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Self-alignment of kinematic couplings: Effects of deformations
Precision Engineering 60 (2019) 348–354
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Precision Engineering journal homepage: http://www.elsevier.com/locate/precision
Self-alignment of kinematic couplings: Effects of deformations Francesco Patti *, J.M. Vogels VDL ETG Technology & Development BV, Eindhoven, the Netherlands
A R T I C L E I N F O
A B S T R A C T
Keywords: Kinematic couplings Self-alignment Self-locking Limiting coefficient of friction Friction force orientation Statically indetermined Statically determined
This paper contains a study of the kinematic coupling, focusing on the self-aligning property. Such a property expresses how the system is able to overcome the friction forces when reaching the nested position: this is why the characterization of these forces is one of the main subjects that will be analyzed. Normally, friction forces are assumed to be oriented against the possible motion even when motion is not present: an alternative approach will be proposed, that is to study the deformations of the involved bodies to identify friction orientations. Since a study of the deformations is not always possible due to lack of information (during the concept phase of the design, for instance) it will be proposed a worst case scenario approach: the effect of deformations will be evaluated without studying them in detail. Such approach can be used to estimate the performance of kinematic couplings in the early phase of the design; moreover, it is also useful to easily compare the two analysis (with and without deformations), helping therefore to understand the importance of this phenomenon with respect to selfalignment.
1. Introduction Kinematic Couplings are interfaces very commonly used in the field of precision mechanics. They can position parts with a repeatability up to submicron level. This paper treats the self-aligning property of the interface, which can be seen as a prerequisite of the repeatability. In fact, while the repeatability expresses the behavior of the system in its nested position, the self-aligning property determines whether the system will be able or not to reach the nested position. Friction plays an important role in the self-alignment, since it will oppose the nesting action. Provided that the centered position is of equilibrium and stable, the system will be able to align itself only if the friction coefficient is smaller than a certain value, the so called limiting coefficient of friction. Studies that explain how to evaluate the limiting coefficient of fric tion are already available (see Ref. [1]). They provide a mathematical model where the friction forces are characterized assuming that the coupling is moving towards the nested position: thus every friction force is oriented in the opposite direction of the relative motion between the parts at the sliding contact. This is certainly true if the interface is actually moving. Supposing to start the alignment of the system with no motion, is it still possible to assume the friction forces oriented as if there were motion? In this paper will be shown that in static condition the friction forces have an orientation that is independent from the motion that is
about to happen, determined only by the way the involved bodies deform. Intuitively, having the friction forces in the opposite direction of the motion works the best for suppressing the motion. However, it will be shown that in case of no motion this is not true. Therefore, not only friction orientation is unrelated to the potential motion, but such assumption could lead to underestimate the friction ability to prevent the motion, thus to predict a system to be self-aligning when it actually isn’t. 2. Problem definition A kinematic coupling is an interface that connects a system to another in an isostatic way: this is achieved by six contacts, in most cases between three spheres and 3 V-grooves (Fig. 1). When the coupling is put together, at first it will be off centered, meaning that not all constraints will be in contact: it will be assumed that the initial position is close enough to the nested position such that the geometry can be considered unchanged. This will allow to write the equilibrium equations of the coupling as if it were in the nested position; the only difference will be the removal of one or more constraints. Moreover, the system will be studied in a state of equilibrium called incipient motion: this is characterized by the fact that there is no motion, but any reduction of the friction coefficient would break the equilib rium. In other words, the system is on the verge of motion.
* Corresponding author. VDL ETG Technology & Development BV, De Schakel 22, 5651, GH, Eindhoven, the Netherlands. E-mail address: [email protected] (F. Patti). https://doi.org/10.1016/j.precisioneng.2019.08.013 Received 14 May 2019; Received in revised form 3 July 2019; Accepted 27 August 2019 Available online 4 September 2019 0141-6359/© 2019 Elsevier Inc. All rights reserved.
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Precision Engineering 60 (2019) 348–354
Fig. 2. Friction force treated as an extra constraint.
2.3. Orientation of friction forces When friction forces are present, the equilibrium problem of a ki nematic coupling is indetermined in case of incipient motion. In fact, the following quantities are unknown:
Fig. 1. An example of conventional kinematic mount.
The question to be answered is: will the system be able to move to wards the nested position? This can be found by studying the equilib rium of the system, in all possible off-center positions. As explained in Ref. [1], it is possible to restrict the study to only 6 cases, the most difficult for motion. In fact, at the beginning of the positioning phase only few constraints will be in contact. Assuming to have a self-aligning system, the system will move towards the centered position, therefore the number of constraints in contact will increase, until all six con straints will be in contact. The tendency of the system will be to move in a way that the minimum amount of energy will be dissipated; the fewer the number of active constraints, the more the freedom of the system to move towards the least resistive path; so, when it will remain only one constraint not engaged, the system will be able to move only in one specific path, which in general will be far from being the least resistive. This line of reasoning brings to the conclusion that if a system is able to move in all of the six paths identified by removing the six constraints one at a time, then it will always be able to move towards the nested position. Therefore there will be six studies of equilibrium, each one with only five active constraints: the most resistive path will be identified by the smaller limiting coefficient of friction, which will measure the selfaligning property of the system.
! - 5 normal reactions1 N i with known direction-> 5 variables; ! - 5 friction reactions T i normal to the previous ones-> 10 variables; - 1 friction coefficient μ -> 1 variable. In total, there are 16 unknown. The equations available instead are: - 3 equations for equilibrium of forces; - 3 equations for equilibrium of moments; - 5 equations defining the friction coefficient NTii ¼ μ. This brings to 16-11 ¼ 5 missing equations: the solution belongs to a five dimensional space: to find the unique solution (when existent) that represents the reality, five more equations are needed. An approach that is often followed is to assign a specific direction for the five friction forces: it is assumed that friction is oriented as if there were motion. In this way, five more equations become available, and the system becomes determined. To know such directions, it is sufficient to evaluate the virtual displacement compatible with the constraints, at each contact point. Although this is often a good approximation as it will be shown later, in reality friction has a different orientation. Instead of assuming a direction for every friction force, these can be actually solved together with the other unknown by following a different approach. Although the kinematic couplings is a statically determined sys tem, if the friction forces are strong enough to prevent motion, the kinematic coupling becomes an hyperstatic system. To see this more clearly, it is sufficient to replace every friction force by two simple constraints,2 as shown in Fig. 2. By doing so, the total number of simple constraints becomes 5 þ 10 ¼ 15; on the other hand, to constrain the system only six simple constraints are required, thus the system is statically inde termined. As explained in Ref. [2], this problem can be solved by considering the involved bodies as solid; the method of displacements can be applied, which leads to nine extra equations (congruence equa tions) expressing that the bodies will deform in a way compatible with the constraints: the mutual distances of the five contact points must be
2.1. Prerequisite for self-alignment A self-aligning system implies that the centering motion aims to the centered position: to make sure this is the case, it must be verified that such position is of equilibrium when μ ¼ 0. 2.2. Model definition: naming convention As stated before, the study will be limited to a very small area around the nested configuration of the system: this means that it will be possible to consider the contact points coincident with the nominal ones, iden tified by the position vector ! r i , where i ¼ 1; 2; …; 6 is the index assigned to the six constraints. Furthermore, every contact will constrain only one direction, iden tified by the unit vector ! n i , which is defined as the normal of the tangent plane of both faces at the contact point. ! The constraint reactions contain a normal component N i ¼ Ni ! n i, ! and a tangential component representing friction T i . ! The force applied to the moving part of the coupling is defined as F e , applied to the point identified by the position vector ! r e. All these quantities will be expressed in a coordinate system CSO ðO; x; y; zÞ whose origin and orientation is arbitrary.
