Analysis of operations of the antiresonance vibration mill of a circular trajectory of chamber vibrations

Paweł Piekaj; Grzegorz Cieplok

doi:10.1515/eng-2025-0121

Article Open Access

Analysis of operations of the antiresonance vibration mill of a circular trajectory of chamber vibrations

Published/Copyright: December 24, 2025

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From the journal Open Engineering Volume 15 Issue 1

Abstract

This work is a continuation of prior research on reducing the forces transferred to the foundation by vibrating mills through a new solution involving antiresonance vibration isolation, based on the operating principle of Frahm’s dynamic vibration absorber. Here, antiresonance phenomena were used to reduce vibrations in two mutually perpendicular directions of motion. The proposed solution enabled lowering the normal and tangential dynamic forces exerted on the foundation by the mill while maintaining the circular trajectory of the chamber movement. This study examined the influence of mill chamber load on the effectiveness of the antiresonance phenomenon for several degrees of chamber filling. The analyses presented in the work were based mainly on experimental research using noncontact (optical) measurement equipment. Studies have shown the effectiveness of the proposed solution, whereby as the degree of filling of the chamber with grinding media and material increased, the vibration isolation effectiveness also increased.

Keywords: antiresonant; vibration mill; grinding; comminution; dynamic vibration absorber; chamber load

1 Introduction

Vibrating mills are widely used in the field of material grinding and are applied in various technological processes, as evidenced by dozens of new publications appearing annually in the literature. They are used, for example, in a wide range of mechanochemical synthesis [1], [2], [3], [4], [5], powder surface activation [6], sample preparation for medical and pharmaceutical research [7], among other processes [8], [9], [10].

On an industrial scale, they are used in most powder manufacturing technologies with widely varying physical properties [11], [12], [13], including recycling [14].

Vibrating mills, in addition to offering many advantages, such as a greater degree of fineness as is possible with much smaller mill dimensions compared to conventional ball mills [12], 13], and these mills also require significantly less input energy [15], reducing their negative impact on the environment. Another one of these negative factors is the dynamic impact of the mill on the foundation [16], [17], [18].

Contemporary research to improve knowledge of the construction and operation of vibrating mills has focusedmainly on the development of mathematical models of mill chamber loads based on discrete element methods (DEMs) [19], [20], [21], [22] and less often on proposals for new design solutions [23], [24], [25]. The subject of reducing these (negative) impacts by improving the vibration isolation of vibrating mills has practically not been addressed in the scientific literature, although the time-varying forces exerted on foundations by vibrating machines are considered a fundamental problem for designers and users of these machines [16], [17], [18].

Therefore, this study developed a new solution for a vibrating mill with a circular trajectory of chamber vibrations and reduced environmental impact using the phenomenon of antiresonance, following techniques of the Frahm dynamic vibration absorber [26].

The phenomenon of antiresonance has not yet been used in vibration mills, while the closest solutions in terms of design, which are based on the principle of operation of a Frahm dynamic vibration absorber, can be found in vibration conveyors, which have experienced a renaissance in recent years [27], [28], [29], [30], [31], [32]. The application of this phenomenon is generally limited to machines with segmented motion trajectories, even in solutions with a single inertial vibrator, where a circular motion trajectory of the machine body should be naturally generated [33], 34].

In the case of vibrating mills, due to the low efficiency of the grinding process, the sectional trajectory of chamber motion is not used in practical solutions, even in the most favourable case, when the trajectory is in the vertical direction [12].

It is therefore valuable to develop a solution to achieve antiresonance on two mutually perpendicular directions of vibration, which can provide a significantly better reduction in the dynamic impacts exerted on the foundation compared to solutions based on conventional vibration isolation.

In the presented solution, which represents a dynamic Frahm vibration absorber, the drive frame acts as the protected mass. In contrast, the dynamic damper represents the mass of the mill chamber. The drive frame and the chamber’s spring system have been designed with radial symmetry to provide uniform stiffness of the chamber suspension in all directions relative to the frame. Additionally, the design ensures that the chamber’s centre of mass coincides with the centre of stiffness of its suspension. This configuration allows the chamber’s motion to be analysed independently in two perpendicular directions and enables the use of the dynamic damper principle in each direction separately. The system is tuned to the desired operating frequency of the mill by adjusting the stiffness of the chamber’s elastic suspension. The elastic suspension for the drive frame, meanwhile, is selected according to standard guidelines for spring design in vibrating machinery, aiming to ensure system stability. However, the elastic suspension of the drive frame plays a less significant role in the mill’s operation compared to that of the chamber, as the vibration amplitudes of the drive frame remain minimal during steady-state operation.

