- Email: [email protected]
Shock wave characteristics of a hydraulic damper for shock test machine
ARTICLE IN PRESS Mechanical Systems and Signal Processing 24 (2010) 1570–1578
Contents lists available at ScienceDirect
Mechanical Systems and Signal Processing journal homepage: www.elsevier.com/locate/jnlabr/ymssp
Shock wave characteristics of a hydraulic damper for shock test machine Sujuan Jiao a,, Yu Wang b, Lei Zhang b, Hongxing Hua a a b
School of Mechanical Engineering, Shanghai Jiao Tong University, 200240 Shanghai, China Naval Research Center, Box 1303-14, 120 CuiWei Road, 100073 Beijing, PR China
a r t i c l e i n f o
abstract
Article history: Received 15 November 2008 Received in revised form 17 December 2009 Accepted 24 December 2009 Available online 7 January 2010
A hydraulic damper is developed to generate the shock wave in this paper. The working principle of the damper is explained and the corresponding mathematical model is established. The shock wave characteristics under different shock velocities are obtained by using the numerical computation and experiment. The results show that the shock wave characteristics directly relate to the sectional area of the exhausted passages. The computational results agree well with the experimental data, which means that the proposed mathematical model can be used for the engineering design. & 2010 Elsevier Ltd. All rights reserved.
Keywords: Hydraulic damper Shock wave characteristic Shock test machine
1. Introduction Shock and vibration cannot only reduce the equipments’ lifetime and working precision, but also lead to structural degradation and component failure, which have been discussed by Wang and Hua [1], Scavuzzo and Pusey [2], Rittweger et al. [3], Yang et al. [4], and Richard et al. [5]. In order to anticipate the anti-shock ability of the equipment or the isolators, various shock test machines are developed. The general configuration of the shock machine is the drop machine, which consists of several vertical guide rods on which the table carrying the test item drops from certain height freely (Fig. 1). When the table strikes the machine base, the impact occurs. In order to obtain an impact characteristic with a given shape, the shock programmer is usually set between the test table and the machine base. The conventional drop machine usually uses the elastic material as the shock programmer, which has three disadvantages. First, the tested mass or impact severity is limited due to the strength of the programmer, and it cannot satisfy the requirements of modern aero and marine industries. Fig. 2 shows three kinds of damaged conventional programmers during our experiments, in which the white one, the red one and the blue one are made of a kind of density felt, polyurethane, and strengthened polyurethane, respectively. Second, the repetition depends on the characteristic of the programmer. Because the permanent deformation usually occurs after several impacts, the shock wave form cannot repeat accurately. Third, the conventional drop machine can only generate single shock wave which is usually true in the air, but in the underwater explosion, there exist dual shock waves. One is the positive shock, the other is the negative shock, which is discussed by many researchers including Wadley and Dharmasena [6], Moyer [7], Li [8], O’Daniel et al. [9], and Chen et al. [10]. In order to simulate this kind of dual shock waves, the machine which can generate dual shock waves should be developed, the first shock is positive shock which accelerates the tested item, the second shock is negative shock which makes the input velocity go to zero and the negative shock wave must be Corresponding author. Tel.: + 86 21 34206813-826; fax: + 86 21 34206813-814.
E-mail address: [email protected] (S. Jiao). 0888-3270/$ - see front matter & 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.ymssp.2009.12.005
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1571
The test item The test table
The programmer The guide rod
The machine base
Fig. 1. Schematic configuration of the shock test machine.
Fig. 2. The damaged programmers during the shock experiments. (a) The density felt; (b) the strengthened polyurethane and (c) another kind of strengthened polyurethane.