1
There are only five active contact points because the coupling is offcentered. 2 This is possible thanks to the assumption of friction forces strong enough to impede motion.
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Assuming that this position is of equilibrium, there will be no sliding at the contact points, since friction forces will act against them. There fore the friction forces will be oriented against such slidings, which are not provoked by rigid movement, but merely by local deformations of the bodies. To see the effect of friction orientation in reaching equilibrium, a simple case in 2D is shown4 (see Fig. 3): the problem is studied with the graphical method explained in Ref. [3]. A body is constrained in two points, having therefore one degree of freedom. Will the external force be able to move it? Considering the hypothesis of friction against the incipient motion, since the motion will be a clockwise rotation around C, the friction angle to be considered is the one in yellow: the system could be in equilibrium only if the external force would intersect the yellow area. Since this is not the case, with the friction coefficients represented, equilibrium cannot be verified. But if friction forces are allowed to be oriented towards the incipient motion, the three areas in green must be added: thanks to this extension, the condition for equilibrium are met. In this particular case, the friction force of constraint 1 is pointing towards the motion, and yet this situa tion is more efficient in terms of preventing the motion, than the case of friction against it (yellow areas). The next paragraphs will examine the equilibrium of the kinematic coupling first assuming friction forces oriented against the incipient motion. After that, this assumption will be removed: friction forces will be considered independent from the incipient motion. 3. Analysis with friction oriented against the incipient motion In this case, the equilibrium of the system is studied considering friction forces parallel and opposed to the motion that would happen if the system were self-aligning. The virtual displacements of the system will be studied; in particular, six different displacements will be considered, each one corresponding to a constraint removed.
Fig. 3. Case of equilibrium with friction towards the incipient motion.
the same for the two bodies of the coupling.3 It is interesting to notice that in this process the possible motion of the system is not involved at all. Moreover, the ratio NTii ¼ μ is not part of the equations: these two considerations bring to the following general conclusions:
3.1. Evaluation of virtual displacements
(1) The friction forces are not oriented against the incipient motion. (2) The limiting friction coefficient is different for each constraint.
Let the constraint number j be removed. A virtual displacement of the system around the nominal position can be expressed using the wellknown formula5:
The conclusion No.1 shows that assuming the friction forces oriented exactly against the motion is not correct (although in many cases is a good approximation); in some circumstances this could result in underestimating the effectiveness of friction in blocking the motion: in other words, in general there are other directions for which friction forces can be more effective, thus being able to block the motion even with a lower coefficient of friction. Conclusion No.2 instead brings to the following: assuming friction coefficient equal for all constraints, the condition of incipient motion is characterized by only one constraint where the reaction lies on the cone of friction, and not all of them. The just explained approach shows that when in equilibrium prob lems friction forces are involved, and there is no relative motion, the deformations of the bodies play an important role: the orientation of friction forces depend on the internal deformations, and it influences the efficiency of friction in contrasting the motion. This behavior can be visualized as follows: when a coupling is put together, at some point the moving body starts touching the counterpart in a number of points, which will generate constraint reactions: the body will deform under this load.
! s O þdθ �! ri d! s i ¼ d!
i ¼ 1; …; 6 except j
(1)
where d! s i is the displacement of the contact point of the body with ! constraint i, d! s O is the displacement of the origin, d θ the angular displacement, and ! r is the vector position of contact point i. i
For each constraint, (1) defines all possible displacements with six degrees of freedom; imposing the compatibility with each of the five active constraints will give one degree of freedom displacements, meaning that only their amplitude is arbitrary. Although (1) is strictly valid for infinitesimal displacements, in this case it will be assigned a certain amplitude and therefore the infinitesimal displacements d! s i will be replaced by the finite Δ! s i . This is a valid step since the only aim to evaluate the virtual displacements here is to know their directions, which are independent of the amplitude. The compatibility equations describe that the displacement at 4
In 2D the friction force has a fixed direction: there are only two possible orientations, against and towards the relative motion. This is perhaps too simple to understand fully the three dimensional case, but allows to have a better understanding of the effect of friction orientation and the relationship with the body deformations. 5 The above formula expresses the velocity field of a rigid motion, thus a relation between velocities: in this case it is multiplied by the infinitesimal time interval dt to give displacements.
3 A simple way to justify the number of equation needed, is to add these five points one at a time, each time adding the minimum number of equations needed to link their positions to the others: for the first point no equations, for the second only one, and so forth, remembering that a point in 3D has 3 DoFs.
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constraint i belongs to the tangent plane at the contact point: Δ! s i ⋅! ni ¼0
(2)
i ¼ 1; 2; …; 6 except j
The ambiguity of the displacement amplitude is removed by assigning the following extra equation: Δ! s j ⋅! nj ¼
(3)
1
which also imposes that the virtual displacement is oriented towards the nested position. Replacing (1) in (2) and (3), a system of six scalar equations in six variables (the components of the two vector variables ! Δ θ , Δ! s O ) is obtained, which allows to find the virtual displacements ! Δ s for each of the six cases obtained removing constraint j: i
�
! ! ! ðΔs! O þ Δ θ � r i Þ⋅ n i ¼ 0 ! !� ! Δs! 1 O þΔθ � r j ⋅nj ¼
i ¼ 1; 2; …; 6 except j
(4)
Introducing the function of Kronecker delta δi;j the above system (4) can be written: ! ! ! ðΔs! O þ Δ θ � r iÞ ⋅ n i ¼
δi;j
i ¼ 1; 2; …; 6
(5)
Fig. 4. Definition of angles.Ωi .
