic and mechanical considerations [5]. Kaliafetis et al. [6] studied the model of an axial piston variable displacement pump with pressure control. The fluctuating aspects of the discharge flow have been addressed in the published literature. Manring [7] derived the closed-form expressions to describe the characteristics of the flow ripple. The ripple height and the pulse frequency of the flow ripple are described. Harrison and Edge [8] achieved a novel timing mechanism to reduce the source flow ripple in hydraulic systems in order to reduce the pressure ripple and system noise. Kojima and Shinada [9] investigated the fluid-borne noise characteristics in the fluid pump. The second source method on model and detection of the fluid noise characteristics have been studied. Some researchers proposed the test method for the measurement of the pump fluid-borne noise characteristics, pump pressure ripple and flow ripple characteristics [10,11]. Edge and Darling [12] proposed a theoretical model of the axial piston pump flow ripple. The dynamic behaviour of the axial piston motor is non-linear and the factors that influenced the performance of the axial piston motor are complicated. The vibration energy transmission characteristics from the cylinder to a swash-plate within an axial piston pump had been studied [13]. Qi and Lu [14] investigated the vibration of oil hydraulic axial piston pumps. Investigations showed that the main source of vibration is the impact between the slipper and swash plate when the piston/slipper moved into a pre-compression process. This impact force may excite a resonant vibration of the pump shell. Bahr et al. [15] developed a mathematical model to investigate the vibration characteristics of the pumping mechanism of oil hydraulic constant-power axial piston pumps with conical cylinder blocks. Nishimura et al. [16] analyzed the dynamic response of a swash-plate type hydraulic piston pump. The purpose of this research work is to analyze the dynamic response of water hydraulic motor. A model consisting of three masses and 14 DOF is developed. This model includes swash plate and outer shell, piston and retaining ring and cylinder block that generate rotary and reciprocal motion. The dynamic kinematic equations are formulated for analyzing the vibration characteristics of this model in numerical simulation. The results of numerical simulation were compared with experimental results to the model developed. It presents the vibration characteristics of the components in water hydraulic motor. 2. Water hydraulic motor description The modern water hydraulic system installed in the School of Mechanical and Production Engineering, Nanyang Technological University, was acquired from Danfoss Inc. [17]. Fig. 1 showed the water hydraulic motor test setup system. The water hydraulic system consists of electrical motor, water hydraulic pump, four kinds of valves, pressure sensors, flow rate sensors and water hydraulic motor. The water hydraulic motor is fixed-displacement swash plate. The water hydraulic pump was 5-piston axial piston pump. The water hydraulic system was connected by the rubber hose to the water hydraulic motor. The output torque and flow rate of the water hydraulic motor were 2 N m and 0.12 l/s. The angular speeds of hydraulic motor and pump were 630 rev/min and 1450 rev/min. The flow rate of the water hydraulic system was 0.12 l/s. The vibration signals were acquired by the accelerometer, which was mounted on the outer shell near the inlet of the water hydraulic motor.

490

H.X. Chen et al. / Mechanism and Machine Theory 41 (2006) 487–504

Fig. 1. Schematic hardware architecture of modern water hydraulic system.

The actuator studied here is a five piston axial piston motor used in a water hydraulic system. A MAH 12.5 water hydraulic motor and water hydraulic system manufactured by Danfoss Inc. were used to measure the vibration and the inlet pressure pulsation of water hydraulic motor. Fig. 2 shows the structure of water hydraulic motor. An axial piston motor consists mainly of a valve port plate with inlet and outlet ports, a swash plate, an outer shell, a cylinder block, pistons with shoes, a bias spring, a port flange and a shaft. The piston fits within bores of the cylinder barrel and is on the same axis as output shaft. The swash plate is positioned at an angle and acts as a surface on which piston shoes travel. The shoes are held in with the swash plate by the retaining ring and the bias spring. The port plate separates incoming fluid from discharging fluid. The output shaft is connected to the cylinder barrel. As the water enters the inlet and exits at the outlet of the hydraulic motor, the pressure in the cylinder hole alternates from high pressure to low pressure. This causes the pressure pulsation to occur. The total cylinder area inside a delivery port is variable as a result of the cycle variations of the piston number ing through the delivery port. It generates the variations of the axial output moment. The variation of force is applied from the piston to the swash plate and the valve cover. The force between the for the swash plate and the valve cover is opposite. The hydraulic motor body vibrates. 3. Characteristics of pressure ripple in water hydraulic system A hydraulic system generates more complicated vibration (noise) on of the interaction between the two pressure pulsations produced by the motor and pump. The fluidborne vibration causes structureborne vibration that has negative effect on the hydraulic motor and pump. Kojima and Shinada [18] and Edge

H.X. Chen et al. / Mechanism and Machine Theory 41 (2006) 487–504

491

Fig. 2. Structural parts of an axial piston motor.

