The rapping acceleration of collecting plates in electrostatic precipitator system determines the dust- rapping performance of electromagnetic vibration exciter. To maximize the acceleration, the resonance phenomena needs to be driven by matching the mechanical natural frequency of the electrostatic precipitator system and the input frequency of electric current which energizes the electromagnetic vibrator. In this paper, the dust collecting plates and the electromagnetic vibration exciter in electrostatic precipitator system are vibration-modeled mathematically to characterize the resonance frequency. The effective mass and stiffness for each mode of the collecting plates are calculated using finite elements analysis and the natural frequency are computed by the method of least error square. In addition, the effective mass and stiffness of the exciter are computed. Then, the whole electrostatic precipitator system is analyzed. A frequency response analysis based on a sine sweep signal experiment is performed on a prototype for verification of calculated theoretical resonance frequency

Byoung-Duk Lim

Je-Hoon Kim

Ji-Hyun Choi

Jin-Ho Kim

English

Journal of Vibroengineering

374 ? VIBROENGINEERING. JOURNAL OF VIBROENGINEERING. MARCH 2013. VOLUME 15, ISSUE 1. ISSN 1392-8716  
955. Mathematical vibration modeling for an 
electrostatic precipitator system 
Ji-Hyun Choi, Je-Hoon Kim, Byoung-Duk Lim, Jin-Ho Kim
955. MATHEMATICAL VIBRATION MODELING FOR AN ELECTROSTATIC PRECIPITATOR SYSTEM.  
JI-HYUN CHOI, JE-HOON KIM, BYOUNG-DUK LIM, JIN-HO KIM 
Ji-Hyun Choi1, Je-Hoon Kim2, Byoung-Duk Lim3, Jin-Ho Kim4 
1, 2, 4Electric Motor and Actuator Laboratory, Department of Mechanical Engineering  
Yeungnam University, Gyeongsan, Gyeongbuk, 712-749, Korea 
3Acoustic and Vibration Laboratory, Department of Mechanical Engineering 
Yeungnam University, Gyeongsan, Gyeongbuk, 712-749, Korea 
4Corresponding author 
E-mail: 1jihyun.m.choi@gmail.com, 2kimjehoon@ynu.ac.kr, 3bdlim@ynu.ac.kr, 4jinho@ynu.ac.kr 
(Received 16 October 2012; accepted 28 February 2013) 
Abstract. The rapping acceleration of collecting plates in electrostatic precipitator system 
determines the dust- rapping performance of electromagnetic vibration exciter. To maximize the 
acceleration, the resonance phenomena needs to be driven by matching the mechanical natural 
frequency of the electrostatic precipitator system and the input frequency of electric current 
which energizes the electromagnetic vibrator. In this paper, the dust collecting plates and the 
electromagnetic vibration exciter in electrostatic precipitator system are vibration-modeled 
mathematically to characterize the resonance frequency. The effective mass and stiffness for 
each mode of the collecting plates are calculated using finite elements analysis and the natural 
frequency are computed by the method of least error square. In addition, the effective mass and 
stiffness of the exciter are computed. Then, the whole electrostatic precipitator system is 
analyzed. A frequency response analysis based on a sine sweep signal experiment is performed 
on a prototype for verification of calculated theoretical resonance frequency. 
Keywords: electrostatic precipitator, electromagnetic vibration exciter, natural frequency, 
resonance, method of the least error square. 
1. Introduction 
There are several methods for dust rapping on the electrostatic precipitators; these include 
motor-driven swing hammers and wet cleaning types. However, there are various problems with 
these methods. The motor-driven swing hammer has good rapping performance, but it occupies 
considerable space and has breakdown risk of collecting plates. The wet cleaning method is 
clean all the time due to continuous water screening during operation, but it may rust, freeze, or 
burst at low temperatures [1]. Consequently, an electrostatic precipitator that uses an 
electromagnetic vibration exciter was proposed to improve the performance of typical rapping 
systems [2]. Figure 1 shows the schematic diagram which consists of the dust collecting plate, 
electromagnetic vibration exciter and a stinger. 
 
