Advances in Systems Science and Applications (2012) Vol.12 No.1 76-88 The Performance Study on Two-Stage Vibration Isolation System with Active Magnetic Suspension Control Xiaojing Liu and Yefa Hu Mechanical and Electronic Engineering School, Wuhan University of Technology, Wuhan 430070, China Abstract Two-stage vibration isolation system is normal equipment on naval vessel, which has good function of vibration isolation. Active control on vibration isolation sys- tem is better than passive control. Active magnetic suspension control has many virtues like alterable control methods and variable stiffness and damping. So that it is applicable to complex multi-interferes and multi-coupling floating raft. This paper set up mathematic models of two-stage vibration isolation with ac- tive magnetic suspension control system and solves the transfer functions. After simulating in MATLAB it analysis the effectiveness of Kp ,τd,τi of PID controller to vibration isolation and the parameters which influence the vibration isolation performance including up and down layer stiffness, damping and mass ratio. The simulation results indicate that active magnetic suspension has good works to vibration isolation. At the same time, on the basis of these results optimize the construction. Keywords Two-stage vibration isolation system, Active control of magnetic sus- pension 1 Introduction Two-stage vibration isolation systems is one kind of device of power equipment for damping vibration, which has better isolation vibration performance com- pared with one layer device. Internal and external research focus on dynamic performance of two-stage vibration isolate system affected by design parame- ters[1], effects of nonlinear spring and damping on two stage vibration isolate system[2], effect of pedestal stiffness on two stage vibration isolate system[3] and study of flexible mass on two stage vibration isolate system[4], and so on. Com- pared with passive control of isolation vibration system, active control can get better effectiveness by adjusting stiffness and damping. On this aspect, there are many researchs about methods[5], the fix position of active vibration isolator[6] and coupling vibration control[7]. Active magnetic suspension control system has many virtues including stiffness and damping controllable and tunable. Fur- thermore, it can apply classical or modern control theories according to different simulation force value and place to get the best effect, which make it become the best control system for multi-turbulent and multi-coupling floatin raft.This paper establish two-stage vibration isolation system with active magnetic suspen- Advances in Systems Science and Applications (2012), Vol.12, No.1 77 sion control equipment and its mathematic models. Then simulate in MATLAB and research how the three parapmeters of PID controller affect the peroformance of the system. Furthermore, study the influence of parameters such as up and down layer stiffness, damping and mass ratio to vibration isolation. Simulation results indicate that active magnetic suspension does good works to vibration isolation. In the end, on the basis of these results optimize the construction. 2 Active Magnetic Suspension Vibration Isolation control system 2.1 Theory of Active Magnetic Suspension Put active magnetic suspension vibration isolation system into the layer between middle mass m2and pedestal m3 .Construction and theory is showed in Fig.1 and Fig.2.there are three masses —m1 is up layer mass,m2 is middle layer mass,m3is pedestal mass.k1and c1 is stiffness and daming of the first stage.k2 and c2 is stiffness and daming of the second stage.U is the system of active magnetic sus- pension.U is the system of active magnetic suspension.f = f0 sinϖt is outer simulate force act on up layer mass m1.x1 is displacement of m1.x2 is placement of m2.m3 is fixed to foundation firmly. So ignore its displacement. Active Fig.1Model of active magnetic suspension in two-stage vibration isolation system magnetic suspension feed back system can provide damping force direct propor- tion to absolute speed of isolated object. When vibration happened and isolated vibration object elasticity mass increase or actual elasticity factor decrease, iso- lation vibration spring damping static deflection still keep constant. Therefore active magnetic suspension feed back isolate vibration system is better than pas- sive isolated vibration system, especially in low frequency area. Middle mass m2 displacement is control variable quantity which is recorded by displacement sensor. After comparing with setting amount, PID controller calculate discrepant value and change electric current to produce magnetic force which help middle mass keep balance position to get the purpose about decrease the force from up layer to base m3. 