Fig. 3. Water hydraulic motor inlet pressures waveform.

et al. [19] investigated the fluidborne vibration in a combination circuit consisting of a pump, a motor and a connecting pipe. The pressure ripple is characteristic of a fundamental component at piston frequency obtained in a general hydraulic system. The pressure ripples in piston pumps and motor are mostly due to an intermittent impulsive back flow into the cylinder chamber in the neighborhood of the bottom and top

492

H.X. Chen et al. / Mechanism and Machine Theory 41 (2006) 487–504

Fig. 4. The spectrum of the pressure ripple in the inlet of hydraulic motor.

dead center. The characteristic frequencies of fluidborne vibration in hydraulic system have two flow and pressure pulsating sources that the pump and motor generate respectively. Fig. 3 shows the typical experimental inlet pressure ripples of water hydraulic motor in water hydraulic system that consists of axial piston pump, swash-plate type water hydraulic motor and connection. The pressure ripple in the inlet of motor is affected predominantly by the fundamental component of the pumpÕs harmonic frequency (fp) and the motorÕs harmonic frequency (fm and f0). Fig. 4 shows the spectrum of the pressure ripple in the inlet of the water hydraulic motor. One revolution of the motor occurs in T0(1/f0). During this period, there are five pulses to be generated as direct result of the unsteady flow produced by the water hydraulic motor. Each pulse produced by each piston of water hydraulic motor occurs in Tm(1/fm). The inter-cylinder pulse variation is negligibly small. The waveform is periodic over each revolution and the period of the waveform covers one cycle of motor. The delivery pulse associated with axial piston pump occurs in Tp(1/fp). There is the small-amplitude and high-frequency decaying oscillation in the pressure ripple. It is directly resulted from the flow pulse. The flow pulsation interacts with the fluid in the delivery line to create a pressure pulsation which propagates at local acoustic velocity. 4. Theoretical model of water hydraulic motor Fig. 5 shows the dynamic model of a swash-plate type water hydraulic motor. The model has three masses and 14 DOF including outer shell and swash plate, cylinder block, piston and retaining ring as well as valve cover (valve port plate and port flange). The overall coordinate system O–XYZ is set on the valve cover. The origin O is set at the intersection of the centerline of the motor shaft and the internal plane of the valve cover. The axis Z is along the centerline of the motor shaft. The axis X is on the perpendicular plane. The axis Y is on the horizontal plane. The coordinate systems of each mass are defined with the subscripts F, P and C. These subscripts are added to the symbols X, Y and Z that represent the translational motion along the x, y and z axial directions. These subscripts are also added to the symbols / and h which represent the circular motion around axes X and Y. These symbols represent the freedom of movement for each mass. 4.1. Exciting force on cylinder block, piston and swash plate The pressure pulsation in the inlet of hydraulic motor produces the exciting force that causes the hydraulic motor to vibrate. The exciting force from the pressure pulsations act on the piston and the inlet port of the cylinder. It creates the force on the surface between the piston and cylinder inside the hydraulic motor in opposing manner. The geometrical relation of the piston and cylinder coordinate system is shown in Fig. 6.

H.X. Chen et al. / Mechanism and Machine Theory 41 (2006) 487–504

493

Fig. 5. Dynamic model of a water hydraulic motor.

δ0

F′Cz

XP

ly

XC

X

a FPz

Y

lx

α

ZP

R Torque arm

ZC

O

Cylinder

Fig. 6. Geometrical relation of the load force and torque in the piston coordinate system and cylinder coordinate system.

The exerting force FCz on the cylinder along the Z-axis, the torque TCx on the cylinder around XC-axis and the torque TCy on the cylinder around YC-axis in the coordinate system OC–XCYCZC can be expressed by

494

H.X. Chen et al. / Mechanism and Machine Theory 41 (2006) 487–504

F Cz ¼ pA;

T Cx ¼ ly F Cz

T Cy ¼ lx F Cz

ð1Þ

where p is the pulsation pressure in the piston as shown in Fig. 3, A is the cross-sectional area of the piston inside the inlet port of hydraulic motor, lx and ly are the equivalent center coordinates in the cylinder block cross-section of the exerting force FCz. The counterforce of F 0Cz on the piston is expressed by F Cz ¼ F 0Cz . There is angle d0 between the coordinate system OC–XCYCZC of the cylinder and OP–XPYPZP of the piston and retaining ring. The force from the pressure pulsation on the cylinder along X-axis equals to zero and the exerted force on the piston from the pressure pulsation along the X-axis equals to zero [20]. The exerting forces FPz and FPx on the piston and retaining ring along the axis Z and X, the torque TPx on the piston and retaining ring around axis XP and the torque TPy on the cylinder block around axis YP can be expressed in the coordinate system OP–XPYPZP by F Pz ¼ F Cz = cos d0

F Px ¼ 0

T Px ¼ T Cx = cos d0

T Py ¼ T Cy

ð2Þ

where d0 is the angle of the swash plate. 4.2. Shaft torque Fig. 6 shows the geometrical relation between the piston and swash plate. The friction force between the piston and swash plate is small and can be neglected. The force produced by the pressure pulse is FPz. The torque arm is R sin a a sin d0 = cos d0 , where R is the radius of cylinder port circle. Therefore, the piston torque is expressed by T P ¼ F Pz ðR sin a þ a sin d0 Þ= cos d0

