Research Article Open Access
Force Adaptation in Robot Transmissions
Konstantin IVANOV1*, Marco Ceccarelli2 and Baurjan Tultaev3
1Almaty University of Power Engineering and Telecommunication, Baytursinov st. 126, Almaty 050013, Kazakhstan
2University of Kassino and South Latium, Italy
3Djoldasbekov Institute of Mechanics and Machine Science MON RK, Kurmangazy st.19, Almaty 050000, Kazakhstan
*Corresponding author: Konstantin IVANOV, Professor, Almaty University of Power Engineering and Telecommunication, Baytursinov st. 126, Almaty 050013, Kazakhstan, Tel: +8-727-248-3052, E-mail: @
Received: July 27, 2017; Accepted: August 07, 2017; Published: August 20, 2017
Citation: IVANOV K, Marco C, Baurjan T (2017) Force Adaptation in Robot Transmissions. Int J Adv Robot Automn.2(2):1-8.
Abstract
In this work the theoretical bases of the force adaptation in robot transmissions are stated. The force adaptation in robot transmissions considers creation, research and designing of robots with adaptive electric drives of modules. The adaptive drive contains the electric motor and the adaptive gear mechanism possessing property independently to change output speed of movement depending on loading. This property of adaptation is named as self-regulation. Self-regulation is carried out only at the expense of mechanics and does not demand a control. The adaptive drive demands smaller power, overcomes emergency overloads, is structurally simple and has small dimensions and weight. These advantages are especially important for intermediate modules of the manipulator. Use of adaptive drives provides high power efficiency of the robot and decrease of the sizes.

Keywords: Adaptive robot transmissions; Electric drive; Force adaptation;
Introduction
The drive of the module of the manipulator is performing the starting-up, the steady-state motion and the braking of a moving part of the module at execution of each operation. Speed of motion varies from null to a maximum and back. The greatest loading in the form of an inertia force takes place in the starting-up of operation. The adequate drive should have the variable transfer ratio in order to overcome a start inertia resistance. However application of the adjustable gear box in the manipulator drive is impossible as the drive connecting gear should have small sizes and weight. Therefore the non controllable drive with the constant transfer ratio is used. The engine of the non controllable drive must overcome the short maximum inertial resistance that reduces power efficiency of the drive because in the rest time a loading will be significantly less. Such drive creates dynamic stresses of motion and vibration. At the present time the variable-frequency electric drives of alternating current are finding the more and more wide application in a robotics. They are expelling the uncontrolled electric drives in many areas and also the drives realizing insufficiently economic control modes of speed [1].

The transfer ratio is changed by means of a control system affecting on electric current parameters. In the frequency-operated electric drive the torque is transferring from the engine to loading through the reducer having the constant transfer ratio. But in a robotics it is not possible to use effectively the electric motor and the reducer with the constant transfer ratio as the electric motor cannot be optimized for all operating modes. The electric motor and the reducer must be selected under the maximum torque moment of loading, and in the basic operating modes the drive will essentially underused on power. Therefore it is rational to combine the frequency-controlled engine with a variator.

At frictional disk variators the torque from the input link to the output link is transferring by a frictional force on surfaces of adjoining bodies, and the special control system is necessary for the transfer ratio change. The disk planetary variator is having the restricted application owing to a number of defects. Their overload capacity does not exceed 2-3 and efficiency sharply decreases with loading growth. The majority of existing variators (for example, a disk variator «Disco» of company “Lenze”) have a small control range of the transfer ratio and do not admit its change in the disconnect state. Therefore application of such variators is restricted to drives with a small control range of speed and torque.

Gulia’s adaptive disk planetary variator allows creating the new class of the electric drives possessing a “soft” external speed-torque characteristic with very high starting torque, overload capacity and possibility of automatic control of the transfer ratio at any changes of loading [1]. However the variator design is very complex and inapplicable for robots.
The toroidal friction variators are no effective for robot transmissions also [2].
The most effective for robots the Ivanov’s gear adaptive variators are [3,4].

The adaptive variator is mechanically simple, has small dimensions and weight and it is working without a control system. These advantages are especially important for intermediate modules of the manipulator. Use of adaptive gear variators provides a high power efficiency of the robot and essential decrease of sizes and weight.

In the electric drive with the adaptive gear variator having a high control range, it is possible to use a usual asynchronous motor. Such drive works without a control system.

The first trying to prove advantage and expediency of application of adaptive drives in a robotics has been made in the work [3]. The researches devoted to creation of adaptive drives of modules of manipulators have been executed. They are allowing a choosing of engine on average (but not on maximal) power of resistance. But this trying was not grounded enough. The adaptive drive has possibility of transition in a “stop” regime with a stop of tool when the engine prolongs to operate [4].

It allows to co-ordinate work of modules at overloading. The “stop” regime can be used also for limiting of force (for example, in grippers). However power advantages of adaptive drives have not been proved by the kinematic and force analysis.

It is expedient to execute comparison of kinematics of the adaptive drive and the usual drive with “rigid” kinematics for an estimation of advantages of the adaptive drive and also to consider additional possibilities of adaptive drives for their use in robot transmissions.
In the present work the adaptive drives of modules of the manipulator are considered. The work and interacting of the adaptive drives is investigated and the basic regularities of interconnection of their parameters are resulted.
Kinematics of a Moving Part of the Module
The kinematics of the module of the manipulator is defined by its drive.
Now the drive with the constant transfer ratio which defines “rigid” constraint between the engine and a moving part of the module translation or a rotary motion (the drive 1) is used. The adaptive drive is alternative to the drive with “rigid” constraint. The adaptive drive has two degrees of freedom and provides motion with the speed depending on loading (the drive 2) [3,4]. We will be to consider the kinematics parameters of both used modules of manipulator: module of translation motion and module of rotation motion on one figure. On figure 1 the schedules of linear parameters for module of translation motion and the schedules of angular parameters for module of rotation are presented. On fig. 1 the linear or angular parameters of displacement (s, φ), speed (v, ω) and acceleration (a, ε) to functions from a time t are presented.
Figure 1:Kinematics of a moving part of the module of the manipulator
Each carried out operation contains three phases of motion: the start or speeding-up (ts - start time), the installed motion (ti - time of installed motion when speed is constant) and the braking (tb - time of braking). A time of performance of operation T= t s + t i + t b MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamivaiabg2 da9iaadshadaWgaaWcbaGaam4CaaqabaGccqGHRaWkcaWG0bWaaSba aSqaaiaadMgaaeqaaOGaey4kaSIaamiDamaaBaaaleaacaWGIbaabe aaaaa@3FE8@
The kinematic schedules for the “rigid” drive 1 are shown by thin lines and for the adaptive drive 2 are shown by contour lines.

Let us to compare the schedules of drive 1 and of drive 2. The comparison of schedules shows the principal advantage of an adaptivedrive 2 - possibility to use a drive with considerably smaller power in comparison with the incontrollable drive having a fixed transfer ratio. Usually in the theory of mechanisms the comparison of various laws of motion is used for cyclic cam mechanisms. Also it is possible to consider the manipulator as the cyclic mechanism. The drive 1 with a fixed transfer ratio has accelerations in the beginning and in the end of the removal phase. These accelerations are equal to infinity theoretically (graphs have been showed by thin arrows). The drive 1 should overcome theoretically infinitely big forces of inertia in these points. The adaptive drive 2 is capable to change independently the speed of motion under the acting of inertia forces of resistance. The initial inertial loading is forcing to increase the motion speed smoothly (not by jerk). The graph of accelerations for an adaptive drive 2 (displayed by a contour line) has only final values which call considerably smaller inertia forces of resistance. Therefore the adaptive drive should have considerably smaller power. For the manipulator the power of a drive is the major power characteristic defining a designer optimality because the sizes and module weight depend on power of its drive.

The drive with “rigid” constraint 1 provides motion with constant speed on all three phases of motion during a time T. Acceleration theoretically attains infinitely great value in the beginning and in the end of operation. The inertial force accepts values conforming to acceleration. Thus the engine of the drive 1 must overcome very big loading at start. Power of an engine must fit to starting loading

The adaptive drive 2 is operating with brand new technology. At start the big inertia force allows to begin motion with a speed equal to null. Further the inertial force decreases, and speed of motion increases. The schedule of speed at start approaches to a cosine curve. The acceleration schedule at start approaches to a sinusoid. The inertial force accepts values conforming to acceleration. Thus the engine of the drive 2 must overcome extremely smaller loading at start. The maximum inertial force will be finite quantity and will occur approximately in the middle of start. Power of an engine should conform to this starting loading. Without account of minor factors acting on a draught resistance it is possible to assume, that the engine of the adaptive drive 2 can have power 10 times less in comparison with the engine of the usual drive 1. Sizes and engine weight conform to its power. The adaptive drive will have small sizes and weight and high power efficiency because of possibility of adaptation to loading.

On the kinematic schedules (figure 1) it is visible a time of performance of each operation by the adaptive drive and by the usual drive equally. Displacements in the end of in stroke and in the end of backward stroke also are equal. The adaptive drive provides the variable transfer ratio inside of cycle at conservation of final displacements of forward stroke and of backward stroke. Therefore accuracy of final positioning of highprecision manipulator with the adaptive drive and with the usual drive will be equal.

Adaptation of the drive to a variable load leads to decrease of dynamic stresses (to unaccented work and elimination of vibrations).
Advantages of the drive 2 with the variable transfer ratio are indisputable. However practical implementation on the basis of existing designs in the form of the operated gear box is an insolvable problem for the robot.

However this problem is solved simply enough if to use an adaptive wheelwork with two degrees of freedom as the drive connecting gear [1, 2]. Such mechanism, first, is extremely simple on the device and, secondly, does not demand a control system. Change of the transfer ratio occurs in the adaptive mechanism independently depending on external loading. Let’s consider concrete circuit designs of adaptive drives of modules of the manipulator
Design of the Adaptive Drive
The adaptive module of translational motion (figure 2) contains following parts:
Figure 2:Adaptive module of translational motion of manipulator
A - module fixed part: 1. Plate. 2. Joint for junction with previous module. 3. Directing for screw nut. 4. Electric motor. 5. Adaptive reducer. 6. Working screw.
B - module moving part: 7. Running screw nut of translational motion. 8. Joint on running screw nut for junction with the subsequent module. The adaptive module of rotary motion (figure 3) contains following parts:
Figure 3:Adaptive module of rotary motion of manipulator
A - module fixed part: 1. Plate. 2. Joint for junction with previous module. 3. Electric motor. 4. Adaptive reducer transferring rotary motion on an exit of module.
B - module moving part (exit of module). 5. Joint for junction with the subsequent module.

