We derive exact dispersion relations for axial and flexural elastic wave
motion in a rod and a beam under finite deformation. For axial motion we
consider a simple rod model, and for flexural motion we employ the
Euler-Bernoulli kinematic hypothesis and consider both a conventional
transverse motion model and an inextensional planar motion model. The
underlying formulation uses the Cauchy stress and the Green-Lagrange strain.
For all models, we consider linear constitutive relations in order to isolate
the effect of finite motion. The proposed methodology, however, is applicable
to problems that also exhibit material nonlinearity. For the rod model, we
obtain the exact analytical explicit solution of the derived finite-deformation
dispersion relation, and compare it with data obtained via numerical simulation
of nonlinear wave motion in a finite rod. For the beam model, we obtain a
solution by numerical root finding. The results allow us to quantify the
deviation in the dispersion curves when exact large deformation is considered
compared to following the assumption of infinitesimal deformation. We show that
incorporation of finite deformation following the chosen definitions of stress
and strain raises the frequency branches for both axial and flexural waves
above the nominal values associated with linear motion. For the beam problem,
only the inextensional planar motion model provides an accurate description of
the finite-deformation response for both static deflection and wave dispersion.
Our findings, which represent the first derivation of finite-strain,
amplitude-dependent dispersion relations for any type of elastic media, draw
attention to 1) the tangible effect of finite deformation on wave dispersion
and consequently on the speed of sound in an elastic medium and 2) the
importance of incorporating longitudinal-transverse motion coupling in
nonlinear analysis of thin structures.Comment: Accepted for publication in Wave Motio
In this paper, we propose a new 4-legged 6-DOF redundantly actuated UPS parallel mechanism. Kinematics, singularity and dynamics analyses of the proposed mechanism are addressed. Inverse kinematics has a closed-form solution, whereas the forward kinematics has a semi-closed form solution. Also, the Jacobian matrix has been determined using the concept of reciprocal screws. Using the principle of virtual work and minimum norm method, a general formulation for the inverse dynamics of redundantly actuated parallel manipulator is presented.Moreover, the proposed redundant mechanism has been compared with a similar but nonredundant mechanism in three aspects: reachable points, and actuation forces under static and dynamic conditions. We show that the redundancy decreases singular points, and dramatically reduces the actuators' forces and torques.
Robotic reduction of long bones is associated with the need for considerable force and high precision. To balance the accuracy, payload, and workspace, we have designed a new six degrees-of-freedom three-legged wide-open robotic system for long-bone fracture reduction. Thanks to the low number of legs and their nonsymmetrical configuration, the mechanism enjoys a unique architecture with a frontally open half-plane. This facilitates positioning the leg inside the mechanism and provides a large workspace for surgical maneuvers, as shown and compared to the well-known Gough–Stewart platform. The experimental tests on a phantom reveal that the mechanism is well capable of applying the desired reduction steps against the large muscular payloads with high accuracy.
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