MECHMT-D-06-00269 Mechanism and Machine Theory (REVISED)

Numerical and Experimental Analysis of Non-Circular

Gears and Cam-Follower Systems as Function Generators

Erika Ottaviano1, Domenico Mundo2, Guido A Danieli2 and Marco Ceccarelli1

1 LARM: Laboratory of Robotics and Mechatronics, University of Cassino, Cassino (Fr) – Italy e-

mail: ottaviano/ceccarelli@unicas.it 2 Department of Mechanics, University of Calabria, Arcavacata di Rende (Cs) – Italy

e-mail: d.mundo/danieli@unical.it

Abstract

The paper shows an analysis of two mechanisms that are typically used as function

generators. The former consists of a pair of non-circular gears, which drives a slider-

crank mechanism, the latter is a cam-follower system. Both mechanisms are designed to

obtain a specific motion law. In this paper the proposed application is to generate a

pulsating blood flow during cardiopulmonary by-pass for cardiac surgery. The

prescribed motion law can be obtained by a volumetric pump, which can be used to

modulate the blood flow in external circulation machines. The reciprocating motion

consists of a quick forward stroke, corresponding to the Systolic phase, and a slow

return stroke, corresponding to the Diastolic phase. The study has been focused on

specific transmission characteristics that are related to a mechanical blood pumping

design. In particular, experimental tests have been analyzed to understand benefits and

drawbacks for using non-circular gears and polynomial cams in pure mechanical

transmissions with limited motion regulation but with specific prescribed motion law.

The contribution of the paper can be recognized in comparing numerically and

experimentally a traditional cam transmission with a non-circular gear solution to show

proper operation feasibility for both solutions without using complex control equipment,

even for a robust/reliable demanding application like a blood pumping system.

Keywords: Mechanism Analysis, Non-Circular Gears, Cam-follower Systems,

Function Generators, Experimental Mechanics

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1 INTRODUCTION

A function generator is a mechanism synthesized to satisfy the optimal Input/Output

relationship for a specified application. In many industrial applications a purely

mechanical device is often desirable to obtain the required motion with robust

operation, like for example for automatic equipment in printing presses, textile industry,

packaging machines, quick-return mechanisms [1-3]. The generation of a specific

motion law is essential also in recreational equipments and rehabilitation devices, in

which the maximization of human output is fundamental.

Several mechanical devices can be designed to obtain a prescribed motion law of the

output element. Among them, cam-follower mechanisms are widely used in modern

machinery. These devices can reproduce almost any characteristics of the follower’s

motion and they are not particularly difficult to design and build. Their manufacturing

costs are rather limited, as well as the number and dimensions of moving parts [4,5].

When a mechanical system is used to obtain a required motion of the output link, non-

circular gears are another possible choice. The application of non-circular gears in

function generating mechanisms has been proposed and discussed in [6,7]. By designing

a pair of non-circular gears, which are able to perform a proper gear ratio function, the

output member of a mechanism can be effectively forced to move according to a

prescribed law of motion, when operated at a constant input-velocity, [8,9]. In typical

arrangements, a pair of variable radius pitch curves is synthesized to drive a slider-crank

mechanism according to a prescribed motion law. For many applications non-circular

gears provide some benefits over cams, although they are more difficult to design and

expensive to manufacture. The main advantages are the lower weight-to-strength ratios

and absence of gross separation or decoupling of moving parts [10].

In this paper, two function generators are designed to drive a flow modulator for

external blood circulation, during cardiopulmonary by-pass for cardiac surgery.

In the first mechanism, the output motion is obtained by using a pair of non-circular

gears. The pitch curves synthesis is formulated by means of the inverse kinematic

analysis of the system. Then, the teeth profiles are generated by assuming the pressure

angle to be constant along each tooth, but variable from tooth to tooth.

