Construction of a novel vibratory valve-injector and its design optimization is presented in the paper. The principle of the system operation is based on the effect of dynamic positioning of a steel ball in a vibrating tube. Theoretical analysis of the stability of this non-linear system is coupled together with the experimental study of an operating valve. Laser holographic interferometry is used for the identification and optimization of working regimes of the syste

A. Palevicius

M. Ragulskis

V. Naginevicius

English

Naginevicius, V.

Ragulskis, M.

Palevicius, A.

JVE International

 331. ANALYSIS OF THE DYNAMICS OF THE VIBRATORY VALVE-INJECTOR V. NAGINEVICIUS1,A, M. RAGULSKIS2,B AND A. PALEVICIUS3,C   © VIBROMECHANIKA.  JOURNAL OF  VIBROENGINEERING.  2008 JANUARY/MARCH,   VOLUME  10,  ISSUE  1,  ISSN 1392-8716  37 331. Analysis of the Dynamics of the Vibratory Valve-Injector  V. Naginevicius1,a, M. Ragulskis2,b and A. Palevicius3,c  Kaunas University of Technology, Kestucio 27-217, LT-44312, Kaunas, Lithuania E-mail: avytenis.naginevicius@ktu.lt, bminvydas.ragulskis@ktu.lt, carvydas.palevicius@ktu.lt  (Received 19  January 2008; accepted 17  March  2008)  Abstract. Construction of a novel vibratory valve-injector and its design optimization is presented in the paper. The principle of the system operation is based on the effect of dynamic positioning of a steel ball in a vibrating tube. Theoretical analysis of the stability of this non-linear system is coupled together with the experimental study of an operating valve. Laser holographic interferometry is used for the identification and optimization of working regimes of the system.  Keywords: Vibratory valve, injector, mathematical modeling, holographic interferometry, fluid dynamics  Introduction  Controlled dosing and spraying of liquid material is applied in different areas ranging from medicine to agriculture [1], [2]. Specific interest exists for elastic catheter pipe type dosing equipment. Complex dynamic processes taking place in such systems are analyzed in [3], [4]. Method examining the dynamic characteristics of a tube under the influence of the internal flow is presented in [3]. Galerkin’s method in conjunction with the method of multiple scales is employed for obtaining the stability of the tube vibration. According to the results, instability can occur under certain conditions of resonance. In [4] it is demonstrated that fluid-elastic effects which are responsible for fluid-elastic instabilities may be directly measured through the analysis of the vibrating motion of a system under flow. Piezoelectric actuators are used to increase the vibratory level when buffeting forces which excite tube vibration are low, and to improve the measurement of fluid-elastic forces.  Application of piezoelectric actuators for the generation of standing waves in the outlet pipe can produce effects which can be exploited for the control of the dosing process. The motion of fluid-suspended particles and fibers in a standing wave field is analyzed in [5], [6]. The dynamics of micron-sized aerosol particles and their agglomeration under the standing wave conditions is analyzed in [7]. Coupling of the dynamic properties of a vibrating tube with the dynamic behavior of a steel ball inside that tube can help producing a new type of smart doser for liquid material which can be effectively controlled by piezoelectric actuators. The unique feature of the tubular vibratory valve consists in the fact that the sealing surface of the seat is facing towards the intake duct and is located in the node of the second natural mode of transverse vibration of the elastic pipe. The vibratory valve for controlling liquid flow 1 (Fig. 1) operates in the following way. Several versions of design of systems for fluid flow control are presented in the following diagrams.  A. The fluid flow controller is presented in Fig. 1.  The controller is composed of tube (1) with input (2) and output (3) channels. A valve consisting from the closing element 5, spring 4 and seat 6 is mounted inside the tube. Vibration exciter 7 is fixed to the one end of the tube. Limiter 9 is placed inside the rigid tube 8 which is fastened to membrane 10. The surface of the tube from the seat 6 up to limiter 9 is cone shape. The feeding generator 11 is controlling the vibrator. The spring 4 presses the limiter 5 into the seat 6 and shuts the flow in the tube when the vibration exciter is off. Standing waves are excited in the tube when the vibrator is on. The excitation frequency coincides with the natural frequency of the system and the fluid flow controller operates in the resonance mode. That enables to reach higher amplitudes and the limiter 5 moves against the spring force into the peak of standing wave and thus opens the fluid flow channel.  B. Fluid sprayer.    Fig. 1. Schematic diagram of fluid flow controller Fig. 2. Schematic diagram of the vibration sprayer    331. ANALYSIS OF THE DYNAMICS OF THE VIBRATORY VALVE-INJECTOR V. NAGINEVICIUS1,A, M. RAGULSKIS2,B AND A. PALEVICIUS3,C   © VIBROMECHANIKA.  JOURNAL OF  VIBROENGINEERING.  2008 JANUARY/MARCH,   VOLUME  10,  ISSUE  1,  ISSN 1392-8716  38       The vibration sprayer (Fig. 2) consists of longitudinal (1) and transverse (2) vibration generators attached to the motionless foundation 3; tube 4 with input (5) and output (6) channels. Limiter 7 and seat 8 are mounted inside the tube. Fluid pressure pushes the limiter 7 into the seat 6 when the vibrator is off. Standing waves are generated in the tube 4 when the vibrator 2 is on. The limiter moves into the peak of the standing wave and opens the fluid valve.      The fluid is sprayed by the oscillating end (6) of the tube (Fig. 3a). The eigenshape of the tube changes when the vibrator 1 is switched on. The standing wave length is shortened and the limiter is pressed into the seat – the spraying process is stopped. Such control mode enables reaching higher control frequencies compared to the control method used in Fig. 1.      (a)  (b)   Fig. 3. Operation diagram of the vibration sprayer: a) valve is open; b) valve is closed  C. Vibration fluid doser (Fig. 4).   Fig. 4. Schematic diagram of the vibration fluid doser  Vibration fluid doser consists from chamber 1 with input (2) and output (3) channels, electromechanical drive 5, controller 6, secondary valve 7 which is pressed by the spring to one of the seats. The chamber is a rigid tube, electromechanical drive serves as an exciter of transient vibrations. The placement of the first (4) and the secondary (7) valves defines the operating principle of the doser. The fluid flows through the input channel and closes the first valve when the vibrator 5 is off. The vibrator excites the third eigenshape vibrations (two standing waves) in the process of fluid dose formation. The valve 4 opens, but the secondary valve 7 closes (Fig. 5a).  Next, the second eigenshape vibrations are generated (Fig. 5b). The first valve 4 gets closed, the secondary valve is still kept closed. The formation of the dose is finished. The dose is ejected when the first eigenshape vibrations are generated (Fig. 5c). The first valve is kept closed, but the secondary valve is opened.          (a) (b) (c) Fig. 5. Operation principle of the vibration fluid doser: a, b and c – the three stages of the doze formation  Dynamics of a steel ball inside a vibrating tube  Motion of a small ball inside a tube performing transverse vibrations may  be  approximated  by  the  non-dimensional    differential equation of motion describing the dynamics of a mass particle on an oscillating profile with an obligatory  condition of contact with the surface [3], [4]:   ( ) Fmgxmxtxxhxmxtxxmxxxmx=++⎟⎟⎠⎞⎜⎜⎝⎛⎟⎠⎞⎜⎝⎛++++⎟⎟⎠⎞⎜⎜⎝⎛⎟⎠⎞⎜⎝⎛+∂∂ζ∂∂ζ∂ζ∂∂∂ζ∂∂ζ∂∂ζ∂∂∂ζ∂ζ∂∂∂ζ22222222121 ,        (1)   331. ANALYSIS OF THE DYNAMICS OF THE VIBRATORY VALVE-INJECTOR V. NAGINEVICIUS1,A, M. RAGULSKIS2,B AND A. PALEVICIUS3,C   © VIBROMECHANIKA.  JOURNAL OF  VIBROENGINEERING.  2008 JANUARY/MARCH,   VOLUME  10,  ISSUE  1,  ISSN 1392-8716  39  where ( )tx,ζζ =  − the shape of the tube; xxx  , ,  − projections of the displacement, velocity and acceleration  of the ball onto the horizontal axis; t − time; m − mass of the ball; h − the coefficient of viscous friction between the ball and the surface of the profile; g − the acceleration of gravity; F − the pressure force of the liquid.  Naturally, Eq.(1) is based on the assumption that the area of the cross section of the tube is small, the fluid flow is laminar, and the pressure of the liquid in the tube does not correlate with the amplitude of the elastic transverse vibrations. Also it is assumed that the mass of the locking ball is sufficient enough to perform the vibration induced motion in the liquid, but not big enough to alter the shape of the tube's vibration due to its relocation.  Such mathematical model of the system enables effective separation of ball’s motion to relatively fast and slow motions. Co-ordinate x represents the horizontal displacement of the ball (slow motion), while the motion of the ball in the vertical direction is defined by the function ( )tx,ζζ = .  When the tube performs steady oscillations of a standing wave type, the form of the tube looks like:  ( ) ( ) ( )wtkxbtx coscos, =ζ ,        (2)  where b − the amplitude of oscillations, k and w − the number and frequency of the standing wave respectively. When the vibrations take place in the horizontal plane x0y (Fig. 1) and no external force is applied to the mass particle, the following conditions are satisfied:  ,0,0,0====xxFgx           (3)  Then, the condition of existence of the steady state solutions is found from Eq. (1):  022=xt ∂∂ζ∂ζ∂,         (4)  or applying Eq. (2) to Eq. (4) :  Ζ∈⋅= sskx   ,π,         (5)  and  Ζ∈⋅+= sskkx   ,2ππ        (6)  Analogously, if the vibrations take place in the vertical plane x0z (Fig. 1), and no external force is applied to the mass particle, the condition of existence of the steady state solutions looks like: 022=+ gxxt ∂∂ζ∂∂ζ∂ζ∂,         (7)  or   Ζ∈= sskx   ,π.         (8)  If the external force is non-zero, there are no steady state solutions satisfying the condition 0== xx .   Stability of solutions when the vibrations take place in the horizontal plane   The stability of the solution described by Eq. (5) is checked constructing the variational equation around skx ⋅=π using the following assumption:  dxskx +=π,         (9)  here dx is a small variation about the steady state solution. After the assumption of approximations  ( ) ( ) ( )( ) ( ) ( )sskdxssdxskkkdxkdxssdxskk1sincoscos1cossinsin−=⋅−≈⎟⎟⎠⎞⎜⎜⎝⎛⎟⎠⎞⎜⎝⎛ +−=⋅+≈⎟⎟⎠⎞⎜⎜⎝⎛⎟⎠⎞⎜⎝⎛ +ππππππ    (10)  and rejection of terms with ( )2dx and ( )2xd , the following variational equation is produced from Eq. (1):  ( )( ) 0cos 2 =++ dxwtbkwxhdxmd .     (11)  Since all coefficients of this ordinary differential equation are positive in the neighborhood of t, the real parts of both roots of the characteristic equation will be negative. Thus the solution of the variation is stable and the position dxkx +=π is also stable. The value of the discriminant of the characteristic equation of Eq. (11) will define the type of the stable attractor. Averaging in time produces the following equation:  ( )022=++ dxmbkwxdmhxd .      (12)  Therefore, the stable attractor will be of a knot type and the roots of the characteristic equation of Eq. (12) will be real, i.e. the discriminant is positive, and the attractor of the  331. ANALYSIS OF THE DYNAMICS OF THE VIBRATORY VALVE-INJECTOR V. NAGINEVICIUS1,A, M. RAGULSKIS2,B AND A. PALEVICIUS3,C   © VIBROMECHANIKA.  JOURNAL OF  VIBROENGINEERING.  2008 JANUARY/MARCH,   VOLUME  10,  ISSUE  1,  ISSN 1392-8716  40 focus type when the roots are complex (discriminant is negative). That produces the following relations:  a)  the attractor of the knot type at mwkhb ≤ ;  b)  the attractor of the focus type at mwkhb > .  