Optimized punch contact action related to control of local structure displacement

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.

1 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

In the assembling process of mechanical elements conducted by a robot gripper, a typical operation is to place one body (cylinder) into a hole of another body. In this case, the cylinder must execute translation ${u}_n^{\ast }$ under increasing axial force. In Fig. 1a, the force F_Q is equal to friction force at the interface between cylinder and hole. Another operation called “pick and place” is related to lifting the cylinder and placing in a new position, Fig. 1b. In this case, the cylinder is compressed along its diameter by two plates inducing normal contact displacements ${u}_n^{\ast }$ non-linearly related to contact forces F_Q, assuring due to friction a fixed cylinder position relative to plates. In both operations, the prescribed force and displacement values are required at the same point.

Fig. 1

Typical robot gripper operations: a placing cylinder into an element hole and b compressing cylinder by required normal forces

Consider a cantilever beam built-in at its end A and loaded at point Q by the force F_Q, Fig. 2, inducing the deflection ${u}_n^{(2)}$. To keep the deflection of Q at the required value ${u}_n^{\ast }$, the discrete or continuous punch action is applied within the specified contact zone region Ω defined as an interval L₁ ≤ x ≤ L₄ and the contact center position x = x₀. The punch is allowed to translate in the normal direction n to the beam exerting contact pressure p_n(x). The desired punch action occurs when the value of punch load F₀ is reached for the contact pressure distribution not exceeding the specified level p_max. The value of F₀ depends on the size and position of the contact zone. The minimal value of F₀ is reached, when the concentrated load acts at x = L₄, but when the constraint p_n ≤ p_max is applied, then the constant pressure zone p_n = p_max at the boundary x = L₄ constitutes the optimal distribution. In our design, the location of the punch and the control function c(x) are prescribed in advance and the value of p_max is controlled by the length of contact zone. The prescribed pressure distribution p_n = c(x) p_max is reached by determination of the initial gap between the punch and beam. The problem is solved in two main steps. First, using the stamp equilibrium condition at a given control function c(x) and constraint of normal displacement at the point Q, the value of p_max is specified, see (6). Second, from contact condition between the stamp and beam, the initial gap between the contacting bodies is determined, see Appendix 1.

Fig. 2

Beam system for transmission of the punch load F₀ to maintain the vertical displacement ${u}_n^{\ast }$ at the point Q loaded by the force F_Q; a discrete action by a set of pins and b continuous punch action

The constraint set on the maximal pressure value can be attained by assuming the following distribution in the contact zone specified by the control function

$$ {p}_n=c(x)\;{p}_{\mathrm{max}},\kern0.5em c(x)\le 1,\kern0.5em for\;x\in \varOmega $$

(1)

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.

Then obviously, the optimal distribution would correspond to c(x) = 1 and constant pressure p_n = p_max; next, this value could be minimized. Figure 3 presents the assumed symmetric pressure distribution for different control functions. Using Hermite functions, two pressure distributions are specified. The formulae of Hermite functions are stated in the caption of Fig. 3.

Fig. 3

Control function c(x) for symmetric contact pressure distribution; for constant pressure: c(x) = const = 1; for Hermite 1: L₁ = 300, L₂ = 350, L₃ = 550, L₄ = 600, $c(x)=0,\kern0.5em 0\le x\le {L}_1;\kern0.5em c(x)=\left\{3{\left(\frac{x-{L}_1}{L_2-{L}_1}\right)}^2-2{\left(\frac{x-{L}_1}{L_2-{L}_1}\right)}^3\right\},\kern0.5em {L}_1\le x\le {L}_2;\kern0.5em c(x)=1,\kern0.5em {L}_2\le x\le {L}_3$$c(x)=\left\{1-3{\left(\frac{x-{L}_3}{L_4-{L}_3}\right)}^2+2{\left(\frac{x-{L}_3}{L_4-{L}_3}\right)}^3\right\},{L}_1\le x\le {L}_4;c(x)=0,{L}_4\le x$; for Hermite 2: L₁ = 300, L₂ = L₃ = 450, L₄ = 600

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 x₀ 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 F₀ 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

Consider first the case of punch action executed by a set of punch pins, as shown in Fig. 2a, of thickness a, width b, and cross-section area A = ab. The forces between punch and beam are specified by (1), thus

$$ {P}_j= Ac\left({x}_j\right){p}_{\mathrm{max}},j=1,2,\dots, kont $$

(2)

Under applied load F₀ on the punch, the maximal pressure results from the equilibrium condition

$$ {p}_{\mathrm{max}}=\frac{F_0}{\sum \limits_{j=1}^{kont}c\left({x}_j\right)\;A} $$

(3)

Using the influence Green function H⁽²⁾(x, s), the beam displacement in the normal direction n along the z-axis equals

$$ {u}_n^{\ast }=\sum \limits_{j=1}^{kont}{H}^{(2)}\left({x}_Q,{x}_j\right){P}_J+{u}_{n, load}^{(2)} $$

(4)

where ${u}_{n, load}^{(2)}<0$ is the displacement at point Q induced by the load F_Q. Using the influence function for calculation ${u}_{n, load}^{(2)}$, we can write ${u}_{n, load}^{(2)}=-{H}^{(2)}\left({x}_Q,{x}_Q\right){F}_Q=-{H}_{Q,Q}^{(2)}\;{F}_Q$. The influence function is defined in Appendix 3. In view of (4), there is

$$ {u}_n^{\ast }=\sum \limits_{j=1}^{kont}{H}^{(2)}\left({x}_Q,{x}_j\right)c\left({x}_j\right)\;A\;{p}_{\mathrm{max}}+{u}_{n, load}^{(2)} $$

(5)

and

$$ {p}_{\mathrm{max}}=\frac{u_n^{\ast }-{u}_{n, load}^{(2)}}{\sum \limits_{j=1}^{kont}{H}^{(2)}\left({x}_Q,{x}_j\right)c\left({x}_j\right)\;A} $$

(6)

Having specified p_max, the punch pin height or punch contact shape can be determined.