Adaptive mechanism (reducer) (figure 4) contains frame 0, input carrier H1, input satellite 2, block of central wheels 1-4, block of epicycle (ring) wheels 3-6 with flywheel on wheel 3, the output satellite 5 and output carrier H2. In operating time links are rotated round the central axis. Sizes of toothed wheels 1, 2, 3, 4, 5, 6 are defined by conforming radiuses r i i=1,2,3,4,5,6 MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOCamaaBa aaleaacaWGPbaabeaakiaaywW7caWGPbGaeyypa0JaaGymaiaacYca caaMe8UaaGOmaiaacYcacaaMe8UaaG4maiaacYcacaaMe8UaaGinai aacYcacaaMe8UaaGynaiaacYcacaaMe8UaaGOnaaaa@4B34@ Radiuses of carriers r H1 = r 1 + r 2 , r H2 = r 4 + r 5 MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOCamaaBa aaleaacaWGibGaaGymaaqabaGccqGH9aqpcaWGYbWaaSbaaSqaaiaa igdaaeqaaOGaey4kaSIaamOCamaaBaaaleaacaaIYaaabeaakiaacY cacaWGYbWaaSbaaSqaaiaadIeacaaIYaaabeaakiabg2da9iaadkha daWgaaWcbaGaaGinaaqabaGccqGHRaWkcaWGYbWaaSbaaSqaaiaaiw daaeqaaaaa@477E@ Number of degrees of freedom of the kinematic chain is equal 2.
Adaptive mechanism (reducer) has variable transfer ratio which depends on loading on output shaft.
Work of the Adaptive Drive
In the drive of module of translational motion (figure 2) electric motor with constant power 4 transfers rotation to working screw 6 which moves a running screw nut 7 and module moving part B. Working screw 6 moves running screw nut 7 together with a module moving part B on
Figure 4:Adaptive mechanism of manipulator module drive
directions 3. The running nut 7 together with attached to it the joint 8 carries out the output translational motion. Adaptive reducer 5 provides rotation of working screw 6 with a speed inversely proportional to moment of resistance which conforms to external loading of the module. The external force of resistance on the joint 8 independently changes speed of its translational motion.

Equations of kinematics of translational motion module define connection of linear motion of running screw nut 7 and output joint 8 with rotation of shaft of electric motor 4.

Speed of motion of running screw nut and moving part of module V 7 = n H2 p MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOvamaaBa aaleaacaaI3aaabeaakiabg2da9iaad6gadaWgaaWcbaGaamisaiaa ikdaaeqaaOGaeyyXICTaamiCaaaa@3EBE@

Here n H2 MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBamaaBa aaleaacaWGibGaaGOmaaqabaaaaa@389D@ revolutions per minute of output carrier H2 of output shaft of adaptive reducer, P- step of nut screw line. Here n H2 = n H1 /u MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBamaaBa aaleaacaWGibGaaGOmaaqabaGccqGH9aqpcaWGUbWaaSbaaSqaaiaa dIeacaaIXaaabeaakiaac+cacaWG1baaaa@3E0B@ ( n H1 MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaad6 gadaWgaaWcbaGaamisaiaaigdaaeqaaaaa@3948@ revolutions per minute of input carrier H1 of electric motor shaft, u- variable transfer ratio of adaptive reducer). Then V 7 = n H1 p/60u. MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOvamaaBa aaleaacaaI3aaabeaakiabg2da9iaad6gadaWgaaWcbaGaamisaiaa igdaaeqaaOGaeyyXICTaamiCaiaac+cacaaI2aGaaGimaiabgwSixl aadwhacaGGUaaaaa@44E0@

Linear moving of running screw nut S 7 = n H1 pt/60u MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4uamaaBa aaleaacaaI3aaabeaakiabg2da9iaad6gadaWgaaWcbaGaamisaiaa igdaaeqaaOGaeyyXICTaamiCaiabgwSixlaadshacaGGVaGaaGOnai aaicdacqGHflY1caWG1baaaa@476E@ , where t - time.
The sensing transducer of linear moving operates by module motion under the set program.
In the drive of rotary motion module (figure 3) electric motor 3 transfers rotation to the adaptive reducer 4, moving part of module B and output Joint 5. The adaptive reducer 4 provides rotation of output joint 5 with a speed inversely proportional to moment of resistance which conforms to external loading of the module. The external moment of resistance on the joint 5 independently changes speed of its rotary motion.

Equations of kinematics of the rotary motion module are equations of reducer 4 [1]. They define connection of angular motion of output joint 5 with rotation of electric motor shaft 4.
The sensing transducer of angular moving operates by module motion under the set program.
Kinematic and Force Analysis of Adaptive Mechanism of Manipulator Module
In presented work unlike the usual mechanism the considered mechanism contains the mobile closed contour which connects input link with output link. This closed contour creates circulation of energy in a contour [3, 4]. Energy circulation represents a connection equation of forces and speeds in the contour. This additional constraint reduces number degreeof freedom to 1 and provides the installed regime of motion. Simultaneously the mobile closed contour of the mechanism creates the property of force adaptation.

Such is essence of the discovery which is the basis of the force adaptation. We will result here the new regularity.

At first we will consider the adaptive mechanism of manipulator module in the condition with two degrees of freedom.

The mechanism looks like the closed gear differential with two degrees of freedom (figure 4). Toothed wheels 1-2-3-6-5-4 are forming a closed four link contour. Input motive force F1 is transferred from input link H1 to point B. Output resistance force R6 is transferred from output link H2 to point K. Application points B and K of contour external forces FH1 and RH2 have external displacements SB, SK. Application points C, E, D, G of contour internal forces (reactions R32, R65, R12, R45) have internal displacements SC, SE, SD, SG. At known external displacements of points B and K the contour internal displacements of points SC, SE, SD, SG are defined as one-valued functions.

For each of contour links 2 and 5 the internal forces R32, R65, R12, R45, can be expressed on statics conditions through active forces FH1 and RH2. Let’s make for links of 2 and 5 a conditions of equilibrium by principle of virtual works accepting the valid displacements for the possible. For contour links 2 and 5 we will express reactions in the kinematic pairs D, C, G, E through external forces FH1, RH2 applied in points B, K

