Steady state motion of a shear deformable beam in contact with a traveling surface
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Abstract
A shear deformable beam moving along a straight path is considered as an idealization of the problem of stationary operation of a belt drive. The partial contact with a traveling surface results in the shear deformation of the beam. The tangential contact force grows near the end of the contact zone. Assuming perfect adhesion of the lower fiber of the beam to the traveling surface (no slip), we analytically demonstrate the necessity of accounting for concentrated contact forces and jump conditions, which is important for modeling the belt–pulley interaction. Along with dynamic effects, we further consider a frictional model with zones of stick and slip contact and demonstrate its convergence to the results with perfect adhesion at growing maximal friction force.
1 Introduction
Belt drive mechanics is an extensively studied research area with applications in power transmissions, conveyors, elevators, and processing of metals and polymers. It is natural to investigate the belt drive mechanics with onedimensional structural models of strings and rods because of the high lengthtothickness ratio; many publications are devoted to this topic, including the monograph [17]. In the contact problem for strings and rods, particular attention should be paid to the transition conditions at the boundaries between the contact and free zones, which we call “touching points” in the following. It is known that concentrated contact forces at these touching points result with Kirchhoff (shear undeformable) rod models even in static problems with frictionless contact; see [2, 5, 6, 11, 19, 32]. The perfect adhesion (noslip) condition between the moving belt and the surface of a rotating pulley also results in concentrated tangential contact forces in the case of a string model of the belt [14, 30, 33]. It is, however, known that the singularity in the normal contact force vanishes with the introduction of shear flexibility of the rod, see contributions [3, 7], in which the tensioning of the belt on the pulleys is treated. It is one of the aims of the present study to answer the question whether shear flexibility eliminates the concentrated tangential contact forces for moving belts as well.
The alternative to the idealized model with perfect adhesion is the admission of Coulomb’s friction law and the corresponding analysis of the developing zones of sticking and sliding contact. This behavior is traditionally called “belt creep”; see, e.g., [27] for an exact analytical solution of the problem of steady state motion of an extensible belt with no bending stiffness. A whole body of publications is related to the transient finite element analysis featuring string and rod models for the belt as well as various simplifications of the friction law; see, e.g., [10, 12] for conventional finite element approaches. Putting the mixed Eulerian–Lagrangian kinematics into the basis of a nonmaterial finite element formulation [28, 29, 31] allows avoiding frequent switching of contact states within single elements. This approach is getting increasingly popular for modeling belt drives and similar structures; see [18, 23, 25, 26]. The latter reference provides also an analytical solution for the transient growth of sliding zones during the operation of an idealized belt drive.

Shear flexibility essentially influences the distribution of the contact pressure between the belt and the pulleys; see [1, 3, 9, 16].

Actual power transmission belts (Vbelts) often feature thin steel fibers embedded in a rubber body. This reinforcement results in relatively high shear flexibility in comparison with extensional and bending ones. The belt creep is then mainly determined by shear deformation, which makes it quite different from the case of flat belts without reinforcement; see [15] for one of the first considerations of this effect.
A straight beam with extension and shear moves along the x axis according to the prescribed rate c of the material influx at the left end and outflux at the right end of the control domain \(0\le x\le L\); see [29, 31] for an indepth discussion of this sort of boundary conditions. The middle segment of the beam \(x_1\le x\le x_2\) is supported by a rough surface moving with the velocity v. The small difference between the “desired” velocity of the beam c and the velocity v, which is imposed on the lower fiber of its middle part, results in the deformation of the beam and a complicated distribution of the contact interactions.

The transverse deflections of the beam are excluded from the consideration, i.e., the beam is assumed to be pressed against a horizontal surface, which is partially frictionless and partially rough and moving (the latter part we denote as “contact segment”).

The small strain theory simplifies the analysis, allowing making no distinction between the derivatives with respect to the material and the spatial coordinates.

Both the quasistatic and dynamic cases are considered under the assumption of steady state motion, when the observed deformations and velocities vary along the considered control domain, but remain unchanged in time at a given point x.