3.2. Evaluation of equilibrium complex. These deformations are strongly dependent on the history of the load application and the order of engaged constraints: in fact, assigned a set of five points of contact, the system can actually reach this configuration in many ways, each of them with a different deformation at the moment of reaching the fifth one. In order to have an estimation of the effect of the deformations, a worst case scenario is proposed. The aim is to evaluate the set of friction directions associated to the smallest limiting coefficient of friction, without studying the particular deformation that is generating them.
For a rigid body, equilibrium can be expressed by the following vector equations: X! Fk ¼ 0 (6) k
X! ! Mk þ ! r k � Fk ¼ 0
(7)
k
In case of a kinematic coupling with one constraint removed, there ! ! are five forces F i applied in five contact points, and an external force F e which provides energy for the alignment. ! Each reaction force is characterized by a normal component N i ¼ ! N! n and a friction component T . If μ is the static friction coefficient, i
i
For each constraint it is introduced an extra parameter Ωi (shown in Fig. 4) which defines the orientation of the friction force with respect to the direction of the incipient motion evaluated with the system (5). �! Given the direction of displacement Δs i and the normal of the ! contact n i , the direction needed to define the plane of the contact is
i
these forces are: ! F i ¼ Ni ! ni
4.1. Equations of equilibrium
Δ! s
μNi !i jΔ s i j
i ¼ 1; 2; …; 6
except j
(8)
! As for the equilibrium of rotations (7), all moments M k can be considered null, since their contribution would be negligible. Equations (6) and (7) can be written as follows: � 8 X� Δ! si ! > > Fe þ n i μN i ! ¼ 0 Ni ! > > jΔ s i j < i6¼j (9) � � > X > Δ! si ! > ! ! ! > r i � N i n i μN i ! ¼ 0 : r e � Fe þ jΔ s i j i6¼j
�! Δs i ! m i ¼ �! � ! ni j Δs i j
(10)
The expression of the general constraint reaction (8) becomes: ! F i ¼ Ni ! ni
�! Δs
μNi �!i cosΩi j Δs i j
�! Δs
μNi �!i � ! n i sinΩi i ¼ 1; 2; …; 6 except j j Δs i j
System (9) becomes:
In the above system (9) the sum must be evaluated for i ¼ 1 to i ¼ 6 except for j, where j is the removed constraint: it is a system of two vector equations, in six variables: μ; Ni i ¼ 1; …; 6 except j. Solving (9) gives the limiting coefficient of friction for the particular simple path resulting by removing constraint j.6 4. Analysis considering deformations As explained before, a more accurate analysis should evaluate the friction forces by studying the deformations of the system. Such approach is not very practical: not only it requires many details of the system that might not be known at concept definition, but it’s also very 6
Fig. 5. Standard V-groove interface.
See 8.1 in the Appendix for more details about the solution of the system. 351
(11)
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Fig. 6. Limiting coefficient of friction for a standard kinematic coupling.
8 X� ! > > þ ni Ni ! F e > > < i6¼j
�! Δs
μNi �!i cosΩi j Δs i j
� > > X > ! > ! :! r e � Fe þ r i � Ni ! ni i6¼j
�
�! Δs
μNi �!i � ! n i sinΩi j Δs i j
�! Δs μNi �!i cosΩi j Δs i j
�! Δs
¼0 �
μNi �!i � ! n i sinΩi ¼ 0 j Δs i j
(12)
line. When instead the angle gets smaller, the difference is not negligible any more: the green line underestimates the efficiency of friction. Moreover, it’s interesting to note the green line doesn’t take into account the phenomenon of self-locking, which is instead included in the blue line: this is because both self-locking and worst case analysis are based on the deformations of the bodies. 6. Conclusions
As shown in the Appendix 8.1 for system (9), it’s easy to prove that for each set fΩ1 ; …; Ω6 g this system has five solutions for μ: the limiting coefficient of friction is the smallest real and positive value. Such value is function of the set fΩ1 ; …; Ω6 g: the worst case is obtained by mini mizing this function with respect to the set fΩ1 ; …; Ω6 g. This is a complex problem, but can be solved relatively easy with a computer algebra system.
This study has shown that analyzing the self-alignment of kinematic couplings by considering friction oriented as if there were motion is a good approximation for the standard kinematic coupling of Fig. 1, but it might not represent the worst case in other situations, e.g. when the half angle of the V groove is reduced. The proposed approach takes into account that, as demonstrated, in case of incipient motion the friction forces won’t be oriented exactly against the relative motion, making friction sometimes more efficient in preventing the motion. This study shows also (paragraph 2.3) that the case of incipient motion is characterized by different values of limiting friction coefficient for each constraint: this has not been taken into account so far, since in equation (11) there is only one coefficient of friction. It would therefore be an interesting development of this study to include the variations of friction coefficients. At the moment this paper is written, a test setup is being developed, to add experimental verification of the theory.
5. Comparison of the two methods The two approaches are now applied to the common case of a ki nematic coupling shown in Fig. 5: the coupling has three V-grooves at 120� ; β, the half angle of the V, is variable; the external force is vertical, applied at the center of the system. Fig. 6 shows the limiting coefficient of friction evaluated in case of friction forces oriented against the incipient motion (curve in green), and oriented in a way that the friction has the best efficiency in pre venting the motion (curve in blue). The line in red instead represents the combination of μ and β for which a V-groove becomes self-locking: above the red line each ball will jam into its groove as soon as it touches it in two points (for more details about self-locking behavior see Appendix 8.2). Around the standard V-groove configuration β ¼ 45� the two ap proaches give almost the same result, therefore it is safe to use the green
Acknowledgements The authors would like to thank Prof. Dannis Brouwer from the University of Twente for his precious advises.
