Kinematic joints are classified in lower pairs and higher pairs. Most multibody modelling techniques focus on lower pairs, because a complete classification in six types is available. Higher pairs are more diverse. In this article, higher pairs that can be exactly modelled by lower pairs are investigated. A complete classification of higher pairs that can be modelled by a chain of five single-degree-of-freedom lower pairs with a central revolute joint at the contact point is proposed. Two-dimensional cases and surfaces with discontinuities are also considered. The equivalent chains can be used for exact and approximate modelling of higher pairs and as design alternatives. Illustrative examples and applications to a bicycle on toroidal wheels and a railway wheelset on a roller rig are shown.

Background: The steep radiation dose gradients in cervical cancer brachytherapy (BT) necessitate a thorough understanding of the behavior of afterloader source cables or needles in the curved channels of (patient-tailored) applicators. Purpose: The purpose of this study is to develop and validate computer models to simulate: (1) BT source positions, and (2) insertion forces of needles in curved applicator channels. The methodology presented can be used to improve the knowledge of instrument behavior in current applicators and aid the development of novel (3D-printed) BT applicators. Methods: For the computer models, BT instruments were discretized in finite elements. Simulations were performed in SPACAR by formulating nodal contact force and motion input models and specifying the instruments’ kinematic and dynamic properties. To evaluate the source cable model, simulated source paths in ring applicators were compared with manufacturer-measured source paths. The impact of discrepancies on the dosimetry was estimated for standard plans. To validate needle models, simulated needle insertion forces in curved channels with varying curvature, torsion, and clearance, were compared with force measurements in dedicated 3D-printed templates. Results: Comparison of simulated with manufacturer-measured source positions showed 0.5–1.2 mm median and <2.0 mm maximum differences, in all but one applicator geometry. The resulting maximum relative dose differences at the lateral surface and at 5 mm depth were 5.5% and 4.7%, respectively. Simulated insertion forces for BT needles in curved channels accurately resembled the forces experimentally obtained by including experimental uncertainties in the simulation. Conclusion: The models developed can accurately predict source positions and insertion forces in BT applicators. Insights from these models can aid novel applicator design with improved motion and force transmission of BT instruments, and contribute to the estimation of overall treatment precision. The methodology presented can be extended to study other applicator geometries, flexible instruments, and afterloading systems.

Humans vary the stiffness in their joints depending on tasks and circumstances. For posture control a high joint stiffness is required to withstand perturbations, whereas for force control a low joint stiffness is required. To investigate how humans vary their joint stiffness precisely for moving an arm, a wearable device is needed that can generate small force perturbations at the wrist while measuring the resulting muscular reactions. The majority of the state-of-the-art devices either offer too little versatility or impede the free movement of the arm. Based on a 3-DoF spatial redundant 4-RUU parallel manipulator applied in an inverted way where the original base with actuators has become the moving platform and the original moving platform is attached to the wrist as a bracelet, a versatile, 0.175 kg lightweight, low impedance, and compact wearable device was developed that can generate perturbation forces in X-, Y-, and Z-direction. The design and a prototype of the device are presented with experimental tests showing controlled perturbations in the order of 4 N with frequencies up to 12 Hz.

The dynamic balancing of flexible mechanisms, that is, the reduction or elimination of shaking forces and shaking moments on the support structure, is considered. Two approaches are pursued: one uses similarity and the other modal balancing. A single rotating link can be balanced by a properly scaled countermass, for which balancing criteria are given. If all balancing conditions are satisfied, shaking force balance can even be achieved if geometric non-linearities are taken into account. This single link can be extended to a translator. Owing to the unscaled pitch of the translator and the asymmetric driving motor, no perfect shaking force balance is achieved, but the results can be considered satisfactory. Then, the dynamic balancing of a four-bar mechanism with a flexible coupler by means of modal balancing is shown. If the coupler is supported at the nodes of the first free vibration mode, this mode can be suppressed in the shaking force and shaking moment response. By supporting the coupler at four points with two whippletree mechanisms, the contribution of the first three symmetric vibration modes can be significantly reduced.

The use of principal points and principal vectors in the formulation of the equations of motion of a general 4R planar four-bar linkage is shown with two kinds of methods, one that opens kinematic loops and one that does not. The opened kinematic loop approach analyses the moving links as a system with a tree connectivity, introducing reaction forces for closing the loops. Compared with the conventional Newton–Euler method, this approach results in fewer equations and constraint forces, whereas the mass matrix entries remain meaningful, but there is a stronger coupling between the equations. Two equivalent mass formulations for the closed kinematic loop approach are presented, which need not open the loop and introduce loop constraint forces in the equations of motion. With the method of complex joint masses, the mass of the links closing the loops is represented by real and virtual equivalent masses, defining the principal points. The principle of virtual work with the inclusion of inertia terms is used to derive the equations of motion. As an example the dynamic balance conditions of the four-bar linkage are derived. With the method of the equivalent mass matrix it is shown how a constant mass matrix can be used to describe the dynamics of binary links with an arbitrary mass distribution. A four-bar linkage could be modelled with only three truss elements instead of the conventional result with three or more beam elements, which gives a significant reduction of the computational complexity.