The second mechanism consists of a cam-follower system, which is commonly used

to achieve complex and very variable transmission laws between input rotative shafts

2

and output linear or rotative motion of the follower. The cam has been designed

according to the prescribed motion law and with suitable values for the base and

follower radii. Both mechanisms have been prototyped by means of a CNC milling

machine. Suitable test-beds have been designed at Department of Mechanical

Engineering, University of Calabria in Arcavacata di Rende and LARM: Laboratory of

Robotics and Mechatronics in Cassino in order to carry out experimental validation. An

analysis and comparison for both mechanical devices have been carried out at

University of Cassino in order to validate the used design methods and to compare main

characteristics of the two mechanical systems, according to the specific proposed

application. Preliminary result of the proposed application has been presented in [11].

2 DESCRIPTION OF THE REQUESTED MOTION LAW

The application task is to generate a pulsating blood flow during cardiopulmonary

by-pass for cardiac surgery. In human body, the blood flow follows a pulsating law

consisting of two phases known as the Systole and the Diastole [12]. During the systolic

phase, the cardiac muscle contracts in order to pump blood into the cardiovascular

circuit, and during the diastolic phase the heart’s relaxation occurs. In physiological

conditions, the contraction phase is quicker than the relaxation one, being the duration

ratio about 1:2.

Starting point of the design process is the definition of the Input/Output relationship,

i.e. the motion that the function generator must satisfy. Therefore, the aim is to design a

mechanical device, which can be able to drive the piston of a pump that is used to

modulate the blood flow during the external circulation [13].

Several mechanisms have been proposed to reproduce these characteristics in the

external blood circulation, as reported in [14]. A roller pump can be used in order to

generate a constant flow, and then to modulate it by means of a volumetric pump. The

periodic motion law for the modulator’s piston can be determined on the basis of emo-

dynamic considerations, [15]. The required law of motion consists of two phases,

corresponding to the forward and return stroke of the piston. The two phases are defined

separately by two polynomial functions. Four boundary conditions, listed in Table 1,

have been properly established at each extreme of the two regions to guarantee

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continuity between the two curves up to the first derivative of the acceleration, so that a

regular dynamic behavior of the proposed mechanisms could be expected. In order to

fulfill the boundary conditions, i.e. displacement, velocity, acceleration and jerk values

at each extreme, the order of such polynomials is set to 7, according to the following

expression

( ) ∑=

=7

0i

ii tats (1)

in which t is the time parameter and the coefficients ai have been determined for both

strokes, by imposing the boundary conditions of Table 1 as listed in Table 2.

The proposed motion law is characterized by a maximum displacement of 40 mm and a

period of 1 second. In order to obtain a blood flow similar to the one that a healthy heart

usually produces, different durations have been assigned to the strokes. In particular, the

return stroke has been assumed with duration of 0.667 sec, and the forward one lasts

0.333 sec. Figure 1 shows the motion law characteristics for the proposed application.

Table 1. Boundary conditions for the polynomial function in Eq.(1)

Time (s) Displacement

s(mm)

Velocity

(mm/s)

Acceleration

(mm/s2)

Jerk

(mm/s3)

0t = 40 0 -800 10000

3/2t = 0 0 800 10000

1t = 40 0 -800 10000

Table 2. Coefficients of the polynomial function in Eq.(1)

Time (s) a7 a6 a5 a4

3/2t0 <≤ 6243.75 -14568.75 13365 -6087.5

1t3/2 <≤ -702000 4095000 -10142280 13822000

Time (s) a3 a2 a1 a0

3/2t0 <≤ 1666.66667 -400 0 40

1t3/2 <≤ -11193533 5388000 -1428000 160853.333

4

Fig. 1. Requested motion law for the output element: a) displacement; b) velocity;

c) acceleration; d) jerk.

The polynomial of 7th order has been imposed to ensure a prescribed behavior even

in term of limiting vibrating output with the aim to reproduce the smooth of the natural

blood pumping.

3 DESIGN FORMULATION FOR NON-CIRCULAR GEARS

In typical arrangements, a pair of non-circular gears is used to drive a slider-crank

mechanism, so that the piston is forced to move according to a specific law of motion.

The design of these mechanisms consists of two phases, namely the synthesis of the

pitch curves, starting from the requested law of motion, and the determination of teeth

profiles.

