Optimized punch contact action related to control of local structure displacement
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Abstract
For a structure under service loads, there is a need to induce precise control of a local displacement by additional punch loading. Such problem exists in design of robot grippers or agricultural tools used in mechanical processing. The punch interaction is assumed to be executed by a discrete set of pins or by a continuously distributed contact pressure. The optimal contact force or pressure distribution and contact shape are specified for both discrete and continuous punch action. Several boundary support conditions are discussed, and their effects on punch action are presented.
Keywords
Contact problem Displacement control Optimal pressure distribution Optimal contact shape1 Introduction
The structural optimization problems are usually related to specification of material parameters, size, shape, or topological variables, loading distribution, supports, element connections, etc. The required stiffness and strength levels are usually imposed as design constraints and the objective function assumed as the volume or cost of materials used, cf. recent books by Banichuk and Neittaanmaki (2010) or Banichuk (2011). A special class of contact optimization problems is characterized by searching for optimal contact traction distribution satisfying strength conditions at the contact interface. The review of contact optimization problems has been presented by Páczelt et al. (2016).
In the paper by Li et al. (2003), the evolutionary structural optimization (ESO) concepts have been applied with a non-gradient procedure presented for incremental shape redesign of contact interfaces. In Zabaras et al.’s (2000) work, a continuum sensitivity analysis has been presented for large inelastic deformations and metal-forming processes.
In the papers by Páczelt and Szabó (1994), Páczelt (2000), Páczelt and Baksa (2002), and Páczelt et al. (2007), the method of contact shape optimization was developed for 2D and 3D problems with the objective to minimize the maximal contact pressure under specified loading conditions. The contact optimization problems for elements in the relative sliding motion with account for wear require special analysis referred to a steady-state wear process. The optimal design objective is to specify the contact shape minimizing the wear rate in the steady-state condition for progressive or periodic sliding. The extensive review of this class of problems has been presented by Páczelt et al. (2015). The monographs of Goryacheva (1998) and Wriggers (2002) provide a solid foundation for analytical and numerical methods of solution of contact problems, including wear analysis. The finite element analysis is most frequently applied in solving contact problems, cf. Szabó and Babuska (1991).
The present paper is devoted to a problem of local displacement control in a structure subjected to service loads, such as an assembling robot gripper (Monkman et al. 2006). Then the precise displacement value should be achieved at the location of robot interaction with an assembled element. It means that the displacement control should be applied at a point transmitting the interaction load. The method of solution of this new class of contact design problems will be discussed in Sect. 2 for the case of beam deflection control with the constraint set on maximal contact pressure. In Sect. 3, the specific examples are discussed, illustrating designs of punch shape for differing beam support constraints. In Sect. 4, the optimal contact pressure distribution and punch shape are determined from the optimization procedure. Applying the strength constraint also punch position and contact zone size can be properly selected. The method presented can be extended to the analysis and design of optimal punch interaction aimed at displacement control at the loaded boundary point of any structural element.
2 Control of beam deflection at the loading point Q for differing support conditions
Further, for technical reasons, it is assumed that the contact pressure distribution is symmetric with respect to center point of contact zone. This allows for punch application in differing support conditions.
Assuming the pressure distributions according to Fig. 3, the problem is reduced to specification of contact surface form represented by the gap function. Several variants of this problem are treated and will be discussed in more detail.
The punch center position x_{0} can be located on both sides of point Q. Depending on the support constraint, it can essentially affect the value of the required punch load F_{0} or the deformation form. The effects of punch position will be discussed in Sect. 4.
2.1 Beam structure deflection control for different support conditions
2.1.1 First variant: cantilever beam built-in at the end a
Having specified p_{max}, the punch pin height or punch contact shape can be determined.