Fig. 1. Schematic diagram of the ESP system 
Dust in the air is collected on steel plates by corona discharger [3]. Usually, these dusts are 
Dust collecting plates 
Electromagnetic vibration exciter Stinger 
955. MATHEMATICAL VIBRATION MODELING FOR AN ELECTROSTATIC PRECIPITATOR SYSTEM.  
JI-HYUN CHOI, JE-HOON KIM, BYOUNG-DUK LIM, JIN-HO KIM 
 ? VIBROENGINEERING. JOURNAL OF VIBROENGINEERING. MARCH 2013. VOLUME 15, ISSUE 1. ISSN 1392-8716 375 
usually fly ash and Particulate Matter-10 (PM-10) and it settles on the collecting plates in over 
1 mm of thickness [4]. To vibrate these collecting plates for dust rapping, electromagnetic 
vibration exciter are used. Figure 2 and Table 1 show schematic diagram of electromagnetic 
vibration exciter, and its specification. A quadratic electromagnetic vibration exciter which is 
shown on Figure 2 has better magnetic flux on air gap, so it has better efficiency. Operating 
principle of this exciter is following steps. When the sinusoidal electric current energizes coil, 
electromagnetic vibration exciter starts to vibrate by Lorentz force. Equation (1) describes the 
Lorentz force. The number of coil turns is ?, the intensity of the magnetic flux in the air gap is 
??, the input current is ?, and the effective coil length is ????: 
????????? ? ???????????? (1) 
The electromagnetic vibration exciter is connected to all collecting plates through a stinger. 
Collected dust on the collecting plates is rapped out by vibration force which came from 
reciprocating motion of the electromagnetic vibration exciter. Generally, it needs the rapping 
acceleration between 20 g and 50 g for effective dust rapping [1, 2]. To maximize the 
acceleration of collecting plate, the resonance phenomena needs to be driven by matching the 
mechanical natural frequency of the collecting plates and the input frequency of electric current 
which energizes the electromagnetic vibration exciter. Therefore, it is important to find the 
mechanical natural frequency of electrostatic precipitator system which is compsoed of the dust 
collecting plates and electromagnetic vibration exciter [5]. 
In this research, a mathematical modeling process for the electrostatic precipitator system is 
introduced using finite element analysis and the method of least squares. For verification of 
mathematical model, a frequency response analysis based on a sine sweep signal experiment is 
performed on the prototype.  
 
Fig. 2. Schematic diagram of the electromagnetic vibration exciter 
Table 1. Specification of the electromagnetic vibration exciter 
Specification of exciter 
Number of coil turns 400 turns 
Input current 1 A ~ 10 A 
Outer magnet Thickness 5 mm Size 100 mm × 50 mm 
Inner magnet Thickness 10 mm Size 50 mm × 50 mm 
Exciter Size 150 mm × 150 mm × 70 mm 
2. Pre-Analysis for Mathematical Modeling 
Figure 3(a) shows the schematic diagram of a collecting plate. To derive effective mass and 
955. MATHEMATICAL VIBRATION MODELING FOR AN ELECTROSTATIC PRECIPITATOR SYSTEM.  
JI-HYUN CHOI, JE-HOON KIM, BYOUNG-DUK LIM, JIN-HO KIM 
376 ? VIBROENGINEERING. JOURNAL OF VIBROENGINEERING. MARCH 2013. VOLUME 15, ISSUE 1. ISSN 1392-8716  
stiffness of this collecting plate, harmonic and modal analyses are performed by ANSYS 
Workbench [6]. The boundary conditions in finite element model are defined on four holes 
around edge as shown Figure 3(b).  
 
(a) 
 
(b) 
Fig. 3. (a) Design of a collecting plate, (b) finite element model of a collecting plate 
Mesh size was 10 mm. 995 ea of SOLID186 element type which has 20 nodes and 8 ea of 
SURF156 element type which has 3 nodes were set as mesh element to finite element model of 
the collecting plate, about 141 seconds was taken to perform a trial of finite element analysis. 
We conducted 5 times of analysis and used average values of them to reduce error of ANSYS 
itself. Through harmonic response analysis, we analyzed the frequency response for steady state 
excitation at the central position of the collecting plates. Then, using modal analysis, we 
obtained nine vibration mode shapes and frequencies of the collecting plates as shown in 
Table 2. 
Table 2. Modal shapes of the collecting plate 
1st mode 2nd mode 3rd mode 
Frequency: 20.3 Hz Frequency: 56.4 Hz Frequency: 85.6 Hz 
   
4th mode 5th mode 6th mode 
Frequency: 113.9 Hz Frequency: 166.9 Hz Frequency: 244.1 Hz 
   
7th mode 8th mode 9th mode 
Frequency: 253.4 Hz Frequency: 334.8 Hz Frequency: 349.3 Hz 
   