The perfect adhesion (noslip) condition between the lower fiber of the beam and the moving surface is imposed in the segment of contact.
2 Quasistatic solution
2.1 Problem formulation
2.2 Segment of sticking contact
2.3 Discussion of continuity
 1.
We release the condition of the conservation of the material length of the control domain and find a unique continuous solution. This may be of theoretical interest, but is not relevant for the present purpose of solving a model problem for the more general case of a moving belt: the belt is closed, its material length is preserved, and experiments with finite element solutions of the belt problem clearly indicate that the concentrated contact interaction needs to be included into the model to obtain reasonable solutions.
 2.We admit a concentrated contact interaction in a single point. Condition of adhesion implies that the strain \(\varepsilon _{\mathrm {bottom}}\) is continuous at the point \(x_1\), in which material particles come into contact. Therefore, we now consider thatin which \(f_0\) is the continuous distribution determined above in Eq. (7), \(\delta \) is the Dirac impulse function, and the concentrated force F (which also results in a concentrated moment) defines jumps in \(\varepsilon \) and \(\theta ^{\prime }\) in the right touching point. This allows satisfying the condition of constant material length, which is similar to the string solution introduced in [14]. Whether the concentrated force will be necessary if not a beam, but rather a 2D continuum is moving with the considered kinematic conditions, remains an open question.$$\begin{aligned} f(x) = f_0(x) + F\delta (xx_2), \end{aligned}$$(9)
 3.
We seek a continuous solution in the presence of slip (creep zone in the terminology of the theory of belt drives). The segment \(x_1 \le x \le x_2\) is divided into a zone of sticking contact \(x_1 \le x \le x_*\), in which the contact force does not exceed some maximal value, \(f < f_*\), and a zone of sliding \(x_*< x \le x_2\), in which we apply \(f=f_*\) and the corresponding distributed moment. The sliding friction force \(f_*\) is prescribed (we can argue that the beam is pressed against the surface so strongly that the normal contact force is almost constant, and use the conventional Coulomb’s friction law), the coordinate \(x_*\) follows from the matching conditions.
2.4 Free segments and boundary conditions in quasistatic case
3 Dynamic solution
3.1 Segment of sticking contact in dynamic case
 1.
the differential equations in the free segments \(0< x < x_1\) and \(x_2< x < L\),
 2.
the boundary conditions at the entry and exit points of the contact domain \(x=0,L\),
 3.
the continuity (matching) and jump conditions between the adjacent segments in the touching points \(x=x_{1,2}\), and
 4.
the conservation of the material length of the beam.
3.2 Free segments and boundary conditions in the dynamic case
As above, we need the condition of conservation of the material length to obtain the value of concentrated force F; condition Eq. (15) is valid here as well. The numerical results for a benchmark example will be given in Sect. 5.
4 Presence of slip
Effects of small sliding at axial motion, which are frequently denoted as elastic microslip [21], were studied in detail for small axial deflections of a beam in contact with a rough surface in [8, 13]. The present analysis is different in the sense that not a single static configuration, but a regime of steady state motion of the beam needs to be determined, which essentially changes the formulation of the problem.
4.1 Equations and boundary conditions
Here, we seek a continuous solution in the presence of slip (creep zone in the terminology of the theory of belt drives). The contact segment \(x_1 \le x \le x_2\) is now divided into zone of sticking contact \(x_1 \le x \le x_*\), in which the contact force does not exceed some maximal value, \(f \le f_*\), and a zone of sliding \(x_*< x \le x_2\), in which we apply constant traction \(f=f_*\) and corresponding distributed moment \(f_*h / 2\). The sliding friction force \(f_*\) is prescribed. This corresponds to the assumptions of the perfect Coulomb’s friction law with constant normal pressure between the beam and the supporting surface, which determines the particular value of \(f_*\). The coordinate \(x_*\) is found from the matching conditions between the zones.
In the stick region \(x_1 \le x \le x_*\), the general solution is the same as in Sect. 3.1 (Eq. (21)); it depends on two yet unknown constants, which are of course different from the above cases.
 1.
the differential equations in the free segments,
 2.
the boundary conditions at the end points of the domain \(x=0,L\),
 3.
the continuity (matching) conditions between the adjacent segments in the touching points \(x=x_{1,2}\), and
 4.
the conservation of the material length of the beam.
4.2 Free segments and boundary conditions in case with slip
At the transition between the stick segment and the left free segment, we again have the continuity conditions for \(\varepsilon \), \(\theta \), and \(\theta ^{\prime }\). Along with the boundary condition at \(x=0\) they result in the same expressions for \(\varepsilon \) and \(\theta \) in the free left segment as above in Sect. 3.2 (Eq. (22)).
5 Numerical results
In this section, we present the numerical results for a benchmark example. Three considered models (quasistatic with perfect adhesion, dynamic with perfect adhesion, and dynamic with the zone of slip) shall be compared such that the influence of the specific effects is made visible.
The model parameters are: \(L=1\,\mathrm {m}\) is the length of the considered part of the belt, coordinates \(x_1 = 1/3\,\mathrm {m}\) and \(x_2=2/3\,\mathrm {m}\) determine the location and length of the traveling contact surface (pulley), \(h=0.1\,\mathrm {m}\) and \(w=0.1\,\mathrm {m}\) are the height and the width of the cross section of the belt, \(E=5 \cdot 10^7\,\mathrm {Pa}\) is the Young’s modulus, \(\nu =0.45\) is the Poisson coefficient, and \(\rho _3=1500\,\mathrm {kg/m^3}\) is the volume material density of the belt.
The elasticity of the belt is characterized by the three stiffness factors: \(a=Ewh^3/12\) is the bending stiffness, \(b_1=Ewh\) is the tension stiffness, and \(b_2=Ekwh/2(1+\nu )\) is the shear stiffness with the shear correction factor \(k=1.1\). The inertia of the belt is prescribed by \(I=\rho _3 h w(h^2 + w^2)/12\), which is the rotary inertia moment, and by \(\rho = \rho _3 h w\), which is the mass per unit length.
Before the considered domain (\(x<0\)) and after it (\(x>L\)), the velocity of the belt is prescribed to have the value \(c= 120\,\mathrm {m/s}\). This high velocity of the belt results in a noticeable effect of inertia, as the dynamic contribution \(\rho c^2=2.16\cdot 10^5\,\mathrm {N}\) in the denominator of the parameter \(\gamma \) (see Eq. (26)) becomes comparable with the tension stiffness \(b_1=5\cdot 10^5\,\mathrm {N}\). The rough contact surface (pulley) is moving with the velocity \(v=121.2\,\mathrm {m/s}\), which results in the positive extension \(\varepsilon \) and positive contact forces f. For the case with slip, we use the prescribed value of the maximal friction force \(f_*=10^5\,\mathrm {N/m}\).
Total friction traction, N
Concentrated part or slipping part in case with slip  Distributed part or sticking part in case with slip  Total  

