Experimental Determination of the Coefficient of Friction on a Screw Joint

Featured Application

The results of this paper can help in lubricant selection when frequently disassembling and assembling a screw joint or when using a screw joint with fine mechanics. By applying the results in practice, it is possible, in certain cases, to minimize the tightening torque as the main parameter for screw joints.

Abstract

This paper deals with the coefficient of determination of screw connection friction between the thread and the matrix. The coefficient of friction was measured using a laboratory device with an M20 screw connection without any grease and, subsequently, plastic grease was added (CX80 silicone, lithium, and copper grease). When grease is added, the friction in the threads and screw heads is limited and consistently retained. When tightening by torque, which represents the prevailing assembly method in standard screwing practice, only part of the torque is effectively used to create axial force and pre-stress. The rest of the torque is employed in friction suppression between threads and converted into heat. In general, the coefficient of friction depends on diverse factors such as the roughness of the thread surface, the gradient angle of the helix, and the grease properties. The tightening torque represents a primary parameter in the experimental measurements, monitored using a digital torque spanner, and generates an axial force in the screw. Based on the aforementioned parameters, the objective of this paper was to monitor changes in the coefficient of friction between the thread of the screw and the matrix in the case of different grease types. The actual coefficient of friction was calculated through the exponential equation of the torque balance. First measured was the load of the bolted joint without the use of grease, where the average value of the coefficient of friction was 0.44732; this value served as a benchmark for comparison to the measurements with the use of grease. The measurements showed that the value of the friction coefficient was reduced by 30.57% when using lithium grease, by 40.56% when using silicone grease, and by 47.64% when using copper-based grease, making the latter the most suitable for the application. Without appropriate greasing, friction suppression was accompanied by extremely high torques, which resulted in insufficient screw prolongation.

1. Introduction

Currently, only minor attention is paid to the given issue; thus, few papers have been published in this field. Most of the friction coefficient research focuses on ball and trapezoidal screws, which are used in machine tool drives. The findings have led authors to design a measuring stand for testing bolted joints, where the primary monitored parameter is the coefficient of friction between the threads.

Bolted joints are a type of dismountable joint, designed to connect diverse components. From a technical point of view, their function is based on threaded coupling. Threads are referred to as functional elements, and they are standardized to a high degree.

The current situation and level of technology in the mechanical coupling of components with thread fasteners are characterized as follows:

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The prevailing results are generally underestimated, based on the fact that the screws and matrices are widely used and, compared to other structural elements, they are the cheapest available.

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Double, even triple, dimensioning, which represents the weakest point of current screwing technology. On the one hand, the structural designer is frightened of bearing potential responsibility for a possible accident, and on the other hand, there is the issue of mastering the theory and practice of screwing.

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Economic pressure to build light structures to achieve material savings [1].

The causes of failures in screw connections have become a subject of interest in many companies and relevant workplaces. The well-known SFK company created a hierarchy of failure causes, showing that the highest number of failures occur in assembly. This is a logical outcome of imprecise tightening.

Bolted joint design includes designing materials and strength dimensioning. At the same time, requirements related to operational load must be taken into consideration. A bolted joint functions as required only if all of the mechanical properties of the individual elements are preserved, along with the degree of precision of the defined tolerated dimensions. In practice, bolted joints are subjected to load tests according to specified calculations. These tests verify the efficiency, different deformations, thread strength, etc., of bolted joints.

All inevitable forces are generated when the respective screw or bolt nut is tightened with the predefined torque. Apart from the force acting in the direction of the screw axis, bolted joints induce other forces perpendicular to their axis, which are caused by friction forces. These forces especially occur in abutting areas of the connected parts. Friction force must act adequately to prevent a shift in the connected parts and consequent shear stress on the screw [2].

From the point of view of modern design engineering, assessment of the carrying capacity, the service life, and the fulfillment of the functional requirements of bolted joints is significantly influenced by the correct definition of axial force F₀ when a screw or a bolt nut is tightened, as well as by practical procedures necessary for their achievement.

