Static Balance Conditions Research Articles

Exact solutions to the motion of two charged particles in lineal gravity

We extend the canonical formalism for the motion of N particles in lineal gravity to include charges. Under suitable coordinate conditions and boundary conditions the Hamiltonian is defined as the spatial integral of the second derivative of the dilaton field which is given as a solution to the constraint equations. For a system of two particles the determining equation of the Hamiltonian (a kind of transcendental equation) is derived from the matching conditions for the dilaton field at the particles' position. The canonical equations of motion are derived from this determining equation. For the equal mass case the canonical equations in terms of the proper time can be exactly solved in terms of hyperbolic and/or trigonometric functions. In electrodynamics with zero cosmological constant the trajectories for repulsive charges exhibit not only bounded motion but also a countably infinite series of unbounded motions for a fixed value of the total energy H0, while for attractive charges the trajectories are simply periodic. When the cosmological constant Λ is introduced, the motion for a given Λ and H0 is classified in terms of the charge-momentum diagram from which we can predict what type of the motion is realized for a given charge. In general the cosmological constant acts on the particles as a repulsive (Λ>0) or an attractive (Λ<0) potential. But for a certain range of Λ<0, small q and small mass the trajectory shows a double peak structure due to an interplay among the induced momentum dependent Λ potential, the gravitational attraction and the relativistic