- Email: [email protected]
A novel heavy-weight shock test machine for simulating underwater explosive shock environment: Mathematical modeling and mechanism analysis
International Journal of Mechanical Sciences 77 (2013) 239–248
Contents lists available at ScienceDirect
International Journal of Mechanical Sciences journal homepage: www.elsevier.com/locate/ijmecsci
A novel heavy-weight shock test machine for simulating underwater explosive shock environment: Mathematical modeling and mechanism analysis Gongxian Wang a,b, Yeping Xiong b,n, Wenzhi Tang a a b
School of Logistics Engineering, Wuhan University of Technology, Wuhan 430063, China Fluid-Structure Interaction Research Group, Faculty of Engineering and the Environment University of Southampton, Highfield, Southampton SO17 1BJ, UK
art ic l e i nf o
a b s t r a c t
Article history: Received 5 May 2012 Received in revised form 25 July 2013 Accepted 12 September 2013 Available online 22 October 2013
A novel heavy-duty shock test machine is developed to satisfy the newly-built shock resistance standard and simulate accurately the actual underwater explosive environments with increased testing capability. The mathematical model for the shock test machine is created to predict its dynamic performance and analyze its mechanism. Then numerical simulation is carried out to evaluate the prospective capability of the shock test machine under different shock velocity inputs. The double protection system incorporating the stroke limit of the accumulator piston and the unloading circuit can effectively prevent the secondary collision in the testing process. The simulation results have demonstrated that the shock test machine proposed in this paper can produce nearly the same shock acceleration waveform as the new shock resistance standard BV043/85 and MIL-S-901D. Moreover, this shock test machine can be regulated conveniently to adjust to a different type of equipment and be extended easily to suit more severe shock environments and heavier equipment. The proposed system configuration and associated mathematical model provide theoretical basis and useful design techniques for practical applications. Crown Copyright & 2013 Published by Elsevier Ltd. All rights reserved.
Keywords: Shock test machine Velocity generator Rapid opening high flow valve Shock test Underwater explosion
1. Introduction Warships and shipboard equipment are often damaged by intense shock waves of the near-field non-contact underwater explosion (UNDEX) during their military service. Therefore the ability to withstand the UNDEX effects has been regarded as an important aspect of the survivability of the new and existing warships, and receives full attention from the navy of all countries. The naval powers have conducted extensive and in-depth research on the methods to examine the shock resistant capability of the key shipboard equipment. For instance, the USA formulated the military specification MIL-S-901D in 1989, and Germany proposed a new military specification defined in BV043/85, specifying the requirements for the high impact test for the shipboard equipment. Although MIL-S-901D is prepared specifically for military applications, the standard is often used for commercial products as well. For marine equipment, in general, shock test machine is a prevailing way to examine the ability to survive an underwater explosion. However, both the light-and medium-weight shock test machines, which have maximum testing capacities of 125 kg and
n
Corresponding author. Tel.: þ 44 2380596619; fax: þ 44 2380597744. E-mail addresses: [email protected], [email protected] (Y. Xiong).
2700 kg respectively and maximum impact velocity less than 3.4 m/s, can merely meet the traditional MIL-S-901 specification. Moreover, these test machines can only generate shock wave in one direction by either dropping or pendulum rotating the impacting mass. Therefore, these types of shock test machines can only meet the MIL-S-901C specification. Challengingly, there is little room to extend the shock level and capability of test payload of these two types of machines due to their nature of energy storage or impact generating patterns as well as their giant sizes. Hence equipment beyond this weight range has to be tested in underwater explosive environment in order to assess its shock resistant capability. However, this is undesirable because of associated high cost, damage to the environment and long test cycle. Nowadays, the trend of continuous improvement of the underwater weapons to enhance accuracy and power has led to an even higher demand for the shock resistant ability of warship equipment. The latest specifications of shock-resistant ability in the world are MIL-S-901D and BV043/85, which have increased the velocity requirement to 5 m/s and even higher in special circumstances such as testing newly developed shipboard equipment. Moreover, the maximum testing capacity of the current shock machines has exceeded the limits of the traditional mediumweight ones (2700 Kg) and it still keeps increasing. Thus, various shock test equipments are developed. The U.S. Team Corporation
0020-7403/$ - see front matter Crown Copyright & 2013 Published by Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijmecsci.2013.09.006
G. Wang et al. / International Journal of Mechanical Sciences 77 (2013) 239–248
developed a new type of shock test system called Subsidiary Component Shock Test System (SSTS), which contains a long stroke hydraulic actuator that can be configured to operate in either a vertical or horizontal position [1,2]. Weidlinger Associates Inc. constructed a system for Non-Explosive Shock Testing of Naval Vessels and Equipment whereby shock test with explosion can be replaced with non-explosive sources based on the very rapid release of high-pressured air from reservoirs very close to the vessel under test [3]. Germany WTD71 of Kiel developed and built a laboratory shock test machine for simulating UNDEX testing which can produce half-sine shock wave according to BV043/85 specification. MTS systems Corporation was contracted by Westinghouse Bettis to design and build a laboratory shock and vibration-testing machine for simulating UNDEX testing of naval vessel components with high load and high fidelity. This test machine can test components up to 10 t [4]. Shanghai Jiaotong University designed and built a laboratory shock testing machine named Heavyweight Dual-wave Shock Test Machine which is being debugged [5,6]. Comparing the current shock test facilities and some new concepts of shock test machines such as MTS Firing Impulse Simulators (FIS) and Full Scale Shock Simulator, it is clearly seen that the development trends of shock test machines towards three major directions. The first trend is that shock test machines are capable to generate both positive and negative shock pulses to simulate real UNDEX environment which is made up of shock wave followed by bubble pulse and structural whipping [7]. The second is that shock test machines can test heavy-weight equipment, and the last is that shock pulses can be controlled and customized conveniently [8]. However, one of the key technologies for developing new shock test machines is how to supply enough energy and dissipate it in a short time, which can be controlled and also user-friendly with good vibration isolation design [9–11]. In this paper, we propose a novel heavy-duty shock test machine and establish its mathematical model for the integrated dynamic system, and also analyze its mechanism and dynamic performance. The key features of this new shock test machine are (i) high energy storage capacity; (ii) instantaneous release of tremendous energy; (iii) rapid dynamic response; (iv) high automaticity; (v) hydraulic damper energy dissipation; (vi) enhanced testing capacity of 5000 kg; and (vii) meets both the latest shock resistance standard of MIL-S-901D and BV043/85.
Test article
Measurement and control system
Table
Damping cylinder
Waveform generator
Impacting mass
Buffer system
Velocity generator
Hydraulic system
Vibration isolation Fig. 1. Schematic of the system of the Heavy-weight Shock Test Machine.
a1
Acceleration/m.s-2
240
τ1
t/ms Fig. 2. The semi-sinusoidal acceleration pulse.
2. System description of the heavy-weight shock test machine Fig. 1 shows a schematic of the system of the proposed heavyweight shock test machine, which produces semi-sinusoidal acceleration pulses vertically as shown in Fig. 2. The shock test machine consists of a hydraulic system made up of a velocity generator and a buffer system, a measurement and control system, and a vibration isolation system. With the test item mounted on the table, the impacting mass, driven by the velocity generator, applies impact load to the waveform generator located at the bottom of the table and produces a semi-sinusoidal acceleration wave required in the specification. Meanwhile, the buffer system, comprised of hydraulic damping cylinders, limits the stroke of the table and prevents it from bouncing off after the impact. The measurement and control system can collect and process data, and monitor important parameters such as pressure of the whole system, velocity of the impacting mass, and acceleration of the table as well as the test item.
3. Mathematical model of the integrated dynamic system The generation of the wave of the shock testing machine is based on the principle of collision, whose working mechanism can be simplified to three degrees of freedom vibrating system as shown in Fig. 3. The system is composed of springs, dampers and masses, in which ks represents the coefficient of elasticity of the waveform generator while C(t) the damping coefficient of the hydraulic damping cylinder. Since the damping cylinder merely serves to prevent the table from flying off too high when hitting the impacting mass, this paper will not study the damping property of the cylinder. Without loss of any important features, the mathematical model of the shock test machine is established based on the following hypotheses: (1) the impacting mass, the table and the foundation mass are rigid bodies, while the waveform generator is regarded as a linear elastic body; (2) on the instant of impact, the compressed gas in the air bag of the accumulator is in heat insulation, i.e. without energy exchange with the outside; (3) only local pressure loss is taken into account due to the shortness of the pipeline of the hydraulic system; (4) the mechanical friction loss and leakage in the motions of piston and spools are ignored; (5) the liquid pressure in each chamber is regarded as evenly distributed; (6) the velocity distribution of each cross-section flow
G. Wang et al. / International Journal of Mechanical Sciences 77 (2013) 239–248
xt
mt ks xi
xh mh
C(t)
F(t) F(t) xf
mf kf
Fig. 3. Mechanical model of the shock test machine.
