Analysis of Power Flow of Floating Raft with Multi-exciters by FEM

FU Jian,WANG Yong-sheng,WEI Ying-san

(College of Naval Architecture and Power Engineering,Naval University of Engineering,Wuhan 430033,China)

1 Introduction

The mechanical noise induced by marine power plant is dominat in underwater acoustic radiation of ships.The vibration isolation technology is effective in decreasing vibratory power transferred from vibrator to foundation,the floating raft vibration isolation system is compact and its isolation effect is excellent,which has become an effective tool to reduce the vibration of power engine in vessel.There are several criterions to evaluate the vibration isolation performance of the floating raft,power flow integrates the phase relationship between force and velocity in one criterion and it is beneficial to the optimization design of the floating raft[1-2].

Many researchers have studied the power flow of complex vibration isolation system and tried to calculate it by FEM[3-7].In this paper,a method to calculate floating raft power flow is proposed based on FEM combining the method of four-end parameters and admittance.There are more than two equipments installed on floating raft,so there will be phase difference between exciters,how does the phase difference affect isolation performance especially the power flow transferred to foundation is an issue which ought to be considered in the design stage of floating raft.If equipments are installed on floating raft symmetrically,it will increase the vibration isolation performance.A symmetrical collocated floating raft is investigated firstly,the power flow transferred to foundation is calculated with equipments installed symmetrically and asymmetrically,and then the influence of phase difference between exciters is investigated.In order to validate the applicability of conclusion obtained from symmetrical collocated floating raft,taking an assembled floating raft as object,the influence of phase difference on power flow transferred to foundation is investigated while the equipments are installed asymmetrically in common plane or installed asymmetrically in different plane.

2 Calculation of power flow based on FEM

The movement equation under harmonic force in finite element analysis is given by:

where:M is mass matrix,K is stiffness matrix,C is damping matrix.andare the matrix of node displacement,velocity and acceleration,respectively,{f（t）}is the matrix of harmonic force.

After dispersing the structure,the matrix of mass,stiffness and damping can be got by finite element analysis,using equation(1),the response of every node can be obtained.

The power flow transferred to machinery is often treated as input power flow,but there are many difficulties in confirming the spectrum and location of machinery excitation,so that referencing the calculation method of vibration level difference,the power flow transferred to upper isolator is considered as input power flow,the power flow transferred to foundation is considered as transmission power flow.

Fig.1 The sketch map of isolator

2.1 Calculation of input power flow

According to the ends displacement of isolators,the transmission matrix equation of isolator can be obtained using four-end parameter method[8].Parameters are shown in Fig.1.The equation can be written as:

where:f1and f2are the force of upper end and lower end of isolator,u1and u2are the velocity of upper end and lower end of isolator,k and c are stiffness and damping of isolator,ω is angular frequency.f1=f2,so that the force and velocity of isolator ends can be written as:

where q1and q2are displacement of upper end and lower end of isolator.Input power flow isandis the conjugate of u1.

2.2 Calculation of transmission power flow

Power flow transferred to foundation can be written asm is the amount of lower isolator,fsiis the force acting on foundation through the ith lower isolator,and the corresponding velocity at junction is vsi,VT*is the conjugate of V.

The admittance of foundation can be written as:H=V/F,that:

If unit excitation force is applied on junction of the ith lower isolator with foundation and perform the harmonic analysis.After the ith analysis we obtain the desired velocity responseand then the admittance H can be obtained,that:

From HF=V and according to the foundation velocity calculated by FEM,the force transferred to foundation can be obtained,and then the transmission power flow can be obtained[7].

3 Validation of the method to calculate power flow

A single freedom vibration isolation system is taken as the object,and the input and the transmission power flow are calculated by analytic method and FEM respectively.The numeration model is in Fig.2.The foundation is simulated by a simply supported beam,and the isolator supports an object whose mass is 1 kg.The stiffness of isolator is k=987 N/m,damping is c=6.28 Ns/m.Density is ρ=7 850 kg/m3,flexible module is E=2.1×1011Pa,the length is L=0.4 m,x0=0.2 m,A=（0.01×0.012）m2is area of the cross section,and the damping factor of beam is β=0.1.

