A numerical survey of nonlinear dynamical responses of discrete pantographic beams

Pantographic lattices are an example of architectured materials having many interesting mechanical peculiarities, see, e.g., the recent review papers [1‐5]. Several synthetic continuum models capable to predict their mechanical behavior in statics and dynamics have been developed in the last years, see, e.g., [6‐12]. Nevertheless, technical references dealing experimentally with their dynamical behavior are rare, one example being [13].

Experience has proven that a fruitful way, alternative to continuum modeling, for predicting the mechanical response of pantographic lattices is the numerical simulation based on a discrete models inspired by Hencky’s idea, see [14]. Indeed, as proven by a series of papers published in the last few years, see, e.g., [15, 16], the complete equilibrium path of pantographic fabrics subjected to mechanical loading can be recovered, in the static regime and for large displacements, by using a stepwise strategy based on Newton’s method as predictor and on Riks’ arc-length as corrector.

When the influence of inertia forces must be necessarily considered, an efficient time-stepping scheme may be used to reconstruct the whole equilibrium path in the case of large displacements and in the elastic regime.¹ Recently, some interesting papers were published analyzing the dynamics of two- , see [17], and three-dimensional, see [18], beams, also in view to applications to robotic arms [19, 20].

In the available literature, there exist several integration schemes for linear and nonlinear mechanical problems. Interested readers can find a large number of references and an extensive discussion thereof in Wriggers’ book [21, Chap. 6]. Here, the integration scheme proposed by Casciaro in 1975 [22] will be used. Such a method is practically unrecognized, even though it has some very interesting peculiarities.

In simple words, Casciaro’s scheme guarantees for a chosen time step length, numerically stability, besides being able of: (i) filtering of the roundoff errors and (ii) avoiding beat phenomenon for large time steps. We remark that for nonlinear analyses, e.g., for large displacements, there is no guarantee that the response produced by Casciaro’s scheme is the best possible response. However, to the Author’s knowledge, this is true for any integration scheme reported in the technical literature.

The paper, after this short Introduction, presents in Sect. 2 the intrinsically discrete model for pantographic beams moving in plane. Successively, in Sect. 3, the employed stepwise solution strategy is briefly discussed. Section 4 reports several numerical simulations, each one being thoroughly discussed. Finally, Sect. 5 presents some concluding remarks and future challenges.

We consider a planar pantographic beam, in brief p-beam, such as that depicted on the top in Fig. 1. The p-beam is made up of two families of extensible Euler–Bernoulli beams connected at their intersecting points by pivots. As already experimented in [16, 23‐25] for pantographic lattices with orthogonal beams, in [26] for non-orthogonal beams and in [27, 28] for pantographic beams, we describe the mechanical behavior of the p-beam by the system of springs depicted on the bottom in Fig. 1.

The interactions consist in extensional and rotational springs to represent in a very simple form the resistance of beams to extension and bending and the torsional stiffness of the pivots. This mechanical description may be improved in a h-refinement way as reported in [29]. Thence, the strain energy of each deformable spring of the p-beam can be written as:²being \(E_a\), \(E_b\), \(E_c\) the strain energies, respectively, stored in extensional springs, in flexural-rotational springs and shear-rotational springs. The corresponding springs are colored in red and blue for extensional and flexural springs and in black for shear-rotational springs, see again the bottom part of Fig. 1. The associated strain measures \(\Delta \ell \) for the stretching, \(\cos \beta \) for the bending and \(\gamma \) for the shear strain are all defined in terms of the current positions of the p-beam nodes or, equivalently, in terms of nodal displacements.³

In particular, we assume as strain measures the following:where \(P_i\) and \(p_i\) are, respectively, the reference and current positions of the ith node (an analogous convention holds for the nodes j, k and m). The interested reader can find more details in [15]. Since for the ith node \(p_i=P_i+\mathbf {u}_i\), the strain energy element contribution depends only on the involved displacement (namely \(\mathbf {u}_i\) and \(\mathbf {u}_j\) for the stretching, \(\mathbf {u}_i\), \(\mathbf {u}_j\) and \(\mathbf {u}_k\) for the bending and \(\mathbf {u}_j\), \(\mathbf {u}_m\) and \(\mathbf {u}_k\) for the shear strain). Stiffnesses of the springs involved in the p-beam are denoted by a, b and c, for extensional, flexural and shear-strain springs, respectively.⁴

Strain energies for each spring of the considered p-beam completely define its mechanical behavior. We remark that different forms for the bending strain energy \(E_b\) are possible, see, e.g., [30, 31]. For instance, the classic form \(E_b=\dfrac{1}{2} \bar{b} \beta ^2\) could equivalently be used where the stiffness parameter \(\bar{b}\), different from b, is introduced. It is easy to prove that for \(\beta \)’s in a small neighborhood of \(\pi \) (i.e. value assumed by \(\beta \) in the reference zero-energy configuration) the two choices for \(E_b\) are equivalent.