is uniform; and (7) the compressibility of liquid is ignored considering its insignificance in comparison with that of gas. The mathematical expression of the system is as follows: 8 m x€ þks ðtÞðxh xt xi Þ ¼ FðtÞ > < h h mt x€ t þ CðtÞðx_ t x_ f Þ þ ks ðtÞðxt þ xi xh Þ ¼ 0 ð1Þ > : m x€ þ CðtÞðx_ x_ Þ þ k x ¼ FðtÞ t f f f f f where the coefficient of elasticity of the waveform generator, ks(t), is formulated as ( ks xh xt Z xi ks ðtÞ ¼ ð2Þ 0 xh xt o xi Here, mt is the mass of the table; mh, that of the impacting mass; mf, of the foundation mass; xf represents the displacement of the foundation mass; xh, that of the impacting mass; xt, of the table. xi is the initial distance between the impacting mass and the waveform generator. ks is the coefficient of elasticity of the spring shown in Fig. 3; kf, that of the coil spring. In order to excite the table with an acceleration pulse shown in Fig. 2, F(t) has to accelerate the impacting mass, mh, to V2 within the given stroke, xi; that is to say, F(t) must satisfy the following equation: 8R < ti FðtÞ=mh dt ¼ ðmh þ mt Þτ1 a1 =ðmh πÞ 0 R R ð3Þ : 0ti ð 0t FðtÞ=mh dtÞdt ¼ xi
241
timely. In order to meet the challenges above, this paper proposes a velocity generator regulated by a rapid opening, high flow valve, the construction and operation of which is illustrated in Fig. 4. The liquid in high-pressure accumulator 1 flows to the lower chamber of hydraulic cylinder barrel 6 through the valve port of the throttle ring upon the moving of spool 2, and the driving impacting mass reaches the requested velocity within the given stroke; wherein, pressure relief valve 3 controls the close of spool 2, control valve 4 monitors the velocity of the system by adjusting the opening of throttle spool 5, vent valve 7 can prevent the piston from colliding with the top of the hydraulic cylinder barrel through a regulation of the gas pressure in the upper chamber of the cylinder barrel, and unloading valve 8 can avoid a second impact of the impacting mass and the table by unloading the lower chamber of the hydraulic cylinder barrel quickly. The mechanical model of valveregulated velocity generator is shown in Fig. 5, and its mathematical model can be deduced as follows. 4.1. The adiabatic state equation of gas The adiabatic state equation of the compressed gas in the accumulator is !k 2 πd1 pg V g0 þ x1 ¼ pg0 V kg0 ð4Þ 4
The adiabatic state equation of the gas in the upper chamber of the hydraulic cylinder barrel is pN ðs4 xh Þk ¼ pN0 sk4
ð5Þ
In the equations above, pg0 and pN0 are the initial pressures in the accumulator and the upper chamber of the hydraulic cylinder barrel respectively; Vg0 the initial volume of the compressed gas in the accumulator; d1 the diameter of the piston of the accumulator; s4 the stroke of the impacting mass; x1 the displacement of the accumulator piston; pg and pN are the gas pressure in the
V
P 7
4
In the equation above, ti is the impacting instant of the impacting mass and the table; τ1 the pulse width of the acceleration wave; a1 the peak value of the acceleration wave.
V
6 5
4. Mathematical model of the velocity generator The impact velocity of the impacting mass and the waveform generator is a critical factor in the realization of the pulse of impact acceleration requested in the specification. Three challenges exist when the impacting mass actuating the system. Firstly, the storage and instantaneous release of high energy in the actuation requires the impacting mass to accelerate from its complete static state to a scheduled velocity within a limited stroke; secondly, the impact velocity must be appropriately regulated; thirdly, a second impact of the impacting mass and the table must be avoided by unloading the actuating system
2 9
3 1
SV
10
8
V
Fig. 4. Schematic of the structure of the velocity generator with: (1) accumulator, (2) spool, (3) pressure relief valve, (4) control valve, (5) throttle spool, (6) hydraulic cylinder barrel, (7) vent valve, (8) unloading valve, (9) spool ring, and (10) limit.
242
G. Wang et al. / International Journal of Mechanical Sciences 77 (2013) 239–248
where D2 is the diameter of spool 2, x2 its displacement, δ the clearance inside its recess hole, and p2 the pressure in its upper chamber; p3 the pressure in the pilot passage of the hydraulic cylinder; ρ fluid density; and C2 flow coefficient, whose value depends on Reynolds number Re. The flow capacity through throttle ring 5 is governed by: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2ðp3 p4 Þ Q 3 ¼ C 3 πD4 xv ð8Þ ρ The unloading flow capacity of pressure relief valve 3 is Qc ¼
πD43 p 128μl c
ð9Þ
and the flow capacity through unloading valve 8 is sffiffiffiffiffiffiffiffi 2pa Q u ¼ C u Au ðtÞ ρ
Au ðtÞ ¼
8 <0 πD2
: 4τuu ðt t i Þ
t o ti t Z ti
ð10Þ
ð11Þ
where C3 is flow coefficient; D4 the inlet diameter of the hydraulic cylinder barrel; xv the opening of spool 5 of the throttle ring; p4 the pressure in the pilot passage of the cylinder barrel; pc the fluid pressure of the lower chamber of spool 2; μ the dynamic viscosity of fluid; l the length of the passage of the hydraulic valve; Cu the flow coefficient of unloading valve 8 and Du, its diameter; pa the fluid pressure of the lower chamber of the hydraulic cylinder barrel; and τu unloading valve 8 response time. The equilibrium equation of the system flow is given by Q 1 ¼ Q 3 þQ C Q3 ¼
1 2 πd x_ þQ u 4 2 h
ð12Þ ð13Þ
4.3. Pressure equation At the inlet of the accumulator, abrupt shrink of the flow area leads to local pressure loss Δp1 as follows: !2 h i Q π 2 1 ð14Þ Δp1 ¼ p1 p2 ¼ 4ρ 1 ðd1 D21 Þ 4 πD21 and at the inlet of the hydraulic cylinder barrel, abrupt expansion of the flow area causes Δp2, i.e., " #2 !2 D24 Q3 ð15Þ Δp2 ¼ p3 p4 ¼ 8ρ 1 4Hd2 πD24 Fig. 5. The mechanical model of the velocity generator. (A) Partial enlarged detail of part A. (B) Partial enlarged detail of part B.
accumulator and upper chamber of cylinder barrel, respectively; k is adiabatic constant of the gas.
In the equations above, D1 is the diameter of the pilot passage of the accumulator; d2 that of the hydraulic cylinder barrel piston; H the distance between the bottom of the piston and that of the cylinder barrel.
4.2. Flow capacity
4.4. The equation of motion The equations of motion of the accumulator piston and spool 2 are respectively
The total flow capacity of the system is Q1 ¼
1 2 πd x_ 1 4 1
and the flow capacity through spool 2 is given by sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2ðp2 p3 Þ Q 2 ¼ C 2 πD2 x22 þ δ2 ρ
ð6Þ
m1 x€ 1 ¼
1 2 πd ðp p1 Þ c1 x_ 1 m1 gμ1 4 1 g
ð16Þ
1 2 πD ðp pc Þ c2 x_ 2 þ m2 g 4 2 3
ð17Þ
and ð7Þ
m2 x€ 2 ¼
G. Wang et al. / International Journal of Mechanical Sciences 77 (2013) 239–248
where m1 is the mass of the accumulator piston; p1 the oil pressure of the accumulator, and c1, the dynamic damping coefficient of its piston; m2 the mass of spool 2, c2 the dynamic damping coefficient of the spool rings. The equation of motion for the impacting mass is written as 1 2 1 2 2 πd p πðd2 d3 ÞpN 4 2 a 4 c3 x_ h ks ðtÞðxh þ xi xt Þ mh g
pN ¼ pN0
pc ¼
s4 s4 xh
p1 ¼ p2 þ ζ 1
( cðtÞ ¼
ð19Þ
xt o disp xt Z disp
0 c
ð20Þ
where d3 the diameter of the piston rod; c3 the dynamic damping coefficient of the piston; c the damping coefficient of the damping cylinder barrel, and disp is the maximum displacement of the table permitted in BV043/85 specification. The equation of motion for the foundation mass is 2 mf x€ f ¼ kf xf þ mf g þpa πd2 =4
ð21Þ
ρ A1 x_ 1 2 2 A2
x€ 1 ¼
A1 ðpg p1 Þ c1 x_ 1 gμ1 m1
x€ 2 ¼
A3 ðp3 pc Þ c2 x_ 2 þg m2
x€ h ¼
A1 ¼ A3 ¼
1 2 πd 4 1
ð22 1Þ
1 2 πD 4 2
A7 pa ðA7 A8 ÞpN c3 x_ h ks ðtÞðxh þxi xt Þ g mh
ð22 2Þ A7 ¼
1 2 1 2 πd ; A8 ¼ πd3 4 2 4
ð22 3Þ x€ t ¼
ks ðtÞðxh xi xt Þ cðtÞx_ t g mt
ð22 4Þ
x€ f ¼
A7 pa kf xf g mf
ð22 5Þ
0 pg ¼ pg0 @
1k V g0 V g0 þ
2
πd1 4
A
ð22 6Þ
x1
ð22 8Þ A2 =2 A2 ¼ πD21 =4; ζ 1 ¼ 1 A1
ρ A1 x_ 1 A3 x_ 2 2 ; 2 C 2 A4
A4 ¼ πD2
p3 ¼ p4 þ
ρ A1 x_ 1 A3 x_ 2 2 ; 2 C 3 A5
A5 ¼ πD4 xv
p4 ¼ pa þ ζ 2
ρ A1 x_ 1 A3 x_ 2 2 ; A6 2
qffiffiffiffiffiffiffiffiffiffiffiffiffiffi x22 þ δ2
ð22 9Þ
ð22 10Þ
ð22 11Þ
A6 2 A6 ¼ πD24 =4; ζ 2 ¼ 1 ; ζ 3 ¼ πd2 H ζ3
ð22 12Þ 8 m ðA x€ A x€ Þ ðA A Þp þ k ðtÞðx x x Þ 3 2 h 1 1 þ 7 8 N As7 h i t > < A27 2 pa ¼ > : 2ρ A1 x_ 1 CAA3 x_ 2ðtÞ A7 x_ h u
Eq. (1) is for the general mechanical model of this shock test machine, whereas Eqs. (18), (19) and (21) are the detailed description of the equations in Eq. (1), respectively. In Eq. (1), the force F(t) acting on the impacting mass and foundation mass is the generalized force which contains gravitational force, impact force and counterforce induced by the velocity generator. They are descripted in Eqs. (18), (19) and (21) in detail. An integrated mechanical model of the shock machine system has 13 variables, i.e., x1, x2, xh, xt, xf,pg, pN, pc, p1, p2, p3, p4 and pa which can be obtained by solving the following set of Eqs. (1)–(21) from which we obtain the derived equations as follows:
ð22 7Þ
p2 ¼ p3 þ
and the equation of motion for the testing table is mt x€ t ¼ ks ðtÞðxh xi xt Þ mt g cðtÞx_ t
k
128μl πD22 x_ 2 πD43 4
mh x€ h ¼
ð18Þ
243
u
t ot i t Zt i
ð22 13Þ
However, the nonlinearity of equations and the complication of boundary conditions make it very difficult, even impossible, to acquire the analytical solutions in reality. So in order to examine the dynamic performance of the system, the method of numerical computation has to be used to obtain the numerical solutions for each unknown variable.