Fig.2 The model of single freedom isolation system

3.1 Calculation of power flow by analytic method

The movement equation of beam is:u2=f2Y,the admittance of beam isthe mode frequency isand the vibration shape isf and u are force and velocity in the junctions,the idiographic meaning can consult Fig.2(b).

If f1is taken as unite excitation force,according to equations(2),(3)and(4),the force and velocity can be obtained,and then the input and the transmission power flow can be calculated.

3.2 Calculation of power flow based on ANSYS and result comparison

The beam,isolator and mass lump are simulated by elements of beam4,combine14 and mass21 respectively in ANSYS.The response from harmonic analysis is used to calculate power flow.The comparison of power flow calculated by FEM and analytic method is shown in Fig.3.

Fig.3 shows that the result calculated by FEM is very close to the result obtained by analytic method.The effectiveness and veracity are validated.

Fig.3 The comparison of power flow results

4 Research on the power flow of floating raft with multi-exciters

The trait of floating raft is that there are more than two equipments installed on raft which can lead the problem of multiple exciters working together.In this paper,a simple floating raft with equipments installed symmetrically is analyzed firstly,and the influence of phase difference between exciters on transmission power flow is investigated,and then the power flow characteristic of a assembly floating raft is also analyzed.The working frequency of every exciter is assumed as the same.

Tab.1 The stiffness parameter of isolators

Fig.4 The finite element model of floating raft(symmetric collocation)

Fig.5 The simplified map of isolator collocation

4.1 Power flow analysis of floating raft with equipments installed symmetrically

The two equipments is installed symmetrically on floating raft,each equipment is connected to raft by four isolators,the raft is connected to foundation by 6 isolators.The material damping factor of raft and foundation is 0.005,density is 7 800 kg/m3,flexible module is 2.1 E11Pa,Poisson’s ratio is 0.3.The stiffness of isolators is shown in Tab.1,K1and K2are the stiffness of upper and lower isolator,and the damping ratio of all isolators is 0.05.The finite element model of floating raft is in Fig.4,and the collocation of isolators is shown in Fig.5.

4.1.1 Mode analysis of floating raft

The mode analysis of structure can reflect its dynamic characteristic,the former 50 modes are calculated,the frequency and vibration shape of partial modes are shown in Tab.2.

Tab.2 The mode analysis of floating raft

4.1.2 Power flow computation of floating raft

The excitation is assumed as unit force.The power flow is calculated and shown in Fig.6.Fig.6 shows that the power flow of floating raft has one more peak compared to the single freedom vibration isolation system in low frequency,the reason is that the existence of raft make floating raft system has one more mode in low frequency.At the same time,the decreasing rate of power flow of floating raft is faster than the rate of single freedom system,when frequency is higher than the frequency of second peak,it can illustrate that the floating raft has excellent isolation performance in high frequency.From further analysis,the fact can be obtained that the two anterior peaks correspond to the 5th and 15th mode,the rest peaks correspond to modes in high frequency.

Fig.6 The power flow with equipments working together

4.1.3 Power flow analysis with multiple exciters

The power flow comparison when single equipment working and two equipments working together is shown in Fig.7(the phase difference is neglected),after that the influence of phase difference on transmission power flow between exciters is analyzed and shown in Fig.8.

Fig.7 Transmission power flow with single and multi-exciters

Fig.7 shows that the transmission power flow curves of single equipment working have more peaks in high frequency than the curve of two equipments working together.The reason is that when two equipments with the same vibration characteristic installed symmetrically,only the modes with symmetrical vibration shape can be inspired,but when single equipment working it can be seen as an asymmetrical excitation and it will inspire more modes.So that when design the floating raft we had better put the equipments with the same vibration characteristic symmetrically and it can improve the isolation performance.Fig.8 shows that when the phase difference is π/2 and 3π/2,the effect on power flow from 0 Hz to 100 Hz is nearly the same,but in 100 Hz to 500 Hz,the effect has great difference.In the frequency of 0 Hz to 50 Hz,when the phase difference is π,power flow is the smallest.The biggest difference is in 40 Hz,the difference can reach to 55 dB.As the whole,the conclusion can be obtained that the effect of phase difference between exciters can not be neglected.