The strain energy E of the whole p-beam can be obtained simply by adding the aforementioned contributions. Collecting the nodal displacements, i.e. the Lagrangian variables, in the vector \(\mathbf {u}\), we can define the structural reaction \(\mathbf {s}\) as the gradient of the strain energy respect to the Lagrangian variablesFurthermore, we can define the stiffness matrix \(\mathbf {K}\) as the Hessian of the strain energy or, equivalently, the gradient of the structural reactionThe gradient and the Hessian of the strain energy are the tools necessary to tackle the incremental-iterative strategy capable to solve the dynamical equilibrium equations.⁵

Besides the strain energy, the study of the dynamical behavior of pantographic beams requires to define the kinetic energy of the system. We distinguish two contributions: that of beams and that of pivots. The first one, for a beam element of length \(\ell _e\), is defined asbeing \(\rho _e\) the mass density, \(A_e\) the area of the beam cross section, \(\mathbf {v}\) the velocity vector written by using the matrix of the blending functions \(\mathbf {B}_e\) and the Lagrangian parameters collected in \(\dot{\mathbf {u}}_e\). The matrix \(\mathbf {B}_e\), using the local coordinate \(0\le x\le \ell _e\), assumes the form:In the case of beams with uniform cross section, some straightforward calculations lead toIn addition, the mass matrix contribution related to the pth pivot can be derived from its kinetic energy defined asbeing \(\rho _p\), \(A_p\), \(\mathbf {v}_p\) and \(\mathbf {I}\) the density, the cross-section area, the velocity of pth pivot and the \(2\times 2\) identity matrix, respectively.⁶ The vector \(\dot{\mathbf {u}}_p\) collects the Lagrangian parameters which represent the velocity of the pth pivot and is connected to the pivot velocity vector \(\mathbf {v}_p = \mathbf {I}\dot{\mathbf {u}}_p\). Consequently, pth pivot mass matrix contribution iswhere \(d_p\) and h are the diameter and the height of the pth cylindrical pivot, respectively.⁷

The mass matrix contribution \(\mathbf {M}_e\) of the eth element connecting nodes i and j is given by Eq. (7). This contribution derives from the ordering chosen for vector of nodal velocities \(\dot{\mathbf {u}}_e\), i.e.  \(x_1\)- and \(x_2\)-velocity for the i-node and \(x_1\)- and \(x_2\)-velocity of the j-node, see Fig. 1. Similarly, the pth pivot mass matrix contribution \(\mathbf {M}_e\) is given by Eq. (9)—always with the same ordering rule for the component velocities, i.e.  first the \(x_1\)- and then the \(x_2\)-velocity. The mass matrix elementary contributions (7) and (9) are the only quantities necessary to assemble the global mass matrix \(\mathbf {M}\).

Kinetic and strain energy, as defined in the immediate foregoing, allow to compute the mass matrix \(\mathbf {M}\) and the structural reaction \(\mathbf {s}\), which are the only quantities strictly necessary to write the system of differential of equations ruling the motion. In the Euler–Lagrange form, see [33], such a system reads⁸where, besides the Lagrangian parameters used to describe the motion, i.e. the displacement vector \(\mathbf {u}\), the velocity vector \(\dot{\mathbf {u}}\) and the acceleration vector \(\ddot{\mathbf {u}}\), also the external force vector \(\mathbf {f}\) appears.

As mentioned in Sect. 1, the technical literature is dense of contributions devoted to solve nonlinear equilibrium equations. Some examples are reported in [37‐39] and references therein. The book [21], updated up to the first years of 2000, provides with a very good introduction on stepwise integration schemes. Starting from a suitable time discretization, possibly non-uniform, these approaches reconstruct the dynamical equilibrium path—i.e. the pair time t and Lagrangian variables collected in the displacement \(\mathbf {u}\) and in the velocity \(\dot{\mathbf {u}}\) vectors—by solving a sequence of initial value problems. Roughly speaking, for each time interval \(\Delta t\) corresponding to the chosen time discretization, the solution at the final point is computed on the basis of the solution at the beginning of the time step. The choice of the time step length is crucial—especially for nonlinear problems—and, generally speaking, it requires both some preliminary analyses and a noteworthy experience of the analyst.