The stability of the solution described by Eq. (6) is checked in analogous way:  dxskkx ++=ππ2,     (13)  here dx is a small variation. Rejection of the non-linear terms and the following approximations   ( )( ) kdxdxskkkdxskkkss12cos12sin−−≈⎟⎟⎠⎞⎜⎜⎝⎛⎟⎠⎞⎜⎝⎛ ++−≈⎟⎟⎠⎞⎜⎜⎝⎛⎟⎠⎞⎜⎝⎛ ++ππππ     (14)  in Eq. (1) lead to:  ( )( )( )( )( )( ) 0coscos12sincos122222222222=−++−++mdxwtkbwxdwtbkhwtwbmkxmdwtbk       (15)  The coefficient of this ordinary differential equation at dx is negative in entire neighborhood of the local time t. Thus, at least one of the roots of the characteristic equation Eq.(15) will be positive, and the solution in Eq.(13) will be unstable. Thus, the positions of the ball at the nodal points of the vibrating profile are unstable. Alternatively, the positions at the peaks of the profile, i.e. at the points which vibrate with the maximum amplitude, are stable. Of course, that is true when the ball cannot jump off the surface of the profile (is placed inside a vibrating tube).   Experimental Analysis of a Tubular Vibratory Valve  A number of experimental studies are needed in order to ensure high dynamic accuracy of operation of the vibrator valves for controlling the flow of liquid substances. In most cases the exciting frequencies of the tubular working tube are fairly high, and the amplitudes corresponding to them are measured in micrometers. Therefore the holographic method can be effectively applied for the visual representation of wave processes taking place in the tubular vibratory valve ([1], [2]). The most effective method for studying the standing wave processes is the method of holographic interferometry with time averaging ([5], [6]). It should be noted that the most clearly expressed bands in the holographic interferograms are those recorded at the positions of minimum amplitudes ([7]). It is important to obtain the distribution of the vibration amplitudes not only in the middle of dark interference bands, but also in arbitrary positions on the surface of the tube. That enables the determination of the location of the steel ball inside the tubular valve. The amplitudes of vibration of the structure are determined using the methodology presented in [8], [9].  Holographic interferograms of the transverse vibrations of a tubular vibratory valve are presented in Fig. 6a, Fig. 6b and Fig. 7a. Fig. 6a and Fig. 6b make it possible to conclude that transverse vibration of the tube is sufficiently uniform (Fig. 6c). Therefore, the seat of the vibratory can be located in a nodal point, regardless of how it is displaced lengthwise in the upper or the lower nodal point of the tube. It should be noted that the frequency of excitation must be selected with care since the best performance of the vibratory valve is taking place at the resonant frequencies. If the excitation of the transverse vibrations is far away from the resonance frequencies of the tube, the operation of the tubular valve turns to be hardly controlled. The obtained results enable to optimize the design of the vibratory valves for controlling and dosing the liquid flow. The following parameters of the system are analyzed and optimized: a) selection of the material of the working tube; b) selection of the area of the transverse cross section of the tubular tube; c) location of the transverse vibration nodes in the tube; d) determination of the transverse vibration amplitudes along the tube. Maximum uniformity of the transverse vibrations in the vibratory valve is achieved due to this optimization which leads to more stable operation of the whole system.      a     b     a. The hologram of the transverse vibrations of the tube at w = 1,3 kHz. Illumination angle of the laser beam π / 4; 6b. The hologram of the tube at the illumination angle  π / 2 c. The interpretation scheme of the transverse vibrations Fig. 6. Holographic interferogram of transverse vibrations of the tube  331. ANALYSIS OF THE DYNAMICS OF THE VIBRATORY VALVE-INJECTOR V. NAGINEVICIUS1,A, M. RAGULSKIS2,B AND A. PALEVICIUS3,C   © VIBROMECHANIKA.  JOURNAL OF  VIBROENGINEERING.  2008 JANUARY/MARCH,   VOLUME  10,  ISSUE  1,  ISSN 1392-8716  41 Concluding Remarks  New model of a tubular vibratory valve is designed using the stabilization effect of a steel ball in the vibrating tube. The methodology of identification of vibration peaks enabled experimental optimization of the working regimes of the system. Such type of analysis could be successfully applied in the design stage of different precise vibratory systems.  References  [1] Cho K. J. and Recinella,  D. K. Pattern of dispersion from a pulse-spray catheter for delivery of thrombolytic agents: Design, theory, and results, J. Academic Radiology, 4(3), 210-216, (1997) [2] Miller P. and Ellis M. Effects of formulation on spray nozzle performance for applications from ground-based sprayers, J. Crop Protection, 19(8-10), 609-615, (2000) [3] Huang Y. and Hsu C. Dynamic behavior of tubes subjected to internal and external cross flows, J. Shock and Vibration, 4(2), 77-91, (1997) [4] Caillaud S. and de Langre E. The measurement of fluidelastic effects at low reduced velocities using piezoelectric actuators, J. of Pressure Vessel Technology – Transactions of the ASME, 121(2), 232-238, (1999) [5] Brodeur P. Motion of fluid-suspended fibres in a standing wave field, J. Ultrasonics, 29, 302-307, (1991) [6] Coakley W. T. and Zamani F. Cell Manipulation in ultrasonic standing wave fields. J. of Chemical Technology and Biotechnology, 44(1), 43-62, (1989) [7] Riera-Franco de Sarabia E. and Gallego-Juarez J. A. Ultrasonic agglomeration of micron aerosols under standing wave conditions, J. of Sound and Vibration, Palevicius, A., and Ragulskis, M., Holographic interference method for investigation of wave transport systems, SPIE Proc. of the 2-nd Intl. Conf. on Vibration Measurements by Laser Techniques, Ancona, Italy, 225-231, (1996)413-427, (1986) [8] Vest C. Holographic Interferometry, Moscow, Mir, (1982) Palevicius, A., and Ragulskis, M., Holographic interference method for investigation of wave transport systems, SPIE Proc. of the 2-nd Intl. Conf. on  Vibration Measurements by Laser Techniques, Ancona,  SItaly, 225-231, (1996) ly, 225-231, (1996)  