In these contact optimization problems, the initial gap (shape form of the contact surface) is the unknown function. In our case, the contact condition will be checked up at the kont points. Supposing contact at each check point, the calculation of the gap can be executed in terms of the special iterative procedure (Páczelt 2000; Páczelt et al. 2016) In Appendices 1 and 2, the discretized equations for determination of the initial gap between the beam and punch are provided.
2.1.2 Second variant: beam allowed to execute displacement along the sliding support A
\( {\mathbf{h}}_Q^{(2),T}=\left[{H}_{1,Q}^{(2)},{H}_{2,Q}^{(2)},...,{H}_{i,Q}^{(2)},...,{H}_{kont,Q}^{(2)}\right] \), e, see (63),
2.1.3 Third variant: free beam is allowed for rigid body translation and rotation
\( {\mathbf{h}}_{Q_s}^{(2),T}=\left({H}_{1,{Q}_s}^{(2)},{H}_{2,{Q}_s}^{(2)},...,{H}_{i,{Q}_s}^{(2)},...,{H}_{iter,{Q}_s}^{(2)}\right) \), s = 1,2, x^{T} = [x_{1}, x_{2}, ..., x_{i}, ..., x_{kont}].
3 Numerical examples
The beam area A_{b} = a_{b}h_{b} = 20 ⋅ 50 = 1000 mm^{2}, inertia moment \( I={a}_b{h}_b^3/12=21833.33\;m{m}^4 \), Young modulus E = 2 ⋅ 10^{5}MPa are assumed and other geometric parameters are:
L_{1} = 300, L_{4} = 600, x_{Q} = 850, L = 900 mm. The punch centre position x_{0} = 450 mm corresponds to the position parameter ξ = x_{0}/x_{Q} = 0.529. The specified vertical displacement at the point Q is \( {u}_n^{\ast }=1\; mm \), and the required force values are F_{Q} = 4 kN, F_{Q} = 5 kN, and F_{Q} = 6 kN.
3.1 Example of the first variant
3.1.1 Discrete model
The punch contact executed by five punch pins of crosssection area A = aa_{b} = 5 ⋅ 20 = 100 mm^{2} (Fig. 2a). The x coordinates of the punch pins are: 383.33, 416.66, 450.00, 483.33, and 516.66 mm.
The initial gaps at discrete points in Fig. 7b illustrate the effect of load values on punch profile at constant contact pressure. It is seen that the height differences become more significant at larger distance from the force F_{Q} application. The maximal height difference is less than 0.15 mm.
3.1.2 Model 2: continuous contact interaction
Consider now the continuous contact interaction for the elastic punch in the plane stress state. The punch interaction is performed in the segment 420 ≤ x ≤ 480 (L_{1} = 420, L_{4} = 480), and its height is 50 mm (see Figure 2b). The uniform pressure distribution, c(x) = 1, was assumed in the numerical solution and the normal displacement \( {u}_n^{(1)}\left(x,{p}_n\right) \) was calculated by the finite element method, using pversion technique, cf. Szabó and Babuska (1991). The punch is loaded by pressure p_{n} = p_{max} at the contact boundary z = 0 and by the same pressure at the upper boundary for z = 50 mm. The elastic boundary value problem has been solved for differing finite element meshes (n_{x} = 4, 8, 12; n_{z} = 5, where n_{x}, n_{z} is number of elements in x and z directions) using polynomial order 3 ≤ p ≤ 8 for quadrilateral finite elements with the shape functions in the trunk space. NDOF at n_{x} = 4, n_{z} = 5, p = 3, 8 is equal to 135, 735 at n_{x} = 8, n_{z} = 5, p = 3, 8 is 270, 1470. The normal displacement distribution on the boundary surfaces exhibits good convergence for increasing p. For p ≥ 6, the numerical results differ very little for all meshes, their difference being smaller than 0.15%.
- 1.
- 2.
Calculation of the vertical beam displacement in contact zone from contact pressure and load F_{Q} (see (65b)) and (65c))
- 3.