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8. Appendix 8.1. Solving the equilibrium equations for friction against incipient motion Solving the system (9) is relatively easy, if done with a computer algebra system. The problem is that (9) has more than one solution: in fact, when converted to scalar equations, it becomes a system of six equations in six variables, but because of the products μNi , it’s not linear. This means that there is more than one solution: computer programs can only find solutions around a given range. So, the risk is that the found solution is not the one that has a physical meaning. To solve this issue, the system (9) can be expressed in a different form, so that it’s easier to find all solutions at once, in order to easily select the right one. For the next steps, the followings will be considered: - The excluded constrain will be j ¼ 6 : all results can be easily applied to the other 5 cases; - If the system is verified for a certain value of external force, it is verified as well for every value of force, provided that the line of force is un changed. This allows to introduce a sixth variable c 2 ℝ which multiplies the external force. System (9) can be written as follows: � 8 X� Δ! si ! ! > > c F N þ n μ N ¼0 e i i i > > jΔ! s ij < i6¼j � � > X > Δ! si ! > ! > r e � cFe þ r i � Ni ! n i μNi ! ¼ 0 : ! jΔ s i j i6¼j
(13)
And moving from vector to scalar equations: 8 Δs1x Δs5x > n1x N1 μN1 þ ⋯ þ n5x N5 μN5 þ c Fex ¼ 0 > ! ! > > jΔ s j jΔ s 5j 1 > > > > > Δs1y Δs5y > > n1y N1 μN1 þ ⋯ þ n5y N5 μN5 þ c Fey ¼ 0 > > ! ! > jΔ s j jΔ s 5j 1 > > > > > Δs1z Δs5z > > > n1z N1 μN1 þ ⋯ þ n5z N5 μN5 þ c Fez ¼ 0 > > jΔ! s 1j jΔ! s 5j < � � Δ! s1 > ! > > ð! n 1 Þx N1 r1� ! μN1 þ ⋯ þ ð! r 5 �! n 5 Þx N5 r 1 �! > > jΔ s 1j x > > > > � � > > Δ! s1 > ! > ð! > r 1 �! n 1 Þy N1 r1� ! μN1 þ ⋯ þ ð! r 5�! n 5 Þy N5 > > jΔ s 1 j y > > > > � � > > > Δ! s1 > ! : ð! r 1 �! n 1 Þz N1 r1� ! μN1 þ ⋯ þ ð! r 5�! n 5 Þz N5 jΔ s j 1
z
� � Δ! s5 ! ! r5� ! μN5 þ cð! r e � F e Þx ¼ 0 jΔ s 5 j x � � Δ! s5 ! ! r5� ! μN5 þ cð! r e � F e Þy ¼ 0 jΔ s 5 j y � � Δ! s5 ! ! r5� ! μN5 þ cð! r e � F e Þz ¼ 0 jΔ s j 5
(14)
z
The above (14) is a system 6 � 6 in N1 ; …; N5 ; c. Introducing the following matrixes: 2
n1x n1y 6 6 n1z 6 A¼6 r 1�! n 1 Þx 6 ð! 6 ! 4ð r 1 � ! n 1 Þy ! ! ðr � n Þ 1
2
1 z
⋅ ⋅ ⋅ ⋅ ⋅
3 Fex Fey 7 7 Fez 7 ! 7 ! ð r e � F e Þx 7 7 ! ð! r e � F e Þy 5 ! ð! r �F Þ
⋅ ⋅ ⋅ ⋅ ⋅
⋅ ⋅ ⋅ ⋅ ⋅
⋅ ⋅
⋅
n5x n5y n5z ð! r 5 �! n 5 Þx ð! r 5 �! n 5 Þy ! ! ðr � n Þ
⋅
⋅ ⋅
Δs5x jΔ! s 5j
⋅
⋅ ⋅
Δs5y jΔ! s 5j
⋅
⋅ ⋅
⋅
⋅ ⋅
⋅
⋅ ⋅
⋅
⋅ ⋅
Δs1x jΔ! s 1j
6 6 6 Δs1y 6 6 ! 6 jΔ s 1j 6 6 6 Δs1z 6 6 jΔ! s 1j 6 6 � � B¼6 Δ! s1 ! 6 r � 1 6 jΔ! s 1j x 6 6 � � 6 ! Δ s1 6 ! r1� ! 6 6 jΔ s 1 j y 6 6 � � 4 Δ! s1 ! r1� ! jΔ s j 1
z
5
5 z
e
(15)
e z
3 0
Δs5z jΔ! s 5j � � Δ! s5 ! r5� ! jΔ s 5 j x � � Δ! s5 ! r5� ! jΔ s 5 j y � � Δ! s5 ! r5� ! jΔ s j 5
7 7 7 7 07 7 7 7 7 07 7 7 7 7 07 7 7 7 7 7 07 7 7 7 5 0
(16)
z
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2
3
N1 6 N2 7 6 7 6 N3 7 7 X¼ 6 6 N4 7 6 7 4 N5 5 c
(17)
the system becomes: (18)
AX þ μBX ¼ 0
This is the well-known generalized problem of eigenvalues: it has six solutions and they can be easily found with the aid of a computer algebra system. Moreover, it can be easily seen that one eigenvalue is always infinite: in fact, dividing the main equation by μ (supposed non null): 1
μ
(19)
X þ BX ¼ 0 If μ→∞ then BX ¼ 0. From equation (16), it is:
(20)
detB � 0
So, the system is always verified by μ→∞. Excluding this one, five solutions are left: they can be both real and complex; the smallest positive real value is the solution with physical meaning: the limiting coefficient of friction for the studied aligning path. The next step is to solve the system (14) for all six aligning paths, and to take the smallest of the solutions as the limiting coefficient of friction of the system. The normal reactions obtained are ! ! associated to external force c F e : the ones associated to the actual external force F e are obtained by dividing them by the coefficient c. 8.2. Self-locking The attribute “self-locking” for a system might refer to several types of systems: here self-locking refers to a situation where the forces that push the contact faces, thus generating friction, are not produced by an external load. In order to be present in a static system, it’s easy to prove that the resulting system of forces (normals and friction) needs to be in equilibrium.
Fig. 7. Force decomposition in a self-locking situation.
An example of this phenomenon is represented in Fig. 7 where a ball is coupled to a vee and the coefficient of friction is such that, for the given angle β, it is possible that the two reactions R1 and R2 are in equilibrium: even without any external force, the ball is “squeezed” by the vee. If the ball is pushed against the vee and then released, the ball remains stuck in its position: the deformations induced by the initial load are generating the normal reactions necessary for friction to restrain the ball from further movement. It is clear that for a kinematic coupling this condition must be avoided.
References
[2] Carpinteri A. Structural mechanics - a unified approach. Taylor and Francis; 1997. [3] Blanding DL. Exact constraint: machine design using kinematic principles. ASME Press; 1999.
[1] Hale LC, Slocum AH. Optimal design techniques for kinematic couplings. Precis Eng 2001;25:114–27.