controllable. Elder and Arden [11] and Hunziker [12] used the servo valve to control the shock wave, which can only test lighter mass and supply mild impact. The hydraulic damper is often used as an absorber to alleviate the violent impacts and attenuate the vibrations. Smooth force characteristic is emphasized in these applications, which are studied by Wang et al. [13], Hou [14], Eyres et al. [15], Jia et al. [16], Hu et al. [17], Lee and Moon [18] and Swevers et al. [19]. However, the hydraulic damper can also be developed to be a programmer, and because the kinetic energy can be absorbed simultaneously, the input velocity can go to zero in the end. By adjusting the parallel passage, the shock wave characteristics can be regulated. The shock wave characteristics of the hydraulic damper are discussed in this paper. This kind of hydraulic damper can be used in the drop machine to generate the serious impact, and it can also be used in the dual chock machine to generate the second shock wave soon after the first wave finishes. Because the shock wave is generated by decreasing or cutting off the fluid passage, the shock velocity and shock severity as well as the tested mass can be increased. 2. Principle of the hydraulic damper Fig. 3 shows the schematic and working principle of the hydraulic damper. The hydraulic oil is filled in the chamber, and the liquid level is a little higher above the top of the chamber. The test item can be connected with the piston (Fig. 3(a)). The piston falls freely from a certain height, before the piston enters into the chamber, a sharp-edged orifice forms between the piston and the chamber (Fig. 3(b)), the fluid exhausts through the sharp-edged orifice, and the orifice pore. After the piston enters into the chamber, the sharp-edged orifice is off, an annular gap forms between the piston and the chamber, the fluid exhausts through the annular gap, and the orifice pore simultaneously (Fig. 3(c)). During this process, because of the compressibility of the oil and the suddenly contracting sectional area, the resulting high pressure in the chamber applies an impact to the piston, then the piston and test item stop. The effective area of the piston can be designed according to the tested mass and shock severity, and the heavier test item can be tested under larger effective area. The falling height of the piston can be changed to obtain different impacting velocities, and faster velocity can be obtained under higher falling height. The orifice pore can also be adjusted to obtain different impact severities. The serious severity can be obtained under smaller sectional area of the orifice pore. 3. Mathematical model and the computational parameters The mathematical model is established based on the following assumptions. First, the influence of oil viscosity is ignored, because our related researches show that comparing to the local pressure difference, the pressure difference
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test item
acceleration transducer
piston rod piston
annular orifice
orifice pore chamber
sharp-edged orifice
Fig. 3. Schematic and working principle of the hydraulic damper. (a) The initial stage; (b) the second stage and (c) the third stage.
induced by the viscosity of working liquid is very small and can be ignored. The temperature measurements in another smaller damper with the same principle indicate that the maximum local temperature change is 20 1C, the average temperature in the cylinder is almost a constant, which shows that the influence of the temperature can be neglected. Therefore, the viscosity which is often affected by the temperature can also be neglected. Second, the entrance effect is ignored. These assumptions simplify the calculation process, and are easy to guide the design of the damper. The starting point of impact is considered as the coordinate origin. 3.1. Mathematical model The force balance equation of the piston and the test table is given by Newton’s second law pp Ap ¼ Mg þ C x_ p þ M x€ p
ð1Þ 2
2
where pp is the pressure in the piston chamber (Pa), Ap = 1/4pD the effective area of the cylinder (m ), D the diameter of the piston (m), it is equal to 125 mm in the paper, M the total mass of the test table and the tested item (kg), g the gravity acceleration, it is 9.8 m s 2 in the paper, C the effective friction coefficient, x_ p the velocity of the test table, it is the differentiation of the displacement of the test table (m s 1), its maximum value is about 5 m s 1 in our work, x€ p the acceleration of the test table, its minimum value is about 1000 m s 2 in our work. According to the working condition, the force induced by the pressure and the inertial acceleration is much bigger than the force induced by the friction. So the second item in the right side of the equation is ignored in order to simplify the calculation. Then acceleration of the piston can be expressed as pp Ap Mg M
ð2Þ
pp ¼ Dpg ¼ Dpb
ð3Þ
x€ p ¼
where Dpg is the pressure loss at the annular gap, Dpb the pressure loss at the orifice pore, Dpg is calculated by
Dpg ¼ zg
ru2g 2
where zg is the loss coefficient and ug the fluid velocity in the necking section of the annular gap. zg is calculated by a zg ¼ 0:5 1 g Ap
ð4Þ
ð5Þ
ug is calculated by ug ¼
Qg ag
ð6Þ
where Qg is the flow rate through the annular gap, ag the sectional area of the annular gap, it is calculated by ag ¼ pDh
ð7Þ
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where h is the width of the annular gap. Dpb is calculated by
Dpb ¼ zb
ru2b
ð8Þ
2
where zb is the loss coefficient and ub the fluid velocity in the orifice pore. They are calculated by the following equations: a ð9Þ zb ¼ 0:5 1 b Ap ub ¼
Qb ab
ð10Þ
where Qb is the flow rate through the orifice pore (m3 s 1), ab the sectional area of the orifice pore, it is calculated by ab ¼
1 2 pd 4
ð11Þ
where d is the diameter of the orifice pore (m). pp is calculated according to the compression flow rate Qp ¼
Ap ðLþ xp Þ dpp be dt
xp r l
Qp ¼
Ap ðLxp Þ dpp be dt
xp Z l
ð12Þ