If the force is extremely low in comparison to the tension tolerated in the screw core, the maximum technical parameters of the structure cannot be achieved. In case of high values, when the superposition of the force, along with the effects of other forces, reach characteristic stress in the thread core due to operational stress, the bolted joint is released or the threads are cut, which represents an unacceptable condition (tightness is insufficient, abutment joints are released, etc.).

Therefore, close attention has been paid to the experimental processes, assuring the pre-stress of screws as required. These processes include the following:

• The use of negative dilatation of a loosely tightened pre-heated screw, which assures pre-stress and the required properties when it is cooled down;
• The application of ultrasound (change in the wave speed along the screw);
• The application of the screw with a dynamometer element (e.g., a resistance strain gauge);
• The application of dynamometer power boards;
• Tightening of the regulated torque, etc.

2. Bolted Joint Design and Testing

Nowadays, a structural designer disposes of a broad assortment of decision-making tools, and just their knowledge and experience are relevant in decision making regarding which variant they opt for. Certainly, a screw connection can serve its function only when correctly installed, i.e., only when tightened to a value prescribed for the tightening torque. As soon as the tightening starts, it changes into a tribological node, in which case, in addition to the mechanical properties, friction plays an important role. Particular attention must be paid to the screw connections for a low coefficient of friction in the course of tightening and for high friction when assembly is completed in order to prevent disengagement of the connection.

Despite the fact that there exist various tables containing recommended torques determined through calculations of material hardness, the final decision is made by a structural designer. The reason is simple: no theoretical calculation can take into consideration the influence of the operating conditions, i.e., the type, magnitude, sense, and direction of stress of the respective structural node during operation. Based on these theoretical facts, experiments were conducted to contribute to the knowledge related to screw connections in practice.

2.1. Ratio of the Forces in Bolted Joints

The ratio of the forces in a screw is the ratio of the forces on the inclined plane. It is the relationship between the axial force F_Q, driving force F₁, and other thread parameters (Figure 1).

The flat thread premise is accepted when the following conditions are met:

• A thread is perpendicular to an axis, causing forces to act tangentially;
• An axis load is evenly distributed along all threads;
• During stroking, the screw rotates and the bolt nut is in the fixed position;
• A calculation is performed with the mean diameter d₂ or d_S.

2.1.1. Frictionless Motion

$$F_1=F_Q·tg\gamma$$

(1)

Here, F₁ is the tangential force acting on the mean diameter of the thread; F_Q is the axial force; and F_N is the normal force.

2.1.2. Frictionless Motion—Tightening

$$F_T=F_N·f, f_{-\mathrm{coefficient} \mathrm{of} \mathrm{friction}}$$

(2)

$$\begin{array}{l}{\displaystyle \sum F_{ix}=0\Rightarrow F_1}=F_N·\sin \gamma +f·F_N·\cos \gamma \\ {\displaystyle \sum F_{iy}=0\Rightarrow F_Q=F_N·\cos \gamma -f·F_N·\sin \gamma }\\ \frac{F_1}{F_Q}=\frac{F_N·\sin \gamma +f·F_N·\cos \gamma }{F_N·\cos \gamma -f·F_N·\sin \gamma }\Rightarrow \\ F_1=F_Q·\frac{\sin \gamma +f·\cos \gamma }{\cos \gamma -f·\sin \gamma }\end{array}$$

(3)

When modified (Figure 2),

$$F_1=F_Q·\frac{tg\gamma +f}{1-f·tg\gamma } resp. F_1=F_Q·\frac{tg\gamma +tg\varphi }{1-tg\gamma ·tg\varphi }=F_Q·tg\left(\gamma +\varphi \right)$$

(4)

2.2. Influencing Factors

The factors influencing the relationship between tightening torque and axial force in a screw can be classified as follows:

• Geometrical—thread shape and contact area between reciprocally rotating parts;
• Tribological—contact area friction [3,4,5].

The value of the coefficient of friction between the area of the matrix thread and the area of the screw thread is influenced by many factors, such as the characteristics of the surface and of the thread grease, as well as the magnitude of the angle of the teeth’s sides. At the same time, the coefficient of friction is a function of the heat treatment of screws and it affects the axial force value. Apart from the aforementioned heat, the friction that occurs in threads causes stress on the screw connection upon torsion, especially in the course of assembling and disassembling, in which case, they become corroded under the long-term influence of a harsh environment [6,7].