R 12 = R 32 =0.5 F H1      (1) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOuamaaBa aaleaacaaIXaGaaGOmaaqabaGccqGH9aqpcaWGsbWaaSbaaSqaaiaa iodacaaIYaaabeaakiabg2da9iaaicdacaGGUaGaaGynaiaadAeada WgaaWcbaGaamisaiaaigdaaeqaaOGaaeiiaiaabccacaqGGaGaaeii aiaabccacaqGOaGaaeymaiaabMcaaaa@46F9@ R 45 = R 65 =0.5 R H2      (2) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOuamaaBa aaleaacaaI0aGaaGynaaqabaGccqGH9aqpcaWGsbWaaSbaaSqaaiaa iAdacaaI1aaabeaakiabg2da9iaaicdacaGGUaGaaGynaiaadkfada WgaaWcbaGaamisaiaaikdaaeqaaOGaaeiiaiaabccacaqGGaGaaeii aiaabccacaqGOaGaaeOmaiaabMcaaaa@4713@ Here
F H1 = M H1 / r H1 ,     (3) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOramaaBa aaleaacaWGibGaaGymaaqabaGccqGH9aqpcaWGnbWaaSbaaSqaaiaa dIeacaaIXaaabeaakiaac+cacaWGYbWaaSbaaSqaaiaadIeacaaIXa aabeaakiaacYcacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabIca caqGZaGaaeykaaaa@4568@
And so on
R 12 = M 12 / r 1 , R 32 = M 32 / r 3 , R H2 = M H2 / r H2 , R 45 = M 45 / r 4 , R 65 = M 65 / r 6 MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOuamaaBa aaleaacaaIXaGaaGOmaaqabaGccqGH9aqpcaWGnbWaaSbaaSqaaiaa igdacaaIYaaabeaakiaac+cacaWGYbWaaSbaaSqaaiaaigdaaeqaaO GaaiilaiaaysW7caWGsbWaaSbaaSqaaiaaiodacaaIYaaabeaakiab g2da9iaad2eadaWgaaWcbaGaaG4maiaaikdaaeqaaOGaai4laiaadk hadaWgaaWcbaGaaG4maaqabaGccaGGSaGaamOuamaaBaaaleaacaWG ibGaaGOmaaqabaGccqGH9aqpcaWGnbWaaSbaaSqaaiaadIeacaaIYa aabeaakiaac+cacaWGYbWaaSbaaSqaaiaadIeacaaIYaaabeaakiaa cYcacaaMe8UaamOuamaaBaaaleaacaaI0aGaaGynaaqabaGccqGH9a qpcaWGnbWaaSbaaSqaaiaaisdacaaI1aaabeaakiaac+cacaWGYbWa aSbaaSqaaiaaisdaaeqaaOGaaiilaiaaysW7caWGsbWaaSbaaSqaai aaiAdacaaI1aaabeaakiabg2da9iaad2eadaWgaaWcbaGaaGOnaiaa iwdaaeqaaOGaai4laiaadkhadaWgaaWcbaGaaGOnaaqabaaaaa@69A2@
M H1 , M H2 MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamytamaaBa aaleaacaWGibGaaGymaaqabaGccaGGSaGaaGjbVlaad2eadaWgaaWc baGaamisaiaaikdaaeqaaaaa@3D49@ -Moments on input and output carriers,
r H1 , r H2 MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOCamaaBa aaleaacaWGibGaaGymaaqabaGccaGGSaGaaGjbVlaadkhadaWgaaWc baGaamisaiaaikdaaeqaaaaa@3D93@ - Radiuses of input and output carriers,
M 12 , M 32 MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamytamaaBa aaleaacaaIXaGaaGOmaaqabaGccaGGSaGaaGjbVlaad2eadaWgaaWc baGaaG4maiaaikdaaeqaaaaa@3D28@ - Moments created on satellite 2 by reactions R 12 , R 32 MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOuamaaBa aaleaacaaIXaGaaGOmaaqabaGccaGGSaGaaGjbVlaadkfadaWgaaWc baGaaG4maiaaikdaaeqaaaaa@3D32@ from toothed wheels 1 and 3,
M 45 , M 65 MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamytamaaBa aaleaacaaI0aGaaGynaaqabaGccaGGSaGaaGjbVlaad2eadaWgaaWc baGaaGOnaiaaiwdaaeqaaaaa@3D34@ Moments created on satellite 5 by reactions R 45 , R 65 MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOuamaaBa aaleaacaaI0aGaaGynaaqabaGccaGGSaGaaGjbVlaadkfadaWgaaWc baGaaGOnaiaaiwdaaeqaaaaa@3D3E@ from toothed wheels 4 and 6,
r i (i=1,2...6) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOCamaaBa aaleaacaWGPbaabeaakiaaysW7caGGOaGaamyAaiabg2da9iaaigda caGGSaGaaGjbVlaaikdacaGGUaGaaiOlaiaac6cacaaI2aGaaiykaa aa@4374@ - Radiuses of wheels.
After substitution of force values in the Eq. 1, Eq. 2 we will receive a formulas for determination of internal moments through the external moments
M 12 =0.5 M H1 r 1 / r H1      (4) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamytamaaBa aaleaacaaIXaGaaGOmaaqabaGccqGH9aqpcaaIWaGaaiOlaiaaiwda caWGnbWaaSbaaSqaaiaadIeacaaIXaaabeaakiaadkhadaWgaaWcba GaaGymaaqabaGccaGGVaGaamOCamaaBaaaleaacaWGibGaaGymaaqa baGccaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabIcacaqG0aGaae ykaaaa@48C2@ M 32 =0.5 M H 1 r 3 / r H1 ,     (5) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamytamaaBa aaleaacaaIZaGaaGOmaaqabaGccqGH9aqpcaaIWaGaaiOlaiaaiwda caWGnbWaaSbaaSqaaiaadIeaaeqaaOWaaSbaaSqaaiaaigdaaeqaaO GaamOCamaaBaaaleaacaaIZaaabeaakiaac+cacaWGYbWaaSbaaSqa aiaadIeacaaIXaaabeaakiaacYcacaaMe8UaaeiiaiaabccacaqGGa GaaeiiaiaabccacaqGOaGaaeynaiaabMcaaaa@4B3A@ M 45 =0.5 M H2 r 4 / r H2      (6) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamytamaaBa aaleaacaaI0aGaaGynaaqabaGccqGH9aqpcaaIWaGaaiOlaiaaiwda caWGnbWaaSbaaSqaaiaadIeacaaIYaaabeaakiaadkhadaWgaaWcba GaaGinaaqabaGccaGGVaGaamOCamaaBaaaleaacaWGibGaaGOmaaqa baGccaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabIcacaqG2aGaae ykaaaa@48CF@ M 65 =0.5 M H2 r 6 / r H2      (7) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamytamaaBa aaleaacaaI2aGaaGynaaqabaGccqGH9aqpcaaIWaGaaiOlaiaaiwda caWGnbWaaSbaaSqaaiaadIeacaaIYaaabeaakiaadkhadaWgaaWcba GaaGOnaaqabaGccaGGVaGaamOCamaaBaaaleaacaWGibGaaGOmaaqa baGccaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabIcacaqG3aGaae ykaaaa@48D4@ Equilibrium equations by a principle of virtual works for satellites 2 and 5 will have the following forms:
R 12 s D + R 32 s C = F H1 s B      (8) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOuamaaBa aaleaacaaIXaGaaGOmaaqabaGccaWGZbWaaSbaaSqaaiaadseaaeqa aOGaey4kaSIaamOuamaaBaaaleaacaaIZaGaaGOmaaqabaGccaWGZb WaaSbaaSqaaiaadoeaaeqaaOGaeyypa0JaamOramaaBaaaleaacaWG ibGaaGymaaqabaGccaWGZbWaaSbaaSqaaiaadkeaaeqaaOGaaeiiai aabccacaqGGaGaaeiiaiaabccacaqGOaGaaeioaiaabMcaaaa@4A93@ R 45 s G + R 65 s E = R H2 s K      (9) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOuamaaBa aaleaacaaI0aGaaGynaaqabaGccaWGZbWaaSbaaSqaaiaadEeaaeqa aOGaey4kaSIaamOuamaaBaaaleaacaaI2aGaaGynaaqabaGccaWGZb WaaSbaaSqaaiaadweaaeqaaOGaeyypa0JaamOuamaaBaaaleaacaWG ibGaaGOmaaqabaGccaWGZbWaaSbaaSqaaiaadUeaaeqaaOGaaeiiai aabccacaqGGaGaaeiiaiaabccacaqGOaGaaeyoaiaabMcaaaa@4ABB@
Here s B , s C , s D , s E , s G , s K MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4CamaaBa aaleaacaWGcbaabeaakiaacYcacaaMe8Uaam4CamaaBaaaleaacaWG dbaabeaakiaacYcacaaMe8Uaam4CamaaBaaaleaacaWGebaabeaaki aacYcacaaMe8Uaam4CamaaBaaaleaacaWGfbaabeaakiaacYcacaaM e8Uaam4CamaaBaaaleaacaWGhbaabeaakiaacYcacaaMe8Uaam4Cam aaBaaaleaacaWGlbaabeaaaaa@4CEE@ - displacements of points B,C,D,E,G,K MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOqaiaacY cacaaMe8Uaam4qaiaacYcacaaMe8UaamiraiaacYcacaaMe8Uaamyr aiaacYcacaaMe8Uaam4raiaacYcacaaMe8Uaam4saaaa@45E4@ The displacements of points expressed through instant angles of turn of links and radiuses will have the following forms: s D = ϕ 1 r 1 , s C = ϕ 3 r 3 , s B = ϕ H1 r H1 , MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4CamaaBa aaleaacaWGebaabeaakiabg2da9iabew9aMnaaBaaaleaacaaIXaaa beaakiaadkhadaWgaaWcbaGaaGymaaqabaGccaGGSaGaaGjbVlaado hadaWgaaWcbaGaam4qaaqabaGccqGH9aqpcqaHvpGzdaWgaaWcbaGa aG4maaqabaGccaWGYbWaaSbaaSqaaiaaiodaaeqaaOGaaiilaiaays W7caWGZbWaaSbaaSqaaiaadkeaaeqaaOGaeyypa0Jaeqy1dy2aaSba aSqaaiaadIeacaaIXaaabeaakiaadkhadaWgaaWcbaGaamisaiaaig daaeqaaOGaaiilaaaa@5394@ s G = ϕ 4 r 4 , s E = ϕ 6 r 6 , s K = ϕ H2 r H2 MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4CamaaBa aaleaacaWGhbaabeaakiabg2da9iabew9aMnaaBaaaleaacaaI0aaa beaakiaadkhadaWgaaWcbaGaaGinaaqabaGccaGGSaGaaGjbVlaado hadaWgaaWcbaGaamyraaqabaGccqGH9aqpcqaHvpGzdaWgaaWcbaGa aGOnaaqabaGccaWGYbWaaSbaaSqaaiaaiAdaaeqaaOGaaiilaiaays W7caWGZbWaaSbaaSqaaiaadUeaaeqaaOGaeyypa0Jaeqy1dy2aaSba aSqaaiaadIeacaaIYaaabeaakiaadkhadaWgaaWcbaGaamisaiaaik daaeqaaaaa@52F6@ ϕ 1 , ϕ 3 , ϕ H1 , ϕ 4 , ϕ 6 , ϕ H2 MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqy1dy2aaS baaSqaaiaaigdaaeqaaOGaaiilaiaaysW7cqaHvpGzdaWgaaWcbaGa aG4maaqabaGccaGGSaGaaGjbVlabew9aMnaaBaaaleaacaWGibGaaG ymaaqabaGccaGGSaGaaGjbVlabew9aMnaaBaaaleaacaaI0aaabeaa kiaacYcacaaMe8Uaeqy1dy2aaSbaaSqaaiaaiAdaaeqaaOGaaiilai aaysW7cqaHvpGzdaWgaaWcbaGaamisaiaaikdaaeqaaaaa@5317@ - instant angles of turn of toothed wheels and carriers. With account ϕ 1 = ϕ 4 , ϕ 3 = ϕ 6 MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqy1dy2aaS baaSqaaiaaigdaaeqaaOGaeyypa0Jaeqy1dy2aaSbaaSqaaiaaisda aeqaaOGaaiilaiaaysW7cqaHvpGzdaWgaaWcbaGaaG4maaqabaGccq GH9aqpcqaHvpGzdaWgaaWcbaGaaGOnaaqabaaaaa@4522@ and a time the equations (8) and (9) will receive following forms: M 12 ω 1 + M 32 ω 3 = M H1 ω H1      (10) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamytamaaBa aaleaacaaIXaGaaGOmaaqabaGccqaHjpWDdaWgaaWcbaGaaGymaaqa baGccqGHRaWkcaWGnbWaaSbaaSqaaiaaiodacaaIYaaabeaakiabeM 8a3naaBaaaleaacaaIZaaabeaakiabg2da9iaad2eadaWgaaWcbaGa amisaiaaigdaaeqaaOGaeqyYdC3aaSbaaSqaaiaadIeacaaIXaaabe aakiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeikaiaabgdacaqG WaGaaeykaaaa@4E63@ M 45 ω 1 + M 65 ω 3 = M H2 ω H2      (11) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamytamaaBa aaleaacaaI0aGaaGynaaqabaGccqaHjpWDdaWgaaWcbaGaaGymaaqa baGccqGHRaWkcaWGnbWaaSbaaSqaaiaaiAdacaaI1aaabeaakiabeM 8a3naaBaaaleaacaaIZaaabeaakiabg2da9iaad2eadaWgaaWcbaGa amisaiaaikdaaeqaaOGaeqyYdC3aaSbaaSqaaiaadIeacaaIYaaabe aakiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeikaiaabgdacaqG XaGaaeykaaaa@4E72@
To receive a condition of parameters interacting of the mechanism as a whole it is necessary to add the made expressions as satellites 2 and 5 are a part of the mechanism as a whole. M 12 ω 1 + M 32 ω 3 + M 45 ω 1 + M 65 ω 3 == M H1 ω H1 + M H2 ω H2      (12) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamytamaaBa aaleaacaaIXaGaaGOmaaqabaGccqaHjpWDdaWgaaWcbaGaaGymaaqa baGccqGHRaWkcaWGnbWaaSbaaSqaaiaaiodacaaIYaaabeaakiabeM 8a3naaBaaaleaacaaIZaaabeaakiabgUcaRiaad2eadaWgaaWcbaGa aGinaiaaiwdaaeqaaOGaeqyYdC3aaSbaaSqaaiaaigdaaeqaaOGaey 4kaSIaamytamaaBaaaleaacaaI2aGaaGynaaqabaGccqaHjpWDdaWg aaWcbaGaaG4maaqabaGccqGH9aqpcqGH9aqpcaWGnbWaaSbaaSqaai aadIeacaaIXaaabeaakiabeM8a3naaBaaaleaacaWGibGaaGymaaqa baGccqGHRaWkcaWGnbWaaSbaaSqaaiaadIeacaaIYaaabeaakiabeM 8a3naaBaaaleaacaWGibGaaGOmaaqabaGccaqGGaGaaeiiaiaabcca caqGGaGaaeiiaiaabIcacaqGXaGaaeOmaiaabMcaaaa@62B8@ In the left side of Eq.12 the sum of powers (corresponding to the sum of works) of contour internal forces takes place.
In the considered mechanism all internal forces are determined through the known superposed forces and all internal displacements are determined through external displacements. Hence work (or power) of internal forces on possible internal displacements is determined. Constraints in kinematic pairs are ideal and fixed. Work of external forces cannot pass in work of internal forces. Hence work (power) of internal forces on possible internal displacements is equal to null M 12 ω 1 + M 32 ω 3 + M 45 ω 1 + M 65 ω 3 =0     (13) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamytamaaBa aaleaacaaIXaGaaGOmaaqabaGccqaHjpWDdaWgaaWcbaGaaGymaaqa baGccqGHRaWkcaWGnbWaaSbaaSqaaiaaiodacaaIYaaabeaakiabeM 8a3naaBaaaleaacaaIZaaabeaakiabgUcaRiaad2eadaWgaaWcbaGa aGinaiaaiwdaaeqaaOGaeqyYdC3aaSbaaSqaaiaaigdaaeqaaOGaey 4kaSIaamytamaaBaaaleaacaaI2aGaaGynaaqabaGccqaHjpWDdaWg aaWcbaGaaG4maaqabaGccqGH9aqpcaaIWaGaaeiiaiaabccacaqGGa GaaeiiaiaabccacaqGOaGaaeymaiaabodacaqGPaaaaa@5553@ The right side of Eq.12 represents the sum of powers (corresponding to the sum of works) of contour external forces. At performance of Eq.33 we will receive from the Eq. 12 a condition of equilibrium for external forces according to principle of possible works M H1 ω H1 + M H2 ω H2 =0     (14) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamytamaaBa aaleaacaWGibGaaGymaaqabaGccqaHjpWDdaWgaaWcbaGaamisaiaa igdaaeqaaOGaey4kaSIaamytamaaBaaaleaacaWGibGaaGOmaaqaba GccqaHjpWDdaWgaaWcbaGaamisaiaaikdaaeqaaOGaeyypa0JaaGim aiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeikaiaabgdacaqG0a Gaaeykaaaa@4AC0@ Equation (14) expresses the additional constraint imposed by a contour on motion of links.
Thus moving four link closed contour imposes a constraint on motion of links.
Additional constraint Eq.14 provides:
1) transformation of the kinematic chain with two degrees of freedom in the mechanism with one degree of freedom that is definability of motion under the action of forces;
2) effect of force adaptation to output loading at the assigned parameters of input power M H1 , ω H1 MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamytamaaBa aaleaacaWGibGaaGymaaqabaGccaGGSaGaaGjbVlabeM8a3naaBaaa leaacaWGibGaaGymaaqabaaaaa@3E43@ and the assigned output moment of resistance M H2 MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamytamaaBa aaleaacaWGibGaaGOmaaqabaaaaa@387C@ According to the Eq.14 with account signs of powers we will receive ω H2 = M H1 ω H1 /M     (15) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyYdC3aaS baaSqaaiaadIeacaaIYaaabeaakiabg2da9iaad2eadaWgaaWcbaGa amisaiaaigdaaeqaaOGaeqyYdC3aaSbaaSqaaiaadIeacaaIXaaabe aakiaac+cacaWGnbGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqG OaGaaeymaiaabwdacaqGPaaaaa@4819@ That is at constant input power the output angular speed ω H2 MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyYdC3aaS baaSqaaiaadIeacaaIYaaabeaaaaa@3977@ is inversely proportional to the variable output resistance moment M H2 . MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamytamaaBa aaleaacaWGibGaaGOmaaqabaGccaGGUaaaaa@3938@ From the equation (13) it is possible to receive ( M 12 + M 45 ) ω 1 +( M 32 + M 65 ) ω 3 =0 MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaad2 eadaWgaaWcbaGaaGymaiaaikdaaeqaaOGaey4kaSIaamytamaaBaaa leaacaaI0aGaaGynaaqabaGccaGGPaGaeqyYdC3aaSbaaSqaaiaaig daaeqaaOGaey4kaSIaaiikaiaad2eadaWgaaWcbaGaaG4maiaaikda aeqaaOGaey4kaSIaamytamaaBaaaleaacaaI2aGaaGynaaqabaGcca GGPaGaeqyYdC3aaSbaaSqaaiaaiodaaeqaaOGaeyypa0JaaGimaaaa @4C97@ With account signs of moments (the motive moments M 12 , M 32 MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamytamaaBa aaleaacaaIXaGaaGOmaaqabaGccaGGSaGaaGjbVlaad2eadaWgaaWc baGaaG4maiaaikdaaeqaaaaa@3D28@ transferred from the input satellite 2 are positive, the resistance moments M 45 , M 65 MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamytamaaBa aaleaacaaI0aGaaGynaaqabaGccaGGSaGaaGjbVlaad2eadaWgaaWc baGaaGOnaiaaiwdaaeqaaaaa@3D34@ transferred from the output satellite 5 are negative) it follows ( M 12 M 45 ) ω 1 +( M 32 M 65 ) ω 3 =0     (16) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaad2 eadaWgaaWcbaGaaGymaiaaikdaaeqaaOGaeyOeI0IaamytamaaBaaa leaacaaI0aGaaGynaaqabaGccaGGPaGaeqyYdC3aaSbaaSqaaiaaig daaeqaaOGaey4kaSIaaiikaiaad2eadaWgaaWcbaGaaG4maiaaikda aeqaaOGaeyOeI0IaamytamaaBaaaleaacaaI2aGaaGynaaqabaGcca GGPaGaeqyYdC3aaSbaaSqaaiaaiodaaeqaaOGaeyypa0JaaGimaiaa bccacaqGGaGaaeiiaiaabccacaqGGaGaaeikaiaabgdacaqG2aGaae ykaaaa@52A0@ Eq.16 represents the equation of works (powers) on intermediate links 1-4 and 3-6. Eq. 16 means the presence of equilibrium on intermediate links 1-4 and 3-6 simultaneously. In the moving closed contour basic new situation takes place: equilibrium in statics is absent separately on each intermediate link but equilibrium of intermediate links simultaneously takes place on the movement all contour.