955. MATHEMATICAL VIBRATION MODELING FOR AN ELECTROSTATIC PRECIPITATOR SYSTEM.  
JI-HYUN CHOI, JE-HOON KIM, BYOUNG-DUK LIM, JIN-HO KIM 
 ? VIBROENGINEERING. JOURNAL OF VIBROENGINEERING. MARCH 2013. VOLUME 15, ISSUE 1. ISSN 1392-8716 377 
3. Theoretical Analysis for Effective Mass and Stiffness 
The system is approximated to vibration system as Figure 4 for mathematical approximation, 
and whole collecting plates are assumed that parallel connected. Effective mass of each 
collecting plate is abbreviated as ??????, effective stiffness of collecting plates as ??????, 
effective mass of the exciter as ????????, and effective stiffness of the exciter as ????????. In this 
chapter, effective mass and stiffness of the electromagnetic vibration exciter were not considered 
to derive precise value of effective mass and stiffness of the collecting plates only. 
 
Fig. 4. Vibration modeling of the ESP system 
At this stage in the analysis, we used the method of least squares on a 1-dof model. The first 
parameter to determine is target error ?? about optimal line which will be properly decreased by 
method of the least error square [7]. Equation (2) is a motion equation of the undamped system 
that is excited by harmonic force, where ??????? is the applied harmonic force on the collecting 
plate: 
????? ? ???? ? ????????? (2) 
Table 3 gives main three physical factors which are derived through the least error square 
method. 
Table 3. Formulas from the least error square method 
Formula Physical factor 
? ? ??? ???? Amplitude ? of the system 
? ????? ? ? ? ??????
?
???
 Estimated error 
?
?
?
?
?
? ?????
?
???
???
?
??? ?
?
?
?
?
?
?
?
?
?
?
?
? ?????
?
???
?????
?
???
? ?????
?
??? ?
?
?
?
?
?
? ??? Effective mass and stiffness matrix 
Table 4 shows?the values of the effective mass and stiffness of the collecting plate for each 
mode by solving effective mass and stiffness matrix from the least error square method. 
955. MATHEMATICAL VIBRATION MODELING FOR AN ELECTROSTATIC PRECIPITATOR SYSTEM.
JI-HYUN CHOI, JE-HOON KIM, BYOUNG-DUK LIM, JIN-HO KIM 
378 ? VIBROENGINEERING. JOURNAL OF VIBROENGINEERING. MARCH 2013. VOLUME 15, ISSUE 1. ISSN 1392-8716  
Table 4. Effective mass and stiffness for each mode?
Mode Effective mass [kg] Effective stiffness [N/m] 
1 0.1272 2.0670 × 103 
2 0.1681 2.1085 × 104 
3 0.1510 4.3686 × 104 
4 0.1392 7.1283 × 104 
5 0.1400 1.5403 × 105 
6 0.1778 4.1829 × 105 
7 0.0453 1.1486 × 105 
8 0.1713 7.5777 × 105 
9 0.0931 4.4820 × 105 
4. Vibrational Experiments 
In order to compare resonance frequencies from our theoretical (mathematical) analysis of 
the collecting plates with experimental results, the prototype was made, and frequency response 
experiment was performed using a 10 to 210 Hz sinusoidal sweep signal input to the 26 EA of 
collecting plates. Figure 5 shows the electrostatic precipitator prototype. The prototype 
precipitator is composed of 26 of collecting plates, cases, 4 fixation bars, insulators, and a 
stinger for connecting each collecting plate. 26 of collecting plates are connected to reference 
shaker. 
 
Fig. 5. Prototype of the ESP with the reference shaker 
 
Fig. 6. Schematic diagram of the experimental setup 
Figure 6 shows a schematic diagram of a vibration measuring experiment setup for the 
electrostatic precipitator. The system was excited by the shaker to find the resonance frequency 
with using a signal analyzer. The generated force and acceleration were measured using an 
Signal analyzer 
Power amplifier 
Shaker 
Charge amplifiers 
Impedance head 
955. MATHEMATICAL VIBRATION MODELING FOR AN ELECTROSTATIC PRECIPITATOR SYSTEM.
JI-HYUN CHOI, JE-HOON KIM, BYOUNG-DUK LIM, JIN-HO KIM 
 ? VIBROENGINEERING. JOURNAL OF VIBROENGINEERING. MARCH 2013. VOLUME 15, ISSUE 1. ISSN 1392-8716 379 
impedance head which is located between the electrostatic precipitator and a shaker. The voltage 
level of each signal was very small, so the signals were amplified by amplifiers. The amplified 
signals were sent to a signal analyzer, and the frequency responses of the dust collecting plates 
were obtained [2]. 
When the input frequency increased, higher plate modes were generated accordingly as 
Figure 7 shows. Table 5 shows the theoretical natural frequency and the experimental natural 
frequency respectively.
 