3.1 Synthesis of the pitch curves

The synthesis of the pitch curves can be formulated by means of the inverse

kinematic analysis of the mechanism. Starting point is the prescribed law of motion of

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the piston s(t). In the proposed application, the purpose is to provide the piston of a

volumetric pump with the specific law of motion that is represented in Fig. 1.

Referring to the schemes in Fig.2, the relation between the displacement of the

connecting-rod end D and the rotational angle δ on the driven gear is

( ) ( ) ( )( )⎥⎦⎤

⎢⎣⎡ δλ−−

λ−δ−=δ 22

D sin111cos1rs (2)

where λ is the crank-to-rod ratio, equal to 0.2 in this application, the crank length r = 20

mm, as it can be assigned on the basis of the stroke value. The constant angular velocity

of the driving gear is ω,while Ω is the variable angular velocity of the driven gear.

By coupling the analytical law sD(δ) and numerical expression sD(t) in Eq.(1), the

time dependence of the rotational angle of the driven gear can be deduced. The

rotational angle θ of the driving gear can be evaluated as

( ) tT2tt π

=ω=θ (3)

where T is the period of the driving gear motion that has been fixed equal to 1 second in

this application. Once δ(t) and θ(t) are known, the variable gear ratio can be evaluated

as a function of θ in the form

Fig. 2. Schematic representation of the mechanism.

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( )( ) ( )( )tdtdt

θδ

=ωΩ

=θτ (4)

Figure 3 shows the computed variable gear ratio that is characterized by a variation

between 0.66 and 1.77, with an average value equal to 1.00.

By imposing that the contact points of the pitch curves have the same velocity, a first

relation between the pitch curves radii can be formulated as

( ) ( ) ( )( )( )θδθ

=ωθΩ

=θτ2

1

RR

(5)

where R1 (θ) and R2 (δ(θ)) are the pitch curve radii at the time t=θ/ω.

In order to determine the shape of the pitch curves a second relation can be obtained

by considering the constant distance rc between the centers of rotation

( ) ( )( ) c21 rRR =θδ+θ (6)

In this application the constant value rc=100 mm has been assumed.

The pitch curves for the requested output curve of Fig. 1 can be obtained by

combining equations 4 and 5. The synthesized curves are shown in Fig. 4.

Fig. 3. Gear ratio as a function of the driving gear angle.

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Fig. 4. Computed pitch curves generating the output curve of Fig. 1.

It is worth to note that the pitch curves represented in Fig. 4 have the same length,

since the average value of the gear ratio function is equal to one. This ensures that each

tooth of the driving gear always meshes with the corresponding tooth on the driven gear.

3.2 Determination of Tooth Profiles

The numerical generation of tooth profiles in non-circular pitch curves is based on

the description of the meshing evolution and can be performed according to several

methods [16]. In this paper the mathematical model of tooth profiles has been based on

the following differential equation

( ) ( ) ( )⎥⎦⎤

⎢⎣⎡ α

θ−θα=

θtan

ddRRcos

ddy 1

12 (7)

where dθ refers to the rotation of the driving gear, and dy is the corresponding

displacement of the contact point, as projected onto the normal direction to the curve

crossing O1 and O2 [17]. The angle α is the constant pressure angle. Equation (7) can be

integrated numerically by starting from the intersection point of a tooth profile with the

pitch curve. Points can be derived as a pair in the conjugate profiles, by means of a

coordinate transformation that is based on two suitable rotation matrices.

8

Once established the contact point between the first teeth pair on the pitch curves and

given the diameter pitch and number of teeth, the intersection point of any profile with

the pitch curves can be determined straightforward .

The integration process stops when one of the following conditions occurs: the radial

dimension of a tooth exceeds the established upper bound; the distance between the

contact point and gear centre is stationary. The last condition refers to undercutting

conditions. By using the above-mentioned procedure for the pitch curves in Fig. 4 and

assuming a number of teeth equal to 30, the profiles are generated as represented in Fig.

5.

Since the tangent line to the pitch curves at the contact points shows variable

direction, the pressure angle is varied from tooth to tooth, so that the angle between the

pressure line and the normal to the pitch curves has always the same prescribed value of

20 degree [18]. Of course, two meshing teeth on different gears have the same pressure

angle.