ð3Þ

Bahr studied the characteristics of the pump mechanism by the partial mathematically modeling and analyzed the dynamic motion of the pump piston [21]. Considering the inertia force of the moving piston, the inertia-related force along the Z-axis is shown as: F inertia ¼ M P x20 R tan d0 sin a þ 2x0 V P

ð4Þ

where x0 is the angular velocity of the hydraulic motor and VP is the velocity of the piston. The first term is the derivative of the piston vector along the z-axis. The second term is the coriolis acceleration and equals to zero. Hence the instantaneous torque of one piston is expressed by T i ¼ ðF Pz þ M P x2 R tan d0 sin a= cos d0 ÞððR sin a þ a sin d0 Þ= cos d0 Þ

ð5Þ

The equation of torque shows that the torque is related to the inlet pressure of hydraulic motor and angular velocity of hydraulic motor. When the inlet pressure and angular velocity of hydraulic motor increases, the torque increases. 4.3. Dynamic model of swash-plate and outer shell The coordinate system of the outer shell and swash plate is defined as OF–XFYFZF. The origin point OF is set on the gravity center of the outer shell and swash plate. The ZF-axis of the coordinate system OF–XFYFZF is along the centerline of the motor shaft. (XF, YF) axes are parallel to the (X, Y) coordinates. The dynamic equilibrium equation of the motion of the outer shell and swash plate along the direction of the XF can be expressed by M F X€ F þ C Sw sin d0 RB2 _ RB2 _ hF hP ðX_ F sin d0 þ Z_ F cos d0 Z_ P Þ RB1 ð/_ F cos d0 /_ P Þ LPF sin d0 þ cos d0 cos d0 þ K Sw sin d0 RB2 RB2 ðX F sin d þ Z F cos d0 Z P Þ RB1 ð/F cos d0 /P Þ LPF sin d0 þ hP hF cos d0 cos d0 þ 4K Box ðX F þ LF2 hF Þ þ K Shx fðX F LF1 hF Þ ðX C LC1 hC Þg þ C Shx fðX F LF1 hF Þ ðX C LC1 hC Þg ¼ 0 ð6Þ

H.X. Chen et al. / Mechanism and Machine Theory 41 (2006) 487–504

495

Pn1 Pn1 where B1 ¼ i¼0 cosðx0 t þ igÞ, B2 ¼ i¼0 sinðx0 t þ igÞ, g = 2p/n. KSw is the stiffness of the piston. CSw is the damping coefficient of the interface between the piston and the swash plate. KBox is the stiffness of the bolt between the outer shell and the valve cover in the direction of the XF-axis. KShx is the stiffness of the motor shaft in direction of the XF-axis. CShx is the damping coefficient of the interface between the shaft, valve cover and outer shell in the direction of XF-axis and n is the number of pistons. The dynamic equation of the motion in the direction of YF is expressed as: M F Y€ F þ 4K Boy ðY F LF2 /F Þ þ K Shy fðY F þ LF1 /F Þ ðY C þ LC1 /C Þg þ C Shy fðY_ F þ LF1 /_ F Þ ðY_ C þ LC1 /_ C Þg ¼ 0

ð7Þ

where KBoy is the stiffness of the bolt between the outer shell and the valve cover in direction of axis YF. KShy is the stiffness of the motor shaft in the direction of the YF-axis. CShy is the damping coefficient of the interface between the shaft and valve cover and outer shell in the direction of the YF-axis. In the direction of the ZF, the equation of motion is M F Z€ F þ C Sw cos d0 RB2 _ RB2 _ hF hP ðX_ F sin d0 þ Z_ F cos d0 Z_ P Þ RB1 ð/_ F cos d0 /_ P Þ LPF sin d0 þ cos d0 cos d0 þ K Sw cos d0 RB2 RB2 ðX F sin d0 þ Z F cos d0 Z P Þ RB1 ð/F cos d0 /P Þ LPF sin d0 þ hP hF cos d0 cos d0 þ 4K Boz Z F ¼ 0

