Sensor-Based Detection of Characteristics of Rubber Springs Used in Vibration Machines

1. Introduction

The current global trend of digitalization referred to as "Industry 4.0" [1,2,3] and the related automation of production places requirements in terms of inspection, diagnostics and monitoring of parts, components and machines as a whole, along with conveying equipment, which are an essential part of production.

The purpose of this study is to monitor, diagnose and fully control the working operation of vibration machines based on sensor measurements of the compressive forces acting on rubber springs (which are digital inputs). The monitoring, detection and analysis of the measured signals by the sensors enables the optimal data inputs to be set up to control the drive of the vibration machine.

Steel [4] or rubber springs are widely used in industrial operations to dampen vibrations transmitted to the substructure of vibration machines. Smooth-running conveying equipment, namely vibratory conveyors with microshafts [5] or vibrating sorters with mechanical exciter [6], electromagnetic vibrators [7,8] or vibration exciters consisting of a pair of vibration electric motors [9,10] use the harmonic vibration of the trough or sorting screens [11,12] to move grains of bulk material.

A list of published works in the field of vibration transport was prepared by the authors M. A. Parameswaran and S. Ganapathy in the article [13].

In the article [14] Z. V. Despotovic et al. presents a mathematical model of a resonant linear vibrating conveyor driven by an electromagnetic excitation force. Derivation of the mathematical model is based on the kinetic and potential energies, dissipative function of the mechanical system, and Lagrangian formulation.

In the article [15], the authors E. M. Sloot and N. P. Kruyt present a method for measuring the coefficient of friction between granular material and a vibratory conveyor. In this paper, a theoretical and experimental study was made of the conveying speed [16] with which granular materials are transported by vibratory conveyors.

In the article [17], a new design solution of a vibrating conveyor was presented and analysed, which, despite the phasing of the vibrators, enables the feed material to be transported along the chute at a constant speed.

In [18] L.L. Howell et al. investigates the spring stiffness equivalent to a pseudo-rigid body and new modeling equations are proposed therein. The result is a simplified method of modelling the force/deflection relationships of large-deflection members in compliant mechanisms.

The articles [19,20] represent a structural design as well as a created 3D model used for subsequent physical implementation of a validation device that allows for measuring the force as well as the shortening of a coiled pressure spring during laboratory experiments.

In the article [21], the author R. Lefanti et al. an evaluation of the reliability of the elastomeric silentblock based on the shape factor and the materials used was developed.

In the article [22], J. Ziobro presents a mechanical calculation of the silentblock model. Based on the model, he performed a 3D numerical analysis using two types of rubber bushings with different hardness.

These studies [23,24,25,26], for example, deal with the analysis of the characteristic properties of silent blocks used in automobiles.

R. K. Luo and W. X. Wu in the article [27] state that the purpose of the FE analysis is to obtain improved fatigue life of the spring. It is shown in the paper that a quasi-static simulation for rubber springs using nonlinear software can provide good indication for product design and failure analysis.

M. Berg in [28] proposed a nonlinear dynamic model of a rubber spring and focused on the representation of the mechanical behavior of rubber suspension components in rail vehicle dynamics.

For the analysis of railway vehicle dynamics, in the article [29] by R. Luo et al. designed a simple and more accurate nonlinear rubber spring model. The characteristics of dynamic stiffness and damping are investigated through both simulations and lab tests with various displacement amplitude and frequency.

In the article [30], H. Shi and P. Wu created a nonlinear dynamic model for a rubber spring and subsequently used it to describe the mechanical behavior of rubber mounts in the suspension system of a railroad vehicle. In this paper, the dynamic stiffness and damping characteristics in relation to the applied displacement amplitude and frequency were investigated by means of simulations and measurements.

J. Austin et al. in [31] report that soft springs provide higher torque resolution at the cost of system bandwidth, whereas stiff springs provide a fast response but lower torque resolution. Nonlinear springs potentially incorporate the benefits of both soft and stiff springs, but such springs are often large. An nonlinear sprin design was recently proposed that combines a variable radius cam with a rubber elastic element, enabling a compact spring design.