5. Dynamic property and system performance Based on the developed mathematical model for the shock machine system in Section 4, we will solve the equation set above using the fourth-order Runge–Kutta method and predict the dynamic performance of the shock machine system and analyze its mechanism. To reveal the dynamic property of the system, parameters for the proposed mathematical model of the shock machine system are carefully selected to accurately describe its mechanical property. The key parameters used in the numerical simulation are listed in Table 1. Moreover, this “virtual test” technology can also lay a solid foundation for the practical construction and adjustment of shock machine systems. 5.1. Response time of the rapid opening high flow valve The response time of the rapid opening, high flow valve refers to the time needed for the upper end of spool ring 2 to migrate from the spool land to the lower part of the pilot passage of the valve body. This is the time needed for the passage from being totally blocked to being completely open, which mainly depends on the velocity of the spool 2. According to Eq. (17), this velocity is
Table 1 Parameters used in simulation. Parameters
Quantity
Parameters
Quantity
Impacting mass (kg) mh Table mass (kg) mt Accumulator piston mass (kg) m1 Spool ring 2 mass (kg) m2 Foundation mass (ton) mf Accumulator piston radius (mm) d1 Accumulator piston stroke (mm) s1 Accumulator inlet radius (mm) D1
5000 5000 10 5 50 250 500 100
Accumulator gas initial pressure (MPa) pg0 Accumulator gas initial volume (m3) Vg0 Piston radius (mm) d2 Piston rod radius (mm) d3 Unloading hole radius (mm) Du Unloading valve 8 response time (ms) τu Spool ring 2 stroke (mm) s2 Pilot passage radius (mm) D4
13 0.1 200 100 40 50 200 50
G. Wang et al. / International Journal of Mechanical Sciences 77 (2013) 239–248
related to the pressure difference between the upper and lower spool ends, and the size as well as the mass of spool 2. Once the initial pressure of the accumulator is set, the pressure difference between the spool ends is determined by the pressure relief velocity, i.e., the passage diameter D3 of pressure relief valve 3. Three cases are shown in Fig. 6 for D3 equals to 10 mm, 15 mm, and 20 mm respectively, and their corresponding response times are approximately 4 ms, 5 ms and 14 ms, which means that the response time can be reduced by enlarging the cross-section surface of the pressure relief passage. Eq. (17) reveals that if the cross-section surface of the spool 2 is enlarged, the driving power will be strengthened, and thus the velocity of the spool ring will be enhanced and its response time will be reduced. In fact, the enlargement of the spool ring’s section surface will not definitely lead to the reduction of its response time, which can be proven by the curves in Fig. 7, which illustrates three different diameters of spool 2, namely, 50 mm, 100 mm and 150 mm. It can be seen that their corresponding response times are approximately 9 ms, 7 ms and 13 ms respectively, which on the other hand reveals that the enlargement of spool ring's crosssection surface cannot definitely reduce its response time. The reason can be explained by Eq. (9), which shows that the enlargement of the spool ring's cross-section surface will possibly increase the flow capacity of the pressure relief hole, and thereby
350 D3 = 10 mm D3 = 15 mm D3 = 20 mm
Dsiplacement/mm
300 250 200 150
X: 4.45 Y: 100.1
X: 14.08 Y: 100
100 X: 5.31 Y: 100.3
50 0
0
5
10
15
20
Time/ms
150 m2 = 5 kg m2 = 10 kg
X: 14.08 Y: 100
m2 = 15 kg
Displacement/mm
244
X: 15.14 Y: 100
100 X: 14.63 Y: 100.1
50
0
0
5
10
15
20
Time/ms Fig. 8. The displacement of spool 2 with different parameter m2 equals 5 kg, 10 kg and 15 kg.
will increase the pressure in sensing chamber. As a result, the pressure difference applied on the spool 2 becomes smaller instead, and then its migration is slackened and hence its response time is prolonged. Fig. 8 shows that an increase in the spool's mass will slightly extend its response time. The acceleration of spool 2 in its opening process is illustrated in Fig. 9, in which the first acceleration peak is formed by the pressure difference exerted on the spool when pressure relief valve 3 is open, and the trough occurs upon the sudden deceleration and brake of the spool 2 when it moves to the end of its stroke and bumps the cushion device. When the acceleration is zero, the spool 2 is in its free travel and hence uniform motion, with the resultant force on it being zero. Figs. 10 and 11 display the spool's corresponding velocity and displacement curves. When opening the valve, the spool ring receives impact from both the acceleration and the deceleration with the acceleration peaks being 350 g and 100 g respectively, which should be considered in the design of cushion devices so as to avoid damage to spool rings. According to the analysis above, when the response time of the valve is less than 10 ms, the maximum flow capacity can reach 15,000 L/min. The most efficient measure to speed up the valve's response is to enlarge the flow area of the pressure relief circuit.
Fig. 6. The displacement of spool 2 with different parameter D3 equals 10 mm, 15 mm and 20 mm.
5.2. Velocity regulating property 300 D2 = 50 mm D2 = 100 mm D2 = 150 mm
Displacement/mm
250 200 150
X: 7.09 Y: 100.1
X: 9.25 Y: 100.1
X: 12.67 Y: 100
100 50 0
0
5
10
15
20
Time/ms Fig. 7. The displacement of spool 2 with different parameter D2 equals 50 mm,100 mm and 150 mm.
The impact velocity of the impacting mass can be regulated by the velocity generator through the control of the flow capacity of the system, i.e. through adjusting the opening of the spool of throttle ring 5. Figs. 12 and 13 show the corresponding velocities of 3.89 m/s, 4.90 m/s and 5.15 m/s of the impacting mass respectively when spool 5 is set at different openings of 10 mm, 20 mm and 30 mm under the prescribed displacement 350 mm for the impacting mass. The corresponding displacement curve of the impacting mass is given in Fig. 14. From the data shown above, it is clearly seen that if the initial distance between the impacting mass and the table is preset as xi which is unchanged, the impacting velocity of the impacting mass can be regulated by adjusting the opening of the spool ring of throttle ring 5. It can also know that the readjustment of the throttle ring will change the instant of the impact between the impacting mass and the table, because if the opening of spool 5 is reduced, then the acceleration time of the impacting mass will be increased, therefore the impact will be delayed. This, however, will not affect the velocity regulating property of the system.
G. Wang et al. / International Journal of Mechanical Sciences 77 (2013) 239–248
600
6
500
5
245
400
Ve1ocity/m.s-1
Acceleration/g
4 300 200 100
3 2
0
1
-100
0
-200
0
10
20
30
40
50
60
-1
Time/ms
xv = 10 mm xv = 20 mm xv = 30 mm 0
50
100
150
200
Displacement/mm
Fig. 9. The acceleration of spool 2.
Fig. 12. The velocity of the impacting mass with different parameter xv equals 10 mm, 20 mm and 30 mm.
8 6 6
Ve1ocity/m.s-1
5
Ve1ocity/m.s-1
4
2
4 3 2
xv = 10 mm xv = 20 mm xv = 30 mm
0 1 -2
0
10
20
30
40
50
60
0
Time/ms
0
50
100
150
200
250
300
350
400
Displacement/mm
Fig. 10. The velocity of spool 2. Fig. 13. Velocity–displacement curve of the impacting mass with different parameter xv equals 10 mm, 20 mm and 30 mm..
250
400 X: 116.4 Y: 350
300 150
Displacement/mm
Displacement/mm
200
100
50
200
100 xv = 10 mm xv = 20 mm xv = 30 mm
0 0
0
10
20
30
40
50
60
Time/ms Fig. 11. The displacement of spool 2.