In the closed contour energy circulation takes place. Eq.16 contains positive and negative members and characterizes equilibrium of powers on intermediate links of a contour. For the considered chain the follow constraint takes place M 45 > M 12 , M 32 > M 65 . MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamytamaaBa aaleaacaaI0aGaaGynaaqabaGccqGH+aGpcaWGnbWaaSbaaSqaaiaa igdacaaIYaaabeaakiaacYcacaaMe8UaamytamaaBaaaleaacaaIZa GaaGOmaaqabaGccqGH+aGpcaWGnbWaaSbaaSqaaiaaiAdacaaI1aaa beaakiaac6caaaa@4500@ Then from Eq.16 it is follows ( M 45 M 12 ) ω 1 +( M 32 M 65 ) ω 3 =0     (17) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOeI0Iaai ikaiaad2eadaWgaaWcbaGaaGinaiaaiwdaaeqaaOGaeyOeI0Iaamyt amaaBaaaleaacaaIXaGaaGOmaaqabaGccaGGPaGaeqyYdC3aaSbaaS qaaiaaigdaaeqaaOGaey4kaSIaaiikaiaad2eadaWgaaWcbaGaaG4m aiaaikdaaeqaaOGaeyOeI0IaamytamaaBaaaleaacaaI2aGaaGynaa qabaGccaGGPaGaeqyYdC3aaSbaaSqaaiaaiodaaeqaaOGaeyypa0Ja aGimaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeikaiaabgdaca qG3aGaaeykaaaa@538E@ From here ( M 45 M 12 ) ω 1 =( M 32 M 65 ) ω 3      (18) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaad2 eadaWgaaWcbaGaaGinaiaaiwdaaeqaaOGaeyOeI0IaamytamaaBaaa leaacaaIXaGaaGOmaaqabaGccaGGPaGaeqyYdC3aaSbaaSqaaiaaig daaeqaaOGaeyypa0Jaaiikaiaad2eadaWgaaWcbaGaaG4maiaaikda aeqaaOGaeyOeI0IaamytamaaBaaaleaacaaI2aGaaGynaaqabaGcca GGPaGaeqyYdC3aaSbaaSqaaiaaiodaaeqaaOGaaeiiaiaabccacaqG GaGaaeiiaiaabccacaqGOaGaaeymaiaabIdacaqGPaaaaa@5106@ Eq.18 reflects an analytical form of circulation of energy in a contour during its motion unknown earlier.
Angular speeds ω 1 , ω 3 MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyYdC3aaS baaSqaaiaaigdaaeqaaOGaaiilaiaaysW7cqaHjpWDdaWgaaWcbaGa aG4maaqabaaaaa@3DA6@ of intermediate links 1-4 and 3-6 are determined through known angular speeds of input and output carriers ω H1 , ω H2 MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyYdC3aaS baaSqaaiaadIeacaaIXaaabeaakiaacYcacaaMe8UaeqyYdC3aaSba aSqaaiaadIeacaaIYaaabeaaaaa@3F3F@ and transfer ratios at the stopped carriers.
Transfer ratios of links of mechanism we will determine through teeth numbers of wheels z i i=1,2,...6 MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOEamaaBa aaleaacaWGPbaabeaakiaaysW7caWGPbGaeyypa0JaaGymaiaacYca caaMe8UaaGOmaiaacYcacaaMe8UaaiOlaiaac6cacaGGUaGaaGjbVl aaiAdaaaa@45ED@ The interconnection of angular speeds of the mechanism is defined by formulas ω 1 ω H1 ω 3 ω H1 = u 13 (H1)      (19) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacq aHjpWDdaWgaaWcbaGaaGymaaqabaGccqGHsislcqaHjpWDdaWgaaWc baGaamisaiaaigdaaeqaaaGcbaGaeqyYdC3aaSbaaSqaaiaaiodaae qaaOGaeyOeI0IaeqyYdC3aaSbaaSqaaiaadIeacaaIXaaabeaaaaGc cqGH9aqpcaWG1bWaa0baaSqaaiaaigdacaaIZaaabaGaaiikaiaadI eacaaIXaGaaiykaaaakiaabccacaqGGaGaaeiiaiaabccacaqGGaGa aeikaiaabgdacaqG5aGaaeykaaaa@50F9@ ω 1 ω H2 ω 3 ω H2 = u 46 (H2) ,     (20) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacq aHjpWDdaWgaaWcbaGaaGymaaqabaGccqGHsislcqaHjpWDdaWgaaWc baGaamisaiaaikdaaeqaaaGcbaGaeqyYdC3aaSbaaSqaaiaaiodaae qaaOGaeyOeI0IaeqyYdC3aaSbaaSqaaiaadIeacaaIYaaabeaaaaGc cqGH9aqpcaWG1bWaa0baaSqaaiaaisdacaaI2aaabaGaaiikaiaadI eacaaIYaGaaiykaaaakiaacYcacaqGGaGaaeiiaiaabccacaqGGaGa aeiiaiaabIcacaqGYaGaaeimaiaabMcaaaa@51AA@ where u 46 (H2) = z 6 / z 4 MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyDamaaDa aaleaacaaI0aGaaGOnaaqaaiaacIcacaWGibGaaGOmaiaacMcaaaGc cqGH9aqpcqGHsislcaWG6bWaaSbaaSqaaiaaiAdaaeqaaOGaai4lai aadQhadaWgaaWcbaGaaGinaaqabaaaaa@420A@ From (19) ω 1 = u 13 (H1) ( ω 3 ω H1 )+ ω H1 .     (21) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGjbVlabeM 8a3naaBaaaleaacaaIXaaabeaakiabg2da9iaadwhadaqhaaWcbaGa aGymaiaaiodaaeaacaGGOaGaamisaiaaigdacaGGPaaaaOGaaiikai abeM8a3naaBaaaleaacaaIZaaabeaakiabgkHiTiabeM8a3naaBaaa leaacaWGibGaaGymaaqabaGccaGGPaGaey4kaSIaeqyYdC3aaSbaaS qaaiaadIeacaaIXaaabeaakiaac6cacaqGGaGaaeiiaiaabccacaqG GaGaaeiiaiaabIcacaqGYaGaaeymaiaabMcaaaa@546F@ From Eq.20 ω 1 = u 46 (H2) ( ω 3 ω H2 )+ ω H2      (22) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGjbVlabeM 8a3naaBaaaleaacaaIXaaabeaakiabg2da9iaadwhadaqhaaWcbaGa aGinaiaaiAdaaeaacaGGOaGaamisaiaaikdacaGGPaaaaOGaaiikai abeM8a3naaBaaaleaacaaIZaaabeaakiabgkHiTiabeM8a3naaBaaa leaacaWGibGaaGOmaaqabaGccaGGPaGaey4kaSIaeqyYdC3aaSbaaS qaaiaadIeacaaIYaaabeaakiaabccacaqGGaGaaeiiaiaabccacaqG GaGaaeikaiaabkdacaqGYaGaaeykaaaa@53C7@ After of subtraction Eq.22 from Eq.21 it follows u 13 (H1) ( ω 3 ω H1 )+ ω H1 u 46 (H2) ( ω 3 ω H2 ) ω H2 =0. MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyDamaaDa aaleaacaaIXaGaaG4maaqaaiaacIcacaWGibGaaGymaiaacMcaaaGc caGGOaGaeqyYdC3aaSbaaSqaaiaaiodaaeqaaOGaeyOeI0IaeqyYdC 3aaSbaaSqaaiaadIeacaaIXaaabeaakiaacMcacqGHRaWkcqaHjpWD daWgaaWcbaGaamisaiaaigdaaeqaaOGaeyOeI0IaamyDamaaDaaale aacaaI0aGaaGOnaaqaaiaacIcacaWGibGaaGOmaiaacMcaaaGccaGG OaGaeqyYdC3aaSbaaSqaaiaaiodaaeqaaOGaeyOeI0IaeqyYdC3aaS baaSqaaiaadIeacaaIYaaabeaakiaacMcacqGHsislcqaHjpWDdaWg aaWcbaGaamisaiaaikdaaeqaaOGaeyypa0JaaGimaiaac6caaaa@5E78@ From here ( u 13 (H1) u 46 (H2) ) ω 3 u 13 (H1) ω H1 + u 46 (H2) ω H2 = ω H2 ω H1 .     (23) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGjbVlaacI cacaWG1bWaa0baaSqaaiaaigdacaaIZaaabaGaaiikaiaadIeacaaI XaGaaiykaaaakiabgkHiTiaadwhadaqhaaWcbaGaaGinaiaaiAdaae aacaGGOaGaamisaiaaikdacaGGPaaaaOGaaiykaiabeM8a3naaBaaa leaacaaIZaaabeaakiabgkHiTiaadwhadaqhaaWcbaGaaGymaiaaio daaeaacaGGOaGaamisaiaaigdacaGGPaaaaOGaeqyYdC3aaSbaaSqa aiaadIeacaaIXaaabeaakiaaysW7cqGHRaWkcaWG1bWaa0baaSqaai aaisdacaaI2aaabaGaaiikaiaadIeacaaIYaGaaiykaaaakiabeM8a 3naaBaaaleaacaWGibGaaGOmaaqabaGccqGH9aqpcqaHjpWDdaWgaa WcbaGaamisaiaaikdaaeqaaOGaeyOeI0IaeqyYdC3aaSbaaSqaaiaa dIeacaaIXaaabeaakiaab6cacaqGGaGaaeiiaiaabccacaqGGaGaae iiaiaabIcacaqGYaGaae4maiaabMcaaaa@6CDD@ Eq.15, Eq.13 and Eq.21 define sequence of acts for definition angular speeds ω H2 , ω 3 , ω 1 MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyYdC3aaS baaSqaaiaadIeacaaIYaaabeaakiaacYcacaaMe8UaeqyYdC3aaSba aSqaaiaaiodaaeqaaOGaaiilaiaaysW7cqaHjpWDdaWgaaWcbaGaaG ymaaqabaaaaa@436F@ of mechanism links. It is necessary to note that at start-up the kinematic chain will move in a condition with one degree of freedom in the absence of mobility in a contour before performance of condition M H2 > M H1 , MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamytamaaBa aaleaacaWGibGaaGOmaaqabaGccqGH+aGpcaWGnbWaaSbaaSqaaiaa dIeacaaIXaaabeaakiaacYcaaaa@3CCE@ Thus all kinematic and force parameters are determined, and all mechanism has the kinematic and static definability.