The main objective of the study was to use sensors to detect the magnitude of the compressive force acting on the rubber spring, thus enabling monitoring, data collection and storage for the purpose of autonomous operation of vibration machines. The ability to detect the actual instantaneous magnitudes of the applied forces by sensors in real time and to use these digital values to communicate with the control unit allows for increasing the safety and reliability of the operation of the vibration machine [31,32] and its key machine parts, which can also include springs. A significant aspect is the reduction of the risk of accidents both during maintenance and working activities and the assurance of functionality without the need for human presence due to the use of automation and sensor monitoring.

2. Materials and Methods

Vibrating conveyors/sorters use the inertial forces on the individual grains of the conveyed material to move/sort the material. The inertial forces are generated by the harmonic oscillating motion of the trough (micro-rotor conveyors), see Figure 1, in which the material particles are separated from the trough surface at a certain phase of the transport time (the vertical component of the acceleration of the oscillating motion is greater than the acceleration due to gravity).

The source of the harmonic oscillations is the exciter, which is usually firmly connected to the trough. The trough is supported/suspended by steel or rubber (silent block) springs. The springs allow the trough to perform harmonic motion and at the same time prevent the transmission of dynamic forces to the substructure, for this reason there is an attempt to select springs with the lowest possible stiffness s_s [N·m^-1]. The stiffness of the springs also determines whether the vibration machine will operate in the sub-resonant (z < 1), resonant (z = 0.85÷0.95) or supra-resonant (z > 1÷5) zone, where z = _w/w0, w = 2-p-f [^s-1] is the operating frequency of the machine, f [Hz] is the frequency of vibration generated by the exciter, _w0 [^s-1] is the natural frequency of the machine (1). The stiffness of the springs supporting the trough of mass m [kg] (including the mass of the conveyed material placed on the trough) can be determined according to (1).

$${\mathsf{\omega }}_0=\sqrt{{\displaystyle \frac{{\mathrm{s}}_{\mathrm{s}}}{\mathrm{m}}}}{[\mathrm{s}}^{-1}]\Rightarrow {\mathrm{s}}_{\mathrm{s}}=\mathrm{m}\cdot {\omega }_0[\mathrm{N}\cdot {\mathrm{m}}^{-1}],$$

(1)

In many springs that are in use, especially in the case of the frequently used helical springs or rubber springs, the dependence of the deformation (compression) x [m] due to the applied force F [N] is almost linear [20], and in graphical expression it is a line. The slope of this line gives the spring stiffness s_s [N·m^-1], which is a constant expressing the force required to deform the spring per unit [4]. In the case of a direct proportionality between force and deflection, such behaviour can be expressed mathematically by Hooke's law (2).

$${\mathrm{F}=\mathrm{s}}_{\mathrm{s}}\cdot \mathrm{x}\left[\mathrm{N}\right],$$

(2)

where F [N] is the spring force, x [m] x [m] is the spring deflection from the resting state.

A body suspended by a spring, or supported by a spring, is one of the simplest mechanical oscillators, i.e. a system that performs oscillations. If damping is neglected, a body of mass m [kg] on a spring of stiffness s_s [N·m^-1] performs harmonic oscillations of natural angular frequency ω₀ [rad·s^-1] (1).

The spring characteristic is generally a curve expressing the dependence between the force acting on the spring and its elastic deformation. The area under the curve corresponds to the delivered work W [J] (3) required for a certain deformation of the spring or the accumulated potential energy E_p [J] of the loaded spring.

$${\mathrm{E}}_{\mathrm{p}}=\mathrm{W}=0.5\cdot \mathrm{F}\cdot \mathrm{x}=0.5\cdot s_s\cdot {\mathrm{x}}^2[\mathrm{J}=\mathrm{kg}\cdot {\mathrm{m}}^2\cdot {\mathrm{s}}^{-2}],$$

(3)

In order to verify the catalogue values [34] of the stiffness of rubber springs, a laboratory device was designed and constructed which allows to measure the elastic deformation of rubber springs, which are called "silent blocks", depending on the magnitude of the applied compressive force, using sensors [35,36] and positioning units with a digital indicator [37]. This was carried out in a laboratory at the Department of Machine and Industrial Design, Faculty of Mechanical Engineering, VSB-Technical University of Ostrava.