Fig.8 The effect of phase difference on transmission power flow between exciters

4.2 Power flow analysis of assembly floating raft

If there are many equipments installed on floating raft,and some are disposed on top of the raft,while the others are disposed in side of the raft,it can be called as assembly floating raft.The conclusion of the former analysis is that the effect of phase difference between exciters can not be neglected.In order to validate correctness and applicability of the conclusion,a assembly floating raft is taken as the new research object.The finite element model of floating raft is in Fig.9 and relative parameters can be found in Ref.[9].The equipments 2 and 1 disposed in common plane have the same disturbance characteristic but equipment 3 disposed in side has different disturbance with equipment 1.According to the measured acceleration in feet of equipments,the effect of phase difference on the transmission power flow between equipments of 1 and 2 is analyzed firstly,which is shown in Fig.10(a),and then the effect of phase difference on transmission power flow between the equipments 3 and 1 is also analyzed,which is shown in Fig.10(b).

Fig.9 The finite element model of assembly floating raft(unsymmetric collocation)

Fig.10 The effect of phase difference on transmission power of assembly floating raft between equipments

The power flow curves corresponding to different phase difference in Fig.10(a)shows that the curves do not have the same frequency in peak.In frequency of 0 Hz to 50 Hz and 500 Hz to 1 000 Hz,the curves have the same variety current;the power flow corresponding to phase difference of π has the smallest value in frequency from 0 Hz to 80 Hz,and difference is evident in low frequency,it is up to 20 dB.Besides that the difference is more than 10 dB in 125 Hz,315 Hz and 800 Hz.The conclusion can be obtained from the former analysis that when the phase difference is π with equipments which have the same disturbance characteristic installed symmetrically,the power flow is the smallest and the influence of phase difference on power flow is complex and notable.The power flow curves corresponding to different phase difference in Fig.10(b)show that different curves have the same frequency in peak,especially they are nearly the same in high frequency.The biggest contrast is in 160 Hz and the difference between curves of 0 and π/2 is 20 dB.

When the influence of phase difference on power flow between equipment 1 and equipment 3 is analyzed,the force acting on equipment 1 is assumed as unchanged,and only the force acting on equipment 3 is changed.The fact can be obtained from further analysis that when equipments 1 and 3 working together,the power flow transferred to foundation is nearly the same with equipment 3 working alone in low frequency,but it is nearly the same with equipment 1 working alone in high frequency;so that the influence of phase difference on transmission power flow is notable in low frequency and is small in high frequency.The conclusion can be obtained that when the equipments with different disturbance characteristic working together,the influence of phase difference on transmission power flow is decided by equipment disturbance characteristic,isolator type and connection mode(rigid or elastic).

5 Conclusions

(1)The input and the transmission power flow of floating raft are calculated using the method of four-end parameter and admittance based on FEM.Validity of the method is proved by a single freedom vibration isolation system.

(2)The exciters with the same disturbance characteristic installed symmetrically will inspire less modes in high frequency than asymmetrical exciters,and it can improve the vibration isolation performance.

(3)When equipments with the same disturbance characteristic working together,the influence of phase difference on transmission power flow between exciters is notable and the power flow does not have the same frequency in peak.The power flow corresponding to phase difference of π has the smallest value in low frequency,and this trait can be used to improve the vibration isolation performance.

(4)When equipments with different disturbance characteristic working together,the influence of phase difference on transmission power flow between exciters is determined by the integration of isolation effect and excitation spectrum under conditions of each equipment working alone.

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