2.3. Coefficient of Friction of the V-Thread

As with friction, the flat thread premise is accepted; from it, the moment equation stems, in which φ or f is substituted by φ′ or f′.

$$\begin{array}{l}{T}_A-F_O·tg\left(\gamma +\varphi \prime \right)·\frac{d_s}{2}=0\\ T_A-F_O·\left(tg\gamma +f\prime \right)·\frac{d_s}{2}=0\end{array}$$

(5)

The axial force in the screw is as follows:

$$F_O=\frac{2·\pi ·F_1·l}{\pi ·f\prime ·d_s+s}$$

(6)

The coefficient of friction in the threads regardless of the inclination angle of the thread is:

$$f\prime =\frac{2\pi ·T_A-F_O·s}{\pi ·F_O·d_s}$$

(7)

The coefficient of friction of the V-thread with α standing for the inclination angle of the thread is:

$$f\prime =\frac{f}{\cos \frac{\alpha }{2}}\Rightarrow f=f\prime ·\cos \frac{\alpha }{2}$$

(8)

The influence of the α angle results in an increase in the coefficient of friction f′ in the case of the V-thread with an increasing α value. In this paper, we used the magnitude of the coefficient of friction f′.

3. Stand and Elements Essential for the Experiment

The experimental monitoring and the measurement of the friction coefficient in the case of the threads of the screw connection were monitored loading (tightening torque) and emerging axial force. In the case of the laboratory measurement, an M20 × 100 STN EN 24,017 bolt was used, due to the current design and capabilities of the measuring equipment. Measurement was preceded by degreasing the screw to prevent errors when measuring the coefficient of friction.

The emerging axial force Fo was monitored in the experimental measurement through controlled tightening (axial) torque. The bolted joint load was monitored using degreased meshing threads and threads greased with MOL Alubia AK2.

A stand (Figure 3) was used for the practical part, designed for monitoring the axial force in the bolted joint. Measurement was performed in the Testing and Monitoring Centre of Technical Systems in the Department of Technical Systems Design and Monitoring in the Faculty of Manufacturing Technologies, Technical University of Kosice. Measurements were performed with M20 × 1.5 threaded bolts; a total of 10 bolts were used and five measurements were carried out for each alternative of the experiment. The measurement required preparation of a measuring stand, including pre-specified parts, the absence of which would make the measurement impossible [8,9]. Measurement was performed with pre-defined values of axial torque. Therefore, the computer software displayed a graph that allowed reading of the axial force with individual axial torque values.

The system designed for monitoring and determining the coefficient of friction in the case of screw connections consisted of a basic structure (I), a membrane force sensor (II), and connection to a computer (III) by means of a converter (IV). The equipment included a replaceable tested screw (V), loaded via tightening torque and monitored by the measuring instrument (VI). Loading using the required torque was a manual process, which consequently generated loading of the screw connection by axial force. Monitoring of the axial force was conducted by means of EMS Center software from Emsyst.

3.1. Digital Electronic Converter EMS650

Laboratory measurement was performed using a digital electronic converter, Emsyst EMS650 (Figure 4), with a USB 2.0 interface. EMS650 is a digital converter designed to process the electrical signals produced by a tensometric sensor of force (e.g., tensile and compression force and torque). The signal produced by the sensor is intensified, adequately modified and driven to the analogue input of the microprocessor. Therefore, it is converted to a digital number and sent to the computer [10] through the USB interface. The converter does not dispose of any control elements, as it has two connectors only. The sensor is connected to a Cannon type connector, and the other connector (micro-USB) is used to connect the converter to the computer. All converter functions are controlled solely through the USB interface [11].