In addition, appropriate synchronization of vibrators in this solution can result in a circular trajectory of the chamber motion.

The schematic diagram of the model is shown in Figure 1 and the dynamic equations of motion of the drive frame and chamber in the form of a system of second-order nonlinear differential equations using equations (1a)–(1l) while the parameters used in the simulations are summarized in Table 1.

(1a) ( 2 m w + m R ) x ¨ C − e m w sin ( φ 2 ) φ ¨ 2 − e m w cos ( φ 2 ) φ ˙ 2 2 − e m w sin ( φ 1 ) φ ¨ 1 − e m w cos ( φ 1 ) φ ˙ 1 2 = Q x C

(1b) ( 2 m w + m R ) y ¨ C + e m w cos ( φ 2 ) φ ¨ 2 − e m w sin ( φ 2 ) φ ˙ 2 2 + e m w cos ( φ 1 ) φ ¨ 1 − e m w sin ( φ 1 ) φ ˙ 1 2 = Q y C

(1c) J c R α ¨ − a w e m w cos ( φ 2 ) φ 2 ¨ + a w e m w sin ( φ 2 ) φ 2 ˙ 2 + a w e m w cos ( φ 1 ) φ 1 ¨ − a w e m w sin ( φ 1 ) φ 1 ˙ 2 + 2 α ¨ a w 2 m w = Q α

(1d) ( 4 m p + m k ) x ¨ C k + − sin ( β 4 ) β ¨ 4 − cos ( β 4 ) β ˙ 4 2 + sin ( β 3 ) β ¨ 3 + cos ( β 3 ) β ˙ 3 2 + sin ( β 2 ) β ¨ 2 + cos ( β 2 ) β ˙ 2 2 − sin ( β 1 ) β ¨ 1 − cos ( β 1 ) β ˙ 1 2 l c m p = Q x C k + ∑ i F n ( i ) x + T ( i ) x

(1e) ( 4 m p + m k ) y ¨ C k + − cos ( β 4 ) β ¨ 4 + sin ( β 4 ) β ˙ 4 2 − cos ( β 3 ) β ¨ 3 + sin ( β 3 ) β ˙ 3 2 + cos ( β 2 ) β ¨ 2 − sin ( β 2 ) β ˙ 2 2 + cos ( β 1 ) β ¨ 1 − sin ( β 1 ) β ˙ 1 2 l c m p = Q y C k + ∑ i F n ( i ) y + T ( i ) y

(1f) J C k β ¨ k + − sin ( β 4 ) β ¨ 4 − cos ( β 4 ) β ˙ 4 2 + sin ( β 3 ) β ¨ 3 + cos ( β 3 ) β ˙ 3 2 − sin ( β 2 ) β ¨ 2 − cos ( β 2 ) β ˙ 2 2 + sin ( β 1 ) β ¨ 1 + cos ( β 1 ) β ˙ 1 2 h k − a k cos ( β 4 ) β ¨ 4 + a k sin ( β 4 ) β ˙ 4 2 + a k cos ( β 3 ) β ¨ 3 − a k sin ( β 3 ) β ˙ 3 2 − a k cos ( β 2 ) β ¨ 2 + a k sin ( β 2 ) β ˙ 2 2 + a k cos ( β 1 ) β ¨ 1 − a k sin ( β 1 ) β ˙ 1 2 l c + 4 β ¨ k h k 2 + 4 a k 2 β ¨ k m p = Q β k + ∑ i M 0 ( T ( i ) ) + M 0 ( i ) tocz

(1g) J c p β ¨ 1 + cos ( β 1 ) l c m p y ¨ C k − sin ( β 1 ) l c m p x ¨ C k + ( β ¨ 1 l c 2 + ( sin ( β 1 ) β ¨ k h k + a k cos ( β 1 ) β ¨ k ) l c ) m p = Q β 1

(1h) J c p β ¨ 2 + cos ( β 2 ) l c m p y ¨ C k + sin ( β 2 ) l c m p x ¨ C k + ( β ¨ 2 l c 2 + ( − sin ( β 2 ) β ¨ k h k − a k cos ( β 2 ) β ¨ k ) l c ) m p = Q β 2