X: 118.8X: 135.5 Y: 350 Y: 350
-100
0
50
100
150
200
Time/ms
5.3. Unloading property
Fig. 14. Displacement of the impacting mass with different parameter xv equals 10mm, 20mm and 30mm.
Understanding the failure mechanism is important for successful design of shock test machine. According to the shock testing specification, the impacting mass is required to impact the table only once in each shock test without secondary impacts, otherwise it would be regarded as a failure. It is understood that when liquid, as a medium, is squeezed by the expansion of the high-pressure
gas, the impacting mass is actuated to impact the table and then it will bound back, and thus the gas will be compressed again and also expanded again to drive the impacting mass to impact the table for a second time resulting a failure. In order to prevent the secondary impact, a double protection system is introduced in the system, which includes both the stroke limit of the accumulator
246
G. Wang et al. / International Journal of Mechanical Sciences 77 (2013) 239–248
600
14 Hydraulic shock in accerleration chamber due to impacting between shock hammer and shock table
12
Displacement/mm
Pressure curve during the opening of unloading valve
10
Pressure/MPa
500
8 Pressure in acceleration chamber during shock hammer accelerating
6
400 300 200
4
0
100
The pressure drop due to limit of piston in accumulator
2
0 0
50
100
150
200
0
50
100
150
200
Time/ms
Time/ms
Fig. 17. The displacement of the accumulator piston.
Fig. 15. The pressure in the lower chamber of the hydraulic cylinder.
6 5 Moment of spool 2 impacting spool ring 9
5 4
Moment of accumulator piston impacting limits
3 Moment of impact
Ve1ocity/m.s-1
Ve1ocity/m.s-1
4
2
3 2 1
1
0 0 -1 -1
0
50
100
150
0
50
200
100
150
200
Time/ms
Time/ms
Fig. 18. The velocity of impacting mass.
Fig. 16. The velocity of the accumulator piston.
600
Impacting mass Table
400
Displacement/mm
piston and a decompression circuit of the accelerating cavity in the shock cylinder barrel. Fig. 15 shows the relationship between the pressure in the lower chamber of the hydraulic cylinder barrel and the time. This figure is comprised of four curves: an accumulative pressure curve of the impacting mass, a pressurizing curve of the hydraulic impact caused by impacting, a depressurizing curve of the unloading circuit in its operation after the impacting, and a prompt dropping curve of the accumulator when the piston moves to the end of its stroke to cut off its pressure supply. The velocity curve of the accumulator piston in the whole shock process is shown in Fig. 16, which explicitly reveals its velocity changes at each stage. When the accumulator piston bumps the limiting stopper, the velocity of the piston will slightly fluctuate at the end of its stroke, but its displacement will remain unchanged, as demonstrated in Fig. 17 when the piston brakes at 500 mm, so the pressure outlet of the accumulator will be blocked. Fig. 18 shows the velocity curve of the impacting mass, from which it can be seen that the velocity of the impacting mass drops to zero instantly when it impacts the table, which causes the table to move upward. The relative displacement of the impacting mass and the table is shown in Fig. 19. It is seen that after the collision, the table moves upward while the impacting mass rebounds and slowly goes back to the piston seat under the gravitational effect. There is no secondary collision occurred due to the introduced prevention system. These results suggest that the secondary
200
0
-200
-400
0
50
108
150
200
Time/ms Fig. 19. The displacements of the impacting mass and the table.
impacting on the table during unloading can be successfully prevented by incorporating the unloading circuit and the stroke limit of the accumulator piston. Fig. 20 shows the displacement changes of the dynamic load of the foundation mass when the impacting mass actuates the system when the vibration isolation frequency is 3 Hz. It can be
G. Wang et al. / International Journal of Mechanical Sciences 77 (2013) 239–248
agreement and the maximum deviation of waveform amplitude is within 5.4% and that of pulse width is less than 4%.
0 vt = 5 m/s vt = 4 m/s vt = 3.3 m/s
Displacement/mm
-5
6. Conclusions
-10
A new heavy-duty shock test machine is developed and its mathematical model is created to analyze its mechanism and dynamic performance. From the theoretical analysis and numerical simulation of the dynamic performance, the following conclusions are drawn:
-15
-20
-25
-30 0
50
100
150
200
Time/ms Fig. 20. The displacement of the foundation mass with different parameter vt equals 5 m/s, 4 m/s and 3.3 m/s.
vt = 5 m/s vt = 4 m/s vt = 3.3 m/s
150
Acceleration/g
247
X: 108.8 Y: 161.2
X: 83.84 Y: 85.2
100 X: 65.3 Y: 52.65
50 X: 59.99 Y: 0
X: 70.31 Y: -1
0
X: 87.62 Y: -1
X: 106.1 Y: 0
X: 80 Y: 0
50
60
70
X: 111.3 Y: -1
80
90
100
110
(1) The rapid opening high flow valve proposed in this paper is quick in response. It can open all valves within 10 ms and its maximum flow capacity can reach up to tens of thousands of liters per minute. Moreover, the exact flow capacity can be regulated if necessary by changing the geometric dimension of the valve structure, which implies good extensibility. It resolves the contradiction between the rapid response of the hydraulic flow valve and the high flow capacity. (2) The velocity generator can well meet the demand of instantaneous energy storage and release in the shock process. The acceleration waveforms of the table and the test item of the system show good agreement with the latest BV043/85 specification. (3) Although the stroke of the impacting mass in this shock machine system is fixed, equipment at different levels can be tested by regulating the impact velocity through an adjustment of the throttle valve, which can be realized conveniently. (4) The double protection of the unloading circuit and the stroke limit of the accumulator piston can effectively prevent the secondary collision in the testing process. (5) The vibration isolation system of large foundation mass can meet the demand of environmental protection through avoiding negative impacts on the surrounding environment effectively.
120
Time/ms Fig. 21. The acceleration pulses of the table at different velocities vt equals 5 m/s, 4 m/s and 3.3 m/s.
Table 2 Comparison of the simulation results and BV043/85 specification. Velocity Waveform parameter
V ¼ 3.3 m/s
V ¼4.0 m/s
V ¼5.0 m/s
Acceleration peak (g) Pulse width (ms)
52.7/50 10.32/10.3
83.8/85 7.62/7.5
161.2/160 5.2/5
seen that the maximum displacement change is only 27 mm, which can fully meet the requirement on the vibration isolation in engineering application.
5.4. Impact acceleration waveform Based on the requirements in the BV043/85 specification on the shock test environments for equipment of different levels, we simulate and examine the prospective performances of the shock systems for three types of equipment by setting the shock velocity of the test article as 3.3 m/s, 4.0 m/s and 5.0 m/s respectively. Fig. 21 shows the acceleration waveforms of the table and the test item under different impact velocities. A comparison between the simulation results and the waveform parameters defined in BV043/85 specification is given in Table 2, which exhibits a good
Acknowledgments The first author would like to thank the National Natural Science Foundation of China for financially supporting this research under the Contract no. 51275369 and also to thank the China Scholarship Council to support related research hosted by Prof. Yeping Xiong at the University of Southampton. References [1] Wang Gong Xian, Hu Yong. Advances in shock testing facilities for naval shipboard equipments. Adv Mater Res 2011;301–303:1220–5. [2] Woyski Bill, Nelson Curt. An accurate, repeatable subsidiary component shock test system for the laboratory. In: Proceedings of the 75th Shock and vibration symposium. Virginia Beach, Virginia; 2004. pp.17–22. [3] Elder Steven, Arden Kevin. Land based full scale shock simulator[C]. In: Proceedings of the 70th shock and vibration symposium-CD. Albuquerque, New Mexico: Shock and Vibration Information Analysis Center HI-TEST Laboratories; 1999. [4] Gram MM, Clark AJ, Hegemier GA, Seible F. Laboratory simulation of blast loading on building and bridge structures[J]. Struct Under Shock Impact IX 2006;87:33–44. [5] Zhi-Yi Zhang, Gong-Xian Wang, Yu Wang. Numerical modeling of dual-pulse shock test machine for simulating underwater explosion shock loads on warship equipments. J Shanghai Jiaotong Univ (Sci) 2009;14:233–40. [6] Zhaodong Wen, Yu Wang, Lei Zhang Jianye Du. Modeling and dynamic simulation of novel dual-wave shock test machine. In: Proceedings of the 2011 international conference on fluid power and mechatronics. 2011. pp. 898–901. [7] Guo-hua Li, Yu-jie LI, Xiao-ci Zhang. Experimental verification of relationship between bubble motion and vibration of ship equipment. J Ship Mech 2005;9:98–105.
248
G. Wang et al. / International Journal of Mechanical Sciences 77 (2013) 239–248
[8] Wan J, Chen B, Wang QT. Sensitivity analysis of repetitive shock machine's vibration energy. Advances in safety, reliability and risk management. In: Proceedings of the European safety and reliability conference. 2012. pp. 2183–8. [9] Piersol AG, Paez TL. Shock testing machines. Harris' Shock and Vibration Handbook. New York: McGraw-Hill; 2010.
[10] Jiao Sujuan, Wang Yu, Zhang Lei, Hongxing Hua. Shock wave characteristics of a hydraulic damper for shock test machine. Mech Syst Signal Process 2010;24:1570–8. [11] Piotr Czop, Sawik Damian. A high-frequency first-principle model of a shock absorber and servo-hydraulic tester. Mech Syst Signal Process 2011;25: 1937–55.