ð8Þ

where KBoz is the stiffness of the bolt between the outer shell and the valve cover in direction of ZF-axis. The dynamic equations of rotation around XF and YF can be respectively expressed as follows: € C Sw cos d0 R B1 ðX_ F sin d0 þ Z_ F cos d0 Z_ P Þ þ RB3 ð/_ cos d0 þ /_ Þ I Fx / F F P RB5 _ RB5 _ hF þ hP þ LPF B1 sin d0 þ K Sw cos d0 R cos d0 cos d0 B1 ðX F sin d0 þ Z F cos d0 Z P Þ þ RB3 ð/F cos d0 þ /P Þ RB5 RB5 þ LPF B1 sin d0 þ hF þ 4K Boy LF2 ðY F LF2 /F Þ cos d0 cos d0 þ 4K Boz L2b /F þ K Shy LF1 fðY F þ LF1 /F Þ ðY C LC1 /C Þg ð9Þ þ C Shy LF1 fðY_ F þ LF1 /_ F Þ ðY_ C LC1 /_ C Þg ¼ 0 Pn1 2 Pn1 where B3 ¼ i¼0 cos ðx0 t þ igÞ, B5 ¼ i¼0 sinðx0 t þ igÞ cosðx0 t þ igÞ. IFx is the inertia of mass of the outer shell and swash plate about the XF-axis. I Fy € hF þ 4K Box LF2 ðX F þ LF2 hF Þ þ 4K Boz L2b hF RB2 RB5 C Sw ðX_ F sin d0 þ Z_ F cos d0 Z_ P Þ LPF sin d0 þ Rð/_ F cos d0 /_ P Þ LPF sin d0 B1 þ cos d0 cos d0 2 2 R B R B 4 4 h_ F RB2 tg d0 LPF þ h_ P L2PF sin2 d0 þ 2LPF RB2 tg d0 þ þ C Sw cos2 d0 cos2 d0 RB2 RB5 K Sw ðX F sin d0 þ Z F cos d0 Z P Þ LPF sin d0 þ Rð/F cos d0 /P Þ LPF sin d0 B1 þ cos d0 cos d0 2 2 R B4 R B4 L2PF sin2 d0 þ 2LPF RB2 tg d0 þ þ K Sw hF RB2 tg d0 LPF þ hP cos2 d0 cos2 d0 ð10Þ K Shx LF1 fðX F LF1 hF Þ ðX C LC1 hC Þg C Shx LF1 fðX_ F LF1 h_ F Þ ðX_ C LC1 h_ C Þg ¼ 0 Pn1 2 where B4 ¼ i¼0 sin ðx0 t þ igÞ. IFy is the inertia of mass of the outer shell and swash plate around the YF-axis.

496

H.X. Chen et al. / Mechanism and Machine Theory 41 (2006) 487–504

4.4. Dynamic model of piston and swash plate OP–XPYPZP is the coordinate system of the piston and swash plate. The origin point OP is set on the gravity center of the piston and swash plate. The ZP-axis of the coordinate system OP–XPYPZP is along the centerline of the motor shaft. (X, Y) coordinates are on (XP, YP)-axis. Because the surface between the piston and swashplate is smooth, the friction between them is very small. As shown in Fig. 5, the swash-plate s the pistons by the shoes. The interface between the swash plate and piston shoe is very smooth. The friction of the interface along the Y-axis is very small. The swash-plate interface is symmetrical about the X-axis. The damping, stiffness and friction coefficients of the interface between the swash plate and piston along the Y-axis are zero. The motion and force in the direction of YP are zero to simplify the model calculation [20]. The dynamic equilibrium equation of the motion of the swash-plate and outer shell along the direction of the XP and ZP can be respectively expressed by n o M P X€ P C HZ sin d0 nðZ_ P cos d0 X_ P sin d0 Z_ C Þ þ RB1 ð/_ P cos d0 þ /_ C Þ RB2 ðh_ P h_ C Þ K HZ sin d0 fnðZ P cos d0 X P sin d0 Z C Þ þ RB1 ð/P cos d0 þ /C Þ RB2 ðhP hC Þg K P sin d0 fnðZ P cos d0 X P sin d0 Z C Þ þ RA1 ð/P cos d0 þ /C Þ RA2 ðhP hC Þg ¼ 0

ð11Þ

Pn Pni where A1 ¼ i¼0 cosðx0 t þ 1:5igÞ, A2 ¼ i¼0 sinðx0 t þ 1:5igÞ. CHZ is the damping coefficient of the interface between the piston and the cylinder port. KHZ is the stiffness in the interface between the piston and the cylinder port and KP is the stiffness of the spring. € M P Z P C Sw nðX_ F sin d0 þ Z_ F cos d0 Z_ P Þ RB1 ð/_ F cos d0 /_ P Þ RB2 _ RB2 _ hF LPF sin d0 þ hP K Sw nðX F sin d0 þ Z F cos d0 Z P Þ cos d0 cos d0 RB2 RB2 RB1 ð/F cos d0 /P Þ LPF sin d0 þ hP hF cos d0 cos d0 n o þ C HZ cos d0 nðZ_ P cos d0 X_ P sin d0 Z_ C Þ þ RB1 ð/_ P cos d0 þ /_ C Þ RB2 ðh_ P h_ C Þ

þ K HZ cos d0 fnðZ P cos d0 X P sin d0 Z C Þ þ RB1 ð/P cos d0 þ /C Þ RB2 ðhP hC Þg þ K P cos d0 fnðZ P cos d0 X P sin d0 Z C Þ þ RB1 ð/P cos d0 þ /C Þ RB2 ðhP hC Þg ¼ F PZ

