Compliant mechanisms

Compliant mechanisms are mechanisms whose function is not enabled by conventional joints or guides (in which surfaces slide or roll on top of each other) but by the deliberate use of elastic strain at points that are explicitly designed for this purpose. For compliant mechanisms with lumped compliance, these are local areas of low stiffness (solid-state joints); for compliant mechanisms with distributed compliance, they are longer bending or torsion springs.

Mechanisms are devices that allow deliberately pronounced deformations in order to perform a specific function. Let's take two hand tools as simple examples: the commercial, pincer-like nutcracker and the tweezers. In both examples, the relative pivoting of two rod-like elements (links) is allowed. The function to be performed thereby is the transfer of hand force to another location. Thereby, due to the principle of leverage, the force is amplified (in the case of the nutcracker) or attenuated (in the case of the tweezers). It is important to observe that apart from the desired deformation necessary for the mechanism to function, the mechanism should remain stiff, ideally rigid. If, for example, the links did not have sufficient stiffness, they would hardly be able to transmit forces. Again, the functional movement of the mechanism should require as little energy as possible. The pivoting of the links in tweezers costs much less energy than, for example, the bending of one of the rods in itself; in the nutcracker, this relationship is even more pronounced, because there (apart from frictional losses) the pivoting of the (unloaded) mechanism takes place without overcoming a counterforce.
As mechanisms, the nutcracker and the tweezers are designed according to two different principles. In the first case, the function of the mechanism is enabled by the inclusion of a pin joint between two rigid elements. In the second case, the elasticity of certain areas of the structure provides the necessary pivoting. This makes the difference between conventional and compliant mechanisms: The nutcracker is a mechanism of the conventional type, while the tweezers are a compliant mechanism.
Many of the advantages of compliant systems are obvious: They are free of backlash, wear and friction. They basically require no lubrication. They can be manufactured in one piece, without complex assembly procedures. They are easier to clean due to the absence of gaps and other discontinuities on the surface. They do not release particles due to the lack of wear. They can be used freely in heavily polluted environments and are also quieter. Another set of advantages results from specific analyses: compliant mechanisms are more precise,easier to miniaturize and lightweight.
The disadvantages of compliant mechanisms include a fundamental limitation on deformation due to limited allowable strain. In addition, the design of compliant mechanisms is much more complex for reasons explained below.

 

Despite the numerous advantages of compliant mechanisms, mechanical engineering is dominated by conventional mechanisms. One of the reasons of this is the above mentioned much higher complexity of the design of compliant mechanisms compared to conventional ones. The staff of our institute, which deals with compliant mechanisms, has partly decades of experience with the topic of their design and continuously develops this focus in targeted research work.
The design of conventional mechanisms benefits from an important fact: Their desired deformations result from relative rigid-body motions of the links to each other. Therefore, they can basically be analyzed without considering forces and elastic deformations. In design, the determination of the desired deformations is a purely topological-geometric task. Only in a subsequent phase are the individual elements (links and joints) dimensioned. The kinematic aspect (i.e. the consideration of the desired, functional deformations) can therefore be completely separated from the static aspect (the consideration of forces and resulting stresses in the material). This is not possible with compliant mechanisms. A compliant system modeled without considering elastic strains would not be able to represent a desired deformation. Thus, the relationship between force and deformation must be considered from the first step in the design of compliant mechanisms. In addition, requirement conflicts between deformability and load capacity are much less problematic to resolve in conventional mechanisms because these properties are assigned to separate components: Links and joints can be stiffened as desired without negatively affecting the kinematics (i.e., the feasibility of realizing the desired deformations). In compliant systems, deformability and loadability must be balanced at every point in the system: If a cross-section is strengthened, this will have a positive effect on load-bearing capacity on the one hand, and a negative effect on deformability on the other. For this reason, the design of compliant systems above a certain complexity is always carried out on the basis of a formally imposed optimization task.
An optimization task consists of different subtasks, for which different special knowledge is required. First (parametrization), in order to mathematically capture the object of the design (given constant material, the unknown geometry of the mechanism) must be parametrized so that it can be expressed as a function of a certain number of variables (design variables). Second (problem formulation), it must be defined which mathematical function of the design variables (objective function) should be minimized or maximized and under which constraints (restrictions) this should be done. And thirdly (search strategy) the algorithm must be selected, by which the optimal solution is to be found.
As far as our research in the field of design of compliant mechanisms is concerned, the main focus is on the study of suitable problem formulations. As far as parameterization is concerned, we specialize in the so-called topology optimization, where the domain in which the geometry is to be defined is partitioned into a fine mesh of elements. The contribution of each element to the stiffness of the system is scaled by a factor (density). The densities of the individual elements then represent the design variables. A special topic addressed at the institute in this context is the integrated design of compliant mechanisms and actuators that provide for the activation of the mechanisms.