where L is the length of the piston chamber (m), it is equal to 75 mm in this paper, be the oil bulk modulus (Pa), it is equal to 0.5 109 Pa in this paper, xp the displacement of the test table, it is the twice integration of the acceleration, l the initial opening of the sharp-edged orifice (m), when the damper is used in the test rig to generate the second shock wave, it is used to improve the quality of the connection of the two shock waves, it is equal to 5 mm in this paper, dpp/dt is calculated using the conventional differentiation pp(k) pp(k 1)/dt, and the value of dt is equal to 1 10 5 s. Continuity equations have different expressions during different working processes, so Qp is calculated by different equations. Before the piston enters into the chamber, Qp is calculated by Qp ¼ Q Q1 Qb
ð13Þ
When the piston enters into the chamber, Qp is calculated by Qp ¼ Q Qg Qb
ð14Þ
where Q is the total flow rate, Q1 the flow rate through the sharp-edged orifice, they are calculated by Q ¼ Ap x_ p Q1 ¼ Cd pDðlxp Þ
ð15Þ sffiffiffiffiffiffiffiffi 2pp
r
xp ol
ð16Þ
where r is the oil density (kg m 3), it is equal to 850 kg m 3 and Cd the flow rate coefficient, it is equal to 0.69 in this paper. 3.2. Computational parameters The computational parameters are as follows: the piston diameter is 125 mm, the oil density is 850 kg m 3, the oil bulk modulus be is 0.5 109 Pa, the oil viscosity is 30 cSt, the total mass of the piston is 253 kg, the maximum falling height is 1130 mm, the impact velocity can be calculated by the free fall law and integration of the acceleration. 4. Results and discussions 4.1. The test rig The test rig is set up to study the shock wave characteristics. Fig. 4 is the photograph of the test rig and the pistons. The hydraulic damper is mounted on a base. The piston is lifted to a certain height, and falls down freely. Because the shock wave characteristics are mainly affected by the sectional area of the exhausted passages, three kinds of pistons and orifices with different dimensions are manufactured, different annular gaps form between different pistons and the chamber. The shock wave characteristics under different velocities and exhausted sectional areas are tested. 4.2. Influence of the annular gap Fig. 5 shows the computational and experimental results under different annular gaps. The peak accelerations increase and the shocking durations decrease with the decrease of annular gap, which means the shock wave can be adjusted by
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Fig. 4. Photograph of the test rig and the pistons. (a) The hydraulic damper and (b) the pistons.
changing the annular gap. When the annular gap is 0.2, 0.3, 0.4 mm, the diameter of the orifice pore is 2 mm, and the impact velocity is 3.32 m s 1, the experimental peak acceleration is 95, 90, 70g, and the corresponding duration is 10, 10, 12 ms, respectively. It may be noted that when the acceleration is equal to 1g, the shocking duration is considered finished. The computational peak acceleration is 100, 70, 50g, and the duration is 8, 10, 15 ms. When the annular gap is 0.2, 0.3, 0.4 mm, the difference in peak acceleration between the computational and experimental results is 5%, 22.2%, 28.6%, respectively. The difference becomes larger with the increase of annular gap. That might be explained as follows: the mathematic model is based on the oil compressibility and the local pressure loss, the peak acceleration is mainly influenced by the compressibility. The duration is mainly influenced by the sectional area, at the instant the impact occurs, and the fluid does not flow actually. The transition process needs further investigation. It can be observed that the piston stops after impacting. The computational and experimental results indicate that the hydraulic damper can produce serious impact, and the shock machine does not need additional brake system. 4.3. Influence of the diameter of the orifice pore According to the mathematic model, the orifice pore is also an important factor affecting the shock wave characteristics, the computational and experimental results under different orifice diameters are shown in Fig. 6. When the annular gap is 0.2 mm, the impact velocity is 3.32 m s 1, and the diameter of the orifice pore is 2, 5, 10 mm, the peak acceleration obtained by experiment is 95, 90, 68g, and the relevant duration is 12, 15, 15 ms, respectively, which means that the impact severity of the hydraulic damper can be adjusted by changing the orifice diameter. The peak acceleration produced by numerical computation is 100, 88, 52g, and the duration time is 9, 10, 15 ms, respectively. The precision of the computation is 95%, 97.8%, 76.4%, which also agrees well with the experiment, especially when the orifice diameter is smaller. The difference becomes larger with the increase of the orifice diameter. 4.4. Shock wave under different velocities The shock wave characteristics under different impact velocities are also investigated. The computational and experimental results under different velocities are shown in Fig. 7. The impact velocity is 3.32, 3.06, 2.79 m s 1. When the annular gap is 0.2 mm, and the diameter of the orifice pore is 2 mm, the peak acceleration obtained by experiment is 95, 82, 70g, and the duration time is 11, 15, 15 ms, respectively, which means the damper can satisfy the requirements of different impact velocities. The corresponding computational result is 100, 90, 78g, respectively, which is a little larger than the test results. The precision of the computation is 95%, 91.1%, 89.7%. Because of the smaller sectional area, the computational precision is higher. The comparisons between the results obtained from calculation and experiment show that the larger area of the annular gap or the orifice results in lower modeling accuracy. So it is better to make a smaller area, and design the area of the orifice to be equivalent to that of the annular gap. But this criterion decreases the regulating range of the acceleration. Just as shown by the experimental results, the shock waves have a little hysteresis, which perhaps is not permitted in some conditions. If this is true, the hysteresis can be attenuated by chamfering the entrance edge of the piston. The angle
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120 experimental
acceleration (g)
100
simulation
80 60 40 20 0 -20 0
5
10
15
20
time (ms) 100
acceleration (g)
80 60 40 20 0 -20 0
5
0
5
10 time (ms)
15
20
80
acceleration (g)
60 40 20 0 -20
10 time (ms)
15
20
Fig. 5. Shock waves under different annular gaps. (a) Annular gap is 0.2 mm; (b) annular gap is 0.3 mm and (c) annular gap is 0.4 mm.