No inertia, no slip  6482  6237  12,719 
With inertia, no slip  2024  5375  7398 
With inertia, with slip  3253  3870  7124 
6 Conclusion
In the present contribution, we considered the steady state motion of a straight shear deformable beam in partial contact with a moving rough surface under the assumptions of the small strain theory. This simplified model is a prototype for the future analysis of frictional belt drive dynamics, the beam representing the belt and the rough surface being the rotating pulley. The idealized formulation and analytical (or semianalytical) solutions allow investigating the effects of inertia and belt creep (sliding between the belt and the pulley). Transition conditions between the free segments, zone of stick contact, and zone of sliding contact play an important role in deriving the solutions.
The research presents a clear evidence for the appearance of concentrated contact interaction in the model with perfect adhesion between the belt and the moving surface of the rotating pulley. This concentrated force acts in the touching point, where the belt leaves the pulley. This has already been demonstrated earlier for the case of an axially moving string [14], but is for the first time shown for a shear deformable beam. Furthermore, we have demonstrated that the concentrated contact interaction is a limiting case of the distributed sliding friction interaction, when the sliding zone collapses into a single point because of the high friction force. Future finite element solutions for the steady state dynamic problem for a closed belt drive as well as the numerical integration of the boundary value problem similar to [4, 32] will be based on the above theoretical results and techniques of modeling.
Footnotes
Notes
Acknowledgements
Open access funding provided by Austrian Science Fund (FWF). The work is carried out in the framework of the joint project of the Russian Foundation for Basic Research (Grant No. 145115001) and the Austrian Science Fund (FWF, Grant No. I 2093N25 International Project).
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