Calculation of the elastic displacement (\( {u}_z^{(1)e}\left(x,z=0\right)=-{u}_n^{(1)}=-{u}_n^{(1)}\left({p}_n,{F}_0\right) \)) in punch from a separate solution (see (70)).
- 4.
From Signorini contact condition: d = 0, we can find initial gap ^{iter}g^{(0)} and rigid body displacement ^{iter}λ (see (72), (65c), (66).
- 5.
As the initial gap modifies punch shape, the iteration process must be used. The steps 3 and 4 are continued until the satisfactory convergence condition for shape modification is reached (cf. Páczelt et al. (2016)].
Maximal contact pressures and punch loads for the optimal solution
F_{Q} (kN) | F_{0} (kN) | p_{max} (MPa) |
---|---|---|
4 | 12.13 | 10.11 |
5 | 15.02 | 12.51 |
6 | 17.90 | 14.92 |
The beam deflection profiles for both discrete and continuous punch actions are very similar (see Figs. 6 and 9). Note that the contact zone for discrete punch action is much larger (300 mm), than that for continuous action (60 mm) at the same centre position x = 450 mm. The force efficiency factors in all cases are practically equal to 3.0.
3.2 Example for the second variant: allowed beam support translation
Punch and beam rigid body displacements and maximal contact pressures for second variant: a_{b} = 20, h_{b} = 50 mm
F_{Q} (kN) | \( {\lambda}_F^{(1)},{\lambda}_F^{(2)} \) (mm) | p_{max} (MPa) |
---|---|---|
4 | 11.402, 13.752 | 8.0 |
5 | 14.003, 16.940 | 10.0 |
6 | 16.603, 20.128 | 12.0 |
3.3 Example for 3rd variant: beam allowed for rigid body translation and rotation
In the first step, the rigid body motion components \( {\lambda}_F^{(2)} \) and \( {\lambda}_M^{(2)} \)are calculated form (24).
The maximal contact pressure and rigid body displacements resulting from solution of variant 3
\( {\displaystyle \begin{array}{l}{F}_0={F}_{Q_1}+{F}_{Q_2}\;\\ {}{F}_{Q_1},{F}_{Q_2}\;(kN)\end{array}} \) | \( {\lambda}_F^{(1)},\kern1.12em {\lambda}_F^{(2)},\kern1.12em {\lambda}_M^{(2)}\cdot {10}^3 \) (mm, mm, rad) | p_{max} (MPa) |
---|---|---|
4.0, 2.0 | 7.671, 4.612, 7.86 | 12.0 |
5.0, 2.5 | 7.765, 4.586, 8.26 | 15.0 |
6.0, 3.0 | 7.860, 4.566, 8.67 | 18.0 |
4 Localized and distributed punch action: effect of design constraints
The numerical examples presented in Sect. 3 illustrate the discrete and continuous punch action in order to control the deflection at point Q (or at several points, as discussed in variant 3). In the design of such control system, the punch position x_{0} and length L_{1 − 4} = L_{4} − L_{1} of the contact zone are important variables, as the applied load value and the maximal contact pressure should be minimized. Also, the maximal beam stress σ_{max} in the loaded state should not exceed the critical stress value, σ_{max} ≤ σ_{u}. Assume first the contact pressure distribution as the design variable subject to optimization.
4.1 Contact pressure distribution: optimization for deflection control
Consider the cantilever beam under punch action on the contact zone L_{1 − 4} = L_{4} − L_{1} inducing the deflection w_{Q} = − u_{z} at the location Q. Apply the unit force F = 1 at Q, inducing the state M^{(q)}(x), κ^{(q)}(x), w^{(q)}(x) (bending moment, curvature, deflection) in the beam The independent punch action induces the state M(x), κ(x), and w(x).
- 1.
Contact force specified: maximize deflection w_{Q}
- 2.
Deflection w_{Q} specified: minimize the punch force F_{0}
- 3.