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Contents lists available at ScienceDirect
Precision Engineering journal homepage: http://www.elsevier.com/locate/precision
Self-alignment of kinematic couplings: Effects of deformations Francesco Patti *, J.M. Vogels VDL ETG Technology & Development BV, Eindhoven, the Netherlands
A R T I C L E I N F O
A B S T R A C T
Keywords: Kinematic couplings Self-alignment Self-locking Limiting coefficient of friction Friction force orientation Statically indetermined Statically determined
This paper contains a study of the kinematic coupling, focusing on the self-aligning property. Such a property expresses how the system is able to overcome the friction forces when reaching the nested position: this is why the characterization of these forces is one of the main subjects that will be analyzed. Normally, friction forces are assumed to be oriented against the possible motion even when motion is not present: an alternative approach will be proposed, that is to study the deformations of the involved bodies to identify friction orientations. Since a study of the deformations is not always possible due to lack of information (during the concept phase of the design, for instance) it will be proposed a worst case scenario approach: the effect of deformations will be evaluated without studying them in detail. Such approach can be used to estimate the performance of kinematic couplings in the early phase of the design; moreover, it is also useful to easily compare the two analysis (with and without deformations), helping therefore to understand the importance of this phenomenon with respect to selfalignment.
1. Introduction Kinematic Couplings are interfaces very commonly used in the field of precision mechanics. They can position parts with a repeatability up to submicron level. This paper treats the self-aligning property of the interface, which can be seen as a prerequisite of the repeatability. In fact, while the repeatability expresses the behavior of the system in its nested position, the self-aligning property determines whether the system will be able or not to reach the nested position. Friction plays an important role in the self-alignment, since it will oppose the nesting action. Provided that the centered position is of equilibrium and stable, the system will be able to align itself only if the friction coefficient is smaller than a certain value, the so called limiting coefficient of friction. Studies that explain how to evaluate the limiting coefficient of fric tion are already available (see Ref. [1]). They provide a mathematical model where the friction forces are characterized assuming that the coupling is moving towards the nested position: thus every friction force is oriented in the opposite direction of the relative motion between the parts at the sliding contact. This is certainly true if the interface is actually moving. Supposing to start the alignment of the system with no motion, is it still possible to assume the friction forces oriented as if there were motion? In this paper will be shown that in static condition the friction forces have an orientation that is independent from the motion that is
about to happen, determined only by the way the involved bodies deform. Intuitively, having the friction forces in the opposite direction of the motion works the best for suppressing the motion. However, it will be shown that in case of no motion this is not true. Therefore, not only friction orientation is unrelated to the potential motion, but such assumption could lead to underestimate the friction ability to prevent the motion, thus to predict a system to be self-aligning when it actually isn’t. 2. Problem definition A kinematic coupling is an interface that connects a system to another in an isostatic way: this is achieved by six contacts, in most cases between three spheres and 3 V-grooves (Fig. 1). When the coupling is put together, at first it will be off centered, meaning that not all constraints will be in contact: it will be assumed that the initial position is close enough to the nested position such that the geometry can be considered unchanged. This will allow to write the equilibrium equations of the coupling as if it were in the nested position; the only difference will be the removal of one or more constraints. Moreover, the system will be studied in a state of equilibrium called incipient motion: this is characterized by the fact that there is no motion, but any reduction of the friction coefficient would break the equilib rium. In other words, the system is on the verge of motion.
* Corresponding author. VDL ETG Technology & Development BV, De Schakel 22, 5651, GH, Eindhoven, the Netherlands. E-mail address: [email protected] (F. Patti). https://doi.org/10.1016/j.precisioneng.2019.08.013 Received 14 May 2019; Received in revised form 3 July 2019; Accepted 27 August 2019 Available online 4 September 2019 0141-6359/© 2019 Elsevier Inc. All rights reserved.
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Precision Engineering 60 (2019) 348–354
Fig. 2. Friction force treated as an extra constraint.
2.3. Orientation of friction forces When friction forces are present, the equilibrium problem of a ki nematic coupling is indetermined in case of incipient motion. In fact, the following quantities are unknown:
Fig. 1. An example of conventional kinematic mount.
The question to be answered is: will the system be able to move to wards the nested position? This can be found by studying the equilib rium of the system, in all possible off-center positions. As explained in Ref. [1], it is possible to restrict the study to only 6 cases, the most difficult for motion. In fact, at the beginning of the positioning phase only few constraints will be in contact. Assuming to have a self-aligning system, the system will move towards the centered position, therefore the number of constraints in contact will increase, until all six con straints will be in contact. The tendency of the system will be to move in a way that the minimum amount of energy will be dissipated; the fewer the number of active constraints, the more the freedom of the system to move towards the least resistive path; so, when it will remain only one constraint not engaged, the system will be able to move only in one specific path, which in general will be far from being the least resistive. This line of reasoning brings to the conclusion that if a system is able to move in all of the six paths identified by removing the six constraints one at a time, then it will always be able to move towards the nested position. Therefore there will be six studies of equilibrium, each one with only five active constraints: the most resistive path will be identified by the smaller limiting coefficient of friction, which will measure the selfaligning property of the system.
! - 5 normal reactions1 N i with known direction-> 5 variables; ! - 5 friction reactions T i normal to the previous ones-> 10 variables; - 1 friction coefficient μ -> 1 variable. In total, there are 16 unknown. The equations available instead are: - 3 equations for equilibrium of forces; - 3 equations for equilibrium of moments; - 5 equations defining the friction coefficient NTii ¼ μ. This brings to 16-11 ¼ 5 missing equations: the solution belongs to a five dimensional space: to find the unique solution (when existent) that represents the reality, five more equations are needed. An approach that is often followed is to assign a specific direction for the five friction forces: it is assumed that friction is oriented as if there were motion. In this way, five more equations become available, and the system becomes determined. To know such directions, it is sufficient to evaluate the virtual displacement compatible with the constraints, at each contact point. Although this is often a good approximation as it will be shown later, in reality friction has a different orientation. Instead of assuming a direction for every friction force, these can be actually solved together with the other unknown by following a different approach. Although the kinematic couplings is a statically determined sys tem, if the friction forces are strong enough to prevent motion, the kinematic coupling becomes an hyperstatic system. To see this more clearly, it is sufficient to replace every friction force by two simple constraints,2 as shown in Fig. 2. By doing so, the total number of simple constraints becomes 5 þ 10 ¼ 15; on the other hand, to constrain the system only six simple constraints are required, thus the system is statically inde termined. As explained in Ref. [2], this problem can be solved by considering the involved bodies as solid; the method of displacements can be applied, which leads to nine extra equations (congruence equa tions) expressing that the bodies will deform in a way compatible with the constraints: the mutual distances of the five contact points must be
2.1. Prerequisite for self-alignment A self-aligning system implies that the centering motion aims to the centered position: to make sure this is the case, it must be verified that such position is of equilibrium when μ ¼ 0. 2.2. Model definition: naming convention As stated before, the study will be limited to a very small area around the nested configuration of the system: this means that it will be possible to consider the contact points coincident with the nominal ones, iden tified by the position vector ! r i , where i ¼ 1; 2; …; 6 is the index assigned to the six constraints. Furthermore, every contact will constrain only one direction, iden tified by the unit vector ! n i , which is defined as the normal of the tangent plane of both faces at the contact point. ! The constraint reactions contain a normal component N i ¼ Ni ! n i, ! and a tangential component representing friction T i . ! The force applied to the moving part of the coupling is defined as F e , applied to the point identified by the position vector ! r e. All these quantities will be expressed in a coordinate system CSO ðO; x; y; zÞ whose origin and orientation is arbitrary.