The positioning problem should be solved by usual method - by using of feedback in a control system of module motion. The feedback sensor prolongs or shortens a time of operation performance. Hence, basic new problems of positioning of the adaptive drive are absent.
Testing of the Adaptive Drive
The adaptive drive has been tested at the stand (figure 5) [5].
Figure 5:Stand for test adaptive drive
At the stand adaptive drive contains electric motor 1 and adaptive- mechanical transfer 2. Electric generator 3 simulates useful output loading. Theoretical regularities of adaptive drive are conforming to test results. Adaptive drive creates effect of force adaptation – at constant power of electric motor 1 angular speed of output shaft of reducer 2 is inverse proportional with variable loading which is creating by electric generator 3. Definiteness of reducer motion takes place both as in condition with two degrees of freedom (in operating regime) and in a condition with one degree of freedom (at start-up).
Work of the adaptive drive conforms to its tractive characteristic (figure 6) which has been gained at the drive experimental research on the test-bed.
Figure 6:Experimental tractive characteristic of adaptive reducer
A - motion in condition with one degree of freedom in the absence of internal mobility into reducer; ABC - motion with two degrees of freedom (operating regime); B - intermediate point; C - operating regime end (maximum moment of resistance and stop).
On figure 6 the experimental tractive characteristic of adaptive drive is presented. It has form of the schedule traction moment (torque) on output shaft of reducer depending on rotational speed of the shaft. Traction moment on reducer output shaft is equal to variable moment of resistance in operating regime. Adaptive reducer is set in motion by the electric motor of constant power at constant rotational speed of motor shaft. In the course of experiment no control means by power of the electric motor and its parameters were used.

In start-up regime driving moment of electric motor sweepingly changes from null to the rating value after switching of electric motor. Tractive characteristic of drive is presented by line which defines a startup regime in state with one degree of freedom. Adaptive reducer is moving as a single whole. The internal relative motion of wheels in the reducer is absent. Drive output shaft is revolution with rotation speed which is equal rated rotation speed of the electric motor. Tractive effort torque on output shaft of adaptive drive (in point of the characteristic) is equal to the moment on electric motor shaft M H2 = M H1 =0.480Nm MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamytamaaBa aaleaacaWGibGaaGOmaaqabaGccqGH9aqpcaWGnbWaaSbaaSqaaiaa dIeacaaIXaaabeaakiabg2da9iaaicdacaGGUaGaaGinaiaaiIdaca aIWaGaaGjbVlaad6eacaWGTbaaaa@441A@ Rotational speed of the output shaft of reducer is equal to rotational speed of motor shaft n H2 = n H1 =460RPM. MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBamaaBa aaleaacaWGibGaaGOmaaqabaGccqGH9aqpcaWGUbWaaSbaaSqaaiaa dIeacaaIXaaabeaakiabg2da9iaaisdacaaI2aGaaGimaiaaysW7ca WGsbGaamiuaiaad2eacaGGUaaaaa@4459@
The operating regime of motion begins in point of curve when the output moment of resistance begins to exceed nominal tractive effort torque M H2 > M H1 =0.480Nm. MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamytamaaBa aaleaacaWGibGaaGOmaaqabaGccqGH+aGpcaWGnbWaaSbaaSqaaiaa dIeacaaIXaaabeaakiabg2da9iaaicdacaGGUaGaaGinaiaaiIdaca aIWaGaaGjbVlaad6eacaWGTbGaaiOlaaaa@44CE@ In this case the reducer passes in state with two degrees of freedom. The power adaptation takes place. The output shaft rotational speed independently changes inversely a moment of resistance. The input moment of motor and its input rotational speed remain without change. They are equal to conforming rating values of parameters of the electric motor.
For example, in point B traction moment on output shaft and conforming moment of resistance has the value M H2 =1.490Nm MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamytamaaBa aaleaacaWGibGaaGOmaaqabaGccqGH9aqpcaaIXaGaaiOlaiaaisda caaI5aGaaGimaiaaysW7caWGobGaamyBaaaa@4086@ the rotational speed of output reducer shaft is equal n H2 =140RPM. MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBamaaBa aaleaacaWGibGaaGOmaaqabaGccqGH9aqpcaaIXaGaaGinaiaaicda caaMe8UaamOuaiaadcfacaWGnbGaaiOlaaaa@409D@ The maximum traction moment on reducer output shaft takes place in point C The maximum traction moment is equal to maximum moment of resistance M H2 =2.500Nm. MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamytamaaBa aaleaacaWGibGaaGOmaaqabaGccqGH9aqpcaaIYaGaaiOlaiaaiwda caaIWaGaaGimaiaaysW7caWGobGaamyBaiaac6caaaa@4131@ The output shaft rotational speed at the approach to point turn a minimum n H2 =8RPM MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBamaaBa aaleaacaWGibGaaGOmaaqabaGccqGH9aqpcaaI4aGaaGjbVlaadkfa caWGqbGaamytaaaa@3E7A@ Then in point occurs a stop of the reducer output shaft n H2 =0. MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBamaaBa aaleaacaWGibGaaGOmaaqabaGccqGH9aqpcaaIWaGaaiOlaaaa@3B19@ Input shaft of reducer continues to rotation with rated rotation speed of electric motor n H1 =470RPM. MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBamaaBa aaleaacaWGibGaaGymaaqabaGccqGH9aqpcaaI0aGaaG4naiaaicda caaMe8UaamOuaiaadcfacaWGnbGaaiOlaaaa@40A2@ The reducer passes in a state with one degree of freedom when the input shaft is rotation, and the output shaft is stop. So-called stop regime takes place.