In order to check for the continuity of power transmission, the contact ratio can be

evaluated through the parameter

1i

1i,i

1i

i,1i1c+

+

−

−

θ∆

θ∆+

θ∆

θ∆+= (8)

Fig. 5. Designed teeth profiles

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where ∆θi-1,i is the rotation of the gear 1 while the meshing evolves from the beginning

of the contact on the ith tooth to the end on the (i-1)th tooth; ∆θi,i+1 corresponds to i th and

(i+1) th teeth; ∆θi-1 is the rotation angle referring to (i-1)th tooth; and ∆θi+1 is for (i+1)th

tooth.

Computing Eq. (8) for the designed gears in Fig. 5, the meshing continuity is

obtained with suitable contact ratio greater than one along the whole pitch curve, as

shown in Fig. 6.

The contact ratio curve shows the lowest values in the region between the 8th and the

12th tooth pair, with a minimum value in correspondence of the 11th tooth pair.

Such a behavior is due to the fact that non-undercutting condition varies along the

pitch curves of non-circular gears, as indicated by Danieli [19]. The non-undercutting

condition is used to limit the extension of each profile, thus it gives a gear design with a

variability of the contact ratio.

The last step of the design process has been the 3D modeling of both gears. In Fig.

7a) the CAD model is shown as useful for CNC manufacture which results are reported

in Fig.7b). In particular, the gears of Fig. 7a) have been manufactured by means of a

CAM software and CNC machine Mazak Nexus 410 A, by using an aluminum alloy.

The prototypes of the gears are shown in Fig. 7b), where it is possible to note that

additional considerations have been used to obtain a mechanical design of the gears for

a dynamic balanced operation.

1,4

1,5

1,6

1,7

1,8

1,9

1 5 9 13 17 21 25 29 33 37 41 45 49 53 57No. profile

Fig. 6. Contact ratio for the designed gears in Fig.5.

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Fig. 7. Non-circular gears: a) a CAD model; b) manufactured prototypes.

4 DESIGN FORMULATION FOR CAM-FOLLOWER SYSTEM

Cam design is usually based upon the definition of a required motion, which is often

given in the form of timing diagram. In fact, a cam profile is defined by the required

output function of the follower. The problem of the profile definition can be formulated

by considering the extreme positions of the follower without any specification of

intermediate path points, or it can be formulated by taking into account the path or its

derivatives over any portion of the operation cycle, [20, 21]. The usual approach

consists of imposition of displacement, velocity, acceleration and jerk of the cam

follower. For the proposed application a specific motion law is given by Eq. (1) and

Tables 1 and 2 and shown Fig. 1, and the design problem for the cam-follower system

consists of determining the base and roller radii.

The synthesis procedure has been based on a traditional technique that has been

described synthetically, since main attention of the paper is addressed to analysis

aspects. The mechanical design of the built cam is also outlined in Figs. 8 and 9 as

referring to a dynamically balanced solution, which has been obtained by using a fairly

simple geometry of the void for an easy machining. The base radius Rb can be designed

in order to obtain a suitable pressure angle that can give a good mechanical

transmission. Let us consider the maximum pressure angle αmax through the expression

( )sb

vmax R

hctan

θ=α (9)

in which cv is the velocity coefficient, which has been set equal to 2 according to the

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specific motion law in Eq.(1); h is the maximum stroke; θs is the rise angle. By

choosing a maximum value of 30 deg for the pressure angle, [22], Eq.(9) gives a value

for Rb equal to 65 mm. A value of 25 mm has been chosen for the roller radius, in

agreement with the dimension of the cam. The mechanical design of the cam for the

proposed application is shown in Fig. 8. A prototype has been built by means of CNC

machine Mazak Nexus 410 A, by using aluminum alloy. The built prototype is shown in

Fig. 9. Additional consideration have been used to obtain a mechanical design of the

cam for a dynamically balanced operation.

Fig. 8. Mechanical design of the designed cam for the proposed application

.

a) b)

Fig. 9. The designed cam in Fig.11: a) a CAD model; b) the manufactured prototype.

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5 EXPERIMENTAL TESTS

Low-cost test-beds have been designed and settled up at LARM in Cassino, Italy.