and the dimension of the chamfer are the important factors for the performance of shock wave. However, in many environments, whether the serious shock wave can be produced is the most important thing, so the hysteresis is not discussed in the paper.
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experimental
100
simulation
accelation (g)
80 60 40 20 0 -20 0
5
10 Time (ms)
0
5
10 Time (ms)
0
5
15
20
100
accelation (g)
80 60 40 20 0 15
20
100
accelation(g)
80 60 40 20 0 10 Time (ms)
15
20
Fig. 6. Shock waves under different orifice diameters. (a) Orifice diameter is 2 mm; (b) orifice diameter is 5 mm and (c) orifice diameter is 10 mm.
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120 100
acceleration (g)
80 60 40 20 0 -20 0
5
10 time (ms)
15
20
0
5
10 time (ms)
15
20
0
5
10 time (ms)
15
20
100
acceleration (g)
80 60 40 20 0 -20
80
acceleration (g)
60 40 20 0 -20
Fig. 7. Shock waves under different velocities. (a) velocity is 3.32 ms 1; (b) velocity is 3.06 ms 1 and (c) velocity is 2.79 ms 1.
5. Conclusions Some conclusions can be drawn according to the computational and experimental results: (1) The peak acceleration of the hydraulic damper can reach 100g, and the duration time is within 15 ms, which means that the hydraulic damper can be used to generate the shock wave and supply violent impact to the tested item.
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(2) The shock wave characteristics are mainly influenced by the exhausted sectional area, the annular gap and the orifice. The shock wave characteristics can be adjusted through changing the sectional area of the orifice, the higher shock level could be obtained using smaller sectional area. (3) The shock severity varies with the mass of the tested item and/or impact velocity, by adjusting the orifice diameter and the falling height, different test requirements could be satisfied. (4) The computational results agree well with the experiment, especially when the sectional area is smaller. It can be drawn that the peak acceleration is mainly influenced by the compressibility and the local pressure loss, and the duration depends on the sectional area.
References [1] Y. Wang, H.X. Hua, Shock Theory and Application of Modern Vehicle, Science Press, Beijing, 2005 (in Chinese). [2] R.J. Scavuzzo, H.C. Pusey, Naval Shock Analysis and Design, The Shock and Vibration Information Analysis Center, Booz-Allen and Hamilton Inc., USA, 2000, pp. 1–46. [3] A. Rittweger, J. Albus, H.P. Mourey, Passive damping devices for aerospace structures, Acta Astronautica 50 (2002) 597–608. [4] P. Yang, Y.H. Tan, J.M. Yang, N. Sun, Measurement, simulation on dynamic characteristics of a wire gauze-fluid damping shock absorber, Mechanical Systems and Signal Processing 20 (2006) 745–756. [5] D.N. Richard, I. Ana, A.J. Starkey, A.A. Rodger, Design and dynamic analysis of a pneumatic impulse generating device for the non-destructive testing of ground anchorages, Mechanical Systems and Signal Processing 21 (2007) 2523–2545. [6] H. Wadley, K. Dharmasena, Compressive response of multilayered pyramidal lattices during underwater shock loading, International Journal of Impact Engineering 35 (2008) 1102–1114. [7] E. T. Moyer, The effects of plasticity on the whipping response of surface ships and submarines, in: Proceedings of the 62th Shock and Vibration Symposium-CD, 1991. [8] Y.J. Li, Experimental study of ship whipping induced by underwater explosive bubble, Journal of Ship Mechanics 5 (6) (2001) 75–83. [9] J.L. O’Daniel, T. Krauthammer, L.K. Kevin, L.H. Strait, An UNDEX response validation methodology, International Journal of Impact Engineering 27 (2002) 919–937. [10] Y. Chen, Z.P. Tong, H.X. Hua, Y. Wang, H.Y. Gou, Experimental investigation on the dynamic response of scaled ship model with rubber sandwich coatings subjected to underwater explosion, International Journal of Impact Engineering 36 (2009) 318–328. [11] S. Elder, K. Arden, Land based full scale shock simulator, in: Proceedings of the 70th Shock and Vibration Symposium-CD, 1999. [12] P. Hunziker, VESPA: A servohydraulic shock testing machine, Fourth International Conference on Structures under Shock and Impact, Udine, 1996, pp. 255–264. [13] Y.B. Wang, M.L. Xu, Y.L. Qiu, Four-linear model of hydraulic damper used in car, Chinese Journal of Applied Mechanics 20 (2003) 60–63. [14] C.Y. Hou, Fluid dynamics and behavior of nonlinear viscous fluid dampers, Journal of Structural Engineering 134 (2008) 