1
There are only five active contact points because the coupling is offcentered. 2 This is possible thanks to the assumption of friction forces strong enough to impede motion.
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Assuming that this position is of equilibrium, there will be no sliding at the contact points, since friction forces will act against them. There fore the friction forces will be oriented against such slidings, which are not provoked by rigid movement, but merely by local deformations of the bodies. To see the effect of friction orientation in reaching equilibrium, a simple case in 2D is shown4 (see Fig. 3): the problem is studied with the graphical method explained in Ref. [3]. A body is constrained in two points, having therefore one degree of freedom. Will the external force be able to move it? Considering the hypothesis of friction against the incipient motion, since the motion will be a clockwise rotation around C, the friction angle to be considered is the one in yellow: the system could be in equilibrium only if the external force would intersect the yellow area. Since this is not the case, with the friction coefficients represented, equilibrium cannot be verified. But if friction forces are allowed to be oriented towards the incipient motion, the three areas in green must be added: thanks to this extension, the condition for equilibrium are met. In this particular case, the friction force of constraint 1 is pointing towards the motion, and yet this situa tion is more efficient in terms of preventing the motion, than the case of friction against it (yellow areas). The next paragraphs will examine the equilibrium of the kinematic coupling first assuming friction forces oriented against the incipient motion. After that, this assumption will be removed: friction forces will be considered independent from the incipient motion. 3. Analysis with friction oriented against the incipient motion In this case, the equilibrium of the system is studied considering friction forces parallel and opposed to the motion that would happen if the system were self-aligning. The virtual displacements of the system will be studied; in particular, six different displacements will be considered, each one corresponding to a constraint removed.
Fig. 3. Case of equilibrium with friction towards the incipient motion.
the same for the two bodies of the coupling.3 It is interesting to notice that in this process the possible motion of the system is not involved at all. Moreover, the ratio NTii ¼ μ is not part of the equations: these two considerations bring to the following general conclusions:
3.1. Evaluation of virtual displacements
(1) The friction forces are not oriented against the incipient motion. (2) The limiting friction coefficient is different for each constraint.
Let the constraint number j be removed. A virtual displacement of the system around the nominal position can be expressed using the wellknown formula5:
The conclusion No.1 shows that assuming the friction forces oriented exactly against the motion is not correct (although in many cases is a good approximation); in some circumstances this could result in underestimating the effectiveness of friction in blocking the motion: in other words, in general there are other directions for which friction forces can be more effective, thus being able to block the motion even with a lower coefficient of friction. Conclusion No.2 instead brings to the following: assuming friction coefficient equal for all constraints, the condition of incipient motion is characterized by only one constraint where the reaction lies on the cone of friction, and not all of them. The just explained approach shows that when in equilibrium prob lems friction forces are involved, and there is no relative motion, the deformations of the bodies play an important role: the orientation of friction forces depend on the internal deformations, and it influences the efficiency of friction in contrasting the motion. This behavior can be visualized as follows: when a coupling is put together, at some point the moving body starts touching the counterpart in a number of points, which will generate constraint reactions: the body will deform under this load.
! s O þdθ �! ri d! s i ¼ d!
i ¼ 1; …; 6 except j
(1)
where d! s i is the displacement of the contact point of the body with ! constraint i, d! s O is the displacement of the origin, d θ the angular displacement, and ! r is the vector position of contact point i. i
For each constraint, (1) defines all possible displacements with six degrees of freedom; imposing the compatibility with each of the five active constraints will give one degree of freedom displacements, meaning that only their amplitude is arbitrary. Although (1) is strictly valid for infinitesimal displacements, in this case it will be assigned a certain amplitude and therefore the infinitesimal displacements d! s i will be replaced by the finite Δ! s i . This is a valid step since the only aim to evaluate the virtual displacements here is to know their directions, which are independent of the amplitude. The compatibility equations describe that the displacement at 4
In 2D the friction force has a fixed direction: there are only two possible orientations, against and towards the relative motion. This is perhaps too simple to understand fully the three dimensional case, but allows to have a better understanding of the effect of friction orientation and the relationship with the body deformations. 5 The above formula expresses the velocity field of a rigid motion, thus a relation between velocities: in this case it is multiplied by the infinitesimal time interval dt to give displacements.
3 A simple way to justify the number of equation needed, is to add these five points one at a time, each time adding the minimum number of equations needed to link their positions to the others: for the first point no equations, for the second only one, and so forth, remembering that a point in 3D has 3 DoFs.
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constraint i belongs to the tangent plane at the contact point: Δ! s i ⋅! ni ¼0
(2)
i ¼ 1; 2; …; 6 except j
The ambiguity of the displacement amplitude is removed by assigning the following extra equation: Δ! s j ⋅! nj ¼
(3)
1
which also imposes that the virtual displacement is oriented towards the nested position. Replacing (1) in (2) and (3), a system of six scalar equations in six variables (the components of the two vector variables ! Δ θ , Δ! s O ) is obtained, which allows to find the virtual displacements ! Δ s for each of the six cases obtained removing constraint j: i
�
! ! ! ðΔs! O þ Δ θ � r i Þ⋅ n i ¼ 0 ! !� ! Δs! 1 O þΔθ � r j ⋅nj ¼
i ¼ 1; 2; …; 6 except j
(4)
Introducing the function of Kronecker delta δi;j the above system (4) can be written: ! ! ! ðΔs! O þ Δ θ � r iÞ ⋅ n i ¼
δi;j
i ¼ 1; 2; …; 6
(5)
Fig. 4. Definition of angles.Ωi .