Analysis of the dynamics of the vibratory valve-injector

Journal of Vibroengineering

 331. ANALYSIS OF THE DYNAMICS OF THE VIBRATORY VALVE-INJECTOR V. NAGINEVICIUS1,A, M. RAGULSKIS2,B AND A. PALEVICIUS3,C 
 
 
© VIBROMECHANIKA.  JOURNAL OF  VIBROENGINEERING.  2008 JANUARY/MARCH,   VOLUME  10,  ISSUE  1,  ISSN 1392-8716 
 
37 
331. Analysis of the Dynamics of the Vibratory Valve-Injector 
 
V. Naginevicius1,a, M. Ragulskis2,b and A. Palevicius3,c  
Kaunas University of Technology, 
Kestucio 27-217, LT-44312, Kaunas, Lithuania 
E-mail: avytenis.naginevicius@ktu.lt, bminvydas.ragulskis@ktu.lt, carvydas.palevicius@ktu.lt 
 
(Received 19  January 2008; accepted 17  March  2008) 
 
Abstract. Construction of a novel vibratory valve-injector and its design optimization is presented in the paper. The 
principle of the system operation is based on the effect of dynamic positioning of a steel ball in a vibrating tube. 
Theoretical analysis of the stability of this non-linear system is coupled together with the experimental study of an 
operating valve. Laser holographic interferometry is used for the identification and optimization of working regimes of the 
system. 
 