Figure 2 presents the first variation of the measurement of compression rubber spring characteristics. The bolt of the tested silent block 3 was attached to the sled of the positioning unit (long positioning table PT7312-PA [37]) 4. The measurements were carried out on a total of 8 types of silent blocks, made of NR/SBR rubber with a hardness of 55°shA, with the bolt having an M6 thread. Four silent blocks were of diameter D = 20 mm and lengths H₀ = 10, 15, 20 and 25 mm, while the remaining four silent blocks were of diameter D = 25 mm and lengths H₀ = 15, 20, 25 and 30 mm. The positioning unit 4 was attached to the steel frame 1 of the laboratory apparatus similarly to digital force gauge DST-220A [35] with a measuring range of 0÷1000 N.

By manually rotating the locking bolt of the positioning unit 4 in the appropriate direction, the shank of the rubber spring bolt 3 was brought closer/farther away from the measuring surface of the digital force gauge 2. At the moment when the shank of the bolt of the rubber spring 3 reached the measuring surface of the digital force gauge 2, the digital indicator of the positioning unit 4 and the digital force gauge 2 were reset. By turning the locking bolt of the positioning unit 4, the sled was moved vertically by the desired amount L_i [m]. The value of the applied compressive force F_com,i [N] was read on the digital force gauge 2 for the compression value L_i [m] (i.e. the length of the rubber spring H_i = H₀ - H_i [m]) of the rubber spring 3.

The procedure was repeated until the predetermined maximum compression _Li = L_i = L_max [m] of the rubber spring 3. By turning the locking bolt of the positioning unit 4 in the reverse direction, the values of the compressive force F_rel,i [N] at the compression value L_i [m] of the rubber spring 3 were measured on the digital force gauge 2. The values of the measured compressive forces by the digital force gauge 2 can be recorded using the Force Logger software [38] and saved on a computer.

Figure 3 presents the second variation of measuring the characteristics of a compression rubber spring. tension pressure sensor C9B 2 [36] with a measuring range of 0÷1000 N was mechanically attached to the sled of the horizontally positioned positioning unit 4.

By manually rotating the locking bolt of the vertically positioned positioning unit 4 in the appropriate direction, the shank of the rubber spring screw 3 was brought closer/farther away from the measuring surface of the tension pressure sensor 2. Using the DEWESoft X2 SP5 software [39], the signals of the measured quantity (pressure forces F_com,i [N]) were recorded, having been detected by the DEWESoft DS-NET measuring apparatus [40], see Figure 4.

The cable of the tension pressure sensor C9B-1 kN, see Figure 4, fitted with a D-Sub 9-pin plug, was connected to the DS NET BR4 module [40]. The RJ45 connectors of the network cable connect the DS GATE module [40] to a PC (ASUS K72JR-TY131) running the DEWESoft X2 SP5 software [39].

The constructed laboratory device that allows detection of the magnitude of compressive forces acting on rubber springs is presented in Figure 5.

3. Results

Compressive forces of compressed rubber springs φ20 mm, with lengths in unloaded state H₀ = 10, 15, 20 and 25 mm and φ25 mm, H₀ = 15, 20, 25 and 30 mm were measured on the laboratory equipment, see Figure 5. All types of rubber springs were made of NR/SBR rubber with a hardness of 55°shA.

The values of n times repeated measurements of compressive forces under the same technical conditions were, for individual compression of rubber springs, displayed in the software Force Logger [38] using the digital force gauge DST-220A or in the software DEWESoft X2 SP5 using the tension pressure sensor C9B and recorded in tables, see 3.1 and 3.2.

3.1. Subsection

Table 1 shows the measured values of the compressive force F_com,i [N] of a silent block φ20 mm with a length in the unloaded state of H₀ = 10 mm during its compression, i.e. deformation L_i [m]. Also presented in Table 1 are the measured values of the compressive force Frel_,i [N] when the silent block is released from the length H_i = H₀ - L_i [m]. The maximum possible compression of the silent block is L_i = L_max [m], from which the minimum silent block length H_N = H₀ – L_.max [m] can be expressed.

