Determination of coefficient of friction for a novel bed mechanism

Designing new products and deciding their specifications and power requirements are always challenging for engineers and designers. Mechanical systems consist of various components like screws, gears, bearings, etc. All these elements cause frictional losses in the drives. Therefore, in any system the actual power or torque requirement are higher than the designed. As a result, after designing and developing mechanical systems it is important that the friction or other losses are found out. It helps in appropriate selection of drive power. It also helps in correcting the design if actual requirements are much higher than the designed ones. In this context, this paper first describes the design of a new bed and then it discusses simple analytical and practical methods to find friction in the drive.

For developing new products, different guidelines, methods, algorithms, and suggestions are available in the literature [1–4]. According to Kaldate et al [1] matrix methods such as House of Quality are the first step in relating product attributes to engineering parameters. They proposed a method to reduce the number of engineering parameters for product design to minimize the product cost. van Kesteren et al [2] have focussed on identifying critical factors influencing material selection process in user centred design projects. Schmitt et al [3] presented a model based on theoretical mechanical engineering design approaches. But, the focus was on extending system boundaries to modern work environment of mechanical design engineers. Three areas of operation relevant to economic product design were included in the model. The areas were: classification of products according to the designated market, the decision-making process and the creation sequence. Sheldon et al [4] discussed a number of practices in manufacturing and construction field in different industrial sectors. They investigated the underlying methodologies and general procedures. It was concluded that future research should recognize generic aspects to fit design for cost into the company infrastructure as a fundamental working principle. Few researchers have designed beds with integrated toilets to address the issue of care and assistance required for bedridden patients [5, 6]. Researchers [7] had surveyed fifty two persons including Doctors, patients & their family members, hospital staff and attendants for deciding the criteria to design a bed for bedridden patients. The survey included personal interaction and filling of a questionnaire by the people involved in managing bedridden patients. Based on the survey, the requirements for the design of the bed were listed. Provision of commode was one of the main suggestions. Based on the product requirements and specifications decided after the survey, various mechanisms were synthesized and analysed [8]. After mechanism synthesis, the bed was designed, fabricated, and tested for its functionality [5, 9]. Drawing of the designed bed is shown in figure 1. An important mechanism in this bed is a transfer mechanism used for moving the platform down and then horizontally away from the opening for commode and vice versa. The transfer mechanism is operated manually by a handle through a screw. This paper discusses the design of this mechanism, the calculation of torque for its operation, and experimental validation. The next section describes, in brief, the mechanism operation and its two different positions.

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1.1. Platform mechanism

The bed has two positions. Usually, the commode (part 2 in figure 1) is away from the platform (part 1), and a mattress on the platform covers the opening in bed. The platform mechanism is actuated by a telescopic screw and nut mechanism (part 4), which is operated by handle/hand wheel (part 5), and it is crucial to estimate the torque required to operate the device. Figures 2 and 3 show the two positions of mechanism along with desired dimensions, which were determined by considering available space below the bed and travel of platform and commode.

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While retracting the platform and lifting the commode for use, the handle (part 5) attached to the screw requires rotation, the platform moves down as stopper 1 (part 8 in figure 2) restricts its horizontal motion. During this downward motion of the platform, the commode mechanism does not operate as the telescopic puller mechanism (between points 10 & 11 in figure 2) does not open to its maximum position till the platform reaches its extreme downward position. On further rotation of the screw, the platform starts moving horizontally to clear an opening in bed. As soon as the platform goes away from the hole in the bed, the telescopic puller mechanism achieves open condition and starts pulling the commode mechanism, thereby lifting the commode to the required top position.

While withdrawing commode after use and lifting platform, the handle rotation is in the reverse direction, and the telescopic screw mechanism pushes the platform towards the opening. This motion allows the telescopic puller mechanism to close, and it starts lowering the commode. The lowering of this commode is complete when the platform is away from the hole. Further motion of screw allows the platform to move horizontally towards the hole. When the platform reaches its position below the hole, further horizontal movement is restricted by a stopper 2 (Part 9 in figure 2). Further action of screw elevates the platform to close the opening.