The unilateral Signorini contact condition takes the form

$$ d=\left({u}_n^{(2)}\left({p}_n\right)+{u}_{n, load}^{(2)}\right)-\left({u}_n^{(1)}\left({p}_n\right)+\lambda \right)+{g}^{(0)}\ge 0,{p}_n\ge 0,\kern1.24em {p}_n\;d=0 $$

(7)

where ${u}_n^{(i)}\left({p}_n\right)=\underset{S_c}{\int }{H}^{(i)}\left(x,s\right){p}_n(s)\; ds$ is the normal displacement of beam or punch induced by the contact pressure, H⁽ⁱ⁾(x, s) is the Green function for normal displacements, ${u}_{n, load}^{(2)}$ is the normal displacement resulted from the load F_Q, g⁽⁰⁾ is the initial gap, λ is the rigid body displacement of the punch, that is d = d(p_n, λ), where the notation ${\lambda}_F^{(1)}=\lambda$ is used. Usually, the contact problem is solved by the principle of modified complementary energy or by the displacement formulation using minimum principle of the total potential energy in the penalty form (cf. Wriggers 2002), (Konyukhov and Schweizerhof (2013). In the last case, we use p-versions of finite element method for discretization (cf. Szabó and Babuska 1991).

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

Let us now refer to the system of Fig. 4, where beam 2 can execute the rigid body displacement $-{u}_{A,z}={\lambda}_F^{(2)}$ along the sliding support A and deform elastically due to load F_Q and punch force F₀ action. Assume the initial beam configuration to correspond to contact engagement at Q with an object (e.g., cylinder).The vertical displacement at the point Q then is

$$ {u}_n^{\ast }={u}_n^{(2)}={\lambda}_F^{(2)}+\sum \limits_{j=1}^{kont}{H}^{(2)}\left({x}_Q,{x}_j\right){P}_J+{u}_{n, load}^{(2)} $$

(8)

Fig. 4

Beam and punch system for transmission of load F₀ interacting with the force F_Q to induce vertical displacement ${u}_n^{\ast }$. Beam 2 is allowed for rigid body displacement ${\lambda}_F^{(2)}$ and elastic deformation

Assuming that there is no force acting at the sliding support A, the equilibrium condition of beam 2 requires that

$$ {F}_0={F}_Q, and\;{p}_{\mathrm{max}}=\frac{F_Q}{\sum \limits_{j=1}^{kont}c\left({x}_j\right)\;A} $$

(9)

as the contact forces are P_j = Ac(x_j)p_max.

Using the influence function for calculation of ${u}_{n, load}^{(2)}$, we can write

$$ {u}_{n, load}^{(2)}=-{H}^{(2)}\left({x}_Q,{x}_Q\right){F}_Q=-{H}_{Q,Q}^{(2)}\;{F}_Q $$

(10)

and the support displacement can be determined from Eq.(8)

$$ {\lambda}_F^{(2)}={u}_n^{\ast }-\left(\sum \limits_{j=1}^{kont}{H}^{(2)}\left({x}_Q,{x}_j\right)\frac{c\left({x}_j\right)}{\sum \limits_{k=1}^{kont}c\left({x}_k\right)}-{H}_{Q,Q}^{(2)}\right)\;{F}_Q $$

(11)

The contact condition between the punch and beam is

$$ {u}_{in}^{(2)}-{u}_{in}^{(1)}+{g}_i^{(0)}=0,i=1,\dots, kont $$

(12)

Using (62a), (12) can be expressed as follows

$$ {\lambda}_F^{(2)}+\left(\sum \limits_{j=1}^{kont}\left({H}^{(1)}\left({x}_i,{x}_j\right)+{H}^{(2)}\left({x}_i,{x}_j\right)\right)\frac{c\left({x}_j\right)}{\sum \limits_{k=1}^{kont}c\left({x}_k\right)}-{H}_{i,Q}^{(2)}\right)\;{F}_Q-{\lambda}_F^{(1)}+{g}_i^{(0)}=0 $$

(13)

or

$$ {\lambda}_F^{(2)}+\left(\sum \limits_{j=1}^{kont}H\left({x}_i,{x}_j\right)-{H}_{i,Q}^{(2)}\right)\;{F}_Q-{\lambda}_F^{(1)}+{g}_i^{(0)}=0 $$

(14)

where

$$ H\left({x}_i,{x}_j\right)=\left({H}^{(1)}\left({x}_i,{x}_j\right)+{H}^{(2)}\left({x}_i,{x}_j\right)\right)\frac{c\left({x}_j\right)}{\sum \limits_{k=1}^{kont}c\left({x}_k\right)} $$

(15)

Discretizing (12)–(15), we can write

$$ \mathbf{d}=\left({}^{iter}\mathbf{H}\mathbf{e}-{\mathbf{h}}_Q^{(2)}\right){F}_Q+\mathbf{e}{\lambda}_F^{(2)}-\mathbf{e}\;{{}^{(iter)}\lambda}_F^{(1)}+{{}^{(iter)}\mathbf{g}}^{(0)}=\mathbf{0} $$

(16)

where

${\mathbf{h}}_Q^{(2),T}=\left[{H}_{1,Q}^{(2)},{H}_{2,Q}^{(2)},...,{H}_{i,Q}^{(2)},...,{H}_{kont,Q}^{(2)}\right]$, e, see (63),

Because F_Q and ${\lambda}_F^{(2)}$ are known (see (11)), then ^iteru can be calculated, namely

$$ {}^{iter}\mathbf{u}=\left({}^{iter}\mathbf{H}\mathbf{e}-{\mathbf{h}}_Q^{(2)}\right){F}_Q+\mathbf{e}{\lambda}_F^{(2)} $$

(17)

and from the following equation

$$ {}^{iter}\mathbf{u}{}-\mathbf{e}\;{{}^{(iter)}\lambda}_F^{(1)}+{{}^{(iter)}\mathbf{g}}^{(0)}=\mathbf{0} $$

(18)

one can easily find ${{}^{iter}\lambda}_F^{(1)}{},{{}^{(iter)}\mathbf{g}}^{(0)}$ supposing ^(iter)g₁⁽⁰⁾ = 0. Then ${{}^{(iter)}\lambda}_F^{(1)}={{}^{(iter)}u}_1$.