Keywords: Vibratory valve, injector, mathematical modeling, holographic interferometry, fluid dynamics 
 
Introduction 
 
Controlled dosing and spraying of liquid material is 
applied in different areas ranging from medicine to 
agriculture [1], [2]. Specific interest exists for elastic 
catheter pipe type dosing equipment. Complex dynamic 
processes taking place in such systems are analyzed in [3], 
[4]. Method examining the dynamic characteristics of a 
tube under the influence of the internal flow is presented in 
[3]. Galerkin’s method in conjunction with the method of 
multiple scales is employed for obtaining the stability of 
the tube vibration. According to the results, instability can 
occur under certain conditions of resonance. In [4] it is 
demonstrated that fluid-elastic effects which are 
responsible for fluid-elastic instabilities may be directly 
measured through the analysis of the vibrating motion of a 
system under flow. Piezoelectric actuators are used to 
increase the vibratory level when buffeting forces which 
excite tube vibration are low, and to improve the 
measurement of fluid-elastic forces.  
Application of piezoelectric actuators for the generation of 
standing waves in the outlet pipe can produce effects which 
can be exploited for the control of the dosing process. The 
motion of fluid-suspended particles and fibers in a standing 
wave field is analyzed in [5], [6]. The dynamics of micron-
sized aerosol particles and their agglomeration under the 
standing wave conditions is analyzed in [7]. Coupling of 
the dynamic properties of a vibrating tube with the dynamic 
behavior of a steel ball inside that tube can help producing 
a new type of smart doser for liquid material which can be 
effectively controlled by piezoelectric actuators. The 
unique feature of the tubular vibratory valve consists in the 
fact that the sealing surface of the seat is facing towards the 
intake duct and is located in the node of the second natural 
mode of transverse vibration of the elastic pipe. The 
vibratory valve for controlling liquid flow 1 (Fig. 1) 
operates in the following way. 
Several versions of design of systems for fluid flow 
control are presented in the following diagrams.  
A. The fluid flow controller is presented in Fig. 1.  
The controller is composed of tube (1) with input (2) and 
output (3) channels. A valve consisting from the closing 
element 5, spring 4 and seat 6 is mounted inside the tube. 
Vibration exciter 7 is fixed to the one end of the tube. 
Limiter 9 is placed inside the rigid tube 8 which is fastened 
to membrane 10. The surface of the tube from the seat 6 up 
to limiter 9 is cone shape. The feeding generator 11 is 
controlling the vibrator. 
The spring 4 presses the limiter 5 into the seat 6 and 
shuts the flow in the tube when the vibration exciter is off. 
Standing waves are excited in the tube when the vibrator is 
on. The excitation frequency coincides with the natural 
frequency of the system and the fluid flow controller 
operates in the resonance mode. That enables to reach 
higher amplitudes and the limiter 5 moves against the 
spring force into the peak of standing wave and thus opens 
the fluid flow channel.  
B. Fluid sprayer.  
 
 
Fig. 1. Schematic diagram of fluid flow controller Fig. 2. Schematic diagram of the vibration sprayer 
 
 
 331. ANALYSIS OF THE DYNAMICS OF THE VIBRATORY VALVE-INJECTOR V. NAGINEVICIUS1,A, M. RAGULSKIS2,B AND A. PALEVICIUS3,C 
 
 
© VIBROMECHANIKA.  JOURNAL OF  VIBROENGINEERING.  2008 JANUARY/MARCH,   VOLUME  10,  ISSUE  1,  ISSN 1392-8716 
 
38 
      The vibration sprayer (Fig. 2) consists of longitudinal 
(1) and transverse (2) vibration generators attached to the 
motionless foundation 3; tube 4 with input (5) and output 
(6) channels. Limiter 7 and seat 8 are mounted inside the 
tube. Fluid pressure pushes the limiter 7 into the seat 6 
when the vibrator is off. Standing waves are generated in 
the tube 4 when the vibrator 2 is on. The limiter moves into 
the peak of the standing wave and opens the fluid valve.  
    The fluid is sprayed by the oscillating end (6) of the tube 
(Fig. 3a). The eigenshape of the tube changes when the 
vibrator 1 is switched on. The standing wave length is 
shortened and the limiter is pressed into the seat – the 
spraying process is stopped. Such control mode enables 
reaching higher control frequencies compared to the 
control method used in Fig. 1.  
 