Theoretically the reducer output power is equal to electric motor input power without a friction loss in operating regime of motion. At experiment conducting it was found out, that the increase in the traction moment leads to some decrease of an output power. It is connected with increase in relative speeds of rotation of wheels in the closed contour and increase in a friction loss.
Calculation of Interconnecting Parameters of Adaptive Reducer
Algorithm of adaptive reducer calculation conforms to earlier gained regularities [5]. Calculation is carried out in the set range of change of variable output torque M H2 MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamytamaaBa aaleaacaWGibGaaGOmaaqabaaaaa@387C@
Algorithm of calculation of the adaptive reducer (figure 4) has next form.
Initial data: input angular speed ω H1 MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyYdC3aaS baaSqaaiaadIeacaaIXaaabeaaaaa@3976@ input driving moment M H1 MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamytamaaBa aaleaacaWGibGaaGymaaqabaaaaa@387B@ and variable output torque M H2 MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamytamaaBa aaleaacaWGibGaaGOmaaqabaaaaa@387C@ numbers of teeth of wheels z i ,i=1,2,3,4,5,6 MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOEamaaBa aaleaacaWGPbaabeaakiaacYcacaaMf8UaamyAaiabg2da9iaaigda caGGSaGaaGjbVlaaikdacaGGSaGaaGjbVlaaiodacaGGSaGaaGjbVl aaisdacaGGSaGaaGjbVlaaiwdacaGGSaGaaGjbVlaaiAdaaaa@4BEC@ module of engagement m=1 MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyBaiabg2 da9iaaigdaaaa@38A8@

It is necessary to determine the next parameters: angular speeds of output carrier and blocks of wheels 1-4 and 3-6. ω H2 , ω 1 , ω 3 MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyYdC3aaS baaSqaaiaadIeacaaIYaaabeaakiaacYcacaaMe8UaeqyYdC3aaSba aSqaaiaaigdaaeqaaOGaaiilaiaaysW7cqaHjpWDdaWgaaWcbaGaaG 4maaqabaaaaa@436F@ and moments of forces acting on wheels 1, 3, 4, 6 M 12 , M 32 , M 45 , M 65 MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamytamaaBa aaleaacaaIXaGaaGOmaaqabaGccaGGSaGaaGjbVlaad2eadaWgaaWc baGaaG4maiaaikdaaeqaaOGaaiilaiaaysW7caWGnbWaaSbaaSqaai aaisdacaaI1aaabeaakiaacYcacaaMe8UaamytamaaBaaaleaacaaI 2aGaaGynaaqabaaaaa@46AE@
Solution.
1. Radiuses of toothed wheels r i =m z i /2. MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOCamaaBa aaleaacaWGPbaabeaakiabg2da9iaad2gacaWG6bWaaSbaaSqaaiaa dMgaaeqaaOGaai4laiaaikdacaGGUaaaaa@3E4C@
2. Radiuses of carriers r H1 =( r 1 + r 3 )/2, r H2 =( r 4 + r 6 )/2. MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOCamaaBa aaleaacaWGibGaaGymaaqabaGccqGH9aqpcaGGOaGaamOCamaaBaaa leaacaaIXaaabeaakiabgUcaRiaadkhadaWgaaWcbaGaaG4maaqaba GccaGGPaGaai4laiaaikdacaGGSaGaaGjbVlaadkhadaWgaaWcbaGa amisaiaaikdaaeqaaOGaeyypa0JaaiikaiaadkhadaWgaaWcbaGaaG inaaqabaGccqGHRaWkcaWGYbWaaSbaaSqaaiaaiAdaaeqaaOGaaiyk aiaac+cacaaIYaGaaiOlaaaa@4F59@
3. Transfer ratios in mechanism u 13 (H1) = z 3 / z 1 , u 46 (H2) = z 6 / z 4 MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyDamaaDa aaleaacaaIXaGaaG4maaqaaiaacIcacaWGibGaaGymaiaacMcaaaGc cqGH9aqpcqGHsislcaWG6bWaaSbaaSqaaiaaiodaaeqaaOGaai4lai aadQhadaWgaaWcbaGaaGymaaqabaGccaGGSaGaaGjbVlaadwhadaqh aaWcbaGaaGinaiaaiAdaaeaacaGGOaGaamisaiaaikdacaGGPaaaaO Gaeyypa0JaeyOeI0IaamOEamaaBaaaleaacaaI2aaabeaakiaac+ca caWG6bWaaSbaaSqaaiaaisdaaeqaaaaa@5059@
4. Angular speed of output carrier ω H2 = M H1 ω H1 / M H2 . MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyYdC3aaS baaSqaaiaadIeacaaIYaaabeaakiabg2da9iaad2eadaWgaaWcbaGa amisaiaaigdaaeqaaOGaeqyYdC3aaSbaaSqaaiaadIeacaaIXaaabe aakiaac+cacaWGnbWaaSbaaSqaaiaadIeacaaIYaaabeaakiaac6ca aaa@4498@

5. Angular speed of block of wheels 3-6 ω 3 = ω H2 (1 u 46 (H2) ) ω H1 (1 u 13 (H1) ) u 13 (H1) u 46 (H2) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyYdC3aaS baaSqaaiaaiodaaeqaaOGaeyypa0ZaaSaaaeaacqaHjpWDdaWgaaWc baGaamisaiaaikdaaeqaaOGaaiikaiaaigdacqGHsislcaWG1bWaa0 baaSqaaiaaisdacaaI2aaabaGaaiikaiaadIeacaaIYaGaaiykaaaa kiaacMcacqGHsislcqaHjpWDdaWgaaWcbaGaamisaiaaigdaaeqaaO GaaiikaiaaigdacqGHsislcaWG1bWaa0baaSqaaiaaigdacaaIZaaa baGaaiikaiaadIeacaaIXaGaaiykaaaakiaacMcaaeaacaWG1bWaa0 baaSqaaiaaigdacaaIZaaabaGaaiikaiaadIeacaaIXaGaaiykaaaa kiabgkHiTiaadwhadaqhaaWcbaGaaGinaiaaiAdaaeaacaGGOaGaam isaiaaikdacaGGPaaaaaaaaaa@5EEA@
6.Angular speed of block of wheels 1-4 ω 1 = u 13 (H1) ( ω 3 ω H1 )+ ω H1 MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyYdC3aaS baaSqaaiaaigdaaeqaaOGaeyypa0JaamyDamaaDaaaleaacaaIXaGa aG4maaqaaiaacIcacaWGibGaaGymaiaacMcaaaGccaGGOaGaeqyYdC 3aaSbaaSqaaiaaiodaaeqaaOGaeyOeI0IaeqyYdC3aaSbaaSqaaiaa dIeacaaIXaaabeaakiaacMcacqGHRaWkcqaHjpWDdaWgaaWcbaGaam isaiaaigdaaeqaaaaa@4C37@
7. Moments on wheels 1, 3, 4, 6: M 12 =0.5 M H1 r 1 / r H1 , M 32 =0.5 M H 1 r 3 / r H1 , MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamytamaaBa aaleaacaaIXaGaaGOmaaqabaGccqGH9aqpcaaIWaGaaiOlaiaaiwda caWGnbWaaSbaaSqaaiaadIeacaaIXaaabeaakiaadkhadaWgaaWcba GaaGymaaqabaGccaGGVaGaamOCamaaBaaaleaacaWGibGaaGymaaqa baGccaGGSaGaamytamaaBaaaleaacaaIZaGaaGOmaaqabaGccqGH9a qpcaaIWaGaaiOlaiaaiwdacaWGnbWaaSbaaSqaaiaadIeaaeqaaOWa aSbaaSqaaiaaigdaaeqaaOGaamOCamaaBaaaleaacaaIZaaabeaaki aac+cacaWGYbWaaSbaaSqaaiaadIeacaaIXaaabeaakiaacYcaaaa@52AF@
M 45 =0.5 M H2 r 4 / r H2 , M 65 =0.5 M H2 r 6 / r H2 MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamytamaaBa aaleaacaaI0aGaaGynaaqabaGccqGH9aqpcaaIWaGaaiOlaiaaiwda caWGnbWaaSbaaSqaaiaadIeacaaIYaaabeaakiaadkhadaWgaaWcba GaaGinaaqabaGccaGGVaGaamOCamaaBaaaleaacaWGibGaaGOmaaqa baGccaGGSaGaamytamaaBaaaleaacaaI2aGaaGynaaqabaGccqGH9a qpcaaIWaGaaiOlaiaaiwdacaWGnbWaaSbaaSqaaiaadIeacaaIYaaa beaakiaadkhadaWgaaWcbaGaaGOnaaqabaGccaGGVaGaamOCamaaBa aaleaacaWGibGaaGOmaaqabaaaaa@51D5@
8. Check of balance of circulating energy. Here
M 65 > M 32 , M 12 > M 45 MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamytamaaBa aaleaacaaI2aGaaGynaaqabaGccqGH+aGpcaWGnbWaaSbaaSqaaiaa iodacaaIYaaabeaakiaacYcacaaMe8UaamytamaaBaaaleaacaaIXa GaaGOmaaqabaGccqGH+aGpcaWGnbWaaSbaaSqaaiaaisdacaaI1aaa beaaaaa@4444@
( M 12 M 45 ) ω 1 =( M 65 M 32 ) ω 3 MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaad2 eadaWgaaWcbaGaaGymaiaaikdaaeqaaOGaeyOeI0IaamytamaaBaaa leaacaaI0aGaaGynaaqabaGccaGGPaGaeqyYdC3aaSbaaSqaaiaaig daaeqaaOGaeyypa0Jaaiikaiaad2eadaWgaaWcbaGaaGOnaiaaiwda aeqaaOGaeyOeI0IaamytamaaBaaaleaacaaIZaGaaGOmaaqabaGcca GGPaGaeqyYdC3aaSbaaSqaaiaaiodaaeqaaaaa@4B07@

If point 8 is carried out then performance of a following step of calculation algorithm at the next value of output torque take place. It is continued till using of end value of the rate.