2.1.3 Third variant: free beam is allowed for rigid body translation and rotation

Referring to Fig. 5, consider the case, when the beam 2 is not constrained and is allowed for a rigid body vertical displacement ${\lambda}_F^{(2)}$ and rotation ${\lambda}_M^{(2)}$ at the beam end A. The displacement control is now required at two points Q₁ and Q₂, where the loads ${F}_{Q_1}$ and ${F}_{Q_2}$ are applied. The initial beam configuration corresponds to simultaneous contact engagement with interacting objects at Q₁ and Q₂.

Fig. 5

Beam system for transmission of the load F₀ interacting with the forces ${F}_{Q_1}$ and ${F}_{Q_2}$ to induce vertical displacements ${u}_{1n}^{\ast }$ and ${u}_{2n}^{\ast }$, when the beam 2 is allowed for rigid body translation ${\lambda}_F^{(2)}$ and rotation ${\lambda}_M^{(2)}$

The vertical displacement at point Q_s is

$$ {u}_{sn}^{\ast }={u}_{sn}^{(2)}={\lambda}_F^{(2)}+{\lambda}_M^{(2)}{x}_{Q_s}+\sum \limits_{j=1}^{kont}{H}^{(2)}\left({x}_{Q_s},{x}_j\right){P}_J+{u}_{sn, load}^{(2)}\kern0.5em s=1,2 $$

(19)

Using the influence functions for calculation ${u}_{sn, load}^{(2)}$, we can write

$$ {u}_{sn, load}^{(2)}=-\sum \limits_{p=1}^2{H}_{Q_s,{Q}_p}^{(2)}\;{F}_{Q_p} $$

(20)

From the force and moment equilibrium equations of the beam, we have

$$ {F}_0={F}_{Q_1}+{F}_{Q_2}\kern0.5em 0={x}_0{F}_0-x{{}_{Q_1}{}F}_{Q_1}-{x}_{Q_2}{F}_{Q_2} $$

(21)

and the position of control force F₀ is equal to

$$ {x}_0=\frac{x_{Q_1}{F}_{Q_1}+{x}_{Q_2}{F}_{Q_2}}{F_{Q_1}+{F}_{Q_2}} $$

Introducing the parameter 0 ≤ ζ ≤ 1 and defining the force ${F}_{Q_2}=\left(1-\zeta \right){F}_{Q_1}$, the position of F₀ is

$$ {x}_0=\frac{x{}_{Q_1}{}+{x}_{Q_2}\left(1-\zeta \right)}{2-\zeta } $$

(22)

when ζ = 0, the forces ${F}_{Q_1}={F}_{Q_2}$ and ${x}_0=\frac{x{}_{Q_1}{}+{x}_{Q_2}}{2}$, when ζ = 1, then ${F}_{Q_2}=0$ and ${x}_0={x}_{Q_1}$.

From the equilibrium equation for punch (3), the maximal contact pressure and the contact forces are

$$ {p}_{\mathrm{max}}=\frac{F_0}{\sum \limits_{j=1}^{kont}c\left({x}_j\right)\;A} $$

(23)

For derivation of the rigid body displacements, we have the following two equations

$$ {\lambda}_F^{(2)}+{\lambda}_M^{(2)}{x}_{Q_s}={u}_{sn}^{\ast }-\left(\sum \limits_{j=1}^{kont}{H}^{(2)}\left({x}_{Q_s},{x}_j\right)\frac{c\left({x}_j\right)}{\sum \limits_{k=1}^{kont}c\left({x}_k\right)}{F}_0-\sum \limits_{p=1}^2{H}_{Q_s,{Q}_p}^{(2)}{F}_{Q_p}\right),s=1,2 $$

(24)

The contact condition between the punch and beam is

$$ {u}_{in}^{(2)}-{u}_{in}^{(1)}+{g}_i^{(0)}=0,i=1,\dots, kont $$

(25)

and it can be expressed in the following form

$$ {\displaystyle \begin{array}{l}{\lambda}_F^{(2)}+{\lambda}_M^{(2)}{x}_i+\left(\sum \limits_{j=1}^{kont}{H}^{(2)}\left({x}_i,{x}_j\right)\frac{c\left({x}_j\right)}{\sum \limits_{k=1}^{kont}c\left({x}_k\right)}{F}_0-\sum \limits_{p=1}^2{H}_{i,{Q}_p}^{(2)}{F}_{Q_p}\right)\\ {}-\left({\lambda}_F^{(1)}-\sum \limits_{j=1}^{kont}{H}^{(1)}\left({x}_i,{x}_j\right)\frac{c\left({x}_j\right)}{\sum \limits_{k=1}^{kont}c\left({x}_k\right)}{F}_0\right)+{g}_i^{(0)}=0\end{array}} $$

(26)

or

$$ {\lambda}_F^{(2)}+{\lambda}_M^{(2)}{x}_i+\left(\sum \limits_{j=1}^{kont}H\left({x}_i,{x}_j\right){F}_0-\sum \limits_{p=1}^2{H}_{i,{Q}_p}^{(2)}{F}_{Q_p}\right)-{\lambda}_F^{(1)}+{g}_i^{(0)}=0 $$

(27)

where

$$ H\left({x}_i,{x}_j\right)=\left({H}^{(1)}\left({x}_i,{x}_j\right)+{H}^{(2)}\left({x}_i,{x}_j\right)\right)\frac{c\left({x}_j\right)}{\sum \limits_{k=1}^{kont}c\left({x}_k\right)} $$

(28)

Discretizing (26)–(28), we can write

$$ \mathbf{d}={}^{iter}\mathbf{H}\mathbf{e}{F}_0+\sum \limits_{p=1}^2{\mathbf{h}}_{Q{}_p}^{(2)}{F}_{Q_p}+\mathbf{e}{\lambda}_F^{(2)}+\mathbf{x}{\lambda}_M^{(2)}-\mathbf{e}\;{{}^{(iter)}\lambda}_F^{(1)}+{{}^{(iter)}\mathbf{g}}^{(0)}=\mathbf{0} $$

(29)

where.

${\mathbf{h}}_{Q_s}^{(2),T}=\left({H}_{1,{Q}_s}^{(2)},{H}_{2,{Q}_s}^{(2)},...,{H}_{i,{Q}_s}^{(2)},...,{H}_{iter,{Q}_s}^{(2)}\right)$, s = 1,2, x^T = [x₁, x₂, ..., x_i, ..., x_kont].