56–63. [15] R.D. Eyres, A.R. Champneys, N.A. Lieven, Modelling and dynamic response of a damper with relief valve, Nonlinear Dynamics 40 (2005) 119–147. [16] J.H. Jia, X.Y. Shen, H.X. Hua, Viscoelastic behavior analysis and application of fractional derivative Maxwell model, Journal of Vibration and Control 13 (2007) 385–401. [17] J.W. Hu, J.H. Zhang, S.L. Hang, Dynamic simulation of the hydraulic shock absorber for sled impact test, Mechanics and Engineering 25 (2003) 17–20. [18] C.T. Lee, B.Y. Moon, Simulation and experimental validation of vehicle dynamic characteristics for displacement-sensitive shock absorber using fluidflow modeling, Mechanical Systems and Signal Processing 20 (2006) 373–388. [19] J. Swevers, C. Lauwerys, B. Vandersmissen, M. Maes, K. Reybrouck, P. Sas, A model-free control structure for the on-line tuning of the semi-active suspension of a passenger car, Mechanical Systems and Signal Processing 21 (2007) 1422–1436.
Contents lists available at ScienceDirect
Mechanical Systems and Signal Processing journal homepage: www.elsevier.com/locate/jnlabr/ymssp
Shock wave characteristics of a hydraulic damper for shock test machine Sujuan Jiao a,, Yu Wang b, Lei Zhang b, Hongxing Hua a a b
School of Mechanical Engineering, Shanghai Jiao Tong University, 200240 Shanghai, China Naval Research Center, Box 1303-14, 120 CuiWei Road, 100073 Beijing, PR China
a r t i c l e i n f o
abstract
Article history: Received 15 November 2008 Received in revised form 17 December 2009 Accepted 24 December 2009 Available online 7 January 2010
A hydraulic damper is developed to generate the shock wave in this paper. The working principle of the damper is explained and the corresponding mathematical model is established. The shock wave characteristics under different shock velocities are obtained by using the numerical computation and experiment. The results show that the shock wave characteristics directly relate to the sectional area of the exhausted passages. The computational results agree well with the experimental data, which means that the proposed mathematical model can be used for the engineering design. & 2010 Elsevier Ltd. All rights reserved.
Keywords: Hydraulic damper Shock wave characteristic Shock test machine
1. Introduction Shock and vibration cannot only reduce the equipments’ lifetime and working precision, but also lead to structural degradation and component failure, which have been discussed by Wang and Hua [1], Scavuzzo and Pusey [2], Rittweger et al. [3], Yang et al. [4], and Richard et al. [5]. In order to anticipate the anti-shock ability of the equipment or the isolators, various shock test machines are developed. The general configuration of the shock machine is the drop machine, which consists of several vertical guide rods on which the table carrying the test item drops from certain height freely (Fig. 1). When the table strikes the machine base, the impact occurs. In order to obtain an impact characteristic with a given shape, the shock programmer is usually set between the test table and the machine base. The conventional drop machine usually uses the elastic material as the shock programmer, which has three disadvantages. First, the tested mass or impact severity is limited due to the strength of the programmer, and it cannot satisfy the requirements of modern aero and marine industries. Fig. 2 shows three kinds of damaged conventional programmers during our experiments, in which the white one, the red one and the blue one are made of a kind of density felt, polyurethane, and strengthened polyurethane, respectively. Second, the repetition depends on the characteristic of the programmer. Because the permanent deformation usually occurs after several impacts, the shock wave form cannot repeat accurately. Third, the conventional drop machine can only generate single shock wave which is usually true in the air, but in the underwater explosion, there exist dual shock waves. One is the positive shock, the other is the negative shock, which is discussed by many researchers including Wadley and Dharmasena [6], Moyer [7], Li [8], O’Daniel et al. [9], and Chen et al. [10]. In order to simulate this kind of dual shock waves, the machine which can generate dual shock waves should be developed, the first shock is positive shock which accelerates the tested item, the second shock is negative shock which makes the input velocity go to zero and the negative shock wave must be Corresponding author. Tel.: + 86 21 34206813-826; fax: + 86 21 34206813-814.