3.2. Evaluation of equilibrium complex. These deformations are strongly dependent on the history of the load application and the order of engaged constraints: in fact, assigned a set of five points of contact, the system can actually reach this configuration in many ways, each of them with a different deformation at the moment of reaching the fifth one. In order to have an estimation of the effect of the deformations, a worst case scenario is proposed. The aim is to evaluate the set of friction directions associated to the smallest limiting coefficient of friction, without studying the particular deformation that is generating them.
For a rigid body, equilibrium can be expressed by the following vector equations: X! Fk ¼ 0 (6) k
X! ! Mk þ ! r k � Fk ¼ 0
(7)
k
In case of a kinematic coupling with one constraint removed, there ! ! are five forces F i applied in five contact points, and an external force F e which provides energy for the alignment. ! Each reaction force is characterized by a normal component N i ¼ ! N! n and a friction component T . If μ is the static friction coefficient, i
i
For each constraint it is introduced an extra parameter Ωi (shown in Fig. 4) which defines the orientation of the friction force with respect to the direction of the incipient motion evaluated with the system (5). �! Given the direction of displacement Δs i and the normal of the ! contact n i , the direction needed to define the plane of the contact is
i
these forces are: ! F i ¼ Ni ! ni
4.1. Equations of equilibrium
Δ! s
μNi !i jΔ s i j
i ¼ 1; 2; …; 6
except j
(8)
! As for the equilibrium of rotations (7), all moments M k can be considered null, since their contribution would be negligible. Equations (6) and (7) can be written as follows: � 8 X� Δ! si ! > > Fe þ n i μN i ! ¼ 0 Ni ! > > jΔ s i j < i6¼j (9) � � > X > Δ! si ! > ! ! ! > r i � N i n i μN i ! ¼ 0 : r e � Fe þ jΔ s i j i6¼j
�! Δs i ! m i ¼ �! � ! ni j Δs i j
(10)
The expression of the general constraint reaction (8) becomes: ! F i ¼ Ni ! ni
�! Δs
μNi �!i cosΩi j Δs i j
�! Δs
μNi �!i � ! n i sinΩi i ¼ 1; 2; …; 6 except j j Δs i j
System (9) becomes:
In the above system (9) the sum must be evaluated for i ¼ 1 to i ¼ 6 except for j, where j is the removed constraint: it is a system of two vector equations, in six variables: μ; Ni i ¼ 1; …; 6 except j. Solving (9) gives the limiting coefficient of friction for the particular simple path resulting by removing constraint j.6 4. Analysis considering deformations As explained before, a more accurate analysis should evaluate the friction forces by studying the deformations of the system. Such approach is not very practical: not only it requires many details of the system that might not be known at concept definition, but it’s also very 6
Fig. 5. Standard V-groove interface.
See 8.1 in the Appendix for more details about the solution of the system. 351
(11)
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Fig. 6. Limiting coefficient of friction for a standard kinematic coupling.
8 X� ! > > þ ni Ni ! F e > > < i6¼j
�! Δs
μNi �!i cosΩi j Δs i j
� > > X > ! > ! :! r e � Fe þ r i � Ni ! ni i6¼j
�
�! Δs
μNi �!i � ! n i sinΩi j Δs i j
�! Δs μNi �!i cosΩi j Δs i j
�! Δs
¼0 �
μNi �!i � ! n i sinΩi ¼ 0 j Δs i j
(12)
line. When instead the angle gets smaller, the difference is not negligible any more: the green line underestimates the efficiency of friction. Moreover, it’s interesting to note the green line doesn’t take into account the phenomenon of self-locking, which is instead included in the blue line: this is because both self-locking and worst case analysis are based on the deformations of the bodies. 6. Conclusions
As shown in the Appendix 8.1 for system (9), it’s easy to prove that for each set fΩ1 ; …; Ω6 g this system has five solutions for μ: the limiting coefficient of friction is the smallest real and positive value. Such value is function of the set fΩ1 ; …; Ω6 g: the worst case is obtained by mini mizing this function with respect to the set fΩ1 ; …; Ω6 g. This is a complex problem, but can be solved relatively easy with a computer algebra system.
This study has shown that analyzing the self-alignment of kinematic couplings by considering friction oriented as if there were motion is a good approximation for the standard kinematic coupling of Fig. 1, but it might not represent the worst case in other situations, e.g. when the half angle of the V groove is reduced. The proposed approach takes into account that, as demonstrated, in case of incipient motion the friction forces won’t be oriented exactly against the relative motion, making friction sometimes more efficient in preventing the motion. This study shows also (paragraph 2.3) that the case of incipient motion is characterized by different values of limiting friction coefficient for each constraint: this has not been taken into account so far, since in equation (11) there is only one coefficient of friction. It would therefore be an interesting development of this study to include the variations of friction coefficients. At the moment this paper is written, a test setup is being developed, to add experimental verification of the theory.
5. Comparison of the two methods The two approaches are now applied to the common case of a ki nematic coupling shown in Fig. 5: the coupling has three V-grooves at 120� ; β, the half angle of the V, is variable; the external force is vertical, applied at the center of the system. Fig. 6 shows the limiting coefficient of friction evaluated in case of friction forces oriented against the incipient motion (curve in green), and oriented in a way that the friction has the best efficiency in pre venting the motion (curve in blue). The line in red instead represents the combination of μ and β for which a V-groove becomes self-locking: above the red line each ball will jam into its groove as soon as it touches it in two points (for more details about self-locking behavior see Appendix 8.2). Around the standard V-groove configuration β ¼ 45� the two ap proaches give almost the same result, therefore it is safe to use the green
Acknowledgements The authors would like to thank Prof. Dannis Brouwer from the University of Twente for his precious advises.