 
 
 
(a) 
 
(b) 
 
 
Fig. 3. Operation diagram of the vibration sprayer: a) valve is open; b) valve is closed 
 
C. Vibration fluid doser (Fig. 4).  
 
Fig. 4. Schematic diagram of the vibration fluid doser 
 
Vibration fluid doser consists from chamber 1 with input 
(2) and output (3) channels, electromechanical drive 5, 
controller 6, secondary valve 7 which is pressed by the 
spring to one of the seats. The chamber is a rigid tube, 
electromechanical drive serves as an exciter of transient 
vibrations. The placement of the first (4) and the secondary 
(7) valves defines the operating principle of the doser. The 
fluid flows through the input channel and closes the first 
valve when the vibrator 5 is off. The vibrator excites the 
third eigenshape vibrations (two standing waves) in the 
process of fluid dose formation. The valve 4 opens, but the 
secondary valve 7 closes (Fig. 5a).  
Next, the second eigenshape vibrations are generated (Fig. 
5b). The first valve 4 gets closed, the secondary valve is 
still kept closed. The formation of the dose is finished. The 
dose is ejected when the first eigenshape vibrations are 
generated (Fig. 5c). The first valve is kept closed, but the 
secondary valve is opened.  
 
 
 
 
 
 
 
 
(a) (b) (c) 
Fig. 5. Operation principle of the vibration fluid doser: a, b and c – the three stages of the doze formation 
 
Dynamics of a steel ball inside a vibrating tube 
 
Motion of a small ball inside a tube performing transverse 
vibrations may  be  approximated  by  the  non-dimensional  
 
 
differential equation of motion describing the dynamics of 
a mass particle on an oscillating profile with an obligatory  
condition of contact with the surface [3], [4]: 
 
 
( ) Fmg
x
m
xt
x
x
hxm
xtx
xm
xx
xm
x
=++⎟⎟⎠
⎞
⎜⎜⎝
⎛ ⎟⎠
⎞⎜⎝
⎛++++⎟⎟⎠
⎞
⎜⎜⎝
⎛ ⎟⎠
⎞⎜⎝
⎛+ ∂
∂ζ
∂
∂ζ
∂
ζ∂
∂
∂ζ
∂
∂ζ
∂∂
ζ∂
∂
∂ζ
∂
ζ∂
∂
∂ζ
2
222
2
2
22
121 ????? ,        (1) 
 
 331. ANALYSIS OF THE DYNAMICS OF THE VIBRATORY VALVE-INJECTOR V. NAGINEVICIUS1,A, M. RAGULSKIS2,B AND A. PALEVICIUS3,C 
 
 
© VIBROMECHANIKA.  JOURNAL OF  VIBROENGINEERING.  2008 JANUARY/MARCH,   VOLUME  10,  ISSUE  1,  ISSN 1392-8716 
 
39 
 
where ( )tx,ζζ =  − the shape of the tube; xxx ???  , ,  − 
projections of the displacement, velocity and acceleration  
of the ball onto the horizontal axis; t − time; m − mass of 
the ball; h − the coefficient of viscous friction between the 
ball and the surface of the profile; g − the acceleration of 
gravity; F − the pressure force of the liquid.  
Naturally, Eq.(1) is based on the assumption that the area 
of the cross section of the tube is small, the fluid flow is 
laminar, and the pressure of the liquid in the tube does not 
correlate with the amplitude of the elastic transverse 
vibrations. Also it is assumed that the mass of the locking 
ball is sufficient enough to perform the vibration induced 
motion in the liquid, but not big enough to alter the shape 
of the tube's vibration due to its relocation.  
Such mathematical model of the system enables effective 
separation of ball’s motion to relatively fast and slow 
motions. Co-ordinate x represents the horizontal 
displacement of the ball (slow motion), while the motion of 
the ball in the vertical direction is defined by the 
function ( )tx,ζζ = .  
When the tube performs steady oscillations of a standing 
wave type, the form of the tube looks like: 
 ( ) ( ) ( )wtkxbtx coscos, =ζ ,        (2) 
 
where b − the amplitude of oscillations, k and w − the 
number and frequency of the standing wave respectively. 
When the vibrations take place in the horizontal plane x0y 
(Fig. 1) and no external force is applied to the mass 
particle, the following conditions are satisfied: 
 
,0
,0
,0
==
=
=
xx
F
g
x
???
          (3) 
 
Then, the condition of existence of the steady state 
solutions is found from Eq. (1): 
 
02
2
=
xt ∂
∂ζ
∂
ζ∂
,         (4) 
 
or applying Eq. (2) to Eq. (4) : 
 
Ζ∈⋅= ss
k
x   ,
π
,         (5) 
 
and 
 
Ζ∈⋅+= ss
kk
x   ,
2
ππ
        (6) 
 
Analogously, if the vibrations take place in the vertical 
plane x0z (Fig. 1), and no external force is applied to the 
mass particle, the condition of existence of the steady state 
solutions looks like: 
02
2
=+ g
xxt ∂
∂ζ
∂
∂ζ
∂
ζ∂
,         (7) 
 
or  
 
Ζ∈= ss
k
x   ,
π
.         (8) 
 
If the external force is non-zero, there are no steady state 
solutions satisfying the condition 0== xx ??? . 
 