3.2. Tensometric Force Sensor

The experimental measurements were performed by means of a universal membrane sensor EMS20 (Figure 5), which was used for sensing the axial force within a range from 0 to 5 kN. This sensor can be used to measure both the tensile and the compression force acting in the direction of the sensor axis, and external threads are employed for attaching [12]. The sphere of application consisted of Testing, the measurement of forces on the machines, dynamometers and electronic scales. The recommended loading mode is tensile or compressive loading with a minimum distance of 2 mm between the tensometric sensor and an articulated head or a silent block [13]. The sensitivity of the sensor was 1.0 ± 2% mV/V, and the temperature coefficient was 0.1% Fn/10 °C. The sensor could be used within a temperature range from −10 °C to 70 °C. The insulation resistance exceeded 5000 Ω, and the bridge resistance was 395 ± 10% Ω at the input and 350 ± 5% Ω at the output.

3.3. Digital Torquemeter WRG3-135

A small digital torquemeter was designed for fast measurement and force control during screw tightening. The torquemeter was installed directly onto a tightening key. The torquemeter was equipped with a digital display, which shows the actual measured values of the axial torque or the maximal value [14] in the case of maximum loading. The digital torquemeter WRG3-135, with a measurement range from 6.8 to 135 Nm (Figure 6), had an accuracy of ±2%. The meter was equipped with a sound signalization unit that was activated upon reaching a pre-specified axial torque value. The meter enabled measuring of the screws with either the right or left thread.

3.4. Software EMS Center

All laboratory measurement results of the controlled axial torque loading and of the monitoring of the emerging axial force in the case of particular bolted joints were evaluated via the EMS Center software. The software version was V1.02, which supports the Windows 7, 8, and 10 operating systems. The software was primarily designed for data collection by means of digital electric units using EMSYST. The software offers graphical plotting of the selected developments and their subsequent storage in files of desired formats [15].

The software includes a record data function, which is activated when the record data button is pressed. The length of individual records is selected according to the number of samples or record time in seconds. All recorded parameters and data can be processed further in Excel, for instance. Therefore, it is advisable to record data in the txt.format. The software includes basic buttons—Start, Stop, and Zero. When the Zero button is pressed, the display is reverted to a zeroized state [16]. In order to create a new profile in the given program, the display must be put in a zeroized state. The Start button activates the measurement, while the Stop button discontinues the measurement.

3.5. Lubricants Used in the Experiment

The bolted joint load was monitored using the degreased meshing threads and the threads greased with MOL Alubia AK2. Samples of the lubricants are shown in Figure 7.

4. Experimental Measurements

4.1. Center Dry Abrasion

The first experimental measurements and the subsequent friction coefficient determination in the case of screw connections were realized without any grease or oils. In the event of such direct contact between the screw threads and the matrix, the coefficient of friction should achieve the highest value, contrary to values from the following measurements in which plastic grease was used.

Table 1. Measured values in dry abrasion.

No. of Measurements	1	2	3	4	5
Loading—tightening torque MT (Nm)	4.0	8.0	12.0	16.0	20.0
Emerging axial force FO (N)	624.8	1218.2	1789.4	2326.7	2792.8

displays the measured values, which were processed and are shown in Figure 8.

The calculated coefficients of friction f′ for the individual tightening torques in dry abrasion are as follows:

$$\begin{gathered} {f\prime }_1=\frac{2\pi ·M_{T1}-F_{O1}·s}{\pi ·F_{O1}·d_s}=\frac{2\pi ·4 Nm-624.8 N·0.0025 m}{\pi ·624.8 N·0.018376 m}=0.6534 \\ {f\prime }_2=\frac{2\pi ·M_{T2}-F_{O2}·s}{\pi ·F_{O2}·d_s}=\frac{2\pi ·8 Nm-1218.2 N·0.0025 m}{\pi ·1218.2 N·0.018376 m}=0.6714 \\ {f\prime }_3=\frac{2\pi ·M_{T3}-F_{O3}·s}{\pi ·F_{O3}·d_s}=\frac{2\pi ·12 Nm-1789.4 N·0.0025 m}{\pi ·1789.4 N·0.018376 m}=0.6865 \\ {f\prime }_4=\frac{2\pi ·M_{T4}-F_{O4}·s}{\pi ·F_{O4}·d_s}=\frac{2\pi ·16 Nm-2326.7 N·0.0025 m}{\pi ·2326.7 N·0.018376 m}=0.7051 \\ {f\prime }_5=\frac{2\pi ·M_{T5}-F_{O5}·s}{\pi ·F_{O5}·d_s}=\frac{2\pi ·20 Nm-2792.8 N·0.0025 m}{\pi ·2792.8 N·0.018376 m}=0.7361 \end{gathered}$$