Because F₀ and ${F}_{Q_p}$p = 1, 2 and ${\lambda}_F^{(2)}$ and ${\lambda}_M^{(2)}$ are known, ^iteru can be calculated from the equation

$$ {}^{iter}\mathbf{u}=\left({}^{iter}\mathbf{H}\mathbf{e}{F}_0+\sum \limits_{p=1}^2{\mathbf{h}}_{Q_p}^{(2)}{F}_{Q_p}\right)+\mathbf{e}{\lambda}_F^{(2)}+\mathbf{x}{\lambda}_M^{(2)} $$

(30)

and from the following equation

$$ \mathbf{d}={}^{iter}\mathbf{u}{}-\mathbf{e}\;{{}^{(iter)}\lambda}_F^{(1)}+{{}^{(iter)}\mathbf{g}}^{(0)}=\mathbf{0} $$

(31)

one can easily find ${{}^{iter}\lambda}_F^{(1)}{},{{}^{(iter)}\mathbf{g}}^{(0)}$ supposing ^(iter)g₁⁽⁰⁾ = 0. Then ${{}^{(iter)}\lambda}_F^{(1)}={{}^{(iter)}u}_1$.

3 Numerical examples

The beam area A_b = a_bh_b = 20 ⋅ 50 = 1000 mm², inertia moment $I={a}_b{h}_b^3/12=21833.33\;m{m}^4$, Young modulus E = 2 ⋅ 10⁵MPa are assumed and other geometric parameters are:

L₁ = 300, L₄ = 600, x_Q = 850, L = 900 mm. The punch centre position x₀ = 450 mm corresponds to the position parameter ξ = x₀/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 mm² (Fig. 2a). The x coordinates of the punch pins are: 383.33, 416.66, 450.00, 483.33, and 516.66 mm.

The beam deflection is presented in Fig. 6a for three control functions and F_Q = 5 kN. It is seen that their effect is very small. The biggest displacement corresponds to the Hermite 2 control function (see Fig. 3). At the point Q, the prescribed displacement ${u}_n^{\ast }=1\; mm$ at x = 850 mm is attained. Figure 6b presents the beam deflection for three values of force F_Q. The heights of punch pins at the start of calculation are assumed to be 50 mm. From the numerical solution the new heights are calculated by (68) and depicted in Fig. 7a.

Fig. 6

Deflection of the beam at aF_Q = 5000 N for different control functions and b beam deflection for c(x) = 1; the prescribed forces are F_Q = 4 kN, F_Q = 5 kN, and F_Q = 6 kN. Position of the punch is denoted by circles at F_Q = 5 kN, see Fig. 6b

Fig. 7

a Heights of the punch pins. b Initial gaps between the punch pins and beam at different load values at c(x) = 1

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.

The distribution of contact tractions is presented in Fig. 8 for F_Q = 5 kN. The maximum value is reached for the control function Hermite 2, the least value for c(x) = const = 1. In the latter case the contact force is the same at each pin: P_j = 2.987 kN and the punch load equals F₀ = 14.935 kN. This means that the present design requires the considerably higher load to execute the control at the efficiency load factor ${f}_0=\frac{F_0}{F_Q}=2.987$. At other force levels F_Q = 4 kN, F_Q = 6 kN, the maximal contact forces (c(x) = const = 1) are P_jmax = 2.41 kN, P_jmax = 3.56 kN, respectively, and the external punch loads in these cases are F₀ = 12.07 kN, F₀ = 17.81 kN, corresponding to load factors f₀ = 3.0 and 2.987.

Fig. 8

Distribution of contact tractions at F_Q = 5 kN

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 (L₁ = 420, L₄ = 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%.

The contact conditions are checked at the Lobatto integral points. The discretized problem is solved in the following steps:

• 1. Calculation of the contact pressure by (6), and F₀ calculated by (3).

• 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 ^iterg⁽⁰⁾ and rigid body displacement ^iterλ (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)].

The rigid body vertical displacement of punch, $\lambda ={\lambda}_F^{(1)}$, is a displacement in the –z direction at point x = 420, z = 50. In our case for F_Q = 5 kN, the final value is ${\lambda}_F^{(1)}=2.511\ \mathrm{mm}$. The deflection and initial gap are presented in Fig. 9a, b for all values of forces F_Q. For the optimal solution, the values the load F₀ and pressure p_max are collected in Table 1.

Fig. 9

Punch is an elastic solid body: a displacements of the beam interacting with punch contact surface (thick line, for punch action F_Q = 5 kN) and b initial gap between punch and beam

Table 1 Maximal contact pressures and punch loads for the optimal solution

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

In this case, the load equilibrium requires that F₀ = F_Q (cf. Figure 4). In the first step, the rigid body displacement ${\lambda}_F^{(2)}$ is calculated from Eq. (11). From contact condition between beam and punch, the rigid body displacement ${\lambda}_F^{(1)}$ is determined from (18). Only several iterations are required to reach the solution. The beam deflection is shown in Fig. 10a. The punch pins profile is presented in Fig. 10b.

Fig. 10

Deflections of the beam (a) and heights of the punch pins/number of pins = 5 (b)

The force efficiency factor f₀ = F₀/F_Q in this case is equal to 1. The rigid body punch and beam displacements are collected in Table 2 at different F_Q.

Table 2 Punch and beam rigid body displacements and maximal contact pressures for second variant: a_b = 20, h_b = 50 mm

3.3 Example for 3rd variant: beam allowed for rigid body translation and rotation

Consider the beam of crosssection dimensions a_b = 20, h_b = 75 mm (cf. Fig. 5). In this case, the load on the punch is ${F}_0={F}_{Q_1}+F{}_{Q_2}$, where ${F}_{Q_2}=\left(1-\zeta \right){F}_{Q_1}$, ${F}_{Q_1}=4,5,6\; kN$, and ζ = 0.5. The prescribed vertical displacements at the points Q₁ and Q₂ are assumed: ${u}_{1n}^{\bullet }=5\; mm$ and ${u}_{2n}^{\bullet }=10\; mm$. Let us note that in the case of contact interaction at Q₁ and Q₂, the displacements are specified from the loadpenetration rules. The punch center coordinate is.

$$ {x}_0=\frac{x{}_{Q_1}{}+{x}_{Q_2}\left(1-\zeta \right)}{2-\zeta }=316.66\; mm, because\;{x}_{Q_1}=50,{x}_{Q_2}=850\; mm. $$

In the first step, the rigid body motion components ${\lambda}_F^{(2)}$ and ${\lambda}_M^{(2)}$are calculated form (24).