(1i) J c p β ¨ 3 − cos ( β 3 ) l c m p y ¨ C k + sin ( β 3 ) l c m p x ¨ C k + ( β ¨ 3 l c 2 + ( sin ( β 3 ) β ¨ k h k + a k cos ( β 3 ) β ¨ k ) l c ) m p = Q β 3

(1j) J c p β ¨ 4 − cos ( β 4 ) l c m p y ¨ C k − sin ( β 4 ) l c m p x ¨ C k + β ¨ 4 l c 2 + ( − sin ( β 4 ) β ¨ k h k − a k cos ( β 4 ) β ¨ k ) l c m p = Q β 4

(1k) e m w cos ( φ 1 ) y ¨ C − e m w sin ( φ 1 ) x ¨ C + J c w + e 2 m w φ ¨ 1 + α ¨ a w e m w cos ( φ 1 ) = M el 1

(1l) e m w cos ( φ 2 ) y ¨ C − e m w sin ( φ 2 ) x ¨ C + J c w + e 2 m w φ ¨ 2 − α ¨ a w e m w cos ( φ 2 ) = M el 2

where: ∑ i M 0 ( T ( i ) ) + M 0 ( i ) tocz , ∑ i F n ( i ) y + T ( i ) y denotes the sum of moments and forces from the frictional and rolling resistance forces of the chamber-grinding media interaction with reference to the center of mass of the chamber.

Figure 1:

Model of the antiresonance vibration mill. Coordinates: x _C, y _C, α – describe the movement of the drive frame, coordinates: x _Ck, y _Ck, β _k – describe the movement of the chamber, coordinates: β ₁, β ₂, β ₃, β ₄ – describe the angular displacement of the chamber’s suspension rods, coordinates: φ ₁, φ ₂ – describe the displacement of unbalanced vibrator masses.

Table 1:

Parameters of the mill model.

Chamber parameters			Drive frame parameters
J _ck	0.16	[kg m²]	J _cR	6.62	[kg m²]
J _cp	0.0063	[kg m²]	J _cw	0.0069	[kg m²]
m _k	8.848	[kg]	m _R	45.283	[kg]
m _p	0.463	[kg]	m _w	4.08	[kg]
a _k	0.115	[m]	a _w	0.335	[m]
h _k	0.115	[m]	h _w	0	[m]
l _c	0.285	[m]	a _R2	0.2953	[m]
l _p	0.434	[m]	h _R2	0.2953	[m]
a _R	0.45	[m]	e	0.0154, 0.022	[m]
h _R	0	[m]	k _yR	18,673	[N/m]
k _ξk	68,115	[N/m]	k _xR	5,240	[N/m]
k _ηk	17,029	[N/m]	b _yR	12.36	[N s/m]
b _ξk	19.28	[N s/m]	b _xR	12.52	[N s/m]
b _ηk	9.64	[N s/m]	m _e	18	[kg]

Due to their complex form and publication constraints, only their generalized form has been presented. Generalized forces and drive torques ( Q x C , Q y C , Q _α, Q x C k , Q y C k , Q _β, Q β 1 , Q β 2 , Q β 3 , Q β 4 , M _el2, M _el2) and their forms are available from the authors upon request from the reader.

It should be noted that the stiffness and damping parameters provided in Table 1 were determined experimentally using a dedicated test rig, based on the free vibration decay method for a single-degree-of-freedom system with a mass-damping-stiffness elastic suspension.

The concept of this solution, along with a laboratory stand for testing the trajectory of the chamber’s movement and a simplified mathematical model, was described in more detail in an earlier paper by these authors [35], 36]. However, key aspects of the effect of the chamber load on the vibration isolation efficiency and chamber motion trajectory are still undiscovered, and the present article focuses on these aspects. The results of laboratory tests show that the chamber load and the degree of filling have a significant effect on the mill’s operation and its effectiveness at vibration isolation.

2 Materials and methods

The experimental research was performed with the use of a laboratory stand equipped with the prototype of the antiresonant vibration mill with the circular trajectory of chamber motion, whose photography is shown in Figure 2.

Figure 2:

Laboratory mill: 1 – chamber, 2 – chamber frame, 3 – electrovibrators, 4 – synchronization belt transmission, 5 – chamber spring suspension, 6 – drive frame, 7 – drive frame spring suspension, and 8 – foundation frame.