(9)

The actual coefficients of friction f for the individual tightening torques in dry abrasion are as follows:

$$\begin{gathered} f_1={f\prime }_1·\cos \frac{\alpha }{2}=0.6534·\cos \frac{60°}{2}=0.4233 \\ {f}_2={f\prime }_2·\cos \frac{\alpha }{2}=0.6714·\cos \frac{60°}{2}=0.4349 \\ {f}_3={f\prime }_3·\cos \frac{\alpha }{2}=0.6865·\cos \frac{60°}{2}=0.4448 \\ {f}_4={f\prime }_4·\cos \frac{\alpha }{2}=0.7051·\cos \frac{60°}{2}=0.4568 \\ {f}_5={f\prime }_5·\cos \frac{\alpha }{2}=0.7361·\cos \frac{60°}{2}=0.4768 \end{gathered}$$

(10)

4.2. Abrasion with the Use of Lubricating CX80 Silicone Grease

Colorless lubricant with high viscosity creates a protective film with excellent resistance; hence, coverage of the friction surfaces with a thin layer of silicone grease leads to an effective reduction in friction and provides long-lasting protection of the equipment, suitable for use on contact surfaces of plastic, metal, ceramics, rubber, etc. It does not stain, does not adhere to dust, and is non-adhesive, facilitating the removal of forms, with an operating temperature of −60 to +230 °C.

It is compatible with all kinds of sanitary equipment, including conventional and lever (gas) valves, seals, connectors, fittings, valves, and joints. It can also be used in the clean lubrication of equipment in the textile, paper, and food industries; electric and power distribution equipment (safety protection of electric circuits); power cases that work under extreme temperatures; and the lubrication of links, locks, hinge screws, and sleeves in the automotive industry. Moreover, it protects against corrosion in printing machines where the lubricant comes into contact with paper [17]. It has also been approved by NSF for use in the food industry. The measured values with silicone grease are presented in

Table 2. Values determined via measurement with lubricant CX80—silicone grease.

No. of Measurements	1	2	3	4	5
Loading—tightening torque MTS (Nm)	4.0	8.0	12.0	16.0	20.0
Emerging axial force FOS (N)	974.2	1945.6	2874.6	3823.2	4681.6

and Figure 9.

The calculated coefficients of friction f′ for the individual tightening torques in abrasion with lubricating silicone grease CX80 are as follows:

$$\begin{gathered} {f\prime }_{S1}=\frac{2\pi ·M_{TS1}-F_{OS1}·s}{\pi ·F_{OS1}·d_s}=\frac{2\pi ·4 Nm-974.2 N·0.0025 m}{\pi ·974.2 N·0.018376 m}=0.4035 \\ {f\prime }_{S2}=\frac{2\pi ·M_{TS2}-F_{OS2}·s}{\pi ·F_{OS2}·d_s}=\frac{2\pi ·8 Nm-1945.6 N·0.0025 m}{\pi ·1945.6 N·0.018376 m}=0.4042 \\ {f\prime }_{S3}=\frac{2\pi ·M_{TS3}-F_{OS3}·s}{\pi ·F_{OS3}·d_s}=\frac{2\pi ·12 Nm-2874.6 N·0.0025 m}{\pi ·2874.6 N·0.018376 m}=0.4110 \\ {f\prime }_{S4}=\frac{2\pi ·M_{TS4}-F_{OS4}·s}{\pi ·F_{OS4}·d_s}=\frac{2\pi ·16 Nm-3823.2 N·0.0025 m}{\pi ·3823.2 N·0.018376 m}=0.4121 \\ {f\prime }_{S5}=\frac{2\pi ·M_{TS5}-F_{OS5}·s}{\pi ·F_{OS5}·d_s}=\frac{2\pi ·20 Nm-4681.6 N·0.0025 m}{\pi ·4681.6 N·0.018376 m}=0.4216 \end{gathered}$$

