Rotation of the end mass is given with note that when

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rotation of the end mass is given with . Note that when changes the entire rigid outer link rotates. The angle is the rigid motor position of the outer link relative to . The Fig. 1. Lumped-mass model of a two link robot with one conventional flexible inner link and one rigid outer link. angle changes with manipulator configura tion but not due to vibration. The flexible link has the three degrees of freedom that Fig. 1 labels , and . The position of the masses is given by (the values that change due to vibration are denoted as functions of time) By differentiating these the Kinetic energy can be determined. These terms will include the mass moments of inertia of each mass about their centroids. These inertias will be called and . The potential energy can be found by determining the strain energy generated by displacements , and . For example, the beam loading is a superposition of an axial load, a transverse load and a pure moment all applied at the free end of the beam. The axial load is independent of the other loads and is uniform along the beam therefore its strain energy is by definition The values and are Young’s modulus and link cross sectional area. To determine the energies due to bending start with an expression for the bending moment at a position from the base in terms of an applied transverse load and a moment Next use some elementary beam formulas [16] to express the beam’s displacement and slope at the free end in terms of applied loads. Solve these for and and substitute into the expression for moment
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EVERETT et al. : DESIGNING FLEXIBLE MANIPULATORS 607 The is area moment of inertia. By definition the strain energy is the following integral: By adding the strain energies we arrive at the potential energy for the beam Pe The stiffness terms are: and . According to Lagrangian mechanics, the equations of mo- tion describing the vibration of the system are (note that the mass and stiffness matrices are symmetric) (1) The terms are nonlinear components composed of Coriolis and centrifugal type terms. A dimensional analysis was performed to write the equa- tions in nondimensional form. Define mass ratio length ratio stiffness ratio and inertia ratio Using these definitions we can write linear part (small angle ) of the original equations as (2) If one ignores then frequencies can be determined exclusively in terms of the nondimensional terms. All results will be the nondimensional quantities unless otherwise stated. The eigenvalues of the system are functions of mass, length, stiffness, and inertia ratios, and relative position . 2 2 The eigenvectors are also functions of these ratios and . Fig. 2. First natural frequency of the conventional flexible robot: . The idea is that if the eigensystem changes significantly, it may be necessary to use a complicated controller to damp vibrations.
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