They have been used for experimental tests and validation of cams, [11; 23-26]. In this

paper, a low-cost test bed for experimental validation is presented for tests on both the

herein designed cam-follower and non-circular gears systems, as shown in the scheme

of Fig. 10. In particular, in order to compare kinematic performances of the two

mechanical transmissions the same acquisition system, servomotor and sensors have

been used for both test-beds. For each test-bed configuration it has been thought

convenient to use LabView software with NI 6024E Acquisition Card, in order to work

with suitable virtual instruments, which can manage commercial sensors. Thus, an

accelerometer S1 has been used to determine the output acceleration of the follower.

Signal conditioner and amplifier denoted by A1 in Fig. 10 have been used in order to

provide suitable power supply to S1 and to reduce the noise in the measured signal. The

same DC motor with servo amplifier, an encoder S2 and a tachometer S3 have been used

in order to give and to monitor the input motion. In particular, the encoder gives the

possibility to monitor the angle of the input shaft, whereas the tachometer is used to

monitor the angular velocity of the input shaft. In addition, three control modes can be

selected by DIP switches on board of the servo amplifier: - speed control using

tachometer signals or encoder signals; - compensated speed control; - torque or current

control. In addition, signal conditioner and amplifier A2 have been used in order to

provide suitable power supply to S2 and S3 and to reduce the noise in the measured

signal. A linear transducer, S4 has been used only in the test-bed configuration with the

non-circular gears.

The test-bed with non-circular gears shown in Fig. 11 consists of a base, on which

the driving and driven shafts are supported. It is fundamental that the distance between

the two axes has the nominal value of 100 mm, otherwise an irregular transmission can

be obtained. On the same base a slider-crank mechanism has been assembled, so that a

crank radius of 20 mm is obtained on the driven gear. Figure 12 shows a particular of

the test-bed with the cam transmission system.

The displacement history of a follower point in the cam system gives the output

function as related to the cam profile. Once the cam transmission operates properly, the

cam profile is the characterizing element of the transmission operation of a cam system.

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Thus, it is not necessary to ensure an accurate cam profile fulfilling the prescribed

motion law. Since the cam has been manufactured by means of a CNC machine to

obtain the prescribed polynomial profile, the output motion of the follower has been

verified by verifying accurately the cam profile of the built prototype before its use.

Even zero eccentricity between cam and follower has been verified in the assembling of

the test-bed in Fig. 12.

Since the aim of the transmission systems is to generate accurately a prescribed

function, operation efficiency can be controlled/ensured mainly by looking at the

fulfillment of the motion characteristics that give the function. Of course, this require a

consideration of dynamic conditions, but they can be synthetically considered by

looking at the actuator torque that gives the prescribed function as output motion. Thus,

the feasibility of the non kinematical aspects can be considered by looking at the

feasibility of the actuation torque both in terms of practical possibility for a commercial

motor and a proper dynamic operation. Both of them can be checked by looking at the

time history of the actuation torque. Thus, the proposed test-beds allow also to develop

considerations about dynamical aspects for both transmission systems in order to take

into account those aspects, the input torque has been also monitored for the proposed

experimental results.

Fig. 10. A scheme for the sensored test-bed for mechanical transmission system testing.

(S1 is an accelerometer; S2 is an encoder; S3 is a tachymeter; S4 is a linear transducer.)

14

Fig. 11. A view of the test-bed with non-circular gears of Fig. 7.

Experimental tests have been carried out on the mechanical test-bed transmission

systems of Figs. 11 and 12 by imposing the input velocity to be a constant value

according to design requirements. Information about acceleration is then obtained by

using an accelerometer; actuation torque has been monitored by measuring the DC

motor current. In Fig. 13 an experimental test is reported with the non-circular gears of

Fig. 11. In particular, the input torque and follower acceleration have been reported.

Fig. 12. A view of the test-bed with the cam in Fig. 9.

15

a) b)

Fig.13. Experimental results of the operation of the non-circular gears system in

Fig.11: a) input torque; b) follower acceleration.