The mill comprises three main subassemblies, i.e., the chamber, drive frame, and foundation. The chamber subassembly is coupled with the use of a spring suspension with a drive frame subassembly. The drive frame subassembly is connected with the foundation subassembly by spring coupling.

The chamber subassembly comprises two components, a chamber frame 2 and a chamber 1 were installed inside the frame, which allows convenient assembly and disassembly of the chamber. Furthermore, the chamber frame can be equipped with additional correction masses. The chamber subassembly is suspended by four jointed rods with springs 5. Rods are installed in the X layout, which ensures that the spring stifnesses and damping coefficients are equal in the vertical and horizontal directions.

The subassembly of the drive frame comprises frame 6, where two electrovibrators 3 are installed. Electrovibrators are synchronized by means of a belt transmission 4. The transmission mechanism ensures synchronization in the case of direction and common phase motion of unbalanced masses. The drive frame subassembly is suspended by spring suspension 7.

The foundation subassembly comprises a fundamental frame 8 equipped with four legs that regulate the height, which provides the possibility for proper levelling of construction and, in case of emergency, anchoring to the ground.

Laboratory research was supported using the GOM ARAMIS vision system [37], which records the position and displacement of selected points. Considering the plane character of motion of the mill, measurements were restricted only to the vertical plane of motion.

The abovementioned system was equipped with two high-speed cameras that could take photos with a maximum resolution of up to 1,936 × 1,216 px. The recording mode was set up to operate at 450 fps in the experiment. Cameras were mounted on the stand beam at a suitable angle, and measurement markers were highlighted by a light connected to the measurement system according to the system requirements [37]. An electrical tachometer was used to measure the electrovibrator rotational velocity within a range of 10–9,999 (0.1 %) RPM. The laboratory stand diagram is presented in Figure 3a, and its photograph is presented in Figure 3b.

Figure 3:

Laboratory stand: 1 – mill, 2 – frequency converter, 3 – tachometer, 4 – tachometer indicator, 5 – high frame-rate camera, 6 – stand with installation beam, 7 – light lamp, and 8 – computational station with screen. (a) Diagram of the laboratory stand. (b) Photography of the laboratory stand.

Measurements were performed for two characteristic pairs of points, whose position on the construction is presented in Figure 4. Markers M1 and M2 were fixed to the driving frame to determine the influence on the foundation. For chamber vibration, markers M3 and M4 were used.

Figure 4:

Layout of markers: M1, M2 – markers on the drive frame; and M3, M4 – markers on the chamber.

The measurements presented in this work cover the steady-state operation of the machine. Measurements were made every 0.5 Hz. After each frequency change (using a frequency converter), the measurement was performed after a delay of 120 s to stabilize the mill operation.

For laboratory measurements, the following values were used: a cylindrical steel chamber with a diameter of 210 mm and a volume of V _ch = 2.15 kg/dm³, steel grinding media in the shape of balls with a diameter of ϕ _ball = 15 ± 0.1 mm, and a bulk density in a loose state ρ _ball = 4.3 kg/dm³ with an alignment factor of i _ball = 0.4. All tests were performed for four chamber filling levels, i.e., b = 0.6, b = 0.7, b = 0.8, and b = 0.9. These filling levels are widely used for industrial purposes and are often discussed in the literature [12], 13], 38].

As the grinding material, quartz sand was selected because it is well known for its stable physical properties that enable repeatable test results to be obtained. The sand of choice, which was produced by KiZPPS Osiecznica sp. z o.o., is characterized by grain size stability, purity, chemical resistance, and nonhygroscopicity. This material is commonly used in research on grinding processes in vibration mills and other types of mills with various designs [24], 25], [39], [40], [41]. The material was obtained by taking a representative sample using a Jones sample divider.

The mass of the grinding media were determined from equation (2). The mass of the feed material were determined from equation (3). The total mass of the chamber load was determined as the sum of the masses of the grinding media and feed material as in equation (4).

(2) m ball = b V c h ρ ball

(3) m mat = i ball b V c h ρ mat

(4) m load = m ball + m mat

To test the quality of the grinding process, quartz sand and steel grinding media were used. Due to the cyclical nature of the mill’s operation, grinding times of 5, 10, 20, and 40 min were used, after which samples were taken for granulometric tests.

A major goal of the present research, in addition to determining the effectiveness of antiresonance vibration isolation, was to demonstrate the occurrence of the grinding process in the proposed solution.