1.2. Mechanical design of platform

The platform lowering/lifting mechanism works on the principle, as explained below. The two extreme positions of the arrangement—platform at the lowest and highest position are shown in figures 4 and 5 respectively. The platform mechanism is attached to the screw mechanism by pin (part 16). The platform mechanism consists of a set of parallelogram links on both sides of the platform. The lower three joints are with rollers (part 13). These rollers are free to move in channel guide (part 3). Out of the three upper joints, the middle one (part 14) is fixed at the center of the platform side along the length of the bed, whereas the other two joints (part 15) have rollers. The upper rollers are free to move in a channel below the platform (part 1). Rotation of screw pushes the platform towards the hole in the bed, and the platform freely moves on rollers (part 13), on the channel (part 3). A stopper (part 9) restricts the horizontal travel of the platform when it reaches the desired position. Rotation of screw after reaching the desired position activates parallelograms and raises the platform. When the direction of the screw rotation reverse, the platform lowers first and then moves away to clear the hole. Various components of this system were designed using standard design procedures.

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These two extreme positions of the platform give two different locations for links. These locations have different force requirements. The maximum force requirement is useful for the design of various components and torque calculations. Hence, the next section discusses the torque calculation for operating the mechanism.

The following data are taken from the Design data book [10] and the fabricated bed to estimate the torque and sizes of machine elements such as pins, rollers, screws, and similar parts.

Material for components: SAE 1015 hot rolled steel

Coefficient of friction (μ) = 0.2

Mass of platform = 5 kg

Maximum weight of person on bed (W) = 120 kg

Effective weight on platform = 80 kg (2/3 of W as other body parts are supported)

Total weight on the platform mechanism = 85 kg

Factor of safety = 1.2.

Load for testing of platform = 85 × 1.2 = 102 kg.

Therefore, the platform was tested for a maximum load of 100 kg, by varying load between 10 to 100 kg with load steps of 10 kg. The maximum force on links (figures 6 and 7) for design and torque estimation comes as; F '= 100 × 9.81 = 981 N.

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The calculated length of the parallelogram link was 70 mm for a 90 mm vertical travel of platform. Referring to figure 6 showing forces on links of parallelograms in the uppermost position of the platform, the minimum angle between link AB and vertical (θ₂) = 18°.

Therefore, force along CB at θ₂ = 18°, F_CB = F' × tan (θ₂) = (100 × 9.81) × tan (18°) = 318.75 N.

Referring to figure 7, showing forces on links of parallelograms in the lowest position of the platform, the maximum angle between link AB and vertical (θ₁) = 72°. Hence, Force along C'B' at θ₁ = 72°, F_C'_B' = F' × tan (θ₁) = (100 × 9.81) × tan (72°) = 3019.2 N

Thus, maximum axial force on screw for 100 kg mass on bed = 3019.21 N, which is at the lowest position of the platform. Considering this axial force as the load on a lead screw having square threads, equation for moving load is applied to find the torque to operate the mechanism. The following dimensions of screw are used for torque calculation after designing the screw.

• Core diameter for screw = 20 mm,
• Outside Diameter = 25 mm,
• Mean diameter (d_mean) = 22.5 mm,
• Lead of thread (L) = 5 mm = Pitch (p) of thread for a single start thread

Now, Torque for moving load by a lead screw,

$$\begin{eqnarray}&&T=\displaystyle \frac{P{d}_{mean}}{2}\,\tan \left(\varnothing +\alpha \right)\end{eqnarray} \tag{ 1 }$$

Where, φ = Angle of friction at screw, and $\tan \left(\varnothing \right)=\tfrac{\mu }{sin\beta }$

β = angle between screw face and axis of screw. β = 90° for square thread

Therefore, sin( β) = 1, and hence $\tan (\varnothing )=\mu .$

α = Helix angle and $\tan \left(\alpha \right)=\tfrac{L}{\pi {d}_{mean}}.$

The torque equation can be written as

$$\begin{eqnarray}&&T=\displaystyle \frac{P{d}_{mean}}{2}\left(\displaystyle \frac{\tan \,\varnothing +\,\tan \,\alpha }{1-\,\tan \,\varnothing \,\tan \,\alpha }\right)\end{eqnarray} \tag{ 2 }$$

Substituting for $\tan \,\varnothing .$ and $\tan \,\alpha$ from the above, we get

$$\begin{eqnarray}&&T=\displaystyle \frac{P{d}_{mean}}{2}\left(\displaystyle \frac{\mu +\tfrac{L}{\pi {d}_{mean}}}{1-\mu \tfrac{L}{\pi {d}_{mean}}}\right)\end{eqnarray} \tag{ 3 }$$

simplifying the above, we get,

$$\begin{eqnarray}&&{\rm{T}}=\displaystyle \frac{{\rm{P}}\times {{\rm{d}}}_{{\rm{mean}}}}{2}\left(\displaystyle \frac{{\rm{L}}+\mu \pi {{\rm{d}}}_{{\rm{mean}}}}{\pi {{\rm{d}}}_{{\rm{mean}}}-\mu {\rm{L}}}\right).\end{eqnarray} \tag{ 4 }$$

Then, for axial load (P) = 3019.2 N and assuming μ = 0.2, ${\rm{T}}=9329\,N-{\rm{mm}}.$

Equation (4) gives torque (T) required to rotate the screw for a load of 100 kg = 9329 N-mm. To verify the calculated value of T, an experimental setup was designed for testing. The next section discuss the experimental validation.