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8. Appendix 8.1. Solving the equilibrium equations for friction against incipient motion Solving the system (9) is relatively easy, if done with a computer algebra system. The problem is that (9) has more than one solution: in fact, when converted to scalar equations, it becomes a system of six equations in six variables, but because of the products μNi , it’s not linear. This means that there is more than one solution: computer programs can only find solutions around a given range. So, the risk is that the found solution is not the one that has a physical meaning. To solve this issue, the system (9) can be expressed in a different form, so that it’s easier to find all solutions at once, in order to easily select the right one. For the next steps, the followings will be considered: - The excluded constrain will be j ¼ 6 : all results can be easily applied to the other 5 cases; - If the system is verified for a certain value of external force, it is verified as well for every value of force, provided that the line of force is un changed. This allows to introduce a sixth variable c 2 ℝ which multiplies the external force. System (9) can be written as follows: � 8 X� Δ! si ! ! > > c F N þ n μ N ¼0 e i i i > > jΔ! s ij < i6¼j � � > X > Δ! si ! > ! > r e � cFe þ r i � Ni ! n i μNi ! ¼ 0 : ! jΔ s i j i6¼j
(13)
And moving from vector to scalar equations: 8 Δs1x Δs5x > n1x N1 μN1 þ ⋯ þ n5x N5 μN5 þ c Fex ¼ 0 > ! ! > > jΔ s j jΔ s 5j 1 > > > > > Δs1y Δs5y > > n1y N1 μN1 þ ⋯ þ n5y N5 μN5 þ c Fey ¼ 0 > > ! ! > jΔ s j jΔ s 5j 1 > > > > > Δs1z Δs5z > > > n1z N1 μN1 þ ⋯ þ n5z N5 μN5 þ c Fez ¼ 0 > > jΔ! s 1j jΔ! s 5j < � � Δ! s1 > ! > > ð! n 1 Þx N1 r1� ! μN1 þ ⋯ þ ð! r 5 �! n 5 Þx N5 r 1 �! > > jΔ s 1j x > > > > � � > > Δ! s1 > ! > ð! > r 1 �! n 1 Þy N1 r1� ! μN1 þ ⋯ þ ð! r 5�! n 5 Þy N5 > > jΔ s 1 j y > > > > � � > > > Δ! s1 > ! : ð! r 1 �! n 1 Þz N1 r1� ! μN1 þ ⋯ þ ð! r 5�! n 5 Þz N5 jΔ s j 1
z
� � Δ! s5 ! ! r5� ! μN5 þ cð! r e � F e Þx ¼ 0 jΔ s 5 j x � � Δ! s5 ! ! r5� ! μN5 þ cð! r e � F e Þy ¼ 0 jΔ s 5 j y � � Δ! s5 ! ! r5� ! μN5 þ cð! r e � F e Þz ¼ 0 jΔ s j 5
(14)
z
The above (14) is a system 6 � 6 in N1 ; …; N5 ; c. Introducing the following matrixes: 2
n1x n1y 6 6 n1z 6 A¼6 r 1�! n 1 Þx 6 ð! 6 ! 4ð r 1 � ! n 1 Þy ! ! ðr � n Þ 1
2
1 z
⋅ ⋅ ⋅ ⋅ ⋅
3 Fex Fey 7 7 Fez 7 ! 7 ! ð r e � F e Þx 7 7 ! ð! r e � F e Þy 5 ! ð! r �F Þ
⋅ ⋅ ⋅ ⋅ ⋅
⋅ ⋅ ⋅ ⋅ ⋅
⋅ ⋅
⋅
n5x n5y n5z ð! r 5 �! n 5 Þx ð! r 5 �! n 5 Þy ! ! ðr � n Þ
⋅
⋅ ⋅
Δs5x jΔ! s 5j
⋅
⋅ ⋅
Δs5y jΔ! s 5j
⋅
⋅ ⋅
⋅
⋅ ⋅
⋅
⋅ ⋅
⋅
⋅ ⋅
Δs1x jΔ! s 1j
6 6 6 Δs1y 6 6 ! 6 jΔ s 1j 6 6 6 Δs1z 6 6 jΔ! s 1j 6 6 � � B¼6 Δ! s1 ! 6 r � 1 6 jΔ! s 1j x 6 6 � � 6 ! Δ s1 6 ! r1� ! 6 6 jΔ s 1 j y 6 6 � � 4 Δ! s1 ! r1� ! jΔ s j 1
z
5
5 z
e
(15)
e z
3 0
Δs5z jΔ! s 5j � � Δ! s5 ! r5� ! jΔ s 5 j x � � Δ! s5 ! r5� ! jΔ s 5 j y � � Δ! s5 ! r5� ! jΔ s j 5
7 7 7 7 07 7 7 7 7 07 7 7 7 7 07 7 7 7 7 7 07 7 7 7 5 0
(16)
z
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2
3
N1 6 N2 7 6 7 6 N3 7 7 X¼ 6 6 N4 7 6 7 4 N5 5 c
(17)
the system becomes: (18)
AX þ μBX ¼ 0
This is the well-known generalized problem of eigenvalues: it has six solutions and they can be easily found with the aid of a computer algebra system. Moreover, it can be easily seen that one eigenvalue is always infinite: in fact, dividing the main equation by μ (supposed non null): 1
μ
(19)
X þ BX ¼ 0 If μ→∞ then BX ¼ 0. From equation (16), it is:
(20)
detB � 0
So, the system is always verified by μ→∞. Excluding this one, five solutions are left: they can be both real and complex; the smallest positive real value is the solution with physical meaning: the limiting coefficient of friction for the studied aligning path. The next step is to solve the system (14) for all six aligning paths, and to take the smallest of the solutions as the limiting coefficient of friction of the system. The normal reactions obtained are ! ! associated to external force c F e : the ones associated to the actual external force F e are obtained by dividing them by the coefficient c. 8.2. Self-locking The attribute “self-locking” for a system might refer to several types of systems: here self-locking refers to a situation where the forces that push the contact faces, thus generating friction, are not produced by an external load. In order to be present in a static system, it’s easy to prove that the resulting system of forces (normals and friction) needs to be in equilibrium.
Fig. 7. Force decomposition in a self-locking situation.
An example of this phenomenon is represented in Fig. 7 where a ball is coupled to a vee and the coefficient of friction is such that, for the given angle β, it is possible that the two reactions R1 and R2 are in equilibrium: even without any external force, the ball is “squeezed” by the vee. If the ball is pushed against the vee and then released, the ball remains stuck in its position: the deformations induced by the initial load are generating the normal reactions necessary for friction to restrain the ball from further movement. It is clear that for a kinematic coupling this condition must be avoided.
References
[2] Carpinteri A. Structural mechanics - a unified approach. Taylor and Francis; 1997. [3] Blanding DL. Exact constraint: machine design using kinematic principles. ASME Press; 1999.
[1] Hale LC, Slocum AH. Optimal design techniques for kinematic couplings. Precis Eng 2001;25:114–27.
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