The evaluation of the grain sample parameters of the feed and grinding products for the adopted grinding times was performed using the diffractometric method on a Malvern Mastersizer 3,000 laser analyser [42]. The measurements were performed via dry measurement method in an air environment, using the dispersing unit Aero S’. The utilized surface dispersion module enables the characterization of granular materials over a wide measurement range from 0.01 to 3,500 μm. In the analysis of results, the device uses two scattering models: the Fraunhofer approximation and the Mie theory. The measurements were carried out five times for each sample, and the results presented in the article are the average of these measurements.

Tests with a filled chamber were conducted at the maximum possible excitation force of the inertial vibrators.

3 Results

Measurements covered the operation of the mill with an empty chamber, with the chamber filled with grinding media only and with the chamber filled with grinding media and feed material. However, only the last case corresponds to the reality of vibrating mill operation.

A comparison of the load masses used for test, for certain chamber filling levels based on the equations (2)–(4) is listed in Table 2.

Table 2:

List of chamber load masses for individual filling levels.

Chamber filling level b [−]	Mass of grinding media m _ball [kg]	Mass of feed material m _mat [kg]	Total mass of the chamber m _load [kg]
0.6	5.51	0.76	6.27
0.7	6.43	0.89	7.32
0.8	7.35	1.01	8.37
0.9	8.27	1.14	9.41

The operation of a mill with an empty chamber was analysed in two cases. The first was based on the utilization of an exciting force equal to 7 % of the maximum possible force. This small value of force allows for the registration of vibrations in the frequency range from 2 to 17 Hz, which allows a safe transition through the drive frame and chamber resonance.

The measurement results for these cases are presented in Figure 5a and b for the vertical and horizontal directions, respectively. The second case involved examining the operation under the influence of the maximum (100 %) excitation force, which allowed for the determination of nominal mill parameters. Measurements were conducted in a reduced frequency range, i.e., 6–13.5 Hz; these frequencies are located between the resonant frequencies of the drive frame and the chamber.

Figure 5:

Amplitude-frequency diagram obtained based on the model and experiment for an empty chamber at 7 % of the maximum possible excitation force. (a) Vertical direction. (b) Horizontal direction.

This approach allows for the verification of computational simulation results of the theoretical model (conducted independently of experimental studies [36]) based on the amplitude-frequency characteristics of the mill (Figures 6 and 7).

Figure 6:

Vibration of the chamber and drive frame for an empty chamber at 7 % of the maximum possible excitation force.

Figure 7:

Vibration of the chamber and drive frame for an empty chamber at the 100 % excitation force in the working point range.

The movement trajectories of the chamber and the drive frame at the operating point at 7 % of the maximum possible excitation force and for 100 % excitation force are shown in Figure 8.

Figure 8:

Movement trajectories of the chamber and the drive frame at the operating point. (a) 7 % of the maximum possible excitation force. (b) 100 % excitation force.

A chamber filled with grinding media only was researched for four chamber filling levels. Example measurement results for a chamber fill level of b = 0.6 between 5 and 14 Hz in the vertical direction are shown in Figure 9a, and examples of those for a horizontal direction are shown in Figure 9b.

Figure 9:

Vibration of the chamber and drive frame in the vertical (a) and horizontal (b) direction for the chamber filled with grinding media only with a filling level b = 0, 6 at 100 % of excitation force. (a) Vertical direction. (b) Horizontal direction.

A comparison of the movement trajectories of the chamber and the drive frame with the chamber filled with grinding media only for selected filling levels at the mill operating point is shown in Figure 10.

Figure 10:

Comparison of the movement trajectories of the chamber and the drive frame with the chamber filled with grinding media only for four selected filling levels at the mill operating point: (a) b = 0.6; (b) b = 0.7; (c) b = 0.8; (d) b = 0.9 at 100 % of excitation force.

An analogous set of measurement results is shown for a chamber filled with a suitable chamber load comprising grinding media and material. Figure 11a and b show the results for the case when the filling level of the chamber was 0.9 for working frequencies ranging from 5 to 12 Hz. A list of the results illustrating the trajectories of the chamber and drive frame movement at the operating points for the selected filling levels of the chamber is presented in Figure 12.

Figure 11:

Vibration of the chamber and drive frame in the vertical and horizontal direction for the chamber filled with grinding media and feed material with a filling level b = 0, 9 at 100 % of excitation force. (a) Vertical direction. (b) Horizontal direction.