From the contact condition between the punch and beam, the rigid body displacement ${\lambda}_F^{(1)}$ is easily determined in the several iterative steps, see (31). The deflection of the beam is shown in Fig. 11a. The heights of the punch pins constituting the contact gap are demonstrated in Fig. 11b. The punch vertical translation ${\lambda}_F^{(1)}$ and the loading force F₀ are therefore required for proper operation of gripper at the points Q₁ and Q₂. The rigid body displacements are collected in Table 3. Let us note that for varying control requirements at the contact points, the sliding or rotating punch can be applied assuring proper load F₀ position at each case.

Fig. 11

Deflections of the beam a and heights of the punch pins b for the solution of variant 3 (see Fig. 5)

Table 3 The maximal contact pressure and rigid body displacements resulting from solution of variant 3

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 x₀ and length L_1 − 4 = L₄ − L₁ 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 = L₄ − L₁ 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).

The virtual work equation is

$$ F{w}_Q=\underset{0}{\overset{L}{\int }}{M}^{(q)}\kappa\;dx=\underset{0}{\overset{L}{\int }}M{\kappa}^{(q)} dx=\underset{L_1}{\overset{L_4}{\int }}{p}_n{w}^{(q)}{a}_b dx $$

(32)

where M = EIκ, M^(q) = EIκ^(q), and M^(q)κ = EIκ^(q)κ = Mκ^(q), p_n is the contact pressure between punch and beam, a_b is the beam crosssection width in direction y, h_b is the beam crosssection height in direction z, $I={a}_b{h}_b^3/12$ inertia moment of the cross section, and E is the Young’s modulus.

From (32), we have

$$ w\left(x={x}_Q\right)={w}_Q=\underset{L_1}{\overset{L_4}{\int }}{p}_n{w}^{(q)}{a}_b dx $$

(33)

Now, several optimization problems can be stated

• 1. Contact force specified: maximize deflection w_Q

• 2. Deflection w_Q specified: minimize the punch force F₀

• 3. Deflection w_Q specified: minimize the maximal pressure p_max

Problem 1:

$$ \max \underset{0}{\overset{L}{\int }}M{\kappa}^{(q)} dx=\underset{L_1}{\overset{L_4}{\int }}{p}_n{w}^{(q)}{a}_b dx\kern0.5em subject\kern0.34em to\kern0.5em \underset{L_1}{\overset{L_4}{\int }}{p}_n{a}_b\; dx-{F}_0=0 $$

(34)

Taking the functional

$$ {L}_{p1}=\underset{L_1}{\overset{L_4}{\int }}{p}_n{w}^{(q)}{a}_b\; dx-\lambda \left(\underset{L_1}{\overset{L_4}{\int }}{p}_n{a}_b\; dx-{F}_0\right) $$

(35)

its variation provides the stationary condition

$$ \delta {L}_{p1}=\underset{L_1}{\overset{L_4}{\int }}\delta {p}_n\left({w}^{(q)}-\lambda \right){a}_b\; dx=0,\kern0.5em that\kern0.34em is\kern0.5em {w}^{(q)}=\lambda = const $$

(36)

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:

Assume the penalty approach and the contact pressure functional

$$ {I}_{p3}=\underset{L_1}{\overset{L_4}{\int }}{\left(\frac{p_n}{p_0}\right)}^m{a}_b\; dx,m>1 $$

(37)

where p₀ is the reference pressure, p₀ ≈ p_max, and p₀ ≤ p_max.

Then

$$ {L}_{p3}=\underset{L_1}{\overset{L_4}{\int }}{\left(\frac{p_n}{p_0}\right)}^m{a}_b\; dx-\lambda \left[\underset{L_1}{\overset{L_4}{\int }}{p}_n{w}^{(q)}{a}_b\; dx-{w}_Q\right] $$

(38)

The variation of L_p3 has the following form

$$ \delta {L}_{p3}=\underset{L_1}{\overset{L_4}{\int }}\left\{m{\left(\frac{p_n}{p_0}\right)}^{m-1}-\lambda {w}^{(q)}\right\}\delta {p}_n\;{a}_b\; dx=0 $$

(39)

and the stationary conditions are as follows

$$ {p}_n={\lambda}^{\ast }{\left({w}^{(q)}\right)}^{\frac{1}{m-1}}\kern0.5em where\kern0.5em {\lambda}^{\ast }={\left(\frac{\lambda {p}_0^m}{m}\right)}^{\frac{1}{m-1}} $$

(40)

The value of λ^∗ is specified from the equality ${w}_Q^0=\underset{L_1}{\overset{L_4}{\int }}{p}_n{w}^{(q)}{a}_b dx$.

Consider now the case of symmetric pressure distribution relative to the contact zone center, assuming that

$$ {p}_n\left({L}_1+\overline{\xi}\right)={p}_n\left({L}_4-\overline{\xi}\right) $$

(41)

where $\overline{\xi}$ specifies the position of two symmetrically located points.

The virtual work relation now is

$$ \underset{0}{\overset{L}{\int }}M{\kappa}^{(q)} dx=\underset{L_1}{\overset{L_4}{\int }}{p}_n{w}^{(q)}{a}_b dx=\underset{0}{\overset{\left({L}_4-{L}_1\right)/2}{\int }}{p}_n\left({L}_1+\overline{\xi}\right)\frac{w^{(q)}\left({L}_1+\overline{\xi}\right)+{w}^{(q)}\left({L}_4-\overline{\xi}\right)}{2}{a}_b\;d\overline{\xi} $$

(42)

Denote

$$ {\tilde{w}}^{(q)}\left(\overline{\xi}\right)=\frac{w^{(q)}\left({L}_1+\overline{\xi}\right)+{w}^{(q)}\left({L}_4-\overline{\xi}\right)}{2} $$

(43)

then we have

$$ \underset{0}{\overset{L}{\int }}M{\kappa}^{(q)} dx=\underset{0}{\overset{\left({L}_4-{L}_1\right)/2}{\int }}{p}_n\left({L}_1+\overline{\xi}\right){\tilde{w}}^{(q)}{a}_b\;d\overline{\xi} $$