(11)

The actual coefficients of friction f for the individual tightening torques in abrasion with lubricating silicone grease CX80 are as follows:

$$\begin{gathered} f_{S1}={f\prime }_{S1}·\cos \frac{\alpha }{2}=0.4035·\cos \frac{60°}{2}=0.2614 \\ {f}_{S2}={f\prime }_{S2}·\cos \frac{\alpha }{2}=0.4042·\cos \frac{60°}{2}=0.2618 \\ {f}_{S3}={f\prime }_{S3}·\cos \frac{\alpha }{2}=0.4110·\cos \frac{60°}{2}=0.2662 \\ {f}_{S4}={f\prime }_{S4}·\cos \frac{\alpha }{2}=0.4121·\cos \frac{60°}{2}=0.2670 \\ {f}_{S5}={f\prime }_{S5}·\cos \frac{\alpha }{2}=0.4216·\cos \frac{60°}{2}=0.2731 \end{gathered}$$

(12)

Based on the given calculations, the coefficients of the friction values were determined. Hence, based on the acquired data, the actual coefficients of friction for the individual tightening torques were calculated.

4.3. Abrasion with the Use of Lubricating Lithium Grease CX80

This is a multi-purpose product; thus, it is used for the lubrication of industrial and agricultural systems and for the systems of motor vehicles. It ensures excellent lubrication, even under the toughest conditions.

It is resistant to water and oxidation, protects against attrition and corrosion, extends the life of bearings [4,5], contains EP additives, is mechanically stable, and, due to its soft consistency, is easy to apply. It passes the SKF EMCOR test, with operating temperatures ranging from −20 to +130 °C. Its density is 0.807 g/cm at 20 °C. The measured values with lithium grease are presented in

Table 3. Values determined via measurement with lubricant CX80—lithium grease.

No. of Measurements	1	2	3	4	5
Loading—tightening torque MTL (Nm)	4.0	8.0	12.0	16.0	20.0
Emerging axial force FOL (N)	836.2	1678.5	2518.4	3322.8	4092.3

and in Figure 10.

The calculated coefficients of friction f′ for the individual tightening torques in abrasion with lubricating lithium grease CX80 are as follows:

$$\begin{gathered} {f\prime }_{L1}=\frac{2\pi ·M_{TL1}-F_{OL1}·s}{\pi ·F_{OL1}·d_s}=\frac{2\pi ·4 Nm-836.2 N·0.0025 m}{\pi ·836.2 N·0.018376 m}=0.4773 \\ {f\prime }_{L2}=\frac{2\pi ·M_{TL2}-F_{OL2}·s}{\pi ·F_{OL2}·d_s}=\frac{2\pi ·8 Nm-1678.5 N·0.0025 m}{\pi ·1678.5 N·0.018376 m}=0.4754 \\ {f\prime }_{L3}=\frac{2\pi ·M_{TL3}-F_{OL3}·s}{\pi ·F_{OL3}·d_s}=\frac{2\pi ·12 Nm-2518.4 N·0.0025 m}{\pi ·2518.4 N·0.018376 m}=0.4752 \\ {f\prime }_{L4}=\frac{2\pi ·M_{TL4}-F_{OL4}·s}{\pi ·F_{OL4}·d_s}=\frac{2\pi ·16 Nm-3322.8 N·0.0025 m}{\pi ·3322.8 N·0.018376 m}=0.4807 \\ {f\prime }_{L5}=\frac{2\pi ·M_{TL5}-F_{OL5}·s}{\pi ·F_{OL5}·d_s}=\frac{2\pi ·20 Nm-4092.3 N·0.0025 m}{\pi ·4092.3 N·0.018376 m}=0.4886 \end{gathered}$$

(13)