Figure 14 shows experimental results that have been obtained with the cam system

in Figs. 8 and 12. In particular, Fig. 14a) shows the input torque versus time; Fig. 14b)

shows the measured acceleration of the follower.

A comparative analysis of the two different mechanical systems shows a good

match with respect to the prescribed motion characteristics. The comparison between

numerical and experimental results for the follower acceleration for both mechanical

transmissions is reported in Fig. 15. Discrepancies between theoretical and

experimental results on acceleration plots can be considered due mainly to dynamical

aspects that are not taken into account in the proposed kinematic analysis.

a) b)

Fig.14. Experimental results of the operation of the cam-follower system in Fig.12:

a) input torque; b) follower acceleration.

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a) b)

Fig. 15. A comparison between theoretical and experimental results for the follower

acceleration: a) non-circular gears; b) cam system. (dotted line is for theoretical

acceleration; continuous line is for experimental acceleration)

The results that are plotted in Fig. 15 have been considered satisfactory as matching

the prescribed data in Fig. 1, since the function shape is properly obtained and

deviations are very limited in the pulsing peaks, as easily observable also in Fig.15.

The relationship between the experimental and theoretical displacement of the slider

is shown in Fig.16. The experimental curve is well fitted on the theoretical one: the

values of 0.081 mm and 0.06 mm are the average and the standard deviation of the

difference between the measured and theoretical curves, respectively. The maximum

deviation between the two curves is equal to 0.3 mm.

By looking at numerical simulations in Fig. 1 and mainly at experimental results in

Figs.13 to 16 a comparison of characteristics can be outlined in term of the following

observations:

- non-circular gears show more vibratory response when the input velocity is not

exactly the prescribed value for the design solution;

- cam transmission has a limited maximum value for the input velocity, which is

due to the condition given by the spring ensuring proper contact between cam

profile and follower contact surface;

- cam transmissions have a compact design with reduced dimensions and they can

ensure a robust repetitive operation because of a robust mechanical design. They

can have limitations because they suffer of noise and vibrations due to the

17

profile itself as combined with dynamic properties of the spring element of the

output slider;

- transmissions with non-circular gears have a suitable operation performance that

can be designed at proper matches for specific applications by even avoiding

springs for the return stroke.

Fig. 16. A comparison between theoretical and experimental results for the follower

displacement for non-circular gears. (dotted line is for theoretical acceleration;

continuous line is for experimental acceleration)

6 CONCLUSION

In this paper two function generators, namely, a pair of non-circular gears and a cam-

follower system have been designed to generate a prescribed motion law for an

application concerning blood external circulation in surgical applications. Suitable

design solutions have been designed and tested. The experimental validation of the

proposed mechanical designs have been carried out by means of suitable test-beds,

which have been designed at the Department of Mechanical Engineering of University

of Calabria in Arcavacata di Rende and LARM: Laboratory of Robotics and

Mechatronics in Cassino. Main characteristics of the proposed mechanical have been

outlined by using both numerical and experimental evaluations. In particular, specific

attention has been addressed to a comparison between geared solution and cam design,

since usually they can be considered as competitive solutions for transmissions even in

many mechanical machines as for example in packaging equipment. Experimental tests

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show that both the two mechanical systems can reproduce very well the prescribed

motion law. Experimental results approximate closely the prescribed design acceleration

diagrams, with small oscillatory deviations in the region of low acceleration values. In

particular, cam transmissions have a compact design with reduced dimensions and they

can ensure a robust repetitive operation because of a robust mechanical design. They

can have limitations because they suffer of noise and vibrations due to the profile itself

as combined with dynamic properties of the spring element of the output slider.

Transmissions with non-circular gears have a suitable operation performance that can be

designed at proper matches for specific applications by even avoiding springs for the

return stroke. But they can suffer of an unknown reliability since they need an average

transmission ratio equal to one or a multiple integer, as due to the periodicity of their

pitch curves. This forces to have an equal number of teeth in the two gears, which is a

potential hazard for an efficient operation. Summarizing, the reported numerical and

experimental analysis shows efficiency of mechanical transmissions with robust design

and accurate behaviour that can be still useful even in non-industrial applications like in

medical equipment with very reliable operation but without complex system integration.

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