The resulting calculated compression force F_com [N] or release force F_rel [N] of the form (4), using Student's distribution [41], is given in Table 1 (through to Table 4)

$$\begin{array}{l}{\mathrm{F}}_{\mathrm{com}}{(\mathrm{F}}_{\mathrm{rel}}{)=\mathrm{F}}_{\mathrm{com}}{(\mathrm{F}}_{\mathrm{rel}})\pm {\kappa }_{\alpha ,\mathrm{n}}{=\mathrm{F}}_{\mathrm{com}}{(\mathrm{F}}_{\mathrm{rel}})\pm {\kappa }_{5\%,5}\left[\mathrm{N}\right]\\ {\kappa }_{\alpha ,\mathrm{n}}{=\mathrm{t}}_{\mathsf{\alpha },\mathrm{n}}\cdot {\mathrm{s}=\mathrm{t}}_{5\%,5}\cdot s\left[\mathrm{N}\right]\end{array}$$

(4)

where F_com (F_rel) [N] is the arithmetic mean of all (n = 5 - number of repeated measurements) measured values of F_com,i (F_rel,i) [N], κ_α,n [-] is the marginal error, t_α,n [-] (t_5%,5 = 2.78) is the Student's coefficient for the risk α [%] (α = 5%) and the confounding coefficient P [%] (P = 95%) [41], s [-] is the sample standard deviation of the arithmetic mean.

Table 2 shows the measured values of the compressive force F_com,i [N] and F_rel,i [N] when compressing and releasing a silent block φ20 mm with length in the unloaded state of H₀ = 15 mm. In [34] the maximum compression L_max = 3.75 mm, the maximum load F_max = 352 N and the stiffness of the silent block s_s = 94 N·mm^-1 are given for this type of silent block.

Table 3 shows the measured values of the compressive force F_com,i [N] and F_rel,i [N] when compressing and releasing a silent block φ20 mm with length in the unloaded state of H₀ = 20 mm. In [34] the maximum compression L_max = 5 mm, the maximum load F_max = 260 N and the stiffness of the silent block s_s = 52 N·mm^-1 are given for this type of silent block.

Table 4 shows the measured values of the compressive force F_com,i [N] and F_rel,i [N] when compressing and releasing a silent block φ20 mm with length in the unloaded state of H₀ = 25 mm. In [34] the maximum compression L_max = 6.25 mm, the maximum load F_max = 310 N and the stiffness of the silent block s_s = 50 N·mm^-1 are given for this type of silent block.

Table 4. Rubber spring D = 20 mm, length when not loaded H₀ = 25 mm.

i	1		2		3		4		5
Li	Fcom,1	Frel,1	Fcom,2	Frel,2	Fcom,3	Frel,3	Fcom,4	Frel,4	Fcom,5	Frel,5	Fcom 1	κ5%,5	Frel 1	κ5%,5	sscom,i
mm	N														N·mm-1
0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	-
1.0	50	36	48	38	49	38	48	38	49	39	48.8	1.1	37.8	1.3	48.8
2.0	95	78	95	81	95	82	93	80	94	82	94.4	1.3	80.6	2.2	47.2
4.0	194	163	190	168	190	171	187	169	189	171	190.0	2.8	168.4	4.0	47.5
6.0	292	258	288	264	286	264	283	262	287	264	287.2	3.9	262.4	3.3	47.9
8.0	402	375	399	380	397	382	392	378	391	380	396.2	6.5	379.0	3.5	49.5
9.0	456	456	459	459	462	462	456	456	459	459	458.4	3.3	458.4	3.3	50.9

¹ see Figure 6

Table 4. Rubber spring D = 20 mm, length when not loaded H₀ = 25 mm.

i	1		2		3		4		5
Li	Fcom,1	Frel,1	Fcom,2	Frel,2	Fcom,3	Frel,3	Fcom,4	Frel,4	Fcom,5	Frel,5	Fcom 1	κ5%,5	Frel 1	κ5%,5	sscom,i
mm	N														N·mm-1
0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	-
1.0	50	36	48	38	49	38	48	38	49	39	48.8	1.1	37.8	1.3	48.8
2.0	95	78	95	81	95	82	93	80	94	82	94.4	1.3	80.6	2.2	47.2
4.0	194	163	190	168	190	171	187	169	189	171	190.0	2.8	168.4	4.0	47.5
6.0	292	258	288	264	286	264	283	262	287	264	287.2	3.9	262.4	3.3	47.9
8.0	402	375	399	380	397	382	392	378	391	380	396.2	6.5	379.0	3.5	49.5
9.0	456	456	459	459	462	462	456	456	459	459	458.4	3.3	458.4	3.3	50.9

¹ see Figure 6

Figure 6. Graph of the dependence of the compressive force acting on a rubber spring φ20 mm with length in the unloaded state H₀ = 10÷25 mm and its elastic deformation. H₀ () 10 mm, () 15 mm, () 20 mm, () 25 mm.