The actual coefficients of friction f for the individual tightening torques in abrasion with lubricating lithium grease CX80 are as follows:

$$\begin{gathered} f_{L1}={f\prime }_{L1}·\cos \frac{\alpha }{2}=0.4773·\cos \frac{60°}{2}=0.3092 \\ {f}_{L2}={f\prime }_{L2}·\cos \frac{\alpha }{2}=0.4754·\cos \frac{60°}{2}=0.3080 \\ {f}_{L3}=f{\prime }_{L3}·\cos \frac{\alpha }{2}=0.4752·\cos \frac{60°}{2}=0.3079 \\ {f}_{L4}={f\prime }_{L4}·\cos \frac{\alpha }{2}=0.4807·\cos \frac{60°}{2}=0.3114 \\ {f}_{L5}={f\prime }_{L5}·\cos \frac{\alpha }{2}=0.4886·\cos \frac{60°}{2}=0.3165 \end{gathered}$$

(14)

4.4. Abrasion with the Use of Lubricating CX80 Copper Grease

This is an anti-adhesive, multi-purpose product for use under the most difficult conditions and within a high temperature range from −30 to 1200 °C. It ensures excellent lubrication, safeguards against attrition, corrosion, and high temperatures, and has a wide temperature range (−30 to +1200 °C). Meanwhile, its density is 0.9 g/cm³ [18].

It is compatible with screws and nuts, threads and all connected surfaces, fittings, spark plugs, wheel and brake discs, battery joints, port installations, revolving discs, highly stressed bearings of all kinds, elements in the industrial and construction fields, other elements of industrial systems, construction, and agricultural and motor vehicles [19,20]. The measured values with copper grease can be seen in

Table 4. Values determined via measurement with lubricant CX80—copper grease.

No. of Measurements	1	2	3	4	5
Loading—tightening torque MTC (Nm)	4.0	8.0	12.0	16.0	20.0
Emerging axial force FOC (N)	1076.4	2154.3	3236.9	4315.7	5318.6

and Figure 11.

The calculated coefficients of friction f′ for the individual tightening torques in abrasion with lubricating copper grease CX80 are as follows:

$$\begin{gathered} {f\prime }_{C1}=\frac{2\pi ·M_{TC1}-F_{OC1}·s}{\pi ·F_{OC1}·d_s}=\frac{2\pi ·4 Nm-1076.4 N·0.0025 m}{\pi ·1076.4 N·0.018376 m}=0.3611 \\ {f\prime }_{C2}=\frac{2\pi ·M_{TC2}-F_{OC2}·s}{\pi ·F_{OC2}·d_s}=\frac{2\pi ·8 Nm-2154.3 N·0.0025 m}{\pi ·2154.3 N·0.018376 m}=0.3608 \\ f{\prime }_{C3}=\frac{2\pi ·M_{TC3}-F_{OC3}·s}{\pi ·F_{OC3}·d_s}=\frac{2\pi ·12 Nm-3236.9 N·0.0025 m}{\pi ·3236.9 N·0.018376 m}=0.3601 \\ {f\prime }_{C4}=\frac{2\pi ·M_{TC4}-F_{OC4}·s}{\pi ·F_{OC4}·d_s}=\frac{2\pi ·16 Nm-4315.7 N·0.0025 m}{\pi ·4315.7 N·0.018376 m}=0.3601 \\ f{\prime }_{C5}=\frac{2\pi ·M_{TC5}-F_{OC5}·s}{\pi ·F_{OC5}·d_s}=\frac{2\pi ·20 Nm-5318.6 N·0.0025 m}{\pi ·538.6 N·0.018376 m}=0.3659 \end{gathered}$$

(15)

The actual coefficients of friction f for the individual tightening torques in abrasion with lubricating copper grease CX80 are as follows:

$$\begin{gathered} f_{C1}={f\prime }_{C1}·\cos \frac{\alpha }{2}=0.3611·\cos \frac{60°}{2}=0.2339 \\ {f}_{C2}={f\prime }_{C2}·\cos \frac{\alpha }{2}=0.3608·\cos \frac{60°}{2}=0.2337 \\ {f}_{C3}={f\prime }_{C3}·\cos \frac{\alpha }{2}=0.3601·\cos \frac{60°}{2}=0.2333 \\ {f}_{C4}={f\prime }_{C4}·\cos \frac{\alpha }{2}=0.3601·\cos \frac{60°}{2}=0.2333 \\ {f}_{C5}={f\prime }_{C5}·\cos \frac{\alpha }{2}=0.3659·\cos \frac{60°}{2}=0.2370 \end{gathered}$$

(16)

5. Results and Discussion

Figure 12 is a graphical representation of the dependence of the emerging axial force in the bolt on the tightening torque for tests without lubricating grease and with different alternatives of the tested grease.