Contents lists available at ScienceDirect
International Journal of Mechanical Sciences journal homepage: www.elsevier.com/locate/ijmecsci
A novel heavy-weight shock test machine for simulating underwater explosive shock environment: Mathematical modeling and mechanism analysis Gongxian Wang a,b, Yeping Xiong b,n, Wenzhi Tang a a b
School of Logistics Engineering, Wuhan University of Technology, Wuhan 430063, China Fluid-Structure Interaction Research Group, Faculty of Engineering and the Environment University of Southampton, Highfield, Southampton SO17 1BJ, UK
art ic l e i nf o
a b s t r a c t
Article history: Received 5 May 2012 Received in revised form 25 July 2013 Accepted 12 September 2013 Available online 22 October 2013
A novel heavy-duty shock test machine is developed to satisfy the newly-built shock resistance standard and simulate accurately the actual underwater explosive environments with increased testing capability. The mathematical model for the shock test machine is created to predict its dynamic performance and analyze its mechanism. Then numerical simulation is carried out to evaluate the prospective capability of the shock test machine under different shock velocity inputs. The double protection system incorporating the stroke limit of the accumulator piston and the unloading circuit can effectively prevent the secondary collision in the testing process. The simulation results have demonstrated that the shock test machine proposed in this paper can produce nearly the same shock acceleration waveform as the new shock resistance standard BV043/85 and MIL-S-901D. Moreover, this shock test machine can be regulated conveniently to adjust to a different type of equipment and be extended easily to suit more severe shock environments and heavier equipment. The proposed system configuration and associated mathematical model provide theoretical basis and useful design techniques for practical applications. Crown Copyright & 2013 Published by Elsevier Ltd. All rights reserved.
Keywords: Shock test machine Velocity generator Rapid opening high flow valve Shock test Underwater explosion
1. Introduction Warships and shipboard equipment are often damaged by intense shock waves of the near-field non-contact underwater explosion (UNDEX) during their military service. Therefore the ability to withstand the UNDEX effects has been regarded as an important aspect of the survivability of the new and existing warships, and receives full attention from the navy of all countries. The naval powers have conducted extensive and in-depth research on the methods to examine the shock resistant capability of the key shipboard equipment. For instance, the USA formulated the military specification MIL-S-901D in 1989, and Germany proposed a new military specification defined in BV043/85, specifying the requirements for the high impact test for the shipboard equipment. Although MIL-S-901D is prepared specifically for military applications, the standard is often used for commercial products as well. For marine equipment, in general, shock test machine is a prevailing way to examine the ability to survive an underwater explosion. However, both the light-and medium-weight shock test machines, which have maximum testing capacities of 125 kg and
n
Corresponding author. Tel.: þ 44 2380596619; fax: þ 44 2380597744. E-mail addresses: [email protected], [email protected] (Y. Xiong).
2700 kg respectively and maximum impact velocity less than 3.4 m/s, can merely meet the traditional MIL-S-901 specification. Moreover, these test machines can only generate shock wave in one direction by either dropping or pendulum rotating the impacting mass. Therefore, these types of shock test machines can only meet the MIL-S-901C specification. Challengingly, there is little room to extend the shock level and capability of test payload of these two types of machines due to their nature of energy storage or impact generating patterns as well as their giant sizes. Hence equipment beyond this weight range has to be tested in underwater explosive environment in order to assess its shock resistant capability. However, this is undesirable because of associated high cost, damage to the environment and long test cycle. Nowadays, the trend of continuous improvement of the underwater weapons to enhance accuracy and power has led to an even higher demand for the shock resistant ability of warship equipment. The latest specifications of shock-resistant ability in the world are MIL-S-901D and BV043/85, which have increased the velocity requirement to 5 m/s and even higher in special circumstances such as testing newly developed shipboard equipment. Moreover, the maximum testing capacity of the current shock machines has exceeded the limits of the traditional mediumweight ones (2700 Kg) and it still keeps increasing. Thus, various shock test equipments are developed. The U.S. Team Corporation
0020-7403/$ - see front matter Crown Copyright & 2013 Published by Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijmecsci.2013.09.006
G. Wang et al. / International Journal of Mechanical Sciences 77 (2013) 239–248
developed a new type of shock test system called Subsidiary Component Shock Test System (SSTS), which contains a long stroke hydraulic actuator that can be configured to operate in either a vertical or horizontal position [1,2]. Weidlinger Associates Inc. constructed a system for Non-Explosive Shock Testing of Naval Vessels and Equipment whereby shock test with explosion can be replaced with non-explosive sources based on the very rapid release of high-pressured air from reservoirs very close to the vessel under test [3]. Germany WTD71 of Kiel developed and built a laboratory shock test machine for simulating UNDEX testing which can produce half-sine shock wave according to BV043/85 specification. MTS systems Corporation was contracted by Westinghouse Bettis to design and build a laboratory shock and vibration-testing machine for simulating UNDEX testing of naval vessel components with high load and high fidelity. This test machine can test components up to 10 t [4]. Shanghai Jiaotong University designed and built a laboratory shock testing machine named Heavyweight Dual-wave Shock Test Machine which is being debugged [5,6]. Comparing the current shock test facilities and some new concepts of shock test machines such as MTS Firing Impulse Simulators (FIS) and Full Scale Shock Simulator, it is clearly seen that the development trends of shock test machines towards three major directions. The first trend is that shock test machines are capable to generate both positive and negative shock pulses to simulate real UNDEX environment which is made up of shock wave followed by bubble pulse and structural whipping [7]. The second is that shock test machines can test heavy-weight equipment, and the last is that shock pulses can be controlled and customized conveniently [8]. However, one of the key technologies for developing new shock test machines is how to supply enough energy and dissipate it in a short time, which can be controlled and also user-friendly with good vibration isolation design [9–11]. In this paper, we propose a novel heavy-duty shock test machine and establish its mathematical model for the integrated dynamic system, and also analyze its mechanism and dynamic performance. The key features of this new shock test machine are (i) high energy storage capacity; (ii) instantaneous release of tremendous energy; (iii) rapid dynamic response; (iv) high automaticity; (v) hydraulic damper energy dissipation; (vi) enhanced testing capacity of 5000 kg; and (vii) meets both the latest shock resistance standard of MIL-S-901D and BV043/85.
Test article
Measurement and control system
Table
Damping cylinder
Waveform generator
Impacting mass
Buffer system
Velocity generator
Hydraulic system
Vibration isolation Fig. 1. Schematic of the system of the Heavy-weight Shock Test Machine.
a1
Acceleration/m.s-2
240
τ1
t/ms Fig. 2. The semi-sinusoidal acceleration pulse.
2. System description of the heavy-weight shock test machine Fig. 1 shows a schematic of the system of the proposed heavyweight shock test machine, which produces semi-sinusoidal acceleration pulses vertically as shown in Fig. 2. The shock test machine consists of a hydraulic system made up of a velocity generator and a buffer system, a measurement and control system, and a vibration isolation system. With the test item mounted on the table, the impacting mass, driven by the velocity generator, applies impact load to the waveform generator located at the bottom of the table and produces a semi-sinusoidal acceleration wave required in the specification. Meanwhile, the buffer system, comprised of hydraulic damping cylinders, limits the stroke of the table and prevents it from bouncing off after the impact. The measurement and control system can collect and process data, and monitor important parameters such as pressure of the whole system, velocity of the impacting mass, and acceleration of the table as well as the test item.
3. Mathematical model of the integrated dynamic system The generation of the wave of the shock testing machine is based on the principle of collision, whose working mechanism can be simplified to three degrees of freedom vibrating system as shown in Fig. 3. The system is composed of springs, dampers and masses, in which ks represents the coefficient of elasticity of the waveform generator while C(t) the damping coefficient of the hydraulic damping cylinder. Since the damping cylinder merely serves to prevent the table from flying off too high when hitting the impacting mass, this paper will not study the damping property of the cylinder. Without loss of any important features, the mathematical model of the shock test machine is established based on the following hypotheses: (1) the impacting mass, the table and the foundation mass are rigid bodies, while the waveform generator is regarded as a linear elastic body; (2) on the instant of impact, the compressed gas in the air bag of the accumulator is in heat insulation, i.e. without energy exchange with the outside; (3) only local pressure loss is taken into account due to the shortness of the pipeline of the hydraulic system; (4) the mechanical friction loss and leakage in the motions of piston and spools are ignored; (5) the liquid pressure in each chamber is regarded as evenly distributed; (6) the velocity distribution of each cross-section flow
G. Wang et al. / International Journal of Mechanical Sciences 77 (2013) 239–248
xt
mt ks xi
xh mh
C(t)
F(t) F(t) xf
mf kf
Fig. 3. Mechanical model of the shock test machine.