Actually, there exists in the technical literature a scheme which does not require, at least in principle, a preliminary study. This numerical integration scheme was developed and published by Casciaro [22], notwithstanding that it was communicated on an important journal such as Meccanica, Casciaro’s paper is practically unknown to the international scientific community.⁹

The keynote idea of Casciaro’s numerical integration scheme is based on the use of two free parameters which are determined by optimizing the dynamical response for the a priori chosen time step length. In practice, Casciaro’s idea stems from the observation that the numerical solution of a problem in time is affected by errors depending on the chosen time step length. These errors may be triggered by choosing the time step length \(\Delta t\) as greater than a fraction of the fundamental (minimum) period of the underlying problem (numerical instability). Furthermore, also the roundoff errors, i.e. when the selected time step length \(\Delta t\) is lower than a specific threshold, can affect significantly the problem solution (instability dues to roundoff errors). Finally, a relatively large time step length \(\Delta t\) might introduce spurious beatings, i.e. beating due to numerical errors, even for unconditionally stable schemes.

Essentially, Casciaro’s algorithm is based on: (i) a parabolic interpolation law for the displacements in the current time interval, using as parameters the values of the displacement and of the velocity at the beginning of the step and of the velocity at the end of the step;¹⁰ (ii) the discrete form of the momentum-impulse relationship.

The displacement vector can be written at the time t within the time step \(t_1-t_0\) in the formwhere \(\mathbf {u}_0\), \(\dot{\mathbf {u}}_0\) and \(\dot{\mathbf {u}}_1\) are the displacement vector, the velocity vector at the beginning of the time step and the velocity vector at the end of the time step, respectively; \(\beta _{0}\) and \(\beta _{1}\) are two weighting factors, hence \(\beta _0+\beta _1=1\), which may be chosen to optimize the integration scheme results. Interpolation (11) when evaluated at the end of the time step, furnishes the vector \(\mathbf {u}_{1}\) collecting the displacements at the end of the time interval:The discrete-form of the momentum-impulse relationship read as:where we notice the momentum change \(\mathbf {M}(\dot{\mathbf {u}}_1-\dot{\mathbf {u}}_0)\) and the average net impulse, written using the weighting factors \(\alpha _0\) and \(\alpha _1\), such that \(\alpha _0+\alpha _1=1\), which, once again, can be tuned to optimize the integration scheme.

Remark that owing to the constraint relationships \(\alpha _0+\alpha _1=1\) and \(\beta _0+\beta _1=1\), only two parameters among \(\alpha _0\), \(\alpha _1\), \(\beta _0\), and \(\beta _1\) are independent, say \(\alpha \) and \(\beta \). These independent parameters can be computed optimally on the basis of the estimates of the first, \(T_{1}\), and of the last, \(T_{n}\), natural periods of the considered structure. Casciaro provided in [22] the analytical expressions of the parameters \(\alpha \) and \(\beta \). In formulae, in the case \(\Delta t < T_{n}/2\):where \(\tau =2\pi \dfrac{\Delta t}{T_{n}}\). The limit of Eq. (14), when \(\Delta t \rightarrow 0\), is \(\beta =-\alpha =\dfrac{1}{\sqrt{6}}\). On the other hand, when \(\Delta t \rightarrow \dfrac{T_{n}}{2}\) we get \(\beta =-\alpha =\dfrac{1}{\pi }\).

In the case \(\Delta t > \dfrac{T_{n}}{2}\), the optimal values are:whereFormulae (15) contain two terms. The first one is a hyperbola branch which matches the optimal values of \(\alpha \) and \(\beta \) at the end of the interval \(\Delta t<T_n/2\). The second one is the ratio between two cubic polynomials which tends—when \(\Delta t\) goes to infinite—to the value \(\frac{1}{2}\); therefore, such a value of \(\alpha \) and \(\beta \) which is able to reproduce a quasi-static solution. Figure 2 shows a plot of the optimal values for \(\alpha \) e \(\beta \) obtained using the aforementioned expressions.

