Analysis of a compliant mechanism using Howell’s pseudo-rigid model and comparison with non-linear FEM analysis.

The following article is an extract from a project carried out for the course of Advanced Techniques for the design of Prosthetic Devices. Held for Medical Engineering at the University of Rome Tor Vergata .

Authors:

Mastrofini Alessandro&Muscedere Erica

**Table of Contents**

*Flexure hinges*

Some apparently rigid mechanisms allow particular movements thanks to their flexibility. This mechanism is called a compliant mechanism. Some areas, where deformations are concentrated, can be identified as flexible hinges. They allow relative movements in a similar way to classical kinematic pairs. These bodies have an elasto-kinematic behavior, ie the kinematics is not unique but is influenced by the loads and by the elasticity itself.

### Approach for the study of a compliant mechanism

There are several approaches. A computationally easier approach is to replace the elasto-kinematic body with a pseudo-rigid mechanism. That is, the body is considered as rigid but segmented into different parts connected by kinematic pairs. In particular, we will consider the Howell model which involves the insertion of a hinge and a spring to restore elasticity.

A different approach, on the other hand, involves carrying out a finite element structural analysis, taking into account the overall elasticity. Often this requires evaluating the analysis in large displacements and therefore this non-linearity leads to a considerably higher computational cost.

## Howell’s pseudo-rigid model

The Howell model is part of a broader category known as concentrated hinge mechanisms.

This model allows to evaluate the elasto-kinematic behavior in the case of a jammed beam loaded with a moment and some variants. The structure in question can be broken down into two cases.

### Fixed beam

The flexible rod on the right can be seen as the Howell model with a beam wedged at one end and the other which is only allowed for translations and loaded with a moment. Then there are the corresponding tables for cases, such as this one in question, in which the load is of the force type. The model envisages dividing the beam into three rigid bodies connected by two hinges at a distance from the edges of:

\begin{equation} d_{d x}=\frac{(1-\gamma) l}{2}=18.5375 \mathrm{~mm} \end{equation}

Furthermore, considering the force at an angle of 90° with respect to the axis of the beam, the following parameters are identified:

*𝛾*= 0*.*8517*𝐾𝜃*= 2*.*65

Then the spring stiffness is estimated as:

\begin{equation} K_{d x}=2 \gamma K_{\Theta} \frac{E I}{l}=104.6122 \frac{\mathrm{N} \mathrm{mm}}{\mathrm{rad}} \end{equation}

### Right beam

For the left rod it does not directly correspond to a Howell model but it is possible, observing the load conditions, to refer to the case of a jammed beam loaded with a force. Then the resulting parameters predict a stiffness spring:

\begin{equation} K_{s x}=\gamma K_{\Theta} \frac{E I}{l}=52.8226 \frac{\mathrm{N} \mathrm{mm}}{\mathrm{rad}} \end{equation}

Obtainable from parameters:

*𝛾*= 0*.*8517*𝐾𝜃*= 2*.*67617

And a hinge at a distance from the lower end of:

\begin{equation} d_{s x}=\gamma l=212.9250 \mathrm{~mm} \end{equation}

This model has been analyzed in Solidworks Motion. The segments of the rods were partially cut with the Split function and then, the different files, reloaded together. The coincidence constraints between the axes that identify the average line of the cutting area have been added (see fig. 2a). Torsional springs have been added taking the axis of the hinge and the second body to which it refers as the axis.

The analysis was set at 2 seconds with a load of 1 N on the left side face of the massive body.

## Non linear FEA

To carry out the finite element analysis, a non-linear analysis was set up on Solidworks Simulation to take into account the elasticity of the system and large displacements. Furthermore, this analysis allows you to move along load steps and gradually arrive at applying the total load. This, in addition to gradually solving the non-linear problem, allows us to identify the yield strength (which we will discuss better in the final section).

The analysis was carried out considering two massive bodies with different heights: a first at 120 mm and one at 30 mm, the heights have been varied in order to simplify the computational complexity. Furthermore, the massive body was considered to be made of a much more rigid material (with properties similar to synthetic diamond) than the harmonic steel of which the rods are made.