 
Stability of solutions when the vibrations take place in 
the horizontal plane 
 
 
The stability of the solution described by Eq. (5) is checked 
constructing the variational equation around s
k
x ⋅= π  
using the following assumption: 
 
dxs
k
x += π ,         (9) 
 
here dx is a small variation about the steady state solution. 
After the assumption of approximations 
 
( ) ( ) ( )
( ) ( ) ( )s
s
kdxssdxs
k
k
kdxkdxssdxs
k
k
1sincoscos
1cossinsin
−=⋅−≈⎟⎟⎠
⎞
⎜⎜⎝
⎛ ⎟⎠
⎞⎜⎝
⎛ +
−=⋅+≈⎟⎟⎠
⎞
⎜⎜⎝
⎛ ⎟⎠
⎞⎜⎝
⎛ +
πππ
πππ
    (10) 
 
and rejection of terms with ( )2dx and ( )2xd? , the following 
variational equation is produced from Eq. (1): 
 
( )( ) 0cos 2 =++ dxwtbkwxhdxmd ??? .     (11) 
 
Since all coefficients of this ordinary differential equation 
are positive in the neighborhood of t, the real parts of both 
roots of the characteristic equation will be negative. Thus 
the solution of the variation is stable and the position 
dx
k
x += π  is also stable. The value of the discriminant of 
the characteristic equation of Eq. (11) will define the type 
of the stable attractor. Averaging in time produces the 
following equation: 
 
( ) 0
2
2
=++ dx
m
bkw
xd
m
h
xd ??? .      (12) 
 
Therefore, the stable attractor will be of a knot type and the 
roots of the characteristic equation of Eq. (12) will be real, 
i.e. the discriminant is positive, and the attractor of the 
 331. ANALYSIS OF THE DYNAMICS OF THE VIBRATORY VALVE-INJECTOR V. NAGINEVICIUS1,A, M. RAGULSKIS2,B AND A. PALEVICIUS3,C 
 
 
© VIBROMECHANIKA.  JOURNAL OF  VIBROENGINEERING.  2008 JANUARY/MARCH,   VOLUME  10,  ISSUE  1,  ISSN 1392-8716 
 
40 
focus type when the roots are complex (discriminant is 
negative). That produces the following relations: 
 
a)  the attractor of the knot type at 
mwk
hb ≤ ; 
 
b)  the attractor of the focus type at 
mwk
hb > . 
 
The stability of the solution described by Eq. (6) is checked 
in analogous way: 
 
dxs
kk
x ++= ππ
2
,     (13) 
 
here dx is a small variation. Rejection of the non-linear 
terms and the following approximations  
 
( )
( ) kdxdxs
kk
k
dxs
kk
k
s
s
1
2
cos
1
2
sin
−−≈⎟⎟⎠
⎞
⎜⎜⎝
⎛ ⎟⎠
⎞⎜⎝
⎛ ++
−≈⎟⎟⎠
⎞
⎜⎜⎝
⎛ ⎟⎠
⎞⎜⎝
⎛ ++
ππ
ππ
     (14) 
 
in Eq. (1) lead to: 
 ( )( )
( )( )( )
( ) 0cos
cos12sin
cos1
222
22222
222
=
−++−
++
mdxwtkbw
xdwtbkhwtwbmk
xmdwtbk
?
??
       (15) 
 
The coefficient of this ordinary differential equation at dx 
is negative in entire neighborhood of the local time t. Thus, 
at least one of the roots of the characteristic equation 
Eq.(15) will be positive, and the solution in Eq.(13) will be 
unstable. Thus, the positions of the ball at the nodal points 
of the vibrating profile are unstable. Alternatively, the 
positions at the peaks of the profile, i.e. at the points which 
vibrate with the maximum amplitude, are stable. Of course, 
that is true when the ball cannot jump off the surface of the 
profile (is placed inside a vibrating tube).  
 