Fig. 7. Nine assumed frequency response peak points of 26 collecting plate, excited at the center 
Table 5. Theoretical and experimental natural frequency of 26 collecting plates 
Mode Theoretical natural 
frequency [Hz] 
Mode Experimental natural 
frequency [Hz] 
1 20.29 1 17.64 
2 56.37 2 50.53 
3 85.61 3 76.06 
4 113.89 4 100.01 
5 166.94 5 146.72 
6 244.11 6 220.28 
7 253.43 7 183.70 
8 334.74 8 300.69 
9 349.21 9 291.56 
5. Modeling of Electrostatic Precipitator with Electromagnetic Exciter 
In this chapter, the resonance frequency of the whole electrostatic precipitator system 
includes the electromagnetic vibration exciter was calculated. In previsous chapters, the natural 
frequency of collecting plates only was calculated and verified by mathematical modeling and 
frequency response experiment. However, the electromagnetic vibration exciter has addictive 
mass and stiffness effects on the resonance frequency of the electrostatic precipitator system 
unlike the reference shaker that transfers the excitation force only. The resonance frequency 
which considers the additive effective mass and stiffness of electromagnetic vibration exciter is 
newly computed and compared with the natural frequency of collecting plates only. 26 EA of 
collecting plates with the electromagnetic vibration exciter can be simplified and modified free 
body diagram of 1 DOF vibration. Figure 8(a), (b) show the prototype of electrostatic 
precipitator with the electromagnetic vibration exciter, and modified its free body diagram. In 
the free body diagram, the total effective mass and the total effective stiffness of the electrostatic 
precipitator (which are 26 of collecting plates) are ???? and ???? while the effective mass and 
stiffness of the electromagnetic vibration exciter are ???????? and ????????. And mass of stinger 
is ????????. 
955. MATHEMATICAL VIBRATION MODELING FOR AN ELECTROSTATIC PRECIPITATOR SYSTEM.  
JI-HYUN CHOI, JE-HOON KIM, BYOUNG-DUK LIM, JIN-HO KIM 
380 ? VIBROENGINEERING. JOURNAL OF VIBROENGINEERING. MARCH 2013. VOLUME 15, ISSUE 1. ISSN 1392-8716  
 
(a) 
 
(b) 
Fig. 8. (a) The prototype of the electrostatic precipitator with the electromagnetic exciter,  
(b) modified free body diagram of the electrostatic precipitator with the electromagnetic exciter 
Before the whole system was analyzed, effective mass and stiffness of the collecting plates 
were used same value from FEM model as chapter 2 and mathematical method for obtaining the 
effective mass and stiffness of the electromagnetic vibration exciter was same with one for the 
collecting plates shown in previous chapters. Table 6 shows effective mass and stiffness of the 
electromagnetic vibration exciter. Effective mass ???? and stiffness ???? were multiplied by 26 
to apply number of collecting plates and these values added to electromagnetic vibration 
exciter’s effective mass ???????? and stiffness ????????? Theoretical ?? of the whole system was 
calculated by an equation (3) with these values. Table 7 shows the natural frequencies of the 
precipitator system (26 of collecting plates with electromagnetic vibration exciter). 
?? ? ?
???????? ? ????? ? ???
???????? ? ????? ? ??? ? ????????
?? (3) 
 