ð12Þ

The rotation around XP and YP can be respectively expressed as below: n o € C HZ R cos d0 B1 ðZ_ P cos d0 X_ P sin d0 Z_ C Þ þ RB3 ð/_ cos d0 þ /_ Þ RB5 ðh_ P h_ C Þ I Px / P P C K HZ R cos d0 fB1 ðZ P cos d0 X P sin d0 Z C Þ þ RB3 ð/P cos d0 þ /C Þ RB5 ðhP hC Þg RB5 _ RB5 _ hF þ þ C Sw R nB1 ðX_ F sin d0 þ Z_ F cos d0 Z_ P Þ þ B3 Rð/_ F þ /_ P Þ þ LPF B1 sin d0 þ hP cos d0 cos d0 RB5 RB5 þ K Sw R nB1 ðX F sin d0 þ Z F cos d0 Z P Þ þ B3 Rð/F þ /P Þ þ LPF B1 sin d0 þ hP hF þ cos d0 cos d0 K P r cos d0 fA1 ðZ P cos d0 X P sin d0 Z C Þ þ A3 ð/P cos d0 þ /C Þ rA5 ðhP hC Þg ¼ T Px ð13Þ Pn1 Pn1 where A3 ¼ i¼0 cos2 ðx0 t þ 1:5igÞ, A5 ¼ i¼0 sinðx0 t þ 1:5igÞ cosðx0 t þ 1:5igÞ. IPx is the inertia of mass of the piston and swash plate around XP-axis.

H.X. Chen et al. / Mechanism and Machine Theory 41 (2006) 487–504

497

hP C HZ R cos d0 fB2 ðZ_ P cos d0 X_ P sin d0 Z_ C Þ þ RB5 ð/_ P cos d0 þ /_ C Þ RB4 ðh_ P h_ C Þg I Py € K HZ R cos d0 fB2 ðZ P cos d0 X P sin d0 Z C Þ þ RB5 ð/P cos d0 þ /C Þ RB4 ðhP hC Þg þ C Sw ðR= cos d0 Þ B2 ðX_ F sin d0 þ Z_ F cos d0 Z_ P Þ þ B5 Rð/_ F cos d0 þ /_ P Þ RB4 _ RB4 _ hF þ þ LPF B2 sin d0 þ hP cos d0 cos d0 þ K Sw ðR= cos d0 Þ B2 ðX F sin d0 þ Z F cos d0 Z P Þ þ B5 Rð/F cos d0 þ /P Þ RB4 RB4 þ LPF B2 sin d0 þ hP hF þ K P r cos d0 fA2 ðZ P cos d0 X P sin d0 Z C Þ cos d0 cos d0 þ rA5 ð/P cos d0 þ /C Þ rA4 ðhP hC Þg ¼ T Py where A4 ¼

Pn1

2 i¼0 sin ðx0 t

ð14Þ

þ 1:5igÞ. IPy is the inertia of mass of the piston and swash plate around the YP-axis.

4.5. Dynamic model of cylinder block The coordinate system of the cylinder block is OC–XCYCZC. The original point OC is at the gravity center of the cylinder block. The ZC-axis is on the centerline of the motor shaft. (XC, YC)-axis are parallel to the (X, Y) coordinates. The dynamic equilibrium equation of the motion of the cylinder block in the direction of the XC, YC and ZC can be respectively expressed as follows: M C X€ C K Shx fðX F LF1 hF Þ ðX C LC1 hC Þg C Shx fðX_ F LF1 h_ F Þ ðX_ C LC1 h_ C Þg ¼ 0

ð15Þ

M C Y€ C K Shy fðY F þ LF1 /F Þ ðY C þ LC1 /C Þg þ K Shy ðY C LC2 /C Þ C Shy fðY_ F þ LF1 /_ F Þ ðY_ C þ LC1 /_ C Þg þ C Shy ðY_ C LC2 /_ C Þ ¼ 0

ð16Þ

M C Z€ C C HZ fnðZ_ P cos d0 X_ P sin d0 Z_ C Þ þ RB1 ð/_ P cos d0 þ /_ C Þ RB2 ðh_ P h_ C Þg K HZ fnðZ P cos d0 X P sin d0 Z C Þ þ RB1 ð/P cos d0 þ /C Þ RB2 ðhP hC Þg K P fnðZ P cos d0 X P sin d0 Z C Þ þ RA1 ð/P cos d0 þ /C Þ RA2 ðhP hC Þg þ C Vaz Z_ C þ K Vaz Z C ¼ F Cz

ð17Þ

where KVa and CVa is the stiffness and damping coefficient in the interface between the valve cover and the cylinder block respectively. The equation of rotation about XC and YC can be respectively expressed by € þ C HZ RfB1 ðZ_ P cos d0 X_ P sin d0 Z_ C Þ þ RB3 ð/_ cos d0 þ /_ Þ RB5 ðh_ P h_ C Þg I Cx / C P C þ K HZ RfB1 ðZ P cos d0 X P sin d0 Z C Þ þ RB3 ð/P cos d0 þ /C Þ RB5 ðhP hC Þg C Vaz 2 _ K Vaz 2 RV / C þ R / 2 2 V C K Shy LC1 fðY F þ LF1 /F Þ ðY C þ LC1 /C Þg C Shy LC1 fðY_ F þ LF1 /_ F Þ ðY_ C þ LC1 /_ C Þg ¼ T Cx ð18Þ þ K P rfA1 ðZ P cos d0 X P sin d0 Z C Þ þ rA3 ð/P cos d0 þ /C Þ rA5 ðhP hC Þg þ

hC þ C HZ RfB2 ðZ P cos d0 X P sin d0 Z C Þ þ B5 ð/P cos d0 þ /C Þ þ B4 ðhP þ hC Þg I Cy € þ K HZ fB2 ðZ P cos d0 X P sin d0 Z C Þ þ B5 ð/P cos d0 þ /C Þ þ B4 ðhP þ hC Þg _ ¼ T Cy þ K Shx LC1 fðX F LF1 hF Þ ðX C LC1 hC Þg þ C Shx LC1 fðX_ F LF1 h_ F Þ ðX_ C LC1 hÞg

ð19Þ

where ICx is the inertia of mass of the cylinder block about the XC-axis. ICy is the inertia of mass of the cylinder block about YC-axis.