78 Xiaojing Liu:The Performance Study on Two-Stage Vibration Isolation System... Fig.2 Theory of magnetic suspension system 2.2 Transfer Function Establish system block plant (See Fig.3).f(s) is outer simulation force,f1(s) is force from system to base. Hk(s) is PID controller transfer function.H2(s) is power amplifier transfer function.H1(s) is the first stage transfer function.H3(s) is the second stage transfer function. Fig.3 Active magnetic suspension vibration isolation system block plant Without magnetic force, system force transfer function is T (s) = f1(s) f(s) = H1(s)H3(s) When apply feed back control, system force transfer function is T1(s) = f1(s) f(s) = H1(s)H3(s) 1 +H2(s)Hk(s) fe = k i2 x2 Linearization force to fe = ki(i− i0)− kx(x− x0) Advances in Systems Science and Applications (2012), Vol.12, No.1 79 Hk(s) = Kp + τi s + τds 1 + Tfs H3(s) = k2 + c2s,H2(s) = ks Substitute all parameters into system differential equation. ( m1 0 0 m2 ) .. x1 .. x2 + ( c1 −c1 −c1 c1 + c2 ) . x1 . x2  + ( k1 −k1 −k1 k1 + k2 )( x1(t) x2(t) ) = ( f fe ) Use Laplace transformation to equations and arrange to get H1(s) = x2(s) F (s) = s+ 440 (s+ 3.62)2(s+ 1.01)2 2.3 simulations In Matlab Simulink build block plant of system transfer function (See Fig4.) System with and without active magnetic suspension control forcetime curves are list at Fig.5. Fig.4 Block plant to simulate in Simulink of Matlab Fig.5 showed that active magnetic suspension system indeed helps to improve the performance of two-stage isolate vibration. Force transfer from up layer outer 80 Xiaojing Liu:The Performance Study on Two-Stage Vibration Isolation System... simulate to base become quit so small apparently and vibration curve become smooth. (a) Nno feedback time-force transfer (b) With feedback time-force transfer Fig.5 Comparation with before and after using magnetic suspension feed back control 3 Influence of PID Controller Parameters to Isolate Vibration 3.1 Effective of Proportion Parameter Kp Without change other parameters, Kp = 100, 500, 1000 Result curve showed at Fig.6 From curves we can see increase Kp will decrease force to base tremedously. Fig.6 Different Kp time-force curves 3.2 Effective of Differential Constant τd Without change other parameters, τd=100,500,1000. Result curve showedat Fig.7. Form Fig.7, τd has a little influnce to force max value and the speed of isolation vibriation. However, increase τd can make smooth of force vibraition and improve the performance of the system. Advances in Systems Science and Applications (2012), Vol.12, No.1 81 Fig.7 Different τd time-force curves 3.3 Effective of Integral Constant τi Without change other parameters,τi=0.5,5,50. Result curve showed at Fig.8. Fig.8 Integral constant τd time-force curves Curve tell that τi has no big influence to max force value but only effect to min force value slightly. 4 Effectiveness of Two-Stage Vibration Isolation System Ather Parameters 4.1 Natural Frequency and Force Coefficient of Transmission Calculation Without thinking of damping, system natural frequency will be calculated by listed formula(1) ωn 2 = k1 + k2 2m1 + k2 2m2 ± √ ( k1 + k2 2m1 + k2 2m2 )2 − k1k2 m1m2 Substitute parameters into the formula: ξ2 = 0.06N/(m/s), ξ1 = 0.04N/(m/s) 82 Xiaojing Liu:The Performance Study on Two-Stage Vibration Isolation System... The natural frequencies are: ωn1 = 78HZ,ωn2 = 2HZ When force excited, the force coefficient of transmission is formula (2) Tf = √ (α2 − 4ξ1ξ2αϖ2 1) 2 +ϖ2 1(2ξ1α 2 + 2ξ2α)2 A2 +B2 In the formula A = ϖ1 4 −ϖ1 2(α2 + 4ξ1ξ2α+ µ+ 1) + α2 B = ϖ1 3(2ξ2α+ 2ξ1µ+ 2ξ1)−ϖ1(2ξ1α 2 + 2ξ2α) µ = m1/m2, ϖ1 = ω/ω1, ϖ2 = ω2/ω1 ω1 2 = k1/m1, ω2 2 = k2/m2 ξ1 = c1/2 √ k1m1, ξ2 = c2/2 √ k2m2 When substitute the parameters into the formula, we get the curve of force- coefficient transmission—simulation frequency (See Fig.9). Fig.9 Curve of force-coefficient transmission-simulation frequency From the curve we can see the influence of simulation frequency to force- coefficient transmission: • After 53HZ, force- coefficient of transmission is near zero that indicates the effectiveness of isolation vibration is good; • Before 53HZ, there are two peak values on resonance vibration that indicates this frequency region is the target we will research. to improve the effectiveness. Advances in Systems Science and Applications (2012), Vol.12, No.1 83 4.2 Influence of Stiffness k2 to Natural Frequency According to formula (1), we can get the influence of stiffness to natural frequency (See Fig.10). From the curve, we can see stiffness is almost linear direct proportion to natural frequency. So, the more stiffness is the more natural frequency is. On different working condition, we can change stiffness aim at natural frequency to avoid resonance vibration. Fig.10 Curve of stiffness to natural frequency Fig.11 Influence of stiffness to force-coefficient of 4.3 Influence of Stiffness k2 to Force-coefficient Transmission According to formula (2), choose three different value of stiffness k2500×102N/m, 500× 104N/m, 510×103N/m.We get three curves about k2 — force-coefficient of trans- mission (See Fig.11). 84 Xiaojing Liu:The Performance Study on Two-Stage Vibration Isolation System... Curves indicates that when stiffness is small the peak value move to left. If change up-layer stiffness, result is on the contrary that means when becomes larger the peak value move to left. So, increase up-layer or decrease down-layer stiffness all is good for isolation vibration. 