This paper is a first approach in finding design principles for the design of shaking force balanced compliant mechanisms. Shaking force balance means that the motions of the mechanism do not create any resultant dynamic reaction forces on the base, eliminating base vibrations.It is found that for a single balanced rotatable flexible link two stiffness related balance conditions exist in addition to the balance condition known for a rigid link. With these conditions the shaking force balance of a planar parallelogram mechanism with flexible links is considered. The case with fully compliant hinges is applied to a planar translator and the results are compared with the case in which the hinges are real revolute joints. Simulations show perfect force balance for the model with revolute joints and a reduced shaking force of 67% for the model with flexible joints. Prototypes of both mechanisms were developed and experimentally tested, showing shaking force reductions of 93% and 97.5%, respectively.

For the Delta robot, a high-speed parallel pick-and-place manipulator, base vibrations are a significant problem. Especially since the Delta robot is suspended above its workpiece, it requires a large, stiff, and heavy base frame for fast and accurate motions. Dynamic balancing of the shaking forces and the shaking moments is a known technique to reduce the dynamic loads on the base frame and to the surroundings. In this paper it is investigated how solely with partial force balancing, dynamic loads and pick-and-place accuracy of a Delta robot-like manipulator can be improved, considering also the compliance of the base frame. This is done since partial force balance solutions can be implemented relatively simply in the current Delta robot designs, whereas full force and moment balance solutions are complex to apply in practice. Numerical simulations with a representative planar model of a Delta robot-like manipulator with a compliant base frame show that with an increasing amount of force balance the shaking moments increase up to 16% for full force balance. The floor contact forces first reduce and then increase with increasing force balance. With 43% force balance the floor contact forces are minimal, giving a 63% reduction. The end-effector accuracy slightly improves with increasing force balance until full force balance yields a 31% accuracy improvement. A further increase of the force (over) balance shows a 59% improvement of end-effector accuracy for 350% force balance. These effects are mainly due to the typical design of the Delta robot base frame and the way the robot is mounted to it.

The motion of a four-bar linkage is considered with the goal to study the use of principal vectors to formulate the equations of motion and to get insight. Firstly, kinematic relations for the positions, velocities and accelerations are derived. Then, the motion of the centre of mass of the system is described with the aid of principal points and principal vectors, for which the mass of one link is replaced with equivalent masses. The condition of dynamic force balance is that the centre of mass is stationary. It is shown that the motion of the centres of mass of the links can be described in terms of the principal vectors. The equations of motion and the expressions for the force and moment on the base are derived with the aid of the principle of virtual work, which directly give conditions for dynamic force and moment balance. The equations of motion show a clear structure in their coefficients. The expression for the reaction force becomes simple, but the expression for the reaction moment remains rather complicated.

Three formulations for a flexible spatial beam element for dynamic analysis are compared: a Timoshenko beam with large displacements and rotations, a fully parametrized element according to the absolute nodal coordinate formulation (ANCF), and an ANCF element based on an elastic line approach. In the last formulation, the shear locking of the antisymmetric bending mode is avoided by the application of either the two-field Hellinger-Reissner or the three-field Hu-Washizu variational principle. The comparison is made by means of linear static deflection and eigenfrequency analyses on stylized problems. It is shown that the ANCF fully parametrized element yields too large torsional and flexural rigidities, and shear locking effectively suppresses the antisymmetric bending mode. The presented ANCF formulation with the elastic line approach resolves most of these problems.

In this paper, an experimental validation of some modelling aspects of an uncontrolled bicycle is presented. In numerical models, many physical aspects of the real bicycle are considered negligible, such as the flexibility of the frame and wheels, play in the bearings, and precise tire characteristics. The admissibility of these assumptions has been checked by comparing experimental results with numerical simulation results. The numerical simulations were performed on a three-degree-of-freedom benchmarked bicycle model. For the validation we considered the linearized equations of motion for small perturbations of the upright steady forward motion. The most dubious assumption that was validated in this model was the replacement of the tires by knife-edge wheels rolling without slipping (non-holonomic constraints). The experimental system consisted of an instrumented bicycle without rider. Sensors were present for measuring the roll rate, yaw rate, steering angle, and rear wheel rotation. Measurements were recorded for the case in which the bicycle coasted freely on a level surface. From these measured data, eigenvalues were extracted by means of curve fitting. These eigenvalues were then compared with the results from the linearized equations of motion of the model. As a result, the model appeared to be fairly accurate for the low-speed low-frequency behaviour.

In the case of small elastic deformations in a exible multi- body system, the periodic motion of the system can be modelled as a superposition of a small linear vibration and a non-linear rigid body motion. For the small deformations this analysis re-sults in a set of linear dierential equations with periodic co-ecients. These equations give more insight in the vibration phenomena and are computationally more ecient than a direct non-linear analysis by numeric integration. The realization of the method in a program for exible multibody systems is discussed which requires, besides the determination of the periodic rigid motion, the determination of the linearized equations of motion. The periodic solutions for the linear equations are determined with a harmonic balance method, while transient solutions are obtained by averaging. The stability of the periodic solution is considered. The method is applied to a pendulum with a circular motion of its support pointand a slider-crank mechanism with exible connecting rod. A comparison is made with previous non-linear results.