### Fixtures

To simulate the design constraints, an interlocking type constraint was placed on the lower edge of the left rod and three carriages (in the three orthogonal directions) on the lower edge of the right rod, such as to block the three ways of translation leaving free the rotations.

The forces of 1 N and 10 N have been applied on the left side face of the maximum body.

The first simulation was carried out with a massive body with a height of 120 mm and an applied force of 1N. As can be seen from fig. 3, the structure does not break: the maximum stress reached (step 17) is about 630 MPa, lower than the yield point of harmonic steel of 1020.

Again from fig. 3 it is possible to have a confirmation of how the positioning of the hinges in the pseudo-rigid model has occurred in the correct way: having a joint in the upper end of the left rod, we can see how the segment immediately below remains approximately straight while immediately after the rod has its maximum curvature; in the same way, in the right rod, having two joints in the upper and lower ends, the maximum curvatures occur after the straight segments attached to the joints themselves.

### Loading

A second analysis was carried out by reducing the height of the upper body and increasing the force applied to 10N. Varying the height of the semi-rigid body does not have a great effect on the behavior of the structure (given the large difference in stiffness between the two materials) but has a great effect in the reduction of calculation times, considering that the linear analysis is completed in no less than 15 steps.

This simulation was carried out mainly to get at least one of the two rods to yield. In fig. 4 it is possible to note that the displacements up to the application of 1 N are the same as in the previous analysis. At step 17 the yield point of harmonic steel was exceeded, reaching a stress of approximately 1025 MPa and an applied force of approximately 5.36 N. The forces applied in the various steps were calculated by multiplying the initial force by the corresponding time at the given step. Note that the quoted time is a fictitious time representing the load fraction reached.

## Analysis of the results

In the figure fig. 5 you can see a comparison between the kinematic analysis of Howell’s pseudo-rigid model and the FEM analysis.

We can see how, considering a force of only 1 N, the two displacements can be considered coincident. Furthermore, the FEM analysis fails to cover the entire shift that covers the pseudo-rigid model. Wanting to further investigate the behavior, the second analysis was carried out with 10 N. The displacement, up to a load of approximately 1 N coincides, continuing further begins to diverge. The displacement of the FEM analysis, with an excessive load, tends to exit the arc of circumference and sees a greater contribution of vertical translation.

### Divergence between the two models

Furthermore, observing the rotations (fig. 7a) we see how, in the part that can be covered with the pseudo-rigid model, the rotations coincide, that is, they are zero. In the following steps the model begins to rotate to satisfy the elastic equilibrium of the two rods subjected to different constraints. Observe how the angle cannot be read directly from the solid mesh (only 3 translational degrees of freedom) but can be calculated by considering the displacements of the extremal points of the base segment of the massive body, passing through 𝑀, for simple trigonometric considerations .

## Plasticization

On the basis of non-linear FEM analysis with a load of 10 N it is possible to determine the yield load.

In particular, considering a yield point for the harmonic steel under consideration of *𝜎𝑦 *= 1200 MPa, the structure leaves the elastic limit between 7.14 N and 8.14 N.

On the other hand, considering a structure with ABS rods, the simulation gives very different results.

The massive body is considered to be in aluminum, in order to respect the high stiffness ratio (body to rods). A load of 1 N is set and the singularity elimination factor parameter changed to 0.5 to allow the model to go further and search for the yield strength, for ABS of 𝜎𝑦 = 30 MPa.

The result, in terms of initial displacements, is very similar to the previous one but the yield point is reached at an excessive deformation. Following the simulation, the limit is reached for a load of 0.5 N but the overall deformation is not acceptable. This is attributable to the low stiffness of the material which, together with the reduced thickness of the side rods, allows for very large deformations. The simulation does not see the yield point reached.

Furthermore, it is very likely that before reaching yield the structure will collapse due to the high deformation which, despite being allowed in numerical simulation, may not be acceptable in reality.