E-mail address: [email protected] (S. Jiao). 0888-3270/$ - see front matter & 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.ymssp.2009.12.005
ARTICLE IN PRESS S. Jiao et al. / Mechanical Systems and Signal Processing 24 (2010) 1570–1578
1571
The test item The test table
The programmer The guide rod
The machine base
Fig. 1. Schematic configuration of the shock test machine.
Fig. 2. The damaged programmers during the shock experiments. (a) The density felt; (b) the strengthened polyurethane and (c) another kind of strengthened polyurethane.
controllable. Elder and Arden [11] and Hunziker [12] used the servo valve to control the shock wave, which can only test lighter mass and supply mild impact. The hydraulic damper is often used as an absorber to alleviate the violent impacts and attenuate the vibrations. Smooth force characteristic is emphasized in these applications, which are studied by Wang et al. [13], Hou [14], Eyres et al. [15], Jia et al. [16], Hu et al. [17], Lee and Moon [18] and Swevers et al. [19]. However, the hydraulic damper can also be developed to be a programmer, and because the kinetic energy can be absorbed simultaneously, the input velocity can go to zero in the end. By adjusting the parallel passage, the shock wave characteristics can be regulated. The shock wave characteristics of the hydraulic damper are discussed in this paper. This kind of hydraulic damper can be used in the drop machine to generate the serious impact, and it can also be used in the dual chock machine to generate the second shock wave soon after the first wave finishes. Because the shock wave is generated by decreasing or cutting off the fluid passage, the shock velocity and shock severity as well as the tested mass can be increased. 2. Principle of the hydraulic damper Fig. 3 shows the schematic and working principle of the hydraulic damper. The hydraulic oil is filled in the chamber, and the liquid level is a little higher above the top of the chamber. The test item can be connected with the piston (Fig. 3(a)). The piston falls freely from a certain height, before the piston enters into the chamber, a sharp-edged orifice forms between the piston and the chamber (Fig. 3(b)), the fluid exhausts through the sharp-edged orifice, and the orifice pore. After the piston enters into the chamber, the sharp-edged orifice is off, an annular gap forms between the piston and the chamber, the fluid exhausts through the annular gap, and the orifice pore simultaneously (Fig. 3(c)). During this process, because of the compressibility of the oil and the suddenly contracting sectional area, the resulting high pressure in the chamber applies an impact to the piston, then the piston and test item stop. The effective area of the piston can be designed according to the tested mass and shock severity, and the heavier test item can be tested under larger effective area. The falling height of the piston can be changed to obtain different impacting velocities, and faster velocity can be obtained under higher falling height. The orifice pore can also be adjusted to obtain different impact severities. The serious severity can be obtained under smaller sectional area of the orifice pore. 3. Mathematical model and the computational parameters The mathematical model is established based on the following assumptions. First, the influence of oil viscosity is ignored, because our related researches show that comparing to the local pressure difference, the pressure difference
ARTICLE IN PRESS 1572
S. Jiao et al. / Mechanical Systems and Signal Processing 24 (2010) 1570–1578
test item
acceleration transducer
piston rod piston
annular orifice
orifice pore chamber
sharp-edged orifice
Fig. 3. Schematic and working principle of the hydraulic damper. (a) The initial stage; (b) the second stage and (c) the third stage.