In the context of the problem formulation issue, we succeeded in setting a milestone with the modal procedure, which is the first example of so-called pseudo-kinematic approaches.
It was mentioned above that a strict separation between kinematics and statics is not possible in the design of compliant systems. This has led the scientific community to completely give up the claim of a kinematic design and to design methods basically load case oriented. That is, a limited number of load cases are considered in the problem formulation and the response of the system to deviating load cases is not considered. In our work, on the other hand, we have looked for ways to maintain the kinematic requirement in the design as much as possible, and to optimize the geometry of the system for tending to respond with strongly similar deformations for a very wide range of load cases. We have called this property (small differences in deformation for pronounced differences in load) selectivity. We basically made high selectivity the main goal of optimization, and then focused on giving the selectivity a mathematical equivalent to translate the maximization of selectivity into appropriate problem formulations. We call this approach pseudo-kinematic.
In conventional mechanisms (as long as they are idealized as systems of rigid bodies and perfect joints), selectivity is given in the absolute sense, because there exists an infinite number of loads that lead to exactly the same deformation. In the case of compliant mechanisms, on the other hand, selectivity is a quantitative variable, since, as mentioned, each deformation is coupled with a very specific load, and thus it is not possible for two different loads to lead to the same deformation. If we deepen this fact mathematically in the simple case of linear systems, we come to the basis of the modal procedure.
Let us first consider a plane conventional mechanism at small deformations. The mechanism is modeled using the finite element method, where the links are no longer considered ideally rigid, but are assigned finite stiffness. One of the links is clamped firmly. The FE method then provides the relationship between displacements and forces in the form of a system of linear equations. Since we have a mechanism, the coefficient matrix of the displacements (the stiffness matrix) is singular. This implies that the system has either no solution at all, or infinitely many solutions. Thus, the unique coupling between load and deformation discussed above cannot exist: Most loads do not lead to a solution, and again there are solutions that are feasible with different loads. If we again model a compliant mechanism in a similar way, the coupling mentioned above is given. The stiffness matrix is regular, and the system is always invertible. For any given load there is a single static solution and vice versa. However, it will happen that with different loads very similar deformations are to be realized. This property (the selectivity) is found in the so-called condition of the stiffness matrix, which in turn is related to the eigenvalues of the stiffness matrix. For singular matrices, at least one eigenvalue is zero. The number of vanishing eigenvalues of a singular matrix is called the rank decay R of the matrix and quantifies the number of rows of the matrix that can be expressed as a linear combination of the others. For solutions to exist, the same ratios must also hold on the right-hand side of the equations, so that R whole equations can be expressed as linear combinations of other equations. It is obvious that a stiffness matrix with an eigenvalue that is very small with respect to the others (ill-conditioned matrix) behaves approximately like a singular matrix. Strictly speaking, there will be different solutions for each imposed load, but they will have approximately the same ratio between the individual displacements. This ratio is given by the eigenvector associated with the very small eigenvalue.
If the design succeeds in influencing the stiffness matrix in such a way that one eigenvalue is small in relation to the others and that the corresponding eigenvector maps the desired deformation, the design of the system is methodically brought close to the kinematic design of conventional systems, since a certain deformation is impressed on the system as the preferred deformation, independent of the load case.
This is the modal procedure, which by its nature is only valid for linear systems in this form. It has been gradually developed with respect to the concrete formulation as well as with respect to algorithmic aspects since the first formulation from 2008 and is currently used as the basis of a more general, pseudo-kinematic procedure, which also applies to nonlinear systems.
Apart from the load case independence, the modal procedure also represents a quantum leap forward in the research on the design of compliant mechanisms due to its application flexibility. It can be applied without extensive formulation changes to systems with kinematics defined over a large number of degrees of freedom, as well as to so-called compliant mechanisms with multiple pseudo-mobility. Compliant mechanisms with multiple pseudo-mobility are the compliant equivalent of conventional mechanisms with multiple mobility, i.e., mechanisms that have more than one degree of freedom when represented as systems of rigid bodies and ideal joints.

In the table shown below, the program codes can be downloaded using the links.

Paper title	First author	Link professorship page	Link Mendely Data
Synthesis of compliant mechanisms with selective compliance - An advanced procedure	Kirmse	www.tu-chemnitz.de/mb/mp/mmt	https://data.mendeley.com/datasets/697dnbtz62/2
How to prestress compliant mechanisms for a targeted stiffness adjustment	Mauser	www.tu-chemnitz.de/mb/mp/sms	-
Topology-optimization based design of multi-degree-of-freedom compliant mechanisms (mechanisms with multiple pseudo-mobility)	Seltmann	www.tu-chemnitz.de/mb/mp/jimss	https://data.mendeley.com/datasets/c5k9xnyvtb/1
Topology optimization of compliant mechanisms with distributed compliance (hinge-free compliant mechanisms) by using stiffness and adaptive volume constraints instead of stress constraints	Seltmann	www.tu-chemnitz.de/mb/mp/mmt22	https://data.mendeley.com/datasets/wzzdvf8nts/2