4.4 Influence of Damping k1 Ratio k1 to Force-coefficient Transmission Without changing other parameters, damping ratio ξ1 = 0 ξ2 = 0 become ξ1 = 0.04 ξ2 = 0.06 We get the curve of simulation frequency—force e-coefficient of transmission (See Fig.12).Fig.12 indicates that damping ratio influence peak value directly but it doesn’t influence the position and tendency of peak value. The smaller damp- ing ratio is the bigger peak value is. When ξ = 0.04, ξ2 = (0.01, 0.06, 0.2, 2) respectively, we get the curves (See Fig.12 ξ1 = 0, ξ2 = 0; ξ1 = 0.04, ξ2 = 0.06 Simulation frequency—force e- coefficient of transmission Fig.13).When damping ratio is less than 1, it can’t influence peak value position and tendency, but influence peak value. The bigger damping ratio is the smaller peak value is, that is good for isolation vibration. While when damping ratio is more than 1, one hand it can decrease the peak value, on the other hand, peak value position move from left to right. Advances in Systems Science and Applications (2012), Vol.12, No.1 85 Fig.13 ξ1 = 0.04,ξ2 = (0.01, 0.06, 0.2, 2) Simulation frequency—force e-coefficient of transmission Fig.14 ξ2 = 0.06,ξ2 = (0.01, 0.04, 0.1, 2, 20) Simulation frequency—force e- coefficient of transmission Fig.15 Change ξ2 and ξ1 Simulation frequency—force e-coefficient of transmission 86 Xiaojing Liu:The Performance Study on Two-Stage Vibration Isolation System... When ξ2 = 0.06,ξ1 = (0.01, 0.04, 0.1, 2, 20) respectively, we get the curves (See Fig.14).When damping ratio is less than 1, it can’t influence peak value position and tendency, but influence peak value. The bigger damping ratio is the smaller peak value is, that is good for isolation vibration. While when damping ration is more than 1, peak value position move from left to right and peak value become bigger when damping ratio is bigger. So, damping ratio than close to 1 is the best value to isolation vibration. 4.5 Influence of Mass Ratio to Force-coefficient Transmission When m1 = 104kg, ξ1 = 0.04, ξ2 = 0.06, µ = 0.04042, 2, 5, 10 respectively, we get the curves showed in Fig.16. Isolation vibration system brings about two resonance points which position are determined by mass ratio. When µ ≥ 1, the first peak value position move to right but not very clearly and the second peak value position move to right quiet obvious. There is small influence to curve tendency in lower frequency region ,such as 40HZ. But in low frequency region, the bigger mass ratio is the smaller peak value is. In middle frequency region, from 40HZ to 180HZ, curve change apparently, the second peak value decrease quickly. It is obviously that mass ratio influence middle frequency greatly. To make sure system work at frequency far away from resonance points we hope the distance between the first peak value and the second peak value is large, which means µ ≥ 1(mass m1 ≥ m1) is good for isolation vibration. Fig.16Different curve of simulation frequency—force e-coefficient of transmission 5 Improve Isolation Vibration System According to above analysis, we take three proposals to improve the system per- formance. Because mass has been chosen, then • Without changing other parameters, decreasing under layer stiffness by re- Advances in Systems Science and Applications (2012), Vol.12, No.1 87 ducing spring from 6 to 4, then stiffness k2 = 430× 103N/m • Without changing other parameters, raise up and under layer damping ξ1 = 0.4N/(m/s), ξ2 = 0.6N/(m/s) • Decrease under layer stiffness and increase up layer stiffness at the same time, k2 = 430× 103N/m, ξ1 = 0.4N/(m/s), ξ2 = 0.6N/(m/s) Results showed in Fig.17, from which we can see that damping is the most important influence factor and only change stiffness is good for middle-high fre- quency isolation vibration. The best solution is decreasing under layer stiffness and increase up and under damping simultaneously. Fig.17 Change stiffness and dampin Simulation frequency—force e-coefficient of transmission 6 Conclusion This paper research on the performance of two-stage vibration isolation system and the effectiveness of system parameters including three parameters of PID controller,stiffness, damping and mass ratio. Conlusion are • Essence of active magnetic suspension control system act on two-stage vibra- tion isolation system is change the second layer stiffness and damping to decrease vibration. • Adjusting stiffness by active magnetic suspension to avoid system nature frequency is a good method to prevent resonance vibration. • Decreasing the second stiffness by adjust active magnetic suspenion control has good effect. 88 Xiaojing Liu:The Performance Study on Two-Stage Vibration Isolation System... • Adjusting damping by active magnetic suspension control to keep damping less than 1is good effect. • Increasing proportion parameter Kp is good for vibration isolation speed. • Increasing differential constant τd ralatively is good for decreasing vibration. • Integral constant τi has no apparent influence to prevent vibration. References [1] Su Ronghua, Peng Chenyu, Ding Wenwen. 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