Figure 12:

Comparison of the movement trajectories of the chamber and the drive frame with the chamber filled with grinding media and material for four selected filling levels at the mill operating point: (a) b = 0.6; (b) b = 0.7; (c) b = 0.8; (d) b = 0.9 at 100 % of excitation force.

The parameters of feed material determined for the tests by diffractometric laser analyser are listed in Table 3.

Table 3:

List of parameters of the tested material.

Parameter	Value	Unit
Specific density	2.66 ± 0.02	[kg/dm³]
Bulk density in a loose state ρ _max	1.50 ± 0.02	[kg/dm³]
Kinietic specific surface	31.6 ± 0.5	[m²/kg]
Grain size d ₉₇	490	[μm]
Grain size d ₉₀	459	[μm]
Grain size d ₅₀	275	[μm]

The results of granulometric tests aimed at assessing grinding quality by comparing the particle size distributions for individual grinding times at a given chamber filling level are presented in Figure 13.

Figure 13:

Particle (grain) size distribution for different grinding times for a chamber filling level of: (a) b = 0.6; (b) b = 0.7; (c) b = 0.8; (d) b = 0.9 at 100 % of excitation force.

The graphs show average values from five measurements, with a measurement error not exceeding 1 %.

During the tests, an increase in the volume of the ground material was observed with increasing grinding time. This phenomenon is illustrated in Figure 14.

Figure 14:

Influence of grinding time on the change in the volume of grinding material for a chamber filling level of b = 0.7.

4 Discussion

This paper presents the results of experimental tests of an antiresonance vibration mill in which the principle of a dynamic vibration absorber was applied in two mutually perpendicular directions of machine movement.

The results of the measurements confirmed the efficiency of operation of the proposed solution, in which the chamber with a drive frame creates the form of a dynamic vibration absorber, where vibrations in two mutually perpendicular directions exerted on the foundation are many times smaller than those in conventional solutions with comparable kinematic parameters of the chamber. Thus this solution allowed for a high degree of reduction in the forces transmitted to the foundation by the machine’s suspension elements.

Experimental examinations revealed that according to the presented solution, an antiresonant effect can be obtained at the same time in both mutually perpendicular directions, allowing us to obtain a circular trajectory of the chamber.

This finding is confirmed by the results of the experiment with an empty and filled chamber, whereas the antiresonance effect was the strongest in the case of an empty chamber (Figure 5a and b), where the resulting operating point of the mill at a frequency of 13 Hz in the obtained antiresonant valley, was characterized by the smallest coordinate values of the drive frame movement, as shown in Figure 8.

As can be observed, the ratio of the y _Ck chamber coordinate to the coordinate for drive frame y _C for the vertical direction and the ratio of the coordinates of the x _Ck chamber to the x _C drive frame in the horizontal direction were equal to 32–39 for the maximum excitation force by electrovibrators and 19–32 for 7 % of the maximum possible excitation force.

These results indicates the high effectiveness of vibroinsulation of the mill from the foundation while maintaining the high amplitude of the vibration of the chamber necessary for the grinding process to occur correctly.

In the case of a chamber filled with grinding media only (Figure 9a and b), the vibroinsulation effect was less significant, but the antiresonance valley and the mill’s operating point were visible.

Studies have shown that as the chamber is filled with grinding media, the antiresonance effect decreases, which can be observed in the list shown in Figure 10. The ratio of the chamber coordinate y _Ck to the drive frame coordinate y _C in the vertical direction and the ratio of the chamber coordinate x _Ck to the drive frame coordinate x _C in the horizontal direction for selected filling levels were as follows: for the filling level 0.6 these ratios were 7.5–8, 5 (Figure 10a); for 0.7, 7–7.5 (Figure 10b); for 0.8, 6–6.5 (Figure 10c); and for 0.9, 4.5–5 (Figure 10d).

Research with a chamber filled with grinding media and material suggests very interesting observations. As shown in Figure 11a and b, adding material to the chamber changes the influence of the filling level on the antiresonance effect and the related vibration isolation of the mill from the foundation to the opposite compared to the case of a chamber filled with grinding media only. As confirmed by the test results, as the degree of filling of the chamber with grinding media and material increased, the vibration isolation effectiveness also increased (Figure 12), which confirms the results obtained for a mill with a segmented movement trajectory [35]. The ratio of the chamber coordinate y _Ck to the drive frame coordinate y _C in the vertical direction and the ratio of the chamber coordinate x _Ck to the drive frame coordinate x _C in the horizontal direction for the selected filling levels were as follows: for the filling level 0.6, these ratios were 6–6.5 (Figure 12a); for 0.7, 6.5–8 (Figure 12b); for 0.8, 10.5–11.5 (Figure 12c); and for 0.9, 16.5–17 (Figure 12d).