Numerical check of algorithm of calculation of an adaptive wheelwork for one of values of output torque It is given: ω H1 =100 s 1 , M H1 =15Nm, M H2 =37.5Nm MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyYdC3aaS baaSqaaiaadIeacaaIXaaabeaakiabg2da9iaaigdacaaIWaGaaGim aiaaysW7caWGZbWaaWbaaSqabeaacqGHsislcaaIXaaaaOGaaiilai aaysW7caWGnbWaaSbaaSqaaiaadIeacaaIXaaabeaakiabg2da9iaa igdacaaI1aGaaGjbVlaad6eacaWGTbGaaiilaiaaysW7caWGnbWaaS baaSqaaiaadIeacaaIYaaabeaakiabg2da9iaaiodacaaI3aGaaiOl aiaaiwdacaaMe8UaamOtaiaad2gaaaa@57CD@ z 1 =20, z 2 =20, z 3 =60, z 4 =80, MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOEamaaBa aaleaacaaIXaaabeaakiabg2da9iaaikdacaaIWaGaaiilaiaaysW7 caWG6bWaaSbaaSqaaiaaikdaaeqaaOGaeyypa0JaaGOmaiaaicdaca GGSaGaaGjbVlaadQhadaWgaaWcbaGaaG4maaqabaGccqGH9aqpcaaI 2aGaaGimaiaacYcacaaMe8UaamOEamaaBaaaleaacaaI0aaabeaaki abg2da9iaaiIdacaaIWaGaaiilaaaa@4F1C@ z 5 =20, z 6 =120,m=1mm. MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOEamaaBa aaleaacaaI1aaabeaakiabg2da9iaaikdacaaIWaGaaiilaiaaysW7 caWG6bWaaSbaaSqaaiaaiAdaaeqaaOGaeyypa0JaaGymaiaaikdaca aIWaGaaiilaiaad2gacqGH9aqpcaaIXaGaaGjbVlaad2gacaWGTbGa aiOlaaaa@4954@ It is necessary to define the next parameters: ω H2 , ω 1 , ω 3 , M 12 , M 32 , M 45 , M 65 MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyYdC3aaS baaSqaaiaadIeacaaIYaaabeaakiaacYcacaaMe8UaeqyYdC3aaSba aSqaaiaaigdaaeqaaOGaaiilaiaaysW7cqaHjpWDdaWgaaWcbaGaaG 4maaqabaGccaGGSaGaaGjbVlaad2eadaWgaaWcbaGaaGymaiaaikda aeqaaOGaaiilaiaaysW7caWGnbWaaSbaaSqaaiaaiodacaaIYaaabe aakiaacYcacaaMe8UaamytamaaBaaaleaacaaI0aGaaGynaaqabaGc caGGSaGaaGjbVlaad2eadaWgaaWcbaGaaGOnaiaaiwdaaeqaaaaa@566F@
Solution. 1. r 1 =m z 1 /2=120/2=10, r 2 =10, r 3 =30, r 4 =40, r 5 =10, r 6 =60, MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGymaiaac6 cacaWGYbWaaSbaaSqaaiaaigdaaeqaaOGaeyypa0JaamyBaiaadQha daWgaaWcbaGaaGymaaqabaGccaGGVaGaaGOmaiabg2da9iaaigdacq GHflY1caaIYaGaaGimaiaac+cacaaIYaGaeyypa0JaaGymaiaaicda caGGSaGaaGjbVlaadkhadaWgaaWcbaGaaGOmaaqabaGccqGH9aqpca aIXaGaaGimaiaacYcacaaMe8UaamOCamaaBaaaleaacaaIZaaabeaa kiabg2da9iaaiodacaaIWaGaaiilaiaadkhadaWgaaWcbaGaaGinaa qabaGccqGH9aqpcaaI0aGaaGimaiaacYcacaaMe8UaamOCamaaBaaa leaacaaI1aaabeaakiabg2da9iaaigdacaaIWaGaaiilaiaaysW7ca WGYbWaaSbaaSqaaiaaiAdaaeqaaOGaeyypa0JaaGOnaiaaicdacaGG Saaaaa@6868@ 2.  r H1 =( r 1 + r 3 )/2=(10+30)/2=20, r H2 =( r 4 + r 6 )/2=50 MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGOmaiaac6 cacaqGGaGaamOCamaaBaaaleaacaWGibGaaGymaaqabaGccqGH9aqp caGGOaGaamOCamaaBaaaleaacaaIXaaabeaakiabgUcaRiaadkhada WgaaWcbaGaaG4maaqabaGccaGGPaGaai4laiaaikdacqGH9aqpcaGG OaGaaGymaiaaicdacqGHRaWkcaaIZaGaaGimaiaacMcacaGGVaGaaG Omaiabg2da9iaaikdacaaIWaGaaiilaiaadkhadaWgaaWcbaGaamis aiaaikdaaeqaaOGaeyypa0JaaiikaiaadkhadaWgaaWcbaGaaGinaa qabaGccqGHRaWkcaWGYbWaaSbaaSqaaiaaiAdaaeqaaOGaaiykaiaa c+cacaaIYaGaeyypa0JaaGynaiaaicdaaaa@5BC2@ 3.  u 13 (H1) = z 3 / z 1 =60/20=3, u 46 (H2) = z 6 / z 4 =120/80=1.5 MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaG4maiaac6 cacaqGGaGaamyDamaaDaaaleaacaaIXaGaaG4maaqaaiaacIcacaWG ibGaaGymaiaacMcaaaGccqGH9aqpcqGHsislcaWG6bWaaSbaaSqaai aaiodaaeqaaOGaai4laiaadQhadaWgaaWcbaGaaGymaaqabaGccqGH 9aqpcqGHsislcaaI2aGaaGimaiaac+cacaaIYaGaaGimaiabg2da9i abgkHiTiaaiodacaGGSaGaamyDamaaDaaaleaacaaI0aGaaGOnaaqa aiaacIcacaWGibGaaGOmaiaacMcaaaGccqGH9aqpcqGHsislcaWG6b WaaSbaaSqaaiaaiAdaaeqaaOGaai4laiaadQhadaWgaaWcbaGaaGin aaqabaGccqGH9aqpcqGHsislcaaIXaGaaGOmaiaaicdacaGGVaGaaG ioaiaaicdacqGH9aqpcqGHsislcaaIXaGaaiOlaiaaiwdaaaa@63A0@ 4.  ω H2 = M H1 ω H1 / M H2 =10015/37.5=40 s 1 MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGinaiaac6 cacaqGGaGaeqyYdC3aaSbaaSqaaiaadIeacaaIYaaabeaakiabg2da 9iaad2eadaWgaaWcbaGaamisaiaaigdaaeqaaOGaeqyYdC3aaSbaaS qaaiaadIeacaaIXaaabeaakiaac+cacaWGnbWaaSbaaSqaaiaadIea caaIYaaabeaakiabg2da9iaaigdacaaIWaGaaGimaiabgwSixlaaig dacaaI1aGaai4laiaaiodacaaI3aGaaiOlaiaaiwdacqGH9aqpcaaI 0aGaaGimaiaaysW7caWGZbWaaWbaaSqabeaacqGHsislcaaIXaaaaa aa@576C@ 5.  ω 3 = ω H2 (1 u 46 (H2) ) ω H1 (1 u 13 (H1) ) u 13 (H1) u 46 (H2) = 40(1+1.5)100(1+3) 3+1.5 =200 s 1 . MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGjbVlaaiw dacaGGUaGaaeiiaiabeM8a3naaBaaaleaacaaIZaaabeaakiabg2da 9maalaaabaGaeqyYdC3aaSbaaSqaaiaadIeacaaIYaaabeaakiaacI cacaaIXaGaeyOeI0IaamyDamaaDaaaleaacaaI0aGaaGOnaaqaaiaa cIcacaWGibGaaGOmaiaacMcaaaGccaGGPaGaeyOeI0IaeqyYdC3aaS baaSqaaiaadIeacaaIXaaabeaakiaacIcacaaIXaGaeyOeI0IaamyD amaaDaaaleaacaaIXaGaaG4maaqaaiaacIcacaWGibGaaGymaiaacM caaaGccaGGPaaabaGaamyDamaaDaaaleaacaaIXaGaaG4maaqaaiaa cIcacaWGibGaaGymaiaacMcaaaGccqGHsislcaWG1bWaa0baaSqaai aaisdacaaI2aaabaGaaiikaiaadIeacaaIYaGaaiykaaaaaaGccqGH 9aqpdaWcaaqaaiaaisdacaaIWaGaaiikaiaaigdacqGHRaWkcaaIXa GaaiOlaiaaiwdacaGGPaGaeyOeI0IaaGymaiaaicdacaaIWaGaaiik aiaaigdacqGHRaWkcaaIZaGaaiykaaqaaiabgkHiTiaaiodacqGHRa WkcaaIXaGaaiOlaiaaiwdaaaGaeyypa0JaaGOmaiaaicdacaaIWaGa aGjbVlaadohadaahaaWcbeqaaiabgkHiTiaaigdaaaGccaGGUaaaaa@7E18@ 6.  ω 1 = u 13 (H1) ( ω 3 ω H1 )+ ω H1 =(3)(200100)+100=200 s 1 MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGOnaiaac6 cacaqGGaGaeqyYdC3aaSbaaSqaaiaaigdaaeqaaOGaeyypa0JaamyD amaaDaaaleaacaaIXaGaaG4maaqaaiaacIcacaWGibGaaGymaiaacM caaaGccaGGOaGaeqyYdC3aaSbaaSqaaiaaiodaaeqaaOGaeyOeI0Ia eqyYdC3aaSbaaSqaaiaadIeacaaIXaaabeaakiaacMcacqGHRaWkcq aHjpWDdaWgaaWcbaGaamisaiaaigdaaeqaaOGaeyypa0JaaGjbVlaa cIcacqGHsislcaaIZaGaaiykaiaacIcacaaIYaGaaGimaiaaicdacq GHsislcaaIXaGaaGimaiaaicdacaGGPaGaey4kaSIaaGymaiaaicda caaIWaGaeyypa0JaeyOeI0IaaGOmaiaaicdacaaIWaGaaGjbVlaado hadaahaaWcbeqaaiabgkHiTiaaigdaaaaaaa@661F@ 7.  M 12 =0.5 M H1 r 1 / r H1 =0.51510/20=3.75Nm, MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaG4naiaac6 cacaqGGaGaamytamaaBaaaleaacaaIXaGaaGOmaaqabaGccqGH9aqp caaIWaGaaiOlaiaaiwdacaWGnbWaaSbaaSqaaiaadIeacaaIXaaabe aakiaadkhadaWgaaWcbaGaaGymaaqabaGccaGGVaGaamOCamaaBaaa leaacaWGibGaaGymaaqabaGccqGH9aqpcaaIWaGaaiOlaiaaiwdacq GHflY1caaIXaGaaGynaiabgwSixlaaigdacaaIWaGaai4laiaaikda caaIWaGaeyypa0JaaG4maiaac6cacaaI3aGaaGynaiaaysW7caWGob GaamyBaiaacYcaaaa@5A6F@ M 32 =0.5 M H 1 r 3 / r H1 =0.51530/20=11.25Nm, MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamytamaaBa aaleaacaaIZaGaaGOmaaqabaGccqGH9aqpcaaIWaGaaiOlaiaaiwda caWGnbWaaSbaaSqaaiaadIeaaeqaaOWaaSbaaSqaaiaaigdaaeqaaO GaamOCamaaBaaaleaacaaIZaaabeaakiaac+cacaWGYbWaaSbaaSqa aiaadIeacaaIXaaabeaakiabg2da9iaaysW7caaIWaGaaiOlaiaaiw dacqGHflY1caaIXaGaaGynaiabgwSixlaaiodacaaIWaGaai4laiaa ikdacaaIWaGaeyypa0JaaGymaiaaigdacaGGUaGaaGOmaiaaiwdaca aMe8UaamOtaiaad2gacaGGSaaaaa@5AD6@ M 45 =0.5 M H2 r 4 / r H2 =0.537.540/50=15Nm, MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamytamaaBa aaleaacaaI0aGaaGynaaqabaGccqGH9aqpcaaIWaGaaiOlaiaaiwda caWGnbWaaSbaaSqaaiaadIeacaaIYaaabeaakiaadkhadaWgaaWcba GaaGinaaqabaGccaGGVaGaamOCamaaBaaaleaacaWGibGaaGOmaaqa baGccqGH9aqpcaaMe8UaaGimaiaac6cacaaI1aGaeyyXICTaaG4mai aaiEdacaGGUaGaaGynaiabgwSixlaaisdacaaIWaGaai4laiaaiwda caaIWaGaeyypa0JaaGymaiaaiwdacaaMe8UaamOtaiaad2gacaGGSa aaaa@59F7@ M 65 =0.5 M H2 r 6 / r H2 =0.537.560/50=22.5Nm MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamytamaaBa aaleaacaaI2aGaaGynaaqabaGccqGH9aqpcaaIWaGaaiOlaiaaiwda caWGnbWaaSbaaSqaaiaadIeacaaIYaaabeaakiaadkhadaWgaaWcba GaaGOnaaqabaGccaGGVaGaamOCamaaBaaaleaacaWGibGaaGOmaaqa baGccqGH9aqpcaaIWaGaaiOlaiaaiwdacqGHflY1caaIZaGaaG4nai aac6cacaaI1aGaeyyXICTaaGOnaiaaicdacaGGVaGaaGynaiaaicda cqGH9aqpcaaIYaGaaGOmaiaac6cacaaI1aGaaGjbVlaad6eacaWGTb aaaa@592F@ 8. (3.7515)(200)=(22.511.25)200,11.25=11.25 MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGioaiaac6 cacaqGGaGaaiikaiaaiodacaGGUaGaaG4naiaaiwdacqGHsislcaaI XaGaaGynaiaacMcacqGHflY1caGGOaGaeyOeI0IaaGOmaiaaicdaca aIWaGaaiykaiabg2da9iaacIcacaaIYaGaaGOmaiaac6cacaaI1aGa eyOeI0IaaGymaiaaigdacaGGUaGaaGOmaiaaiwdacaGGPaGaeyyXIC TaaGOmaiaaicdacaaIWaGaaGjbVlaacYcacaaIXaGaaGymaiaac6ca caaIYaGaaGynaiabg2da9iaaigdacaaIXaGaaiOlaiaaikdacaaI1a aaaa@5E56@ Further there is a repetition of calculation with new value of the output torque.
Adaptive Gripper
Gripping and holding of objects are key tasks for robotic manipulators. The development of universal grippers able to pick up unfamiliar objects of widely varying shapes and surfaces is a very challenging task. The authors of the letter Petcovic D and Pavlovic ND [6] have presented the new principle of a new universal gripper with adaptive shape morphing surfaces. The adaptive surfaces have the controllability by a compliant system with embedded actuators and sensors. The main sensing system has to be made of a conductive silicone rubber or foam. These are carbon-blank filled silicone materials with good senses properties whose electrical resistance is changed by compression. The implemented controllable system is able to morph shapes of the gripper to accommodate different objects. But such system has low reliability, efficiency and high complexity. It is offered to use for observed situations adaptive gripper.