is uniform; and (7) the compressibility of liquid is ignored considering its insignificance in comparison with that of gas. The mathematical expression of the system is as follows: 8 m x€ þks ðtÞðxh xt xi Þ ¼ FðtÞ > < h h mt x€ t þ CðtÞðx_ t x_ f Þ þ ks ðtÞðxt þ xi xh Þ ¼ 0 ð1Þ > : m x€ þ CðtÞðx_ x_ Þ þ k x ¼ FðtÞ t f f f f f where the coefficient of elasticity of the waveform generator, ks(t), is formulated as ( ks xh xt Z xi ks ðtÞ ¼ ð2Þ 0 xh xt o xi Here, mt is the mass of the table; mh, that of the impacting mass; mf, of the foundation mass; xf represents the displacement of the foundation mass; xh, that of the impacting mass; xt, of the table. xi is the initial distance between the impacting mass and the waveform generator. ks is the coefficient of elasticity of the spring shown in Fig. 3; kf, that of the coil spring. In order to excite the table with an acceleration pulse shown in Fig. 2, F(t) has to accelerate the impacting mass, mh, to V2 within the given stroke, xi; that is to say, F(t) must satisfy the following equation: 8R < ti FðtÞ=mh dt ¼ ðmh þ mt Þτ1 a1 =ðmh πÞ 0 R R ð3Þ : 0ti ð 0t FðtÞ=mh dtÞdt ¼ xi
241
timely. In order to meet the challenges above, this paper proposes a velocity generator regulated by a rapid opening, high flow valve, the construction and operation of which is illustrated in Fig. 4. The liquid in high-pressure accumulator 1 flows to the lower chamber of hydraulic cylinder barrel 6 through the valve port of the throttle ring upon the moving of spool 2, and the driving impacting mass reaches the requested velocity within the given stroke; wherein, pressure relief valve 3 controls the close of spool 2, control valve 4 monitors the velocity of the system by adjusting the opening of throttle spool 5, vent valve 7 can prevent the piston from colliding with the top of the hydraulic cylinder barrel through a regulation of the gas pressure in the upper chamber of the cylinder barrel, and unloading valve 8 can avoid a second impact of the impacting mass and the table by unloading the lower chamber of the hydraulic cylinder barrel quickly. The mechanical model of valveregulated velocity generator is shown in Fig. 5, and its mathematical model can be deduced as follows. 4.1. The adiabatic state equation of gas The adiabatic state equation of the compressed gas in the accumulator is !k 2 πd1 pg V g0 þ x1 ¼ pg0 V kg0 ð4Þ 4
The adiabatic state equation of the gas in the upper chamber of the hydraulic cylinder barrel is pN ðs4 xh Þk ¼ pN0 sk4
ð5Þ
In the equations above, pg0 and pN0 are the initial pressures in the accumulator and the upper chamber of the hydraulic cylinder barrel respectively; Vg0 the initial volume of the compressed gas in the accumulator; d1 the diameter of the piston of the accumulator; s4 the stroke of the impacting mass; x1 the displacement of the accumulator piston; pg and pN are the gas pressure in the
V
P 7
4
In the equation above, ti is the impacting instant of the impacting mass and the table; τ1 the pulse width of the acceleration wave; a1 the peak value of the acceleration wave.
V
6 5
4. Mathematical model of the velocity generator The impact velocity of the impacting mass and the waveform generator is a critical factor in the realization of the pulse of impact acceleration requested in the specification. Three challenges exist when the impacting mass actuating the system. Firstly, the storage and instantaneous release of high energy in the actuation requires the impacting mass to accelerate from its complete static state to a scheduled velocity within a limited stroke; secondly, the impact velocity must be appropriately regulated; thirdly, a second impact of the impacting mass and the table must be avoided by unloading the actuating system
2 9
3 1
SV
10
8
V
Fig. 4. Schematic of the structure of the velocity generator with: (1) accumulator, (2) spool, (3) pressure relief valve, (4) control valve, (5) throttle spool, (6) hydraulic cylinder barrel, (7) vent valve, (8) unloading valve, (9) spool ring, and (10) limit.
242
G. Wang et al. / International Journal of Mechanical Sciences 77 (2013) 239–248
where D2 is the diameter of spool 2, x2 its displacement, δ the clearance inside its recess hole, and p2 the pressure in its upper chamber; p3 the pressure in the pilot passage of the hydraulic cylinder; ρ fluid density; and C2 flow coefficient, whose value depends on Reynolds number Re. The flow capacity through throttle ring 5 is governed by: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2ðp3 p4 Þ Q 3 ¼ C 3 πD4 xv ð8Þ ρ The unloading flow capacity of pressure relief valve 3 is Qc ¼
πD43 p 128μl c
ð9Þ
and the flow capacity through unloading valve 8 is sffiffiffiffiffiffiffiffi 2pa Q u ¼ C u Au ðtÞ ρ
Au ðtÞ ¼
8 <0 πD2
: 4τuu ðt t i Þ
t o ti t Z ti
ð10Þ
ð11Þ
where C3 is flow coefficient; D4 the inlet diameter of the hydraulic cylinder barrel; xv the opening of spool 5 of the throttle ring; p4 the pressure in the pilot passage of the cylinder barrel; pc the fluid pressure of the lower chamber of spool 2; μ the dynamic viscosity of fluid; l the length of the passage of the hydraulic valve; Cu the flow coefficient of unloading valve 8 and Du, its diameter; pa the fluid pressure of the lower chamber of the hydraulic cylinder barrel; and τu unloading valve 8 response time. The equilibrium equation of the system flow is given by Q 1 ¼ Q 3 þQ C Q3 ¼
1 2 πd x_ þQ u 4 2 h
ð12Þ ð13Þ
4.3. Pressure equation At the inlet of the accumulator, abrupt shrink of the flow area leads to local pressure loss Δp1 as follows: !2 h i Q π 2 1 ð14Þ Δp1 ¼ p1 p2 ¼ 4ρ 1 ðd1 D21 Þ 4 πD21 and at the inlet of the hydraulic cylinder barrel, abrupt expansion of the flow area causes Δp2, i.e., " #2 !2 D24 Q3 ð15Þ Δp2 ¼ p3 p4 ¼ 8ρ 1 4Hd2 πD24 Fig. 5. The mechanical model of the velocity generator. (A) Partial enlarged detail of part A. (B) Partial enlarged detail of part B.
accumulator and upper chamber of cylinder barrel, respectively; k is adiabatic constant of the gas.
In the equations above, D1 is the diameter of the pilot passage of the accumulator; d2 that of the hydraulic cylinder barrel piston; H the distance between the bottom of the piston and that of the cylinder barrel.
4.2. Flow capacity
4.4. The equation of motion The equations of motion of the accumulator piston and spool 2 are respectively
The total flow capacity of the system is Q1 ¼
1 2 πd x_ 1 4 1
and the flow capacity through spool 2 is given by sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2ðp2 p3 Þ Q 2 ¼ C 2 πD2 x22 þ δ2 ρ
ð6Þ
m1 x€ 1 ¼
1 2 πd ðp p1 Þ c1 x_ 1 m1 gμ1 4 1 g
ð16Þ
1 2 πD ðp pc Þ c2 x_ 2 þ m2 g 4 2 3
ð17Þ
and ð7Þ
m2 x€ 2 ¼
G. Wang et al. / International Journal of Mechanical Sciences 77 (2013) 239–248
where m1 is the mass of the accumulator piston; p1 the oil pressure of the accumulator, and c1, the dynamic damping coefficient of its piston; m2 the mass of spool 2, c2 the dynamic damping coefficient of the spool rings. The equation of motion for the impacting mass is written as 1 2 1 2 2 πd p πðd2 d3 ÞpN 4 2 a 4 c3 x_ h ks ðtÞðxh þ xi xt Þ mh g
pN ¼ pN0
pc ¼
s4 s4 xh
p1 ¼ p2 þ ζ 1
( cðtÞ ¼
ð19Þ
xt o disp xt Z disp
0 c
ð20Þ
where d3 the diameter of the piston rod; c3 the dynamic damping coefficient of the piston; c the damping coefficient of the damping cylinder barrel, and disp is the maximum displacement of the table permitted in BV043/85 specification. The equation of motion for the foundation mass is 2 mf x€ f ¼ kf xf þ mf g þpa πd2 =4
ð21Þ
ρ A1 x_ 1 2 2 A2
x€ 1 ¼
A1 ðpg p1 Þ c1 x_ 1 gμ1 m1
x€ 2 ¼
A3 ðp3 pc Þ c2 x_ 2 þg m2
x€ h ¼
A1 ¼ A3 ¼
1 2 πd 4 1
ð22 1Þ
1 2 πD 4 2
A7 pa ðA7 A8 ÞpN c3 x_ h ks ðtÞðxh þxi xt Þ g mh
ð22 2Þ A7 ¼
1 2 1 2 πd ; A8 ¼ πd3 4 2 4
ð22 3Þ x€ t ¼
ks ðtÞðxh xi xt Þ cðtÞx_ t g mt
ð22 4Þ
x€ f ¼
A7 pa kf xf g mf
ð22 5Þ
0 pg ¼ pg0 @
1k V g0 V g0 þ
2
πd1 4
A
ð22 6Þ
x1
ð22 8Þ A2 =2 A2 ¼ πD21 =4; ζ 1 ¼ 1 A1
ρ A1 x_ 1 A3 x_ 2 2 ; 2 C 2 A4
A4 ¼ πD2
p3 ¼ p4 þ
ρ A1 x_ 1 A3 x_ 2 2 ; 2 C 3 A5
A5 ¼ πD4 xv
p4 ¼ pa þ ζ 2
ρ A1 x_ 1 A3 x_ 2 2 ; A6 2
qffiffiffiffiffiffiffiffiffiffiffiffiffiffi x22 þ δ2
ð22 9Þ
ð22 10Þ
ð22 11Þ
A6 2 A6 ¼ πD24 =4; ζ 2 ¼ 1 ; ζ 3 ¼ πd2 H ζ3
ð22 12Þ 8 m ðA x€ A x€ Þ ðA A Þp þ k ðtÞðx x x Þ 3 2 h 1 1 þ 7 8 N As7 h i t > < A27 2 pa ¼ > : 2ρ A1 x_ 1 CAA3 x_ 2ðtÞ A7 x_ h u
Eq. (1) is for the general mechanical model of this shock test machine, whereas Eqs. (18), (19) and (21) are the detailed description of the equations in Eq. (1), respectively. In Eq. (1), the force F(t) acting on the impacting mass and foundation mass is the generalized force which contains gravitational force, impact force and counterforce induced by the velocity generator. They are descripted in Eqs. (18), (19) and (21) in detail. An integrated mechanical model of the shock machine system has 13 variables, i.e., x1, x2, xh, xt, xf,pg, pN, pc, p1, p2, p3, p4 and pa which can be obtained by solving the following set of Eqs. (1)–(21) from which we obtain the derived equations as follows:
ð22 7Þ
p2 ¼ p3 þ
and the equation of motion for the testing table is mt x€ t ¼ ks ðtÞðxh xi xt Þ mt g cðtÞx_ t
k
128μl πD22 x_ 2 πD43 4
mh x€ h ¼
ð18Þ
243
u
t ot i t Zt i
ð22 13Þ
However, the nonlinearity of equations and the complication of boundary conditions make it very difficult, even impossible, to acquire the analytical solutions in reality. So in order to examine the dynamic performance of the system, the method of numerical computation has to be used to obtain the numerical solutions for each unknown variable.