Fig. 9. The electromagnetic vibration exciter for the experiment 
Table 6. Effective mass and stiffness of the electromagnetic vibration exciter 
Mode Effective mass [kg] Effective stiffness [N/m] 
1 0.639 1.4626 × 104 
2 0.523 1.2942 × 106 
3 0.021 9.9063 × 104 
As mentioned earlier, among vibration modes of the exciter, only 1st mode values were 
considered as effective results because the 2nd and 3rd mode were generated by non-axial 
direction motion, not the axial direction of the stinger which does not generate our intended dust 
rapping [2]. For verification by the experiment, the electrostatic precipitator system with the 
electromagnetic exciter was connected to the reference shaker as Figure 10(a). Frequency 
response experiment was performed as previous chapter. Figure 10(b) shows its results. 
955. MATHEMATICAL VIBRATION MODELING FOR AN ELECTROSTATIC PRECIPITATOR SYSTEM.
JI-HYUN CHOI, JE-HOON KIM, BYOUNG-DUK LIM, JIN-HO KIM 
 ? VIBROENGINEERING. JOURNAL OF VIBROENGINEERING. MARCH 2013. VOLUME 15, ISSUE 1. ISSN 1392-8716 381 
Table 7. Natural frequencies of the precipitator system (electrostatic precipitator equipped with the exciter) 
Mode of 
collecting 
plates 
Natural frequency [Hz] 
Mode of exciter 
1st mode 2nd mode 3rd mode 
1 19.35 83.50 29.67 
2 50.06 88.49 54.80 
3 74.54 105.74 79.12 
4 97.89 123.66 102.42 
5 143.29 160.13 148.19 
6 215.47 222.97 220.87 
7 174.49 198.02 185.85 
8 294.13 296.67 300.87 
9 281.13 286.52 291.91 
 
 
(a) 
 
(b) 
Fig. 10. (a) The electrostatic precipitator with the electromagnetic vibration exciter system for experiment,  
(b) nine assumed frequency response peak points of the system, excited at the center 
6. Conclusions 
In this thesis, mathematical modeling of the electrostatic precipitator system using resonance 
phenomena is proposed. Natural frequency of the electrostatic precipitator was calculated by the 
955. MATHEMATICAL VIBRATION MODELING FOR AN ELECTROSTATIC PRECIPITATOR SYSTEM.  
JI-HYUN CHOI, JE-HOON KIM, BYOUNG-DUK LIM, JIN-HO KIM 
382 ? VIBROENGINEERING. JOURNAL OF VIBROENGINEERING. MARCH 2013. VOLUME 15, ISSUE 1. ISSN 1392-8716  
method of least error square, harmonic response and modal analysis. Furthermore, this 
mathematical modeling was verified by comparison with experimental results which were from 
frequency response experiments. The mathematical modeling showed about 90 % of preciseness 
in natural frequency through this comparison at principal modes from 1st to 3rd which is utilized 
for vibration dust rapping. About 10 % of error was caused by the experimental setup such as a 
gap between the insulators and fixation bars for the insulation and support of each plate.  
Although there were slight errors on natural frequency, the mathematical model will be 
helpful for finding or predicting approximate natural frequency of the electrostatic precipitator 
systems for effective rapping.  
Acknowledgment 
This research was supported by the Urban Railroad Technology Development Program 
(Grant 09 Urban Railroad A-01), funded by Ministry of Land, Transport and Maritime Affairs of 
the Korean Government. The authors gratefully acknowledge this support. 
References 
[1] Salbert Oglesby, Grady B. Nicholas Electrostatic Precipitation. Dekker Marcel Inc., New York, 
1978. 
[2] J. H. Kim, J. H. Kim, S. H. Jeong, B. W. Han Design and experiment of an electromagnetic 
vibration exciter for the rapping of an electrostatic precipitator. Journal of Magnetics, Vol. 17, No. 1, 
2012, p. 61-67. 
[3] J. S. Chang Corona Discharge Processes. IEEE Journals, Vol. 19, Issue 6, 1991, p. 1152-1166. 
[4] Paul E. Morrow, Thomas T. Mercer A point-to-plane electrostatic precipitator for particle size 
sampling. American Industrial Hygiene Association Journal, Vol. 25, Issue 1, 1964, p. 8-14. 
[5] Jaehee Kim, Junhyung Kim, Jinho Kim Robust design of air compress-driving quadratic linear 
actuator in fuel cell BOP system using Taguchi method. Journal of Magnetics, Vol. 17, No. 4, 2012, 
p. 275-279. 
[6] D. J. Mead, Y. Yaman The harmonic response of rectangular sandwich plates with multiple 
stiffening: a flexible wave analysis. Journal of Sound and Vibration, Vol. 145, Issue 3, 1991, 
p. 409-428. 
[7] Jack R. McDonald, Wallace B. Smith, Herbert W. Spencer III, Leslie E. Sparks A mathematical 
model for calculating electrical conditions in wire?duct electrostatic precipitation devices. Journal of 
Applied Physics, Vol. 48, Issue 6, 1977, p. 2231-2243. 


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Mathematical vibration modeling for an electrostatic precipitator system

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