Experimental Analysis of a Tubular Vibratory Valve 
 
A number of experimental studies are needed in order to 
ensure high dynamic accuracy of operation of the vibrator 
valves for controlling the flow of liquid substances. In most 
cases the exciting frequencies of the tubular working tube 
are fairly high, and the amplitudes corresponding to them 
are measured in micrometers. Therefore the holographic 
method can be effectively applied for the visual 
representation of wave processes taking place in the tubular 
vibratory valve ([1], [2]). The most effective method for 
studying the standing wave processes is the method of 
holographic interferometry with time averaging ([5], [6]). 
It should be noted that the most clearly expressed bands in 
the holographic interferograms are those recorded at the 
positions of minimum amplitudes ([7]). It is important to 
obtain the distribution of the vibration amplitudes not only 
in the middle of dark interference bands, but also in 
arbitrary positions on the surface of the tube. That enables 
the determination of the location of the steel ball inside the 
tubular valve. The amplitudes of vibration of the structure 
are determined using the methodology presented in [8], [9].  
Holographic interferograms of the transverse vibrations of 
a tubular vibratory valve are presented in Fig. 6a, Fig. 6b 
and Fig. 7a. Fig. 6a and Fig. 6b make it possible to 
conclude that transverse vibration of the tube is sufficiently 
uniform (Fig. 6c). Therefore, the seat of the vibratory can 
be located in a nodal point, regardless of how it is 
displaced lengthwise in the upper or the lower nodal point 
of the tube. It should be noted that the frequency of 
excitation must be selected with care since the best 
performance of the vibratory valve is taking place at the 
resonant frequencies. If the excitation of the transverse 
vibrations is far away from the resonance frequencies of the 
tube, the operation of the tubular valve turns to be hardly 
controlled. 
The obtained results enable to optimize the design of the 
vibratory valves for controlling and dosing the liquid flow. 
The following parameters of the system are analyzed and 
optimized: a) selection of the material of the working tube; 
b) selection of the area of the transverse cross section of 
the tubular tube; c) location of the transverse vibration 
nodes in the tube; d) determination of the transverse 
vibration amplitudes along the tube. Maximum uniformity 
of the transverse vibrations in the vibratory valve is 
achieved due to this optimization which leads to more 
stable operation of the whole system. 
 
 
 
  a 
 
 
  b 
 
 
 
 
a. The hologram of the transverse vibrations of the tube at w = 
1,3 kHz. Illumination angle of the laser beam π / 4; 6b. The 
hologram of the tube at the illumination angle  π / 2 
c. The interpretation scheme of the transverse vibrations 
Fig. 6. Holographic interferogram of transverse vibrations of the tube 
 331. ANALYSIS OF THE DYNAMICS OF THE VIBRATORY VALVE-INJECTOR V. NAGINEVICIUS1,A, M. RAGULSKIS2,B AND A. PALEVICIUS3,C 
 
 
© VIBROMECHANIKA.  JOURNAL OF  VIBROENGINEERING.  2008 JANUARY/MARCH,   VOLUME  10,  ISSUE  1,  ISSN 1392-8716 
 
41 
Concluding Remarks 
 
New model of a tubular vibratory valve is designed using 
the stabilization effect of a steel ball in the vibrating tube. 
The methodology of identification of vibration peaks 
enabled experimental optimization of the working regimes 
of the system. Such type of analysis could be successfully 
applied in the design stage of different precise vibratory 
systems. 
 
References 
 
[1] Cho K. J. and Recinella,  D. K. Pattern of dispersion from a 
pulse-spray catheter for delivery of thrombolytic agents: 
Design, theory, and results, J. Academic Radiology, 4(3), 
210-216, (1997) 
[2] Miller P. and Ellis M. Effects of formulation on spray 
nozzle performance for applications from ground-based 
sprayers, J. Crop Protection, 19(8-10), 609-615, (2000) 
[3] Huang Y. and Hsu C. Dynamic behavior of tubes subjected 
to internal and external cross flows, J. Shock and Vibration, 
4(2), 77-91, (1997) 
[4] Caillaud S. and de Langre E. The measurement of 
fluidelastic effects at low reduced velocities using 
piezoelectric actuators, J. of Pressure Vessel Technology – 
Transactions of the ASME, 121(2), 232-238, (1999) 
[5] Brodeur P. Motion of fluid-suspended fibres in a standing 
wave field, J. Ultrasonics, 29, 302-307, (1991) 
[6] Coakley W. T. and Zamani F. Cell Manipulation in 
ultrasonic standing wave fields. J. of Chemical Technology 
and Biotechnology, 44(1), 43-62, (1989) 
[7] Riera-Franco de Sarabia E. and Gallego-Juarez J. A. 
Ultrasonic agglomeration of micron aerosols under standing 
wave conditions, J. of Sound and Vibration, Palevicius, A., 
and Ragulskis, M., Holographic interference method for 
investigation of wave transport systems, SPIE Proc. of the 2-
nd Intl. Conf. on Vibration Measurements by Laser 
Techniques, Ancona, Italy, 225-231, (1996)413-427, (1986) 
[8] Vest C. Holographic Interferometry, Moscow, Mir, (1982) 
Palevicius, A., and Ragulskis, M., Holographic interference 
method for investigation of wave transport systems, SPIE 
Proc. of the 2-nd Intl. Conf. on  Vibration Measurements by 
Laser Techniques, Ancona,  SItaly, 225-231, (1996) ly, 225-
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Analysis of the dynamics of the vibratory valve-injector

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