Adaptive gripper (figure 7) has adaptive drive in the form of lever mechanism with the closed contour.
Figure 7:Adaptive gripper of the robot
Adaptive gripper contains basis 1 with joint 2, adaptive drive 3-4-5 and mechanism of gripper 6. Product in gripper is shown by a dot line. Adaptive drive contains the electric solenoid 3 (or the hydro cylinder) and the adaptive lever mechanism with closed contour 4.

The solenoid 3 moves input link of adaptive mechanism 4 to the right. The closed contour 4 transfers motion through output link 5 to the mechanism gripper 6 which provides a dripping of product. Force of dripping interferes with motion of link 5 and stops it at achievement of set value. Adaptive mechanism of drive passes in a condition with one degree of freedom. Then the position pickup switches out drive 3 and fixes it in final position. Product dripping by the adaptive mechanism occurs in state with two degrees of freedom without use of tactile sensing transducers. Gripper disclosing occurs at motion of input link of mechanism in the opposite direction.
Adaptive Robot Transmissions in Extreme Working Conditions
In many cases the robot is used for work in extreme conditions which create unpredictable and unexpected disturbances to motion in the form of repeatedly increased forces of resistance. Disturbances to motion can be called by environment (for example, space): the high temperature drop, vacuum, pressure, external mechanical affecting, etc. Disturbances call deformations of links, interacting conditions, break greasing conditions, lead to wedging. Disturbances are especially dangerous to the independent robots working without possibility of operative service. For work in extreme conditions the drive of the module with “rigid” kinematic constraint should be counted for the maximum possible draught resistance that leads to substantial growth of weight and sizes. The adaptive drive with adaptive kinematic constraint is capable to resist to the failures connected with an unforeseen overloading, without increase in power of an engine. For reliable overcoming of unforeseen disturbances motion is offered to use the vibrating adaptive drive which transfers vibration to moving links [7,8]. Vibration allows overcoming the increased forces of resistance at considerable decrease in power expenses.

The adaptive vibrating drive contains the adaptive planetary reducer (figure 4) in which the entrance satellite 2 has unbalanced weight, and blocks of wheels 1 – 4 and 3 – 6 have elastic constraints between wheels. During motion the unbalanced inertial force is transferred to the target carrier and calls torsion oscillations in the form of circular vibrations.

The adaptive vibrating drive has high power efficiency, small weight and sizes
Conclusions
Regularity of interconnection of force and kinematic parameters of the robot in which each module has the adaptive electric drive are developed. The adaptive electric drive contains the electric motor and the adaptive connecting gear with two degrees of freedom. The adaptive mechanism provides motion of module moving part with a speed inverse to force loading at constant power of electric motor. The adaptive mechanism has the variable transfer ratio which depends on loading. Presence of definability of motion of the adaptive electric drive is confirmed by the drive experimental research. The experimental tractive characteristic containing a regime of start-up and operating regime is gained. The tractive characteristic conforms to the theoretical characteristic.

The adaptive drive is extremely simple in design. It changes the transfer ratio depending on loading independently only at the expense of mechanical properties and does not demand control of the transfer ratio. Power adaptation allows reducing sizes and weight of the drive at the expense of decrease of a required power of the electric motor. The equations of interconnection of parameters of motion of the adaptive mechanism are developed. The adaptive drive of the module of the robot provides a stop regime of motion at achievement of the maximum value of variable resistance. In this moment the module moving part appears as motionless and the electric motor continues motion with former speed. Stop regime allows to avoid an overloading and to overcome emergencies. In the robot with adaptive drives of modules stop regime can be overcome at the expense of motion of other modules leading to decrease of resistance.

In the article the demonstration of possibility of decrease of a required power of the engine is resulted. The adaptive module of translational motion and the rotary motion module are presented.

The use of the adaptive hydraulic drive allowing adapting for admitted force (for example, in gripper) is considered. The numerical instance of the drive calculation for confirming the force adaptation existence is resulted. The stand for testing of the adaptive drive is presented.
For reliable overcoming of unforeseen disturbances motion is offered to use the vibrating adaptive drive which transfers vibration to moving links. Vibration allows overcoming the increased forces of resistance at considerable decrease in power expenses.
All these materials should show the principal newness and efficiency of researches which are based on the science discovery. The adaptive drive of the robot provides a high efficiency and has new abilities.
ReferencesTop
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  2. Fuchs RD, Hasuda Y, James IB. Full Toroidal IVT Variator Dynamics. SAE 2002 World Congress & Exhibition. 2002.  
  3. Ivanov KS, Ualiev G, Tultaev B. Kinematic and Force Analysis of Robot with Adaptive Electric Drives. OPTIROB 2014. Springer. 2014:56-63. 
  4. Ivanov KS. Action of Robot with Adaptive Electric Drives of Modules. Advances on Theory and Practice of Robots and Manipulators. 2014;22:563-569.
  5. Ceccarelli M, Balbaev G, Ivanov K. An Experimental Test Validation of a New Planetary Transmission. International Journal of Mechanics and Control. 2014;15(2):1-7.
  6. Petkovic D, Pavlovic ND. A New Principle of Adaptive Compliant Gripper. Mechanism and Machine Science 3. Mechanisms, Transmissions and Applications. 2012:3:143-150.
  7. Ivanov KS, Koilybayeva RK, Ualiev GU. Creation of Vibration Gear Continuously Variable Transmission (CVT). 11th International Conference on Vibration Problems (2013 ICOVP). 2013:91.
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