Figure 6. Graph of the dependence of the compressive force acting on a rubber spring φ20 mm with length in the unloaded state H₀ = 10÷25 mm and its elastic deformation. H₀ () 10 mm, () 15 mm, () 20 mm, () 25 mm.

Figure 7. Graph of the dependence of the compressive force acting on a rubber spring φ25 mm with length in the unloaded state H₀ = 15÷30 mm and its elastic deformation. H₀ () 15 mm, () 20 mm, () 25 mm, () 30 mm.

Figure 7. Graph of the dependence of the compressive force acting on a rubber spring φ25 mm with length in the unloaded state H₀ = 15÷30 mm and its elastic deformation. H₀ () 15 mm, () 20 mm, () 25 mm, () 30 mm.

Table 5 shows the measured values of the compressive force F_com,i [N] of a silent block φ25 mm with a length in the unloaded state of H₀ = 15 mm during its compression, i.e. deformation L_i [m]. Also presented in Table 5 are the measured values of the compressive force F_rel,i [N] when the silent block is released from the length H_i = H₀ - L_i [m]. In [34] the maximum compression L_max = 3.75 mm, the maximum load F_max = 687 N and the stiffness of the silent block s_s = 183 N·mm^-1 are given for this type of silent block.

Table 6 shows the measured values of the compressive force F_com,i [N] and F_rel,i [N] when compressing and releasing a silent block φ25 mm with length in the unloaded state of H₀ = 20 mm. In [34] the maximum compression L_max = 5 mm, the maximum load F_max = 602 N and the stiffness of the silent block s_s = 120 N·mm^-1 are given for this type of silent block.

Table 7 shows the measured values of the compressive force F_com,i [N] and F_rel,i [N] when compressing and releasing a silent block φ25 mm with length in the unloaded state of H₀ = 25 mm. In [34] the maximum compression L_max = 6.25 mm, the maximum load F_max = 675 N and the stiffness of the silent block s_s = 108 N·mm^-1 are given for this type of silent block.

Table 8 shows the measured values of the compressive force F_com,i [N] and F_rel,i [N] when compressing and releasing a silent block φ25 mm with length in the unloaded state of H₀ = 30 mm. In [34] the maximum compression L_max = 7.5 mm, the maximum load F_max = 562 N and the stiffness of the silent block s_s = 75 N·mm^-1 are given for this type of silent block.

Figure 8 presents the equations of the trend lines of the measured forces F_com [N] (see Table 1 to Table 8) in an XY dot chart in Microsoft Excel. The reliability value R measures the reliability of the trend line - the closer R² is to 1, the better the trend line fits the data.

3.2. Subsection

The following tables (Table 9 to Table 12) show the measured values of the compressive forces F_com,i [N] of the silent blocks (φ20 mm and φ25 mm with unloaded length H₀ = 10 to 30 mm) during their compression, i.e. deformation L_i [m]. The tables show the values of F_com [N], which is the arithmetic mean of all (n = 3 number of replicate measurements) measured values of F_com,i [N] and κ_α,n [-], which is the marginal error. t_α,n [-] t_5%,3 = 4.30) is the Student's coefficient for the risk α [%] (α = 5%) and the confounding coefficient P [%] (P = 95%) [41].

Figure 10 and Figure 11 present the values of the measured compressive forces, randomly selected from Table 9. In case of interest, it is possible, upon written request by e-mail to the authors of this paper, to receive all measured data in data files with DXD extension (DEWESoft software) or XLS, XLSX (Microsoft Excel software). The same applies to the following presented tables.

Figure 11 Measured values of the compressive force F_com,i [N] during compression L_i [mm] of a rubber spring of diameter D = 20 mm, length H₀ (a) 15 mm, (b) 20 mm.