498

H.X. Chen et al. / Mechanism and Machine Theory 41 (2006) 487–504

5. Generalized system equations of motion The equation of motion for the hydraulic motor can be written in matrix form as _ þ ½Kfqg ¼ fQðtÞg ½Mf€ qg þ ½Cfqg

ð20Þ

where T

fqg ¼ fX F ; Y F ; Z F ; /F ; hF ; X P ; Z P ; /P ; hP ; X C ; Y C ; Z C ; /C ; hC g ; fQðtÞg ¼ f0; 0; 0; 0; 0; 0; F Pz ; T Px ; T Py ; 0; 0; F Cz ; T Cx ; T Cy g. By incorporating the equivalent viscous modal damping, the equation in the form of modal components is expressed as fqg ¼ ½~ ufgðtÞg, where ½~ u is the orthonormal modal matrix. The generalized force is given by T fN ðtÞg ¼ ½~ u fQðtÞg. By applying the orthogonal condition to simplify the above couple equations and transpose of the orthonormal modal matrix, the following modal forced vibration equations can be obtained. € gr þ 2nr pr g_ þ p2r gr ¼ N r ðtÞ

ð21Þ

where nr is the equivalent modal damping ratio. The applicable Matlab [22] simulation program has been developed based on the above approaches. 6. Hydraulic motor numerical simulation and vibration analysis The numerical simulation result of this dynamic model can be obtained using the parameter value in Table 1. The stiffness of the spring is defined as K = Gd4/8ND3. The surfaces between the cylinder block and outer shell, and between the cylinder block and valve cover can be consider as the fluid film bearings [23], which were modeled by the damping and stiffness coefficients. LeeÕs book [23] shows the stiffness coefficients and damping coefficients for the rotational speed dependent bearing properties. The stiffness coefficient is 1.04 · 107 N/m and the damping coefficient is 1.04 · 107 N s/m when the angular speed of the fluid film bearing was 630 rpm. Nishimura et al. [20] defined and provided the other stiffness and damping coefficients in the model of the swash-plate type hydraulic piston pump such as swash-plate stiffness KSwz, KSwx and KSwy, bolt stiffness KBox, KBoy and KBoz, interface stiffness between piston and cylinder port KHZ, interface damping coefficient between piston and cylinder port CHZ, and interface damping coefficient between valve

Table 1 Parameters used in the model Parameter value A CHZ CShx CShy CSw CVa ICx ICy IFx IFy IPx IPy KBox KBoy KBoz KHZ KP KShx

Parameter value 4

2

2 · 10 (m ) 3.4 · 104 (N s/m) 1.0 · 105 (N s/m) 1.0 · 105 (N s/m) 1.8 · 106 (N s/m) 3.6 · 106 (N s/m) 9.4 · 104 kg m2 9.4 · 104 kg m2 2.26 · 103 kg m2 2.26 · 103 kg m2 1.291 · 104 kg m2 1.291 · 104 kg m2 6.5 · 108 (N/m) 6.5 · 108 (N/m) 2.3 · 109 (N/m) 1.2 · 107 (N/m) 4.444 · 103 (N/m) 1.04 · 107 (N/m)

KShy KSw KVa lx ly Lb LC1 LC2 LF1 LF2 LSh MC MF MP n r RV R d0

1.04 · 107 (N/m) 4.3 · 107 (N/m) 9.8 · 108 (N/m) 2.1 · 102 (m) 4 · 105 (m) 3.6 · 102 (m) 7.7 · 102 (m) 4 · 102 (m) 5 · 102 (m) 5.5 · 102 (m) 1.17 · 101 (m) 1.515 kg 1.012 kg 3.53 · 101 kg 5 1.2 · 102 (m) 3 · 102 (m) 2.2 · 102 (m) 17