(44)

and from (40), there is

$$ {p}_n={\lambda}^{\ast }{\left({\tilde{w}}^{(q)}\right)}^{\frac{1}{m-1}} $$

(45)

The analysis presented characterizes the optimal contact pressure distribution. Figure 12 presents the optimal pressure distribution for varying value of the exponent m and minimization of the functional I_p3. It is seen that for m = 4, the pressure tends to the uniform distribution for both unconstrained and symmetry constrained cases. If there is no constraint set on the maximal pressure value, then the concentrated load action is optimal. Its value and position are determined by requiring the deflection ${u}_z=-{u}_n^{\ast }$ at Q and accounting for the beam stress constraint σ_max ≤ σ_u. When the distributed punch action is considered, and the integral constraint is set on the contact pressure, then its distribution is uniform on the whole contact zone. The required punch load for specified deflection ${u}_z=-{u}_n^{\ast }$ will then be higher from the concentrated load value.

Fig. 12

Contact pressure distribution obtained from the optimization procedure: a unconstrained distribution and b symmetry constrained distribution

4.2 Concentrated and distributed cantilever beam control

Referring to Fig. 2, consider the cantilever beam under the action of concentrated load F_Q and the punch action is simulated by the concentrated load F₀. Assume first that the load positions satisfy the inequality x₀ ≤ x_Q. The bending moment in the beam is expressed as follows

$$ M={M}_b+\left({F}_Q-{F}_0\right)x\kern2em 0\le x\le {x}_0,\kern0.5em M=-{F}_Q\left({x}_Q-x\right)\kern2em {x}_0\le x\le {x}_Q $$

(46)

where M_b =  − F_Qx_Q + F₀x₀ is the boundary moment.

The normal deflection at the point Q (along—zaxis) due to force F_Q (can be expressed from the formulae of Appendix 3) equals

$$ {w}_Q=-\frac{F_Q}{3 EI}{x}_Q^3 $$

(47)

The deflection ${w}_Q^0$ at Q due to load F₀ is

$$ {w}_Q^0=\frac{F_0}{3 EI}\left(\frac{3}{2}{x}_Q{x}_0^2-\frac{1}{2}{x}_0^3\right) $$

(48)

Introducing the nondimensional variables

$$ \frac{F_0}{F_Q}={f}_0,\kern0.5em \frac{x_0}{x_Q}=\xi, \kern0.5em \frac{M_b}{F_Q{x}_Q}={\tilde{M}}_b=-1+{f}_0\xi, \kern0.5em \frac{w_t}{\mid {w}_Q\mid }={\tilde{w}}_t $$

(49)

Now, we can express the total deflection ${w}_t={w}_Q+{w}_Q^0$ at Q as follows

$$ {\tilde{w}}_t=-1+\frac{1}{2}\left(3{\xi}^2-{\xi}^3\right){f}_0\kern0.5em \mathrm{for}\;\xi \le 1 $$

(50)

For the load F₀ acting on the right side of Q, x_Q ≤ x₀, the total deflection at Q equals

$$ {w}_Q^0=\frac{F_0}{3 EI}\left(\frac{3}{2}{x}_Q^2{x}_0-\frac{1}{2}{x}_Q^3\right),\kern0.5em {w}_Q=-\frac{F_Q}{3 EI}{x}_Q^3=-\mid {w}_Q\mid, \kern0.5em {\tilde{w}}_t=-1+\frac{1}{2}\left(3\xi -1\right){f}_0\kern0.5em \mathrm{for}\;\xi \ge 1 $$

(51)

For specified ${\tilde{w}}_t$, the required value of load F₀ is expressed as follows

$$ {f}_0^{-}=\frac{F_0}{F_Q}=\frac{2\left(1+{\tilde{w}}_t\right)}{3{\xi}^2-{\xi}^3}\kern0.5em \mathrm{for}\xi \le 1\kern0.5em {f}_0^{+}=\frac{F_0}{F_Q}=\frac{2\left(1+{\tilde{w}}_t\right)}{3\xi -1}\kern0.5em \mathrm{for}\xi \ge 1 $$

(52)

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 f₀(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 f₀ = F₀/F_Q strongly depends on the punch load F₀ position. Usually, the value of ξ = x₀/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, f₀ ≈ 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 ${\tilde{w}}_t\approx 0.04$.

Consider the strength constraint. The critical bending moment results from the stress constraint $M\le {M}_c={a}_b{h}_b^2{\sigma}_u/6$, where σ_u is the ultimate stress. Denoting $\tilde{M}=M/\left({F}_Q{x}_Q\right)$, the maximal bending moment for ξ ≤ 1 is reached at the support x = 0 and its absolute value equals

$$ \mid {\tilde{M}}_b\mid =-1+{f}_0\xi $$

(53)

For ξ ≥ 1, the maximal bending moment is reached at Q and its absolute value is

$$ \mid {\tilde{M}}_Q\mid ={f}_0\left(\xi -1\right) $$

(54)

Satisfying the strength conditions $\mid {\tilde{M}}_b\mid \le {\tilde{M}}_c$ and $\mid {\tilde{M}}_Q\mid \le {\tilde{M}}_c$, in view of (52), (53), and (54), the limits of applicable load positions are expressed by the inequalities

$$ \xi \ge {\xi}_c^{-}=\frac{1}{2}\left[3-\sqrt{9-8\beta}\right]\kern0.5em where\kern0.5em \beta =\frac{1+{\tilde{w}}_t}{1+{\tilde{M}}_c} $$

(55)

$$ \xi \le {\xi}_c^{+}=\frac{2\overline{\beta}-1}{2\overline{\beta}-3}\kern0.5em where\kern0.5em \overline{\beta}=\frac{1+{\tilde{w}}_t}{{\tilde{M}}_c} $$

(56)

Figure 13 shows the admissible segment of punch action satisfying the constraints (55) and (56). Calculated values are ${\xi}_c^{-}=0.5012$, ${f}_0^{-}=3.31$, ${\xi}_c^{+}=14.76$, and ${f}_0^{+}=0.048$. At another beam cross section a_b = 20, h_b = 75 mm, there are ${\xi}_c^{-}=0.5613$, ${f}_0^{-}=2.96$, ${\xi}_c^{+}=5.572$, and ${f}_0^{+}=0.144$. For the distributed punch action, the contact zone length parameter ${\tilde{L}}_{1-4}={L}_{1-4}/{x}_Q=\left({L}_4-{L}_1\right)/{x}_Q$ affects the resultant punch load relative to the concentrated load action at the same position. The required punch load value is higher for the same value of induced displacement ${\tilde{w}}_t$.