Deflection w_{Q} specified: minimize the maximal pressure p_{max}
It is seen that the problem formulation of optimal pressure distribution has no extremum, since the displacement w^{(q)} is not constant, but λ is assumed constant. There is only the singular optimal solution for the concentrated contact force action located at w_{Qmax} in the contact zone.
Problem 2:
The same result as for problem 1.
Problem 3:
The value of λ^{∗} is specified from the equality \( {w}_Q^0=\underset{L_1}{\overset{L_4}{\int }}{p}_n{w}^{(q)}{a}_b dx \).
4.2 Concentrated and distributed cantilever beam control
At \( {\tilde{w}}_t=0 \) and \( {w}_Q^0+{w}_Q=0 \), this means that there is no interaction of gripper with the assembled element. For contact interaction, the relative penetration depth equals δ = w_{t} and \( \mid {w}_Q\mid <{w}_Q^0 \). The diagrams f_{0}(ξ) for positive values of δ = w_{t} > 0 can be plotted. The region above the curve for \( {\tilde{w}}_t=0 \) (see Fig. 13) corresponds to contact interaction. The load efficiency factor f_{0} = F_{0}/F_{Q} strongly depends on the punch load F_{0} position. Usually, the value of ξ = x_{0}/x_{Q} is selected in a specific design case with account for contact pressure and beam strength constraints. For instance, assume the punch load to belong to segments 0.4 ≤ ξ ≤ 0.7 or 1.2 ≤ ξ ≤ 1.5. Then the point ξ = 450/850 = 0.529, f_{0} ≈ 3, marked in Fig. 13 (point B) represents the design discussed in Sect. 3.1, cf. (Fig. 9), for F_{Q} = 5 kN, \( {w}_t={u}_n^{\ast }=1\; mm \), \( \mid {w}_Q\mid =\mid {u}_{n, load}^{(2)}\mid =24.56\; mm \), and \( {\tilde{w}}_t\approx 0.04 \).
4.3 Translating beam control under concentrated loads, variant 2
Consider the beam of Fig. 4 with the sliding support at the left end, loaded by the concentrated load F_{0} = F_{Q} at the distance x_{0} and the specified contact force F_{Q} at Q located at the distance x_{Q}.
5 Concluding remarks
The optimal design of punch action in order to control normal displacement at a loaded boundary point in a structural element has been discussed in the paper and illustrated by the specific examples of beam deflection control. It was demonstrated that the support constraint can affect essentially the punch load and the beam deflected form. This new class of problems is characterized by a set of design variables, such as contact pressure distribution p_{n}(x), punch resultant load F_{0}, punch centre position x_{0}, and the size of contact zone L_{1 − 4}. The minimization of punch load or the maximal contact pressure, required for displacement control, has been discussed, also the effect of punch position and contact zone size on the punch load value was considered.
The present problem formulation can be extended to more advanced control problems. First, for the gripper operation requiring varying load and displacement control at Q, or following the loading path \( {F}_0={F}_0\left({u}_n^{\ast}\right) \), the punch action should be executed for properly varying load F_{0}. The other extension is related to control of both the deflection and its slope at Q. Such control can be performed by varying loads of two punches. Applying such punch action to plate elements, the normal displacement and the orientation angles of the normal vector to the deflected surface at Q can be controlled. Any tool attached transversely at Q to the plate could then execute both normal and inplane displacements. Such advanced control problems will be discussed in a separate paper.
6 Replication of results
The computer codes are written in FORTRAN and MATLAB in the research form. To use them without additional comments could be complicated. If anybody is interested in the programs, please write to István Páczelt.
Notes
Acknowledgments
The present research was partially supported by the Hungarian Academy of Sciences, by the grant National Research, Development and Innovation Office—NKFIH: K115701.
Funding Information
Open access funding provided by University of Miskolc (ME).
Compliance with ethical standards
Conflict of interest
The authors declare that they have no conflict of interest.
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