H.X. Chen et al. / Mechanism and Machine Theory 41 (2006) 487–504

499

P plate and cylinder block CVaz. The moment of inertia of the whole shape is given by the equation I ¼ mi r2i , where it is the sum of all the elemental particles masses multiplied by their distance from the rotational axis squared [24]. In coordinate system OF–XFYFZF of the outer shell, the outer shell is circular as shown in Fig. 5. The upper part of the outer shell in the direction of +XF is more sensitive to the vibration signal than the other directions of the outer shell in the X and Y axes, which was excited by the inlet pressure of the motor. The structure of the out shell in the exact direction of +XF is very strong and the vibration of the outer shell in the exact direction of +XF is not sensitive. The accelerometer is mounted on the equivalent center coordinates (R0 cos #; R0 sin #Þ of the outer shell, where R0 is the radius of the accelerometer position of the outer shell and # is the anger between the radius and +YF-axis, the anger is about 60. The accelerometer position is the nearest point of the outer shell to the inlet point of the motor. The place is most sensitive to the vibration signal, which is excited by the inlet pressure. In order to confirm the validity of the dynamic model, the numerical simulation of the vibration waveform in the equivalent center coordinates (R0 cos #; R0 sin #) of the outer shell was compared with the experimental vibration waveform. Fig. 7 shows the comparison result of the radial acceleration waveform of the equivalent center coordinates (R0 cos #; R0 sin #) between the simulated signal and the experimental signal. It can be seen that there is a high degree of correlation between the numerical analysis results and experimental results for

Fig. 7. The vibration waveform comparison between the simulation results and the experimental results.

Fig. 8. Comparison of vibration spectrum between the simulation results and experimental results.

500

H.X. Chen et al. / Mechanism and Machine Theory 41 (2006) 487–504

both the cycle period and amplitude of the vibration. Fig. 8 shows the spectrum comparison of the simulated acceleration signal and the experimental signal. The fundamental frequencies are approximately same which are the hydraulic motor rotational speed (f0 and fm) and hydraulic pump speed (fp). The integer multiples of the fundamental frequency are also approximately the same. The result of the above comparison confirms the validity of the dynamic analysis presented. In order to clarify the vibration response characteristics for the total water hydraulic motor, the vibration responses from each mass of the hydraulic motor were obtained from the above simulated dynamic mode. Fig. 9 shows the result of the acceleration of the outer shell in the direction of XF, YF and ZF axes. The amplitudes of the acceleration in XF-axis are the largest among the three axes. The amplitudes of the acceleration in YF are smallest. This was as a result of the external force produced by the pressure pulse in the ZF-axis. The external force tends to warp the swash-plate and the outer shell in a downward direction in the XF-axis and pushes the outer shell in the opposite direction in the ZF-axis. Because the stiffness of the outer shell in the ZF-axis is larger than the stiffness in the XF-axis, the magnitudes of the vibration acceleration in the ZF-axis are smaller than that in the XF-axis. Fig. 10 shows the angular acceleration around the XF-axis and the YF-axis. The vibration characteristics around the XF and YF axes are similar. Fig. 11 shows the acceleration of the piston and swash plate in the direction of the XP-axis and the ZP-axis. The vibration characteristics in the XP-axis are different from that in the ZP-axis. The vibration characteristics in the XP-axis are as a result of the piston frequency and the vibration characteristics in the ZP-axis are as a result of the external load force including the hydraulic motor frequency and pump frequency. Compared with the vibration characteristics of outer shell and swash plate, the amplitudes of the vibration in the pistons are large. This was due to the direct external force on the pistons from the pressure pulse and the small stiffness of the piston coordinate system.

Fig. 9. Simulated acceleration of the outer shell in the direction of XF, YF and ZF axes.

H.X. Chen et al. / Mechanism and Machine Theory 41 (2006) 487–504

501

Fig. 10. Simulated angular acceleration of the outer shell XF and YF axes.

Fig. 11. Simulated acceleration of the piston and swash plate in the directions of XP and ZP axes.

Fig. 12 shows the results of the acceleration of the cylinder in the direction of the XC-axis, the YC-axis and the ZC-axis. The vibration in the ZC-axis was generated by the direct pressure pulse in the inlet of hydraulic motor. The amplitudes of the vibration in the ZC-axis are higher than those in the XC-axis and the YC-axis. Because the stiffness in the XC-axis is larger than that in the YC-axis, the influence of the pressure pulse on the vibration of the cylinder XC-axis is smaller that in the YC-axis. The amplitudes of the vibration of the cylinder in the XC-axis are lowest. Fig. 13 shows the simulated acceleration of the radial vibration of the outer shell under the different output torque of hydraulic motor. When the output torque of hydraulic motor increased from 2 N m to 5 N m, the amplitude of vibration clearly increased. The vibration characteristics are the same.

502

H.X. Chen et al. / Mechanism and Machine Theory 41 (2006) 487–504

Fig. 12. Simulated acceleration of the cylinder in the direction of XC, YC and ZC axes.

Fig. 13. Simulated acceleration of the radial vibration of the outer shell. (a) Torque = 2 N m. (b) Torque = 5 N m.