Before fabricating experimental setup, we studied various analytical methods and experiment structures described in the literature for the determination of frictional torque. The following paragraphs discuss some such works.

Marques et al [11] reviewed the modelling and analysis of frictional effects in multibody sysms. It was concluded that in most of the cases a static friction model is a suitable choice for use. Benavides et al [12] presented analysis of friction systems on non-accelerating shafts under dry friction in various engineering applications. They proposed equations to analyse static and dynamic friction under different conditions. Few authors [13] have proposed methods to determine frictional torques in high precision rotating equipment such as air bearings and gyroscopes. They have used an experimental method also. The experiment methods in this paper involved bringing the rotating cylinder up to operating speed, then cutting the power and allowing the Coulomb and viscous drag to decay the cylinder speed. They calculated the desired values with the measured quantities and proposed techniques. The authors have provided two methods for calculation, but none is preferred. Some groups of researchers [14, 15] have also discussed the experiment set up to find frictional torque. They developed a hydraulic test machine to load and articulate a large diameter pin held in four bushes, typically found in earthmoving machinery and aircraft landing gear. The rig was capable of applying realistic loads and speeds of articulation while measuring torque and angular displacement. The coefficient of friction was between 0.02 and 0.12. This friction was mostly independent of load but depended strongly on speed as more lubricant entered into the contact. In some works related to friction measurement, Olofsson and Holmgren [16] have carried out friction measurement at low sliding speed with a servo-hydraulic tension—torsion machine. de Crevoisier al. [17] have determined in situ frictional properties of bolted assemblies by using image correlation. Ise et al [18] have measured friction coefficients of tire grounding surface under high load. Alaci et al [19] proposed a simple, cheap and rapid method based on motion of a conical pendulum, for measuring coefficient of rolling friction between a steel ball and a plate of similar or dissimilar materials.

The mechanisms in the above works [13–15] were either for high power, had well lubricated single pin joints, or the machines and techniques used were sophisticated [16–18]. However, the mechanism discussed in this paper is complex in nature and runs at a low speed. It has multiple pin joints with less lubrication, rolling contacts, and a power screw. Hence, the authors designed a simple test for the complete mechanism to determine in situ friction.

For the experimental load test, the bed frame was used (figure 8). The horizontal movement before the frame pipe was restricted. The friction between the platform and stoppers was reduced by providing rollers on the platform, which were able to slide on stoppers (figure 9). Different sizes of plates were used to apply different loads (figure 10). The crank (handle) of the driving screw was replaced by a cycle wheel of 570 mm diameter, and it had a wire rope wound on it. The load was applied to the wire rope with sandbag attached to it. Sand was used to know the correct weight and also to have gradual increase in the weight of the bag. A total of ten readings were recorded for 10 to 100 kg load on the platform. The experiments were repeated for three sets of readings.

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The test results obtained, after following the procedure described earlier, are presented in table 1. The torque was calculated for all the loads on bed. Table 1 shows the three values of measured torque obtained from the weights required to move the platform and calculated values of the coefficient of friction for operating the mechanism. The coefficient of friction was calculated by putting the measured torque value, axial load and screw parameters in the equation 4. Table 1 shows that for every load on the bed, the weight needed to start moving the load is almost same in all the three sets of readings. Thus, the torque values are consistent. Also, these values are such that the torque can be applied manually. Also, the actual coefficient of friction shows a decreasing trend with load on the bed (figure 11). Maximum friction (μ = 0.3364) is observed at the lowest load of 10 kg and minimum friction (μ = 0.2133) is at the maximum load of 100 kg. The average value of coefficient of friction is 0.2396. It is also observed that initially when the load is increased from 10 kg to 20 kg, there is a sudden drop in the coefficient of friction. This is because at low load, the static deformations because of load goes into covering the joint clearances before the start of link motions. Further, it is seen that after a load of 50 kg, the coefficient of friction is almost constant as all the clearances are immediately covered due to higher loads. Similar relationship between the friction torque and angular velocity was reported by Pawan Kumar Hari et al [20] for a sensor less vector controlled induction motor drive. A graph of the weights necessary to move the load on the platform is in figure 12. It shows that the weight and hence the torque required for the movement of the platform is directly proportional to the load on the bed platform. In a similar work on a height adjustment mechanism in a hospital bed, Malujda et al [21] determined relation between various loads on the bed deck and the minimum force required to drive the mechanism. The results indicated that the starting force decreases with the increase in the height of the mechanism. However, their aim was to position the mechanism parts to get lowest possible driving force.