5. Dynamic property and system performance Based on the developed mathematical model for the shock machine system in Section 4, we will solve the equation set above using the fourth-order Runge–Kutta method and predict the dynamic performance of the shock machine system and analyze its mechanism. To reveal the dynamic property of the system, parameters for the proposed mathematical model of the shock machine system are carefully selected to accurately describe its mechanical property. The key parameters used in the numerical simulation are listed in Table 1. Moreover, this “virtual test” technology can also lay a solid foundation for the practical construction and adjustment of shock machine systems. 5.1. Response time of the rapid opening high flow valve The response time of the rapid opening, high flow valve refers to the time needed for the upper end of spool ring 2 to migrate from the spool land to the lower part of the pilot passage of the valve body. This is the time needed for the passage from being totally blocked to being completely open, which mainly depends on the velocity of the spool 2. According to Eq. (17), this velocity is
Table 1 Parameters used in simulation. Parameters
Quantity
Parameters
Quantity
Impacting mass (kg) mh Table mass (kg) mt Accumulator piston mass (kg) m1 Spool ring 2 mass (kg) m2 Foundation mass (ton) mf Accumulator piston radius (mm) d1 Accumulator piston stroke (mm) s1 Accumulator inlet radius (mm) D1
5000 5000 10 5 50 250 500 100
Accumulator gas initial pressure (MPa) pg0 Accumulator gas initial volume (m3) Vg0 Piston radius (mm) d2 Piston rod radius (mm) d3 Unloading hole radius (mm) Du Unloading valve 8 response time (ms) τu Spool ring 2 stroke (mm) s2 Pilot passage radius (mm) D4
13 0.1 200 100 40 50 200 50
G. Wang et al. / International Journal of Mechanical Sciences 77 (2013) 239–248
related to the pressure difference between the upper and lower spool ends, and the size as well as the mass of spool 2. Once the initial pressure of the accumulator is set, the pressure difference between the spool ends is determined by the pressure relief velocity, i.e., the passage diameter D3 of pressure relief valve 3. Three cases are shown in Fig. 6 for D3 equals to 10 mm, 15 mm, and 20 mm respectively, and their corresponding response times are approximately 4 ms, 5 ms and 14 ms, which means that the response time can be reduced by enlarging the cross-section surface of the pressure relief passage. Eq. (17) reveals that if the cross-section surface of the spool 2 is enlarged, the driving power will be strengthened, and thus the velocity of the spool ring will be enhanced and its response time will be reduced. In fact, the enlargement of the spool ring’s section surface will not definitely lead to the reduction of its response time, which can be proven by the curves in Fig. 7, which illustrates three different diameters of spool 2, namely, 50 mm, 100 mm and 150 mm. It can be seen that their corresponding response times are approximately 9 ms, 7 ms and 13 ms respectively, which on the other hand reveals that the enlargement of spool ring's crosssection surface cannot definitely reduce its response time. The reason can be explained by Eq. (9), which shows that the enlargement of the spool ring's cross-section surface will possibly increase the flow capacity of the pressure relief hole, and thereby
350 D3 = 10 mm D3 = 15 mm D3 = 20 mm
Dsiplacement/mm
300 250 200 150
X: 4.45 Y: 100.1
X: 14.08 Y: 100
100 X: 5.31 Y: 100.3
50 0
0
5
10
15
20
Time/ms
150 m2 = 5 kg m2 = 10 kg
X: 14.08 Y: 100
m2 = 15 kg
Displacement/mm
244
X: 15.14 Y: 100
100 X: 14.63 Y: 100.1
50
0
0
5
10
15
20
Time/ms Fig. 8. The displacement of spool 2 with different parameter m2 equals 5 kg, 10 kg and 15 kg.
will increase the pressure in sensing chamber. As a result, the pressure difference applied on the spool 2 becomes smaller instead, and then its migration is slackened and hence its response time is prolonged. Fig. 8 shows that an increase in the spool's mass will slightly extend its response time. The acceleration of spool 2 in its opening process is illustrated in Fig. 9, in which the first acceleration peak is formed by the pressure difference exerted on the spool when pressure relief valve 3 is open, and the trough occurs upon the sudden deceleration and brake of the spool 2 when it moves to the end of its stroke and bumps the cushion device. When the acceleration is zero, the spool 2 is in its free travel and hence uniform motion, with the resultant force on it being zero. Figs. 10 and 11 display the spool's corresponding velocity and displacement curves. When opening the valve, the spool ring receives impact from both the acceleration and the deceleration with the acceleration peaks being 350 g and 100 g respectively, which should be considered in the design of cushion devices so as to avoid damage to spool rings. According to the analysis above, when the response time of the valve is less than 10 ms, the maximum flow capacity can reach 15,000 L/min. The most efficient measure to speed up the valve's response is to enlarge the flow area of the pressure relief circuit.
Fig. 6. The displacement of spool 2 with different parameter D3 equals 10 mm, 15 mm and 20 mm.
5.2. Velocity regulating property 300 D2 = 50 mm D2 = 100 mm D2 = 150 mm
Displacement/mm
250 200 150
X: 7.09 Y: 100.1
X: 9.25 Y: 100.1
X: 12.67 Y: 100
100 50 0
0
5
10
15
20
Time/ms Fig. 7. The displacement of spool 2 with different parameter D2 equals 50 mm,100 mm and 150 mm.
The impact velocity of the impacting mass can be regulated by the velocity generator through the control of the flow capacity of the system, i.e. through adjusting the opening of the spool of throttle ring 5. Figs. 12 and 13 show the corresponding velocities of 3.89 m/s, 4.90 m/s and 5.15 m/s of the impacting mass respectively when spool 5 is set at different openings of 10 mm, 20 mm and 30 mm under the prescribed displacement 350 mm for the impacting mass. The corresponding displacement curve of the impacting mass is given in Fig. 14. From the data shown above, it is clearly seen that if the initial distance between the impacting mass and the table is preset as xi which is unchanged, the impacting velocity of the impacting mass can be regulated by adjusting the opening of the spool ring of throttle ring 5. It can also know that the readjustment of the throttle ring will change the instant of the impact between the impacting mass and the table, because if the opening of spool 5 is reduced, then the acceleration time of the impacting mass will be increased, therefore the impact will be delayed. This, however, will not affect the velocity regulating property of the system.
G. Wang et al. / International Journal of Mechanical Sciences 77 (2013) 239–248
600
6
500
5
245
400
Ve1ocity/m.s-1
Acceleration/g
4 300 200 100
3 2
0
1
-100
0
-200
0
10
20
30
40
50
60
-1
Time/ms
xv = 10 mm xv = 20 mm xv = 30 mm 0
50
100
150
200
Displacement/mm
Fig. 9. The acceleration of spool 2.
Fig. 12. The velocity of the impacting mass with different parameter xv equals 10 mm, 20 mm and 30 mm.
8 6 6
Ve1ocity/m.s-1
5
Ve1ocity/m.s-1
4
2
4 3 2
xv = 10 mm xv = 20 mm xv = 30 mm
0 1 -2
0
10
20
30
40
50
60
0
Time/ms
0
50
100
150
200
250
300
350
400
Displacement/mm
Fig. 10. The velocity of spool 2. Fig. 13. Velocity–displacement curve of the impacting mass with different parameter xv equals 10 mm, 20 mm and 30 mm..
250
400 X: 116.4 Y: 350
300 150
Displacement/mm
Displacement/mm
200
100
50
200
100 xv = 10 mm xv = 20 mm xv = 30 mm
0 0
0
10
20
30
40
50
60
Time/ms Fig. 11. The displacement of spool 2.