H.X. Chen et al. / Mechanism and Machine Theory 41 (2006) 487–504

503

7. Conclusion A mathematical model was developed to investigate the vibration characteristics of the swash-plate water hydraulic motor in a water hydraulic system. The three mass and 14 DOF model was applied to represent the complete hydraulic motor which consisted of components, such as a outer shell and a swash plate, piston and retaining ring and cylinder block. The dynamic equilibrium equations of motion were formulated to analyze the response with the pressure pulsation as the excitation force. The obtained simulation results revealed the vibration characteristics of each component in the water hydraulic motor. The validity of the analysis was confirmed by comparing it with experimental results. The results showed that the frequency of the harmonic is at the hydraulic motor rotational speed and hydraulic pump rotational speed. The frequencies of the higher harmonics are integer multiples of the hydraulic motor and pump rotational speed. The study includes investigation of the influence of torque on the vibration of the outer shell in the water hydraulic motor. The simulation results showed that the amplitude of vibration increased with the increase of output torque of the water hydraulic motor and the frequencies are almost similar. Acknowledgments The authors wish to thank the School of Mechanical and Production Engineering at Nanyang Technological University for providing the funding and technical for this research. References [1] P. Sorensen, News and trends by the industrial application of water hydraulics, in: The Sixth Scandinavian International Conference on Fluid Power, 1999, pp. 651–674. [2] G.W. Krutz, P.S.K. Chua, Water Hydraulics—Theory and Applications 2004, Workshop on Water Hydraulics, Agricultural Equipment Technology Conference (AETCÕ04), 2004. [3] Y.B. He, P.S.K. Chua, G.H. Lim, A.C.H. Tan, Fault diagnosis of loaded water hydraulic actuators by online testing with labview, Journal of Testing and Evaluation, ASTM 31 (5) (2003) 378–387. [4] N.D. Manring, The torque on the input shaft of an axial-piston swash-plate type hydrostatic pump, Journal of Dynamic Systems, Measurement, and Control (1998) 57–62. [5] G. Zeiger, A. Akers, Torque on the swashplate of an axial piston pump, Transactions of the ASME (1985) 220–226. [6] P. Kaliafetis, T.H. Costopoulos, Modelling and simulation of an axial piston variable displacement pump with pressure control, Mechanism and Machine Theory 30 (4) (1995) 599–612. [7] N.D. Manring, The discharge flow ripple of an axial-piston swash-plate type hydrostatic pump, Journal of Dynamic Systems, Measurement, and Control, ASME (2000) 263–268. [8] A.M. Harrison, K.A. Edge, Reduction of axial piston pump pressure ripple, Proceedings of the Institution of Mechanical Engineers 214 (1) (2000) 53–63. [9] E. Kojima, M. Shinada, Characteristics of fluidborne noise generated by a fluid power pump, Bulletin of JSME 29 (258) (1986) 4147– 4155. [10] K.A. Edge, D.N. Johnston, The secondary source method for the measurement of pump pressure ripple characteristics, Part 1— Description of method, Proceedings of the Institution of Mechanical Engineers 204 (1990) 33–40. [11] D.N. Johnston, K.A. Edge, A test method for the measurement for pump fluid-borne noise characteristics, SAE Paper 911761, International Off-highway and Powerplant Congress and Exposition, Milwaukee, USA, 1991. [12] K.A. Edge, J. Darling, A theoretical model of axial piston pump flow ripple, in: Proceedings of the First Bath International Fluid Power Workshop, 1998, pp. 113–133. [13] Z.L. Qiu, Z.N. Zhen, Y.Z. Lu, Study on vibrational energy transmission characteristics from cylinder to swashplate within an axial piston pump, in: Proceedings of the 9th International Symposium on Fluid Power, 1990, pp. 261–269. [14] Z.G. Qi, Y.X. Lu, Vibration source, transmission path response analysis and condition monitoring of hydraulic pumps, The Journal of Fluid Control 21 (1) (1991) 61–69. [15] M.K. Bahr, J. Svoboda, R.B. Bhat, Vibration analysis of constant power regulated swash plate axial piston pumps, Journal of Sound and Vibration 259 (5) (2003) 1225–1236. [16] T. Nishimura, T. Umeda, T. Tsuta, M. Fujimara, M. Kamakami, Dynamic response analysis of a swash-plate type hydraulic piston pump, dynamic fracture, failure, and deformation, ASME (1995) 145–155. [17] Danfoss, Inc., Denmark, Available from: . [18] E. Kojima, M. Shinada, Characteristics of fluidborne noise generated by a fluid power pump, Bulletin of JSME 29 (258) (1986) 4147– 4155. [19] K.A. Edge, O.P. Boston, S. Xiao, J. Longvill, C.R. Burrows, Pressure pulsations in reciprocating pump piping systems, Proceedings of Institution of Mechanical Engineers, Part I—Journal of System Control Engineer (1997) 229–250.

504

H.X. Chen et al. / Mechanism and Machine Theory 41 (2006) 487–504

[20] T. Nishimura, T. Umeda, T. Tsuta, M. Fujiwara, M. Kawakami, Dynamic response analysis of a swash-plate type hydraulic piston pump, dynamic fracture, failure and deformation, ASME (1995) 145–155. [21] M.K. Bahr Khalil, J. Svoboda, R.B. Bhat, Modeling of swash plate axial piston pumps with conical cylinder blocks, Journal of Mechanical Design, V 126 (1) (2004) 196–200. [22] D.M. Etter, Engineering Problem Solving with MATLAB, Prentice-Hall, Englewood Cliffs, NJ, 1993. [23] C.W. Lee, Vibration Analysis of Rotors, Kluwer Academic Publishers, 1993. [24] A.P. Boresi, Advanced Mechanics of Materials, John Wiley & Sons, 2003.