Table 1. Measured torque and coefficient of friction based on load test (L = 5 mm and d_mean = 22.5 mm).

		Weight required to move platform up (kg)			Torque required, T (N-mm)			Calculated coefficient of friction (μ) $\mu =\tfrac{2\pi T{d}_{mean}-PL{d}_{mean}}{P\pi {d}_{mean}^{2}+2TL}$
Load on bed (kg)	Max. Axial load, P (N)	Trial 1	Trial 2	Trial 3	1	2	3	1	2	3	Avg. μ
10	301.92	0.50	0.51	0.51	1398	1426	1426	0.3312	0.3390	0.3390	0.3364
20	603.84	0.84	0.83	0.84	2349	2321	2349	0.2685	0.2645	0.2685	0.2672
30	905.76	1.17	1.17	1.18	3271	3271	3299	0.2447	0.2447	0.2474	0.2456
40	1207.68	1.50	1.50	1.51	4194	4194	4222	0.2329	0.2329	0.2349	0.2335
50	1509.60	1.83	1.84	1.84	5117	5145	5145	0.2258	0.2273	0.2273	0.2268
60	1811.52	2.17	2.17	2.18	6067	6067	6095	0.2223	0.2223	0.2236	0.2227
70	2113.45	2.50	2.50	2.51	6990	6990	7018	0.2187	0.2187	0.2198	0.2191
80	2415.37	2.83	2.84	2.84	7913	7941	7941	0.2160	0.2170	0.2170	0.2167
90	2717.29	3.17	3.17	3.18	8863	8863	8891	0.2148	0.2148	0.2157	0.2151
100	3019.21	3.50	3.50	3.51	9786	9786	9814	0.2130	0.2130	0.2138	0.2133
Average of column								0.2388	0.2394	0.2407	0.2396

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The above results are for a fixed length and diameter of the screw because the testing is done on the actual product (bed). It was not possible to alter the length or diameter of screw as it would involve modification in the entire assembly. However, if the screw length is changed, there will not be any change in friction as the size of nut and hence the frictional contact remains unchanged. In case of increase in the diameter of the screw, it will result in increase in the frictional contact for the same length of nut and hence the friction will increase. It is to be noted that the mechanism used in this work is a slow speed, manually operated mechanism needing less than a minute to complete forward/reverse motion. Therefore, we do not anticipate any significant rise in temperature of the contact surfaces. This was confirmed during testing also. Therefore, the temperature dependent material properties are not considered here.

Table 2 shows the error in torque calculated by using the mean value of coefficient of friction (μ = 0.2396), with respect to actual torque. The table indicates that assuming a constant value of coefficient of friction, the error is less than 10% at all loads except the lowest value of 10 kg. Thus, a constant value of coefficient of friction can safely be assumed to estimate the torque required to operate the mechanism. Figure 13 shows the variation of error in the calculated values of torque with the load on the bed. It is observed that the error curve also follows the same trend as that of variation in the coefficient of friction shown in figure 11.

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This paper has presented results of calculation of coefficient of friction (μ) for operating a new platform mechanism in a novel bed with commode. The friction coefficient is calculated using the torque required for operating the mechanism. The operating torque was obtained by providing an attachment to rotate the mechanism in place of the conventional handle used for its operation. The work involves testing on the final product (bed) itself. Thus, the results obtained can be directly applied to the actual product. Based on the experiments performed, the conclusion is that the coefficient of friction (μ) to operate the mechanism does not remain same and it decreases with the load on the bed. The value of μ is decreasing from a maximum of 0.3364 to a minimum of 0.2133. The mean value of the coefficient μ was found to be 0.2396. The estimated torque values assuming the mean value of μ are found to be within 10% of the actual values of torque. From the obtained values of torque, it can be concluded that the bed mechanism can be conveniently operated by manual means. These toque and friction values would also help in the determination of crank size for manual turning of screw and motor rating for automatic operation of the bed. Thus, the paper provides a practical way for estimating in situ friction in complex mechanisms.

There is no funding received for the work presented in this manuscript

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