induced by the viscosity of working liquid is very small and can be ignored. The temperature measurements in another smaller damper with the same principle indicate that the maximum local temperature change is 20 1C, the average temperature in the cylinder is almost a constant, which shows that the influence of the temperature can be neglected. Therefore, the viscosity which is often affected by the temperature can also be neglected. Second, the entrance effect is ignored. These assumptions simplify the calculation process, and are easy to guide the design of the damper. The starting point of impact is considered as the coordinate origin. 3.1. Mathematical model The force balance equation of the piston and the test table is given by Newton’s second law pp Ap ¼ Mg þ C x_ p þ M x€ p
ð1Þ 2
2
where pp is the pressure in the piston chamber (Pa), Ap = 1/4pD the effective area of the cylinder (m ), D the diameter of the piston (m), it is equal to 125 mm in the paper, M the total mass of the test table and the tested item (kg), g the gravity acceleration, it is 9.8 m s 2 in the paper, C the effective friction coefficient, x_ p the velocity of the test table, it is the differentiation of the displacement of the test table (m s 1), its maximum value is about 5 m s 1 in our work, x€ p the acceleration of the test table, its minimum value is about 1000 m s 2 in our work. According to the working condition, the force induced by the pressure and the inertial acceleration is much bigger than the force induced by the friction. So the second item in the right side of the equation is ignored in order to simplify the calculation. Then acceleration of the piston can be expressed as pp Ap Mg M
ð2Þ
pp ¼ Dpg ¼ Dpb
ð3Þ
x€ p ¼
where Dpg is the pressure loss at the annular gap, Dpb the pressure loss at the orifice pore, Dpg is calculated by
Dpg ¼ zg
ru2g 2
where zg is the loss coefficient and ug the fluid velocity in the necking section of the annular gap. zg is calculated by a zg ¼ 0:5 1 g Ap
ð4Þ
ð5Þ
ug is calculated by ug ¼
Qg ag
ð6Þ
where Qg is the flow rate through the annular gap, ag the sectional area of the annular gap, it is calculated by ag ¼ pDh
ð7Þ
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where h is the width of the annular gap. Dpb is calculated by
Dpb ¼ zb
ru2b
ð8Þ
2
where zb is the loss coefficient and ub the fluid velocity in the orifice pore. They are calculated by the following equations: a ð9Þ zb ¼ 0:5 1 b Ap ub ¼
Qb ab
ð10Þ
where Qb is the flow rate through the orifice pore (m3 s 1), ab the sectional area of the orifice pore, it is calculated by ab ¼
1 2 pd 4
ð11Þ
where d is the diameter of the orifice pore (m). pp is calculated according to the compression flow rate Qp ¼
Ap ðLþ xp Þ dpp be dt
xp r l
Qp ¼
Ap ðLxp Þ dpp be dt
xp Z l
ð12Þ
where L is the length of the piston chamber (m), it is equal to 75 mm in this paper, be the oil bulk modulus (Pa), it is equal to 0.5 109 Pa in this paper, xp the displacement of the test table, it is the twice integration of the acceleration, l the initial opening of the sharp-edged orifice (m), when the damper is used in the test rig to generate the second shock wave, it is used to improve the quality of the connection of the two shock waves, it is equal to 5 mm in this paper, dpp/dt is calculated using the conventional differentiation pp(k) pp(k 1)/dt, and the value of dt is equal to 1 10 5 s. Continuity equations have different expressions during different working processes, so Qp is calculated by different equations. Before the piston enters into the chamber, Qp is calculated by Qp ¼ Q Q1 Qb
ð13Þ
When the piston enters into the chamber, Qp is calculated by Qp ¼ Q Qg Qb
ð14Þ
where Q is the total flow rate, Q1 the flow rate through the sharp-edged orifice, they are calculated by Q ¼ Ap x_ p Q1 ¼ Cd pDðlxp Þ
ð15Þ sffiffiffiffiffiffiffiffi 2pp
r
xp ol
ð16Þ
where r is the oil density (kg m 3), it is equal to 850 kg m 3 and Cd the flow rate coefficient, it is equal to 0.69 in this paper. 3.2. Computational parameters The computational parameters are as follows: the piston diameter is 125 mm, the oil density is 850 kg m 3, the oil bulk modulus be is 0.5 109 Pa, the oil viscosity is 30 cSt, the total mass of the piston is 253 kg, the maximum falling height is 1130 mm, the impact velocity can be calculated by the free fall law and integration of the acceleration. 4. Results and discussions 4.1. The test rig The test rig is set up to study the shock wave characteristics. Fig. 4 is the photograph of the test rig and the pistons. The hydraulic damper is mounted on a base. The piston is lifted to a certain height, and falls down freely. Because the shock wave characteristics are mainly affected by the sectional area of the exhausted passages, three kinds of pistons and orifices with different dimensions are manufactured, different annular gaps form between different pistons and the chamber. The shock wave characteristics under different velocities and exhausted sectional areas are tested. 4.2. Influence of the annular gap Fig. 5 shows the computational and experimental results under different annular gaps. The peak accelerations increase and the shocking durations decrease with the decrease of annular gap, which means the shock wave can be adjusted by
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Fig. 4. Photograph of the test rig and the pistons. (a) The hydraulic damper and (b) the pistons.