Figure 10. Measured values of the compressive force F_com,i [N] during compression L_i [mm] of a rubber spring of diameter D = 20 mm, length H₀ = 10 mm. Compressive force F_com,1 (a) 52.3 N, (b) 108.1 N.

Figure 10. Measured values of the compressive force F_com,i [N] during compression L_i [mm] of a rubber spring of diameter D = 20 mm, length H₀ = 10 mm. Compressive force F_com,1 (a) 52.3 N, (b) 108.1 N.

Figure 11. Measured values of the compressive force F_com,i [N] during compression L_i [mm] of a rubber spring of diameter D = 20 mm, length H₀ = 10 mm. Compressive force F_com,1 (a) 15 mm, (b) 20 mm.

Figure 11. Measured values of the compressive force F_com,i [N] during compression L_i [mm] of a rubber spring of diameter D = 20 mm, length H₀ = 10 mm. Compressive force F_com,1 (a) 15 mm, (b) 20 mm.

Figure 12. Measured values of the compressive force F_com,i [N] during compression L_i [mm] of a rubber spring of diameter D = 20 mm and 25 mm, length H₀ (a) 25 mm, (b) 15 mm.

Figure 12. Measured values of the compressive force F_com,i [N] during compression L_i [mm] of a rubber spring of diameter D = 20 mm and 25 mm, length H₀ (a) 25 mm, (b) 15 mm.

Figure 13. Measured values of the compressive force F_com,i [N] during compression L_i [mm] of a rubber spring of diameter D = 25 mm, length H₀ (a) 20 mm, (b) 55 mm.

Figure 13. Measured values of the compressive force F_com,i [N] during compression L_i [mm] of a rubber spring of diameter D = 25 mm, length H₀ (a) 20 mm, (b) 55 mm.

Figure 14 shows the comparison of measured compressive forces F_com,i [N] by digital force gauge DST-220A and tension pressure sensor C9B during compression of rubber springs of diameter D = 20 mm with lengths in unloaded state of H₀ = 10÷25 mm. The numerical values of the compressive forces F_com [N], see Table 1 to Table 4, given for the solid curves correspond to measurement variant 1 (see Chapter 3.1), the values of the compressive forces F_com [N], see Table 9 and Table 10, given for the broken curves correspond to measurement variation 2 (see Chapter 3.2).

Figure 15 shows the comparison of measured compressive forces F_com,i [N] with the digital force gauge DST-220A and the tension pressure sensor C9B during compression of rubber springs of diameter D = 25 mm with lengths in the unloaded state H₀ = 15÷30 mm. The numerical values of the pressure forces F_com,i [N], see Table 5 to Table 8, given for the solid curves correspond to measurement variation 1 (see Chapter 3.1), the values of the pressure forces F_com, [N], see Table 11 and Table 12, given for the dashed curves correspond to measurement variation 2 (see Chapter 3.2).

4. Discussion

Rubber springs made of elastic material (rubber) are used for various mechanical applications because of their ability to be deformed and then return to their original shape. They can effectively absorb and dampen shocks or vibrations, making them suitable for use in automotive, machinery or conveying equipment where vibration needs to be eliminated.

The measurements of the compressive forces acting on the rubber springs presented in Figure 6 and Figure 7 confirmed that the rubber springs have a progressive characteristic [42]. Progressive rubber springs [43,44] change their resistance to compression slowly at first and then more steeply, i.e. parabolically, with increasing load.

In Table 1 to Table 8, the stiffnesses of the rubber springs s_scom,i [N·mm^-1] φ20 mm and φ25 mm are calculated from the measured values of the compressive forces with the digital force gauge DST-220A acting on the rubber spring F_com,i [N] and its elastic deformation L_i [m] during compression.

From the measured values, the calculated arithmetic average of the spring stiffness φ20 mm and the unloaded length H₀ = 15 mm, see Table 2, takes the value s_scom = 83.7 N·mm^-1, in [34] it is stated that s_s = 94 N·mm^-1, which implies that s_scom [N·mm^-1] is 89.0% of s_s [N·mm^-1].