Fig. 13

Diagram f₀ − ξ of punch load dependence on its position for fixed deflection ${\tilde{w}}_t$ at the contact point interaction Q. The strength constraints ${\xi}_c^{-}\le \xi \le {\xi}_c^{+}$ provide the admissible segment for punch action. Beam cross section a_b = 20, h_b = 50 mm, ${\xi}_c^{-}=0.5012$, and ${\xi}_c^{+}=14.76$

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 F₀ = F_Q at the distance x₀ and the specified contact force F_Q at Q located at the distance x_Q.

Consider first the case x₀ ≤ x_Q or ξ ≤ 1. At the support, the transverse force equals F_b = 0 and the moment is $\mid {\tilde{M}}_b\mid =1-{f}_0\xi$. The deflection form now is expressed as follows

$$ w=-\frac{F_Q}{2 EI}\left({x}_Q-{x}_0\right){x}^2+{w}_b,\kern1em 0\le x\le {x}_0,\kern0.5em w=-\frac{F_Q}{2 EI}\left[\left({x}_Qx-{x}_0^2\right)x+\frac{1}{3}\left({x}_0^3-{x}^3\right)\right]+{w}_b,\kern1em {x}_0\le x\le {x}_Q $$

(57)

For x₀ ≥ x_Q or ξ ≥ 1, we have

$$ w=\frac{F_Q}{2 EI}\left({x}_0-{x}_Q\right){x}^2+w{}_b,\kern1em 0\le x\le {x}_Q,\kern0.5em w=\frac{F_Q}{2 EI}\left[\left({x}_0x-{x}_Q^2\right)x-\frac{1}{3}\left({x}^3-{x}_Q^3\right)\right]+{w}_b,\kern1em {x}_Q\le x\le {x}_0 $$

(58)

where ${w}_b={\lambda}_F^{(2)}$ denotes the beam translation at the support. Requiring w = w_t at Q, the beam translation w_b is determined, thus

$$ {\tilde{w}}_b={\tilde{\lambda}}_F^{(2)}={\tilde{w}}_t+1-\frac{1}{2}\left(3{\xi}^2-{\xi}^3\right),\kern1em \xi \le 1\kern0.5em \mathrm{and}\kern0.5em {\tilde{w}}_b={\tilde{\lambda}}_F^{(2)}={\tilde{w}}_t-\frac{3}{2}\left(\xi -1\right),\kern0.5em \xi \ge 1 $$

(59)

where

$$ {\tilde{w}}_b={\tilde{\lambda}}_F^{(2)}=\frac{\lambda_F^{(2)}}{\mid {w}_Q\mid }={\lambda}_F^{(2)}\frac{3 IE}{F_Q{x}_Q^3},\kern1em {\tilde{w}}_t=\frac{w_t}{\mid {w}_Q\mid }. $$

(60)

The diagram ${\tilde{w}}_b=\tilde{\lambda}={\lambda}_F^{(2)}\left({\tilde{w}}_t,\xi \right)$can easily be constructed, see Fig. 14. It is seen that the sliding displacement changes sign at ξ = 1.

Fig. 14

Diagram for $\tilde{\lambda}={\tilde{\lambda}}_F^{(2)}\left({\tilde{w}}_t,\xi \right)$. (Beam cross-section sides are a_b = 20, h_b = 75 mm)

The present sliding support case differs essentially from the previous builtin case. The applied punch force is constant and does not depend on the position, but the deflection form radically changes when passing from ξ ≤ 1 to ξ ≥ 1. Figure 15 presents both deflection forms for concentrated (Fig. 15a, b) and distributed (Fig. 15c) punch action. The present case is convenient for more precise control at Q by requiring both deflection and slope to be kept at specified values and allowing for sliding punch action. In Fig. 15a, c the solutions of the optimization problems satisfying stress constraint are presented. In Fig. 15 the optimal solution point is marked by o.

Fig. 15

Beam deflection forms: a for ξ ≤ 1, b for ξ ≥ 1, and c for distributed punch action at ξ ≤ 1 (beam cross-section sides: a_b = 20, h_b = 75 mm, ${\lambda}_c^{\sim }={\lambda}_F^{(2)}$)

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 F₀, punch centre position x₀, 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 F₀. 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.

Appendices

Appendix 1

The starting point of numerical analysis is related to discretization of (7)

$$ d=\left({u}_n^{(2)}\left({p}_n\right)+{u}_{n, load}^{(2)}\right)-\left({u}_n^{(1)}\left({p}_n\right)+\lambda \right)+{g}^{(0)}=0 $$

The displacements induced by the contact tractions are

$$ {u}_n^{\left(\alpha \right)}\left({p}_n\right)={u}_n^{\left(\alpha \right)}\left(x,{p}_n\right)\Rightarrow {u}_n^{\left(\alpha \right)}\left({x}_i,{P}_j\right)=\sum \limits_{j=1}^{kont}{H}^{\left(\alpha \right)}\left({x}_i,{s}_j\right){P}_j,\kern0.5em {x}_i,i=1,..., kont,\kern0.5em \alpha =1,2 $$

(62a)

and from the given load F_Q, the displacement of body 2 is

$$ {u}_{n, load}^{(2)}(x)\Rightarrow {u}_{n, load}^{(2)}\left({x}_i\right),i=1,..., kont $$

(62b)

Introduce the vectors (contact force, displacement from F_Q, initial gap).

p^T = [P₁...P_j...P_kont]^T, ${\mathbf{u}}_{n,\mathrm{l} oad}^{(2)T}={\left[{u}_1^{(2)}...\kern0.24em {u}_i^{(2)}...\kern0.24em {u}_{load}^{(2)}\right]}_{n, load}^T$, ${\mathbf{g}}^{(0)T}={\left[{g}_1^{(0)}...\kern0.24em {g}_i^{(0)}...\kern0.24em {g}_{kont}^{(0)}\right]}^T$.