The dependence of the calculated actual friction coefficients on the tightening torque for the different test variants with or without lubricating grease is shown in Figure 13.

The calculated coefficients of friction in the thread were slightly distorted. This could be due to many factors, including the running-in of the screw connection. This caused an increase in the axial force in the screw threads, given by law, which was the main cause of the scatter of the measured values. Therefore, after each measurement, we completely unscrewed the screw and screwed it in again so that the same threads did not meet again.

Unlike linear theory, the actual dependence of the axial force on the tightening torque is progressive.

Based on the results of the verification experiments, we recommend a linear formula for estimating the lower limit of the axial force F_O when applying the tightening torque M_T, considering the following:

• The actual F_O value can be up to 50% higher, especially at large M_T values, which are suitable for screws with a high yield strength, because the thread surfaces grind against one another. The actual course of F_O is progressive.
• In order to achieve reproducible values on a set of the same screws, attention must be paid to the same method of lubrication [21].

We determined the coefficients of friction in the thread during dry and lubricated friction. By using better lubricants, the value of the coefficient can be minimized to values close to 0.2, which is the standard value of the steel-steel shear coefficient.

In conclusion, it is advisable to use modern hydraulic tightening machines for demanding applications due to their better accuracy and capability for operation without harmful effects on the operator.

For better-quality measurement, it would be competent to use modern hydraulic tightening machines to derive a more accurate value of the tightening torque. These machines have a small deviation value [22].

Due to the dimensional diversity of the screw connection, it would also be appropriate to adapt the measuring device to a more universal format:

• By making inserts for fastening screws of smaller dimensions. The design of the inserts is of great importance. By making them, the coefficient of friction could also be measured in other types of threads.
• By modifying the device so that it is possible to carry out a second method of measurement, in which the nut under the washer is tightened. With this measurement, another quantity would enter our calculations—the friction coefficient between the nut and the washer. In this method, it is necessary to ensure that the screw head is guided to ensure that the screw does not rotate but only moves in the axial direction.
• By adjusting the device frame or its parts so that screws with smaller and larger lengths can be used.

6. Conclusions

The characteristic of the influence of geometrical factors such as thread shape is predominantly deterministic; however, tribological factors, which include friction occurring in the areas of contact, cause particular variance in the result. Therefore, it is necessary to pay attention to its global effects on screw connections from an experimental point of view. A derived semi-empirical formula appeared to be appropriate for those screws with a metric thread.

The laboratory measurements conducted using a testing stand proved that the coefficient of friction is considerably influenced by the loads of the factors, along with well-run screw connections. This causes axial force enlargement after tightening several times and represents the main reason for the variance in the measured values. Therefore, each measurement was followed by complete unscrewing and re-screwing of the screw to avoid encountering identical threads. Contrary to the linear theory, the actual dependence of the axial force on the tightening torque has a progressive characteristic. At the same time, to achieve reproducible values in a set of uniform screws, it is necessary to pay particular attention to ensuring their identical greasing. The results of the experimental measurements and necessary calculations were presented and plotted in individual figures and graphs.

Under laboratory conditions, the copper-based lubricant had the greatest effect on the coefficient of friction between the threads, where the decrease compared to the measurements without lubricating grease was 47.64%. An important factor in the lubrication of bolted joints is the service life or degradation of the lubricating grease. Lubricant degradation is mainly influenced by ambient conditions such as weathering, temperature gradients, and environmental cleanliness. Therefore, further experiments in this field are needed to verify the achieved laboratory results in technical practice.