In the paper [22], Casciaro does not give many details on the proposed optimization scheme. This may be one of the causes implying the lack of success of his method. Actually, a more detailed description of the integration scheme is reported [41], although both in [22] and in [41] there are some typos in the formulas. The more recent work [42] contains some insights into the method used to build the optimal formulas (14) and (15). We have now almost all the tools for the solution of dynamic nonlinear problems involving multi-degree-of-freedom systems: a simple, yet effective, mechanical model such as that described in Sect. 2 and an algorithm to tackle the time step solution in the dynamic case such as that described in this section. We only need a solution strategy which takes into account the geometrical nonlinearities of the p-beam response. Hence, we can express the structural reaction \(\mathbf {s}_1\) by using the first-order Taylor expansionhaving used the stiffness matrix \(\mathbf {K}_0\), i.e. the stiffness matrix computed at the beginning of the considered step, see Eq. (4):Considering again Eqs. (12) and (13), we can eliminate the unknown displacement variables, i.e. \(\mathbf {u}_1\) by using Eq. (12). The immediately above equations along with Eqs. (12)–(13) are enough to build a Newton-like algorithm for our nonlinear problem.

In a synthetic way, we can start from the definition of the rest computed at the jth iterationand compute the new estimates of the solution in a Newton-like fashionhaving introduced the iteration matrix \(\mathbf {H}\) defined asWe have thus completed the introduction of the tools of the numerical strategy used to solve the nonlinear equilibrium problem in the considered time step. At this point, some remarks are deemed useful:

In this section, we begin the exploration of the dynamical behavior of p-beams under harmonic load by means of a parametric study. In synthesis, we are interested to study: To this end, we selected four representative datasets characterized by the stiffnesses reported in Table 1. The choice of these datasets considers both experimental data obtained from testing of 3D-printed pantographic specimens and simplified models currently employed in the literature for analyzing pantographic lattices. The spring stiffnesses of the first dataset derive from an identification process described in [44], which was performed on a 3D-printed specimen obtained from selective laser sintering of polyamide powder having a density \(\rho =930\) kg/m\(^3\). The specimen is constituted by beams having square cross section with side equal to 1 mm and by cylindrical pivots with height equal to 2 mm and circular cross section with diameter equal to 0.9 mm. Starting from the dataset based on experimental data—dataset 1— we built datasets for three special cases: inextensible beams— dataset 2—inextensible and inflexible beams—dataset 3—and perfect pivots—dataset 4. In datasets 2 and 3, a factor 1000 for increasing the rigidified stiffnesses is considered. Dataset 4 accounts for a special case extensively described in [29]: the pivots are designed in such a way that the corresponding torsional rigidities are practically zero and, therefore, the corresponding stiffness parameter c of each pivot is set to zero.

We consider the case of a clamped-clamped p-beam, therefore constraints are set on the nodes A, B, C and D, see Fig. 1. More specifically, vertical displacement is prevented for all the four nodes A, B, C and D, while the horizontal displacement for nodes A and D only. The considered excitations are two: a transversal force on the p-beam midpoint M—symmetric excitation—and a transversal force on the point N—unsymmetrical excitation, see again Fig. 1.

In this work, we have dealt with the dynamical response in large displacement regime of pantographic beams subjected to harmonic loads. In particular, we analyzed four different stiffness datasets and computing, among the other things, the corresponding frequency-response curves by means of a stepwise numerical approach. Differences with what is expected in small displacement regime were thoroughly discussed in order to build a solid base foundation for successive explorations. The simple mechanical model and the stepwise analysis strategy used in the present investigation have proven their capability to provide a first set of results. The peculiar results obtained within this work could help all researchers concerned with the dynamics of pantographic structures and materials (see [45]), and with metamaterials in general—see, e.g., the very recent work [46]—to face many open or still unexplored problems.

We believe that future research lines of this work shall include: (i) the enhancement of the mechanical model used for the p-beam modeling such as proposed in [47]; furthermore, the three-dimensional discrete beam formulation proposed in [48] could be very fruitful to consider out-of-plane instability phenomena; (ii) the generalization of the stepwise strategy in order to handle the case of non-conservative forces such as those deriving from damping; (iii) the formulation of continuum, hence synthetic, models capable to forecast the main mechanical behaviors of pantographic beams, see, e.g., [7]; (iv) the extension of the utilized discrete mechanical approach to micropolar media [49], thin plates with surface stresses [50] and strongly anisotropic surface elasticity [51]; (v) the application of the utilized time-integration scheme to continuous problems discretized, for instance, by means of B-spline and NURBS, see, e.g., [52] or for models dealing with granular media, e.g., [53, 54].