X: 118.8X: 135.5 Y: 350 Y: 350
-100
0
50
100
150
200
Time/ms
5.3. Unloading property
Fig. 14. Displacement of the impacting mass with different parameter xv equals 10mm, 20mm and 30mm.
Understanding the failure mechanism is important for successful design of shock test machine. According to the shock testing specification, the impacting mass is required to impact the table only once in each shock test without secondary impacts, otherwise it would be regarded as a failure. It is understood that when liquid, as a medium, is squeezed by the expansion of the high-pressure
gas, the impacting mass is actuated to impact the table and then it will bound back, and thus the gas will be compressed again and also expanded again to drive the impacting mass to impact the table for a second time resulting a failure. In order to prevent the secondary impact, a double protection system is introduced in the system, which includes both the stroke limit of the accumulator
246
G. Wang et al. / International Journal of Mechanical Sciences 77 (2013) 239–248
600
14 Hydraulic shock in accerleration chamber due to impacting between shock hammer and shock table
12
Displacement/mm
Pressure curve during the opening of unloading valve
10
Pressure/MPa
500
8 Pressure in acceleration chamber during shock hammer accelerating
6
400 300 200
4
0
100
The pressure drop due to limit of piston in accumulator
2
0 0
50
100
150
200
0
50
100
150
200
Time/ms
Time/ms
Fig. 17. The displacement of the accumulator piston.
Fig. 15. The pressure in the lower chamber of the hydraulic cylinder.
6 5 Moment of spool 2 impacting spool ring 9
5 4
Moment of accumulator piston impacting limits
3 Moment of impact
Ve1ocity/m.s-1
Ve1ocity/m.s-1
4
2
3 2 1
1
0 0 -1 -1
0
50
100
150
0
50
200
100
150
200
Time/ms
Time/ms
Fig. 18. The velocity of impacting mass.
Fig. 16. The velocity of the accumulator piston.
600
Impacting mass Table
400
Displacement/mm
piston and a decompression circuit of the accelerating cavity in the shock cylinder barrel. Fig. 15 shows the relationship between the pressure in the lower chamber of the hydraulic cylinder barrel and the time. This figure is comprised of four curves: an accumulative pressure curve of the impacting mass, a pressurizing curve of the hydraulic impact caused by impacting, a depressurizing curve of the unloading circuit in its operation after the impacting, and a prompt dropping curve of the accumulator when the piston moves to the end of its stroke to cut off its pressure supply. The velocity curve of the accumulator piston in the whole shock process is shown in Fig. 16, which explicitly reveals its velocity changes at each stage. When the accumulator piston bumps the limiting stopper, the velocity of the piston will slightly fluctuate at the end of its stroke, but its displacement will remain unchanged, as demonstrated in Fig. 17 when the piston brakes at 500 mm, so the pressure outlet of the accumulator will be blocked. Fig. 18 shows the velocity curve of the impacting mass, from which it can be seen that the velocity of the impacting mass drops to zero instantly when it impacts the table, which causes the table to move upward. The relative displacement of the impacting mass and the table is shown in Fig. 19. It is seen that after the collision, the table moves upward while the impacting mass rebounds and slowly goes back to the piston seat under the gravitational effect. There is no secondary collision occurred due to the introduced prevention system. These results suggest that the secondary
200
0
-200
-400
0
50
108
150
200
Time/ms Fig. 19. The displacements of the impacting mass and the table.
impacting on the table during unloading can be successfully prevented by incorporating the unloading circuit and the stroke limit of the accumulator piston. Fig. 20 shows the displacement changes of the dynamic load of the foundation mass when the impacting mass actuates the system when the vibration isolation frequency is 3 Hz. It can be
G. Wang et al. / International Journal of Mechanical Sciences 77 (2013) 239–248
agreement and the maximum deviation of waveform amplitude is within 5.4% and that of pulse width is less than 4%.
0 vt = 5 m/s vt = 4 m/s vt = 3.3 m/s
Displacement/mm
-5
6. Conclusions
-10
A new heavy-duty shock test machine is developed and its mathematical model is created to analyze its mechanism and dynamic performance. From the theoretical analysis and numerical simulation of the dynamic performance, the following conclusions are drawn:
-15
-20
-25
-30 0
50
100
150
200
Time/ms Fig. 20. The displacement of the foundation mass with different parameter vt equals 5 m/s, 4 m/s and 3.3 m/s.
vt = 5 m/s vt = 4 m/s vt = 3.3 m/s
150
Acceleration/g
247
X: 108.8 Y: 161.2
X: 83.84 Y: 85.2
100 X: 65.3 Y: 52.65
50 X: 59.99 Y: 0
X: 70.31 Y: -1
0
X: 87.62 Y: -1
X: 106.1 Y: 0
X: 80 Y: 0
50
60
70
X: 111.3 Y: -1
80
90
100
110
(1) The rapid opening high flow valve proposed in this paper is quick in response. It can open all valves within 10 ms and its maximum flow capacity can reach up to tens of thousands of liters per minute. Moreover, the exact flow capacity can be regulated if necessary by changing the geometric dimension of the valve structure, which implies good extensibility. It resolves the contradiction between the rapid response of the hydraulic flow valve and the high flow capacity. (2) The velocity generator can well meet the demand of instantaneous energy storage and release in the shock process. The acceleration waveforms of the table and the test item of the system show good agreement with the latest BV043/85 specification. (3) Although the stroke of the impacting mass in this shock machine system is fixed, equipment at different levels can be tested by regulating the impact velocity through an adjustment of the throttle valve, which can be realized conveniently. (4) The double protection of the unloading circuit and the stroke limit of the accumulator piston can effectively prevent the secondary collision in the testing process. (5) The vibration isolation system of large foundation mass can meet the demand of environmental protection through avoiding negative impacts on the surrounding environment effectively.
120
Time/ms Fig. 21. The acceleration pulses of the table at different velocities vt equals 5 m/s, 4 m/s and 3.3 m/s.
Table 2 Comparison of the simulation results and BV043/85 specification. Velocity Waveform parameter
V ¼ 3.3 m/s
V ¼4.0 m/s
V ¼5.0 m/s
Acceleration peak (g) Pulse width (ms)
52.7/50 10.32/10.3
83.8/85 7.62/7.5
161.2/160 5.2/5
seen that the maximum displacement change is only 27 mm, which can fully meet the requirement on the vibration isolation in engineering application.
5.4. Impact acceleration waveform Based on the requirements in the BV043/85 specification on the shock test environments for equipment of different levels, we simulate and examine the prospective performances of the shock systems for three types of equipment by setting the shock velocity of the test article as 3.3 m/s, 4.0 m/s and 5.0 m/s respectively. Fig. 21 shows the acceleration waveforms of the table and the test item under different impact velocities. A comparison between the simulation results and the waveform parameters defined in BV043/85 specification is given in Table 2, which exhibits a good
Acknowledgments The first author would like to thank the National Natural Science Foundation of China for financially supporting this research under the Contract no. 51275369 and also to thank the China Scholarship Council to support related research hosted by Prof. Yeping Xiong at the University of Southampton. References [1] Wang Gong Xian, Hu Yong. Advances in shock testing facilities for naval shipboard equipments. Adv Mater Res 2011;301–303:1220–5. [2] Woyski Bill, Nelson Curt. An accurate, repeatable subsidiary component shock test system for the laboratory. In: Proceedings of the 75th Shock and vibration symposium. Virginia Beach, Virginia; 2004. pp.17–22. [3] Elder Steven, Arden Kevin. Land based full scale shock simulator[C]. In: Proceedings of the 70th shock and vibration symposium-CD. Albuquerque, New Mexico: Shock and Vibration Information Analysis Center HI-TEST Laboratories; 1999. [4] Gram MM, Clark AJ, Hegemier GA, Seible F. Laboratory simulation of blast loading on building and bridge structures[J]. Struct Under Shock Impact IX 2006;87:33–44. [5] Zhi-Yi Zhang, Gong-Xian Wang, Yu Wang. Numerical modeling of dual-pulse shock test machine for simulating underwater explosion shock loads on warship equipments. J Shanghai Jiaotong Univ (Sci) 2009;14:233–40. [6] Zhaodong Wen, Yu Wang, Lei Zhang Jianye Du. Modeling and dynamic simulation of novel dual-wave shock test machine. In: Proceedings of the 2011 international conference on fluid power and mechatronics. 2011. pp. 898–901. [7] Guo-hua Li, Yu-jie LI, Xiao-ci Zhang. Experimental verification of relationship between bubble motion and vibration of ship equipment. J Ship Mech 2005;9:98–105.
248
G. Wang et al. / International Journal of Mechanical Sciences 77 (2013) 239–248
[8] Wan J, Chen B, Wang QT. Sensitivity analysis of repetitive shock machine's vibration energy. Advances in safety, reliability and risk management. In: Proceedings of the European safety and reliability conference. 2012. pp. 2183–8. [9] Piersol AG, Paez TL. Shock testing machines. Harris' Shock and Vibration Handbook. New York: McGraw-Hill; 2010.
[10] Jiao Sujuan, Wang Yu, Zhang Lei, Hongxing Hua. Shock wave characteristics of a hydraulic damper for shock test machine. Mech Syst Signal Process 2010;24:1570–8. [11] Piotr Czop, Sawik Damian. A high-frequency first-principle model of a shock absorber and servo-hydraulic tester. Mech Syst Signal Process 2011;25: 1937–55.