# The distance between the two flexible links measured

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The distance between the two flexible links measured at the base (joint assembly 1; see Fig. 3) is the same as the dimension at joint assembly 2. The rocker and the follower (side beams) are elastic and have equal length. Unlike a typical four-bar linkage, the side beams are clamped (fixed), not pinned, at the joint assembly 1 of the PFBL link. The side beams are connected via pin joints to a short rigid coupler at joint assembly 2. As in the conventional manipulator, the mass of the PFBL link is lumped at the coupler. The PFBL model has the same three degrees of freedom at joint assembly 2 as did the conventional. 1) Coupler motion transverse to the side beams measured with . This is caused by common mode beam-like deformation of the side beams. 2) Coupler motion compressing the side beams measured by . This is caused by common mode axial deformation of the side beams. 3) An undesirable rotation of the coupler measured by . This is partially caused by uneven deformation of the side beams. Rotation might also result from nonparallel side beams. Nonparallel side beams might result from manufacturing tolerances or errors. The former cause of rotation is discussed in the model results, the later is discussed in a later section. The position, velocity and hence kinetic energy in the PFBL are exactly the same as the conventional manipulator so they will not be repeated. To compute the potential energy due to axial deformation, use a formula similar to the conventional manipulator. The only difference is the angle contributes to the deformation. In one beam adds to and in the other beam it subtracts as shown at the bottom of this page. The bending deformation is caused exclusively by the displacement therefore it is much easier to find the strain energy The 2 in front of the integral is due to the fact that there are two beams bending. According to Lagrangian mechanics, the equations of mo- tion of the system are (remember the mass and stiffness matrices are symmetric) (3) Notice that the mass matrix is the same as the conventional manipulator but the stiffness differs. There are solutions to the general equations that modify the inertia matrix but they are the subject of future work. It would not be a fair comparison to match the PFBL against the conventional if both use the same values of and in their formula. For example, using the same in the PFBL would mean it has twice the material as the conventional and one might expect it to perform better simply because of this fact. To make a more reasonable comparison, the value of and used in the PFBL will be half of the values used in the conventional. In essence we are saying that the PFBL gets to use two beams by splitting the single conventional one in half. Again, we write the equations in nondimensional form. First introduce the aspect ratio Also notice that since the PFBL links have half the area moment of inertia as the conventional link, the value of for the PFBL will be half of the value for the conventional.

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