Next the vector e and the gap d after deformation are

$$ {\mathbf{e}}^T={\left[\mathrm{1...}\kern0.24em \mathrm{1...}\kern0.24em 1\right]}^T,\kern0.5em {\mathbf{d}}^T={\left[{d}_1...\kern0.24em {d}_i...\kern0.24em {d}_{kont}\right]}^T $$

(63)

the influence matrices are

$$ {}^{(iter)}\mathbf{H}={{}^{(iter)}\mathbf{H}}^{(1)}+{{}^{(iter)}\mathbf{H}}^{(2)} $$

(64)

the discretized gap after deformation can be written in the following form

$$ \mathbf{d}={}^{(iter)}\mathbf{H}\mathbf{p}+{\mathbf{u}}_{n, load}^{(2)}+{{}^{(iter)}\mathbf{g}}^{(0)}-\mathbf{e}{}^{(iter)}\lambda $$

(65a)

The displacements due to contact tractions are

$$ {{}^{(iter)}\mathbf{g}}^{(0)\ast }={}^{(iter)}\mathbf{H}\mathbf{p} $$

(65b)

and from (65a) we get

$$ {{}^{(iter)}\mathbf{g}}^{(0)\ast }=-{\mathbf{u}}_{n, load}^{(2)}-{{}^{(iter)}\mathbf{g}}^{(0)}+\mathbf{e}{}^{(iter)}\lambda $$

(65c)

In this equation ^(iter)g⁽⁰⁾, ^(iter)λare unknown. Assume that ${{}^{(iter)}g}_{j\min}^{(0)\ast }=\min {{}^{(iter)}\mathbf{g}}^{(0)\ast }$, that is minimum value is obtained at the jmin component. Choosing the initial gap ${{}^{(iter)}g}_{j\min}^{(0)}=0$, that is at the point j = jmin, the initial gap is assumed to vanish.

Then, according to (65c) for jmin component we get

$$ {}^{(iter)}\lambda ={{}^{(iter)}\mathrm{g}}_{j\min}^{(0)\ast }+{{}^{(iter)}u}_{n, load;j\min}^{(0)} $$

(66)

After calculation of ^(iter)λ, the gap vector is (see (65c))

$$ {{}^{(iter)}\mathbf{g}}^{(0)}=-{\mathbf{u}}_{n, load}^{(2)}-{{}^{(iter)}\mathbf{g}}^{(0)\ast }+\mathbf{e}{}^{(iter)}\lambda $$

(67)

The lengths of punch pins are

$$ {l}_i={l}_i^{(0)}-{{}^{(iter)}\mathrm{g}}_i^{(0)},\kern0.5em i=1,\dots, kont $$

(68)

and the influence matrix element is

$$ {{}^{(iter)}H}_{i,i}^{(1)}=\frac{{{}^{(iter)}l}_i}{AE} $$

(69)

where A and E denote the crosssection area and Young’s modulus. Usually, the convergent solution is obtained in the fast iterative process.

Appendix 2

Consider now the case, when the punch (body 1) deforms elastically in the plane stress state. In this case the discretisation is performed by the finite element technique. The control of contact conditions is made at the Lobatto integral points (Páczelt et al. 2015, 2016). The force F₀ and the contact pressure p_n(x) are assumed as given. The value of p_max is obtained from the equilibrium equation, cf. (3). The punch displacement is assumed to be composed of a rigid body translation λ and an elastic translation ${u}_z^{(1)e}={u}_n^{(1)}\left({p}_n,{F}_0\right)$ reaching its value $\lambda +{u}_z^{(1)e}$ on the contact surface. For calculation of this normal elastic displacement ${u}_z^{(1)e}\left(x,z=0\right)=-{u}_n^{(1)}\left({p}_n,{F}_0\right)$, the elastic boundary value problem is solved with the condition ${u}_z^{(1)e}=0$ at the upper punch boundary point, Fig. 16. Discretizing the elastic problem by finite element technique, we get the normal displacement ${u}_n^{(1)}$ on the contact surface. Between the punch (body 1) and the beam (body 2) the initial gap g⁽⁰⁾ results from the punch contact surface shape. The geometrical contact condition is

$$ d=\left({u}_n^{(2)}\left({p}_n\right)+{u}_{n, iload}^{(2)}\right)-\left({u}_n^{(1)}\left({p}_n,{F}_0\right)+\lambda \right)+{g}^{(0)}=0 $$

(70)

Fig. 16

Punch with its load F₀ and contact pressure p_n(x): elastic displacement constraint ${u}_z^{(1)e}=0$

After discretization, using the earlier defined influence matrix H⁽²⁾, the initial gap vector g⁽⁰⁾ is specified. As the initial gap changes in consecutive iterations, the discretized equation has the following form

$$ \mathbf{d}={{}\mathbf{H}}^{(2)}\mathbf{p}-{{}^{(iter)}\mathbf{u}}_n^{(1)}+{\mathbf{u}}_{n, load}^{(2)}+{}^{(iter)}\mathbf{g}{}^{(0)} - \mathbf{e}{}^{(iter)}\lambda =\mathbf{0}. $$

(71)

After definition

$$ {}^{(iter)}\mathbf{g}{}^{(0)\ast }={\mathbf{H}}^{(2)}\mathbf{p}-{{}^{(iter)}\mathbf{u}}_n^{(1)} $$

(72)

the steps (65c)–(67) should be repeated.

Appendix 3

Using Betti theorem, the influence (Green) function for cantilever beam (Fig. 17) has the form.

$$ {H}^{(2)}\left({x}_i,{x}_j\right)={H}_{i,j}^{(2)}=\frac{1}{6 EI}\left\{\left(3{x}_j{x}_i^2-{x}_i^3\right)+{\left\langle {x}_i-{x}_j\right\rangle}^3\right\}, $$

where ${\left\langle {x}_i-{x}_j\right\rangle}^3=\Big\{{\displaystyle \begin{array}{l}{\left({x}_i-{x}_j\right)}^3, if\;{x}_i>{x}_j\\ {}0,\kern3.12em if\;{x}_i\le {x}_j\end{array}}$, I is the inertia moment of cross section, and E is the Young’s modulus.

Fig. 17

Load for calculation of the Influence function of the cantilever beam