The differences in the vertical y _Ck and horizontal x _Ck coordinate ratios for the mill chamber indicate that the mill chamber trajectory is not a circular trajectory in the literal sense of that meaning, which is consistent with observations from previous work for empty chamber [36]. However, the nomenclature is acceptable for vibratory mills, which are used interchangeably with the quasicircular trajectory.

Moreover, the authors made a number of observations:

The results of experimental tests for chambers filled with grinding media and material have shown that with increasing chamber filling level, the frequency of the operating point (in the antiresonance valley) decreases; however, the ratio of the chamber amplitude to the drive frame increases.
The results of the grinding quality examination showed that the proposed solution can be used to milling granular material and can be used as a vibration mill.
The presented structure of the laboratory antiresonance mill did not require anchoring to the ground.
The presence of material inside a chamber has a positive effect on reducing the self-vibrations of their system excited by grinding media impacting the chamber walls.
The volume of quartz sand increases as the grinding time increases.
An increase in the volume of the material during milling increased the efficiency of the antiresonance vibroinsulation.
The filling level b = 0.9 for the material used should not be used in practice due to the phenomenon of increasing the volume of material during grinding. After some time of milling, the entire volume of the chamber was filled, and the intensity of the grinding media movement inside the chamber was substantially reduced. Which is consistent with observations from other work [12], 39].

Studies have shown the occurrence of the grinding process for all chamber filling levels. An increase in the volume of material during grinding was also observed with a change in the size of the particles (grains) during grinding, which was shown in the example of milling for a chamber filling level of 0.7, as illustrated in Figure 14.

During the research, changes in the value of the chamber and frame amplitude were also observed during the mill’s operation, especially for higher filling levels in the first minutes of operation. With time, the chamber amplitude increased, whereas the drive frame amplitude decreased. These changes could be attributed to the abovementioned effect of the increase in the volume of raw material during grinding, which may result in the phenomena of “binding” the material to the chamber and the approach of the weight of the chamber and its load to the form of one rigid body.

The results of the experiments indicate that at all tested filling levels of the chamber, the grinding time affects the size of the particles obtained, and as the grinding time increases, the size of the grains decreases. Figure 13a–d show the dependencies of the particle size distribution on the grinding time for individual filling levels of the chamber.

Grinding quality tests were conducted to demonstrate the suitability of the presented solution as a vibrating mill. However, the results obtained from the laser analyser did not differ significantly from the results obtained by other researchers using machines with similar chamber movement and grinding parameters when grinding quartz sand [25]. This result was expected because in the proposed solution, only the method of chamber movement changed. This was done while maintaining its kinematics known from classic vibrating mill solutions, such that the results for the same grinding and chamber parameters were expected to be the same. Moreover, the presented solutions make it possible to simultaneously reduce the negative impact of device operation on the surrounding environment.

The vibrators, which in classical vibrating mill solutions are mounted on the chamber, in the presented solution have been relocated and mounted on the driving frame. This enables reduction of the chamber subassembly mass, which can increase the amplitude of vibration of the chamber subassembly while maintaining the same unbalanced masses, or to reduce these unbalanced masses to a size that allows the chamber subassembly to maintain the same amplitude of operation as in classical vibrating mills, with the result that it is possible to reduce the drive units of these vibrators.

In additions, decrease in the vibrator unbalanced mass ensures a decrease in the vibration amplitude of the mill when passing through the resonance zone during the start-up and run-down of the machine, simultaneously reducing the maximum impact exerted on the foundation. For the prototype of the vibrating mill presented in this paper, there was no need to even anchor the machine to the foundation. This fact may be important in reducing the weight of foundations for large mills, thus contributing to a measurable reduction in the costs of installation mills.

The reduction of dynamic forces transmitted to the ground depends largely on the stiffness of the elastic suspension between the mill and the ground. Therefore, differences in mass between the classical mill and the proposed solution must also be considered. In simple terms, the total mass of a conventional vibrating mill mainly comprises the chamber, frame, and vibrators. In contrast, the proposed solution includes additional mass due to the drive frame and extra elastic suspension required to fulfil its function. In both cases, the spring suspension is selected according to standard design principles to ensure structural stability. As a result, the stiffness of the spring suspension increases proportionally with the system mass. Based on the parameters in Table 1, it can be observed that although the system mass nearly doubles, the stiffness of the elastic suspension also doubles accordingly. However, the proposed solution achieves a significantly greater reduction in vibration amplitude compared to the classical design. Consequently, the resulting decrease in dynamic forces transmitted to the ground is highly effective. (It should be noted that in Table 1, the drive frame mass m _R includes both the mass of the drive frame itself and the vibrator motors, listed in Table 1 as m _e).

As can be seen, no direct measurements of the forces transmitted to the foundation were carried out on the experimental laboratory stand. However, these forces can be accurately estimated using an indirect method, by multiplying the equivalent stiffness coefficient of the drive frame suspension by the vibration amplitude of the frame in the given direction of motion. Any discrepancy between the actual force and the estimated value would mainly result from the omission of dissipative forces, represented by the damping coefficient. However, these forces are significantly smaller than the elastic forces, and in the case considered, the resulting error would be approximately 1 %.

The proposed solution may be an alternative to the currently used mills because, in industrial practice, it does not require interference with the technology of the milling material. The same parameters of chamber movement are maintained, and only the method of propulsion and elastic suspension of the mill are changed. These features can be more attractive to potential recipients who are looking for solutions to limit the transfer of dynamic forces to the ground through mills without changing the technological parameters used.

5 Conclusions

The experimental results presented in this study confirm the effectiveness of the proposed antiresonance vibration mill design, which incorporates a dynamic vibration absorber principle in two mutually perpendicular directions. The findings demonstrate a significant reduction in the forces transmitted to the foundation while maintaining the necessary vibration amplitude for efficient grinding.

The antiresonance effect was most pronounced in an empty chamber, where the vibration isolation from the foundation was at its highest. However, as the chamber was filled with grinding media, the antiresonance effect diminished. Interestingly, when both grinding media and material were present, the antiresonance effect and vibration isolation efficiency increased with the degree of filling, contrary to the case of grinding media alone. Notably, only this condition where both grinding media and material are present, represents a practical application of this solution.

Furthermore, the study confirms that the proposed solution allows for effective milling of granular material without the need for anchoring the machine to the ground. Observations indicate that the vibration isolation properties improve with increased material volume during milling, leading to a reduction in self-induced vibrations. However, an excessive filling level (b = 0.9) was found to be impractical due to the expansion of material volume, which significantly reduced the intensity of grinding media movement.

The grinding quality assessment indicates that the proposed mill design performs comparably to conventional vibrating mills with similar chamber movement and grinding parameters. The relocation of vibrators from the chamber to the drive frame contributed to chamber subassembly mass reduction, enabling either an increase in chamber amplitude while maintaining the same unbalanced masses or a reduction in unbalanced masses while sustaining operational efficiency. This modificationalso reduced the impact of vibrations on the foundation, potentially lowering installation costs for industrial applications.

In conclusion, the proposed antiresonance vibration mill design provides an alternative to traditional vibrating mills by reducing dynamic force transmission to the ground while maintaining technological parameters. These advantages make the solution attractive for industrial applications where vibration isolation and energy efficiency are critical considerations.

Corresponding author: Paweł Piekaj, Faculty of Mechanical Engineering and Robotics, AGH University of Krakow, al. A. Mickiewicza 30, 30–059 Krakow, Poland, E-mail: piekaj@agh.edu.pl

Acknowledgments

General scientific research methods are practiced in this research.

Funding information: Authors state no funding involved.
Authors contribution: All authors have accepted responsibility for the entire content of this manuscript and consented to its submission to the journal, reviewed all the results and approved the final version of the manuscript. P.P: Conceptualization, Conceived and conduct the experiment, Formal analysis, Visualization, Writing – original draft and review. G.C: Conceived the experiment, Formal analysis, Writing – review and editing, Supervision.
Conflict of interest: Authors state no conflict of interest.
Data availability statement: The datasets generated during and/or analysed during the current study are available from the corresponding author on reasonable request.

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Received: 2024-09-16

Accepted: 2025-05-27

Published Online: 2025-12-24

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

https://doi.org/10.1515/eng-2025-0121

Keywords for this article

antiresonant; vibration mill; grinding; comminution; dynamic vibration absorber; chamber load

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