changing the annular gap. When the annular gap is 0.2, 0.3, 0.4 mm, the diameter of the orifice pore is 2 mm, and the impact velocity is 3.32 m s 1, the experimental peak acceleration is 95, 90, 70g, and the corresponding duration is 10, 10, 12 ms, respectively. It may be noted that when the acceleration is equal to 1g, the shocking duration is considered finished. The computational peak acceleration is 100, 70, 50g, and the duration is 8, 10, 15 ms. When the annular gap is 0.2, 0.3, 0.4 mm, the difference in peak acceleration between the computational and experimental results is 5%, 22.2%, 28.6%, respectively. The difference becomes larger with the increase of annular gap. That might be explained as follows: the mathematic model is based on the oil compressibility and the local pressure loss, the peak acceleration is mainly influenced by the compressibility. The duration is mainly influenced by the sectional area, at the instant the impact occurs, and the fluid does not flow actually. The transition process needs further investigation. It can be observed that the piston stops after impacting. The computational and experimental results indicate that the hydraulic damper can produce serious impact, and the shock machine does not need additional brake system. 4.3. Influence of the diameter of the orifice pore According to the mathematic model, the orifice pore is also an important factor affecting the shock wave characteristics, the computational and experimental results under different orifice diameters are shown in Fig. 6. When the annular gap is 0.2 mm, the impact velocity is 3.32 m s 1, and the diameter of the orifice pore is 2, 5, 10 mm, the peak acceleration obtained by experiment is 95, 90, 68g, and the relevant duration is 12, 15, 15 ms, respectively, which means that the impact severity of the hydraulic damper can be adjusted by changing the orifice diameter. The peak acceleration produced by numerical computation is 100, 88, 52g, and the duration time is 9, 10, 15 ms, respectively. The precision of the computation is 95%, 97.8%, 76.4%, which also agrees well with the experiment, especially when the orifice diameter is smaller. The difference becomes larger with the increase of the orifice diameter. 4.4. Shock wave under different velocities The shock wave characteristics under different impact velocities are also investigated. The computational and experimental results under different velocities are shown in Fig. 7. The impact velocity is 3.32, 3.06, 2.79 m s 1. When the annular gap is 0.2 mm, and the diameter of the orifice pore is 2 mm, the peak acceleration obtained by experiment is 95, 82, 70g, and the duration time is 11, 15, 15 ms, respectively, which means the damper can satisfy the requirements of different impact velocities. The corresponding computational result is 100, 90, 78g, respectively, which is a little larger than the test results. The precision of the computation is 95%, 91.1%, 89.7%. Because of the smaller sectional area, the computational precision is higher. The comparisons between the results obtained from calculation and experiment show that the larger area of the annular gap or the orifice results in lower modeling accuracy. So it is better to make a smaller area, and design the area of the orifice to be equivalent to that of the annular gap. But this criterion decreases the regulating range of the acceleration. Just as shown by the experimental results, the shock waves have a little hysteresis, which perhaps is not permitted in some conditions. If this is true, the hysteresis can be attenuated by chamfering the entrance edge of the piston. The angle
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acceleration (g)
100
simulation
80 60 40 20 0 -20 0
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time (ms) 100
acceleration (g)
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0
5
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acceleration (g)
60 40 20 0 -20
10 time (ms)
15
20
Fig. 5. Shock waves under different annular gaps. (a) Annular gap is 0.2 mm; (b) annular gap is 0.3 mm and (c) annular gap is 0.4 mm.
and the dimension of the chamfer are the important factors for the performance of shock wave. However, in many environments, whether the serious shock wave can be produced is the most important thing, so the hysteresis is not discussed in the paper.
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experimental
100
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accelation (g)
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0
5
15
20
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accelation (g)
80 60 40 20 0 15
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accelation(g)
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Fig. 6. Shock waves under different orifice diameters. (a) Orifice diameter is 2 mm; (b) orifice diameter is 5 mm and (c) orifice diameter is 10 mm.
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15
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100
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80 60 40 20 0 -20
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acceleration (g)
60 40 20 0 -20
Fig. 7. Shock waves under different velocities. (a) velocity is 3.32 ms 1; (b) velocity is 3.06 ms 1 and (c) velocity is 2.79 ms 1.
5. Conclusions Some conclusions can be drawn according to the computational and experimental results: (1) The peak acceleration of the hydraulic damper can reach 100g, and the duration time is within 15 ms, which means that the hydraulic damper can be used to generate the shock wave and supply violent impact to the tested item.
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(2) The shock wave characteristics are mainly influenced by the exhausted sectional area, the annular gap and the orifice. The shock wave characteristics can be adjusted through changing the sectional area of the orifice, the higher shock level could be obtained using smaller sectional area. (3) The shock severity varies with the mass of the tested item and/or impact velocity, by adjusting the orifice diameter and the falling height, different test requirements could be satisfied. (4) The computational results agree well with the experiment, especially when the sectional area is smaller. It can be drawn that the peak acceleration is mainly influenced by the compressibility and the local pressure loss, and the duration depends on the sectional area.
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