The arithmetic diameter of the spring stiffness φ20 mm and the length in the unloaded state H₀ = 20 mm, see Table 3, takes the value s_scom = 80.8 N·mm^-1, in [34] it is given as s_s = 52 N·mm^-1, which implies that s_scom [N·mm^-1] is 155.4% of s_s [N·mm^-1].

The arithmetic diameter of the spring stiffness φ20 mm and the length in the unloaded state H₀ = 25 mm, see Table 4, takes the value s_scom = 48.6 N·mm^-1, in [34] it is given as s_s = 50 N·mm^-1, which implies that s_scom [N·mm^-1] is 97.2% of s_s [N·mm^-1].

The arithmetic diameter of the spring stiffness φ25 mm and the length in the unloaded state H₀ = 15 mm, see Table 5, takes the value s_scom = 135.9 N·mm^-1, in [34] it is given as s_s = 183 N·mm^-1, which implies that s_scom [N·mm^-1] is 74.3% of s_s [N·mm^-1].

The arithmetic diameter of the spring stiffness φ25 mm and the length in the unloaded state H₀ = 20 mm, see Table 6, takes the value s_scom = 87.9 N·mm^-1, in [34] it is given as s_s = 120 N·mm^-1, which implies that s_scom [N·mm^-1] is 73.3% of s_s [N·mm^-1].

The arithmetic diameter of the spring stiffness φ25 mm and the length in the unloaded state H₀ = 25 mm, see Table 7, takes the value s_scom = 71.7 N·mm^-1, in [34] it is given as s_s = 108 N·mm^-1, which implies that s_scom [N·mm^-1] is 66.4% of s_s [N·mm^-1].

The arithmetic diameter of the spring stiffness φ25 mm and the length in the unloaded state H₀ = 30 mm, see Table 8, takes the value s_scom = 57.3 N·mm^-1, in [34] it is given as s_s = 75 N·mm^-1, which implies that s_scom [N·mm^-1] is 76.4% of s_s [N·mm^-1].

In Table 9 to Table 12, the stiffnesses of the rubber spring s_scom,i [N·mm^-1] φ20mm and φ25 mm are calculated from the measured values of the compressive forces of the tension pressure sensor C9B acting on the rubber spring F_com,i [N] and its elastic deformation L_i [m] during compression.

The arithmetic average of the spring stiffness φ20 mm and the unloaded length H₀ = 15 mm calculated from the measured values, see Table 9, takes the value s_scom = 83.3 N·mm^-1, and in [34] it is given as s_s = 94 N·mm^-1, which implies that s_scom [N·mm^-1] is 88.6% of s_s [N·mm^-1].

The arithmetic diameter of the spring stiffness φ20 mm and the length in the unloaded state H₀ = 20 mm, see Table 10, takes the value s_scom = 88.6 N·mm^-1, in [34] it is given as s_s = 52 N·mm^-1, which implies that s_scom [N·mm^-1] is 171% of s_s [N·mm^-1].

The arithmetic diameter of the spring stiffness φ20 mm and the length in the unloaded state H₀ = 25 mm, see Table 10, takes the value s_scom = 52.2 N·mm^-1, in [34] it is given as s_s = 50 N·mm^-1, which implies that s_scom [N·mm^-1] is 104.4% of s_s [N·mm^-1].

The arithmetic diameter of the spring stiffness φ25 mm and the length in the unloaded state H₀ = 15 mm, see Table 11, takes the value s_scom = 162.1 N·mm^-1, in [34] it is given as s_s = 183 N·mm^-1, which implies that s_scom [N·mm^-1] is 88.6% of s_s [N·mm^-1].

The arithmetic diameter of the spring stiffness φ25 mm and the length in the unloaded state H₀ = 20 mm, see Table 11, takes the value s_scom = 94.6 N·mm^-1, in [34] it is given as s_s = 120 N·mm^-1, which implies that s_scom [N·mm^-1] is 78.8% of s_s [N·mm^-1].

The arithmetic diameter of the spring stiffness φ25 mm and the length in the unloaded state H₀ = 25 mm, see Table 12, takes the value s_scom = 75 N·mm^-1, in [34] it is given as s_s = 108 N·mm^-1, which implies that s_scom [N·mm^-1] is 96.4% of s_s [N·mm^-1].

The arithmetic diameter of the spring stiffness φ25 mm and the length in the unloaded state H₀ = 30 mm, see Table 11, takes the value s_scom = 65.3 N·mm^-1, in [34] it is given as s_s = 75 N·mm^-1, which implies that s_scom [N·mm^-1] is 87.1% of s_s [N·mm^-1].

The measured values given in Table 1 to Table 12 show that the stiffnesses of rubber springs s_scom,i [N·mm^-1]are lower (except for the rubber spring φ20 mm in length in the unloaded state H₀ = 20 mm, see Table 3 and Table 10 and H₀ = 25 mm, see Table 11) than the stiffness values of rubber springs of identical dimensions given in [34].

It is known that in the case of a linear spring, the resistance of the spring to compression increases linearly with increasing load. Progressive springs, which include rubber springs, first change their resistance to compression slowly with increasing load and then more steeply, i.e. parabolically. It follows that the stiffness of a progressive spring is not a constant, but changes (increases) depending on its compression.

It is therefore not possible to define the stiffness of a rubber spring as a constant, as is stated for example in [45]. From the measured values of the compressive forces (linear data set) F_com,i [N] during compression of the rubber springs, linear trend lines were plotted in Microsoft Excel, see Figure 8. The trend line is the function that best describes the data, it is a linear function, the graphical presentation of which is a straight line. The trend line function is found by the least squares method. It can be seen from Figure 8 that the smaller diameter rubber springs (D = 20 mm) exhibit lower stiffness relative to the larger diameter rubber springs (D = 25 mm). It is also true that shorter length rubber springs have a higher stiffness than longer length rubber springs of the same diameter. Rubber springs of shorter lengths can be subjected to less maximum compression and more maximum load than rubber springs of longer length of the same diameter.

Sensors applied to an actual working vibration machine are able to control its working activity or to control certain technical parameters to a certain extent. In the case of vibration machines, sensors can, for example, detect the load applied to the springs that support the trough/sorting screens. The correct operation of a particular vibration machine can be monitored automatically from the detection of spring stiffnesses determined precisely in advance in the laboratory (e.g. according to the procedure described in Chapter 3 of this paper). In vibrating machines, the instantaneous values of the compressive forces acting on the springs can be measured during their working operations and the amount of conveyed/sorted material on the trough/sorting surface can be tracked according to the applied load.

5. Conclusions

Measuring the characteristics of rubber springs is an important process that involves testing not only stiffness and elasticity, but also behaviour under long-term or cyclic loading, which is essential to ensure the reliability and serviceability of springs in various applications.

In this paper, the compressive forces acting on rubber springs during their compression were measured on laboratory equipment using sensors. The relationship between the force acting on the spring and its elastic deformation can be used as an input value for control, diagnosis and monitoring of vibrating machines. Vibrating machines in the form of vibrating conveyors or sorters, using an electromagnetic vibration exciter as the source of excitation force, transmit high frequency vibrations to the substrate. By supporting the trough of vibratory conveyors with rubber springs of suitable stiffness, these springs can capture the high frequency oscillatory motion and attenuate the noise due to the high self-attenuation capability of the rubber.

The measurements carried out in a laboratory at the Department of Machine and Industrial Design, Faculty of Mechanical Engineering, VSB-Technical University of Ostrava confirmed the progressivity of the rubber springs. The progressivity of a spring can be defined by the fact that its stiffness increases with increasing compression of the spring; the characteristic of the spring is not a straight line but an exponential. It has been shown that at half the compression, a rubber spring has less stiffness than a steel coil spring.

It is an attempt to select the most optimal spring from the actual stiffnesses of the measured rubber springs and use it as a spring to support the trough of a laboratory vibrating conveyor, which uses a single-phase asynchronous vibration motor to excite the trough vibrations. On the vibrating conveyor used in the study, sensors detect and the DEWESoft DS-NET measuring instrument monitor the pressure force acting on a selected number (or all) of the rubber springs supporting the trough of the vibrating conveyor. The electrical signal obtained by the tension pressure sensor will be used as an input parameter (digital input) to administer the automated operation and control of the vibrating conveyor, which fits with the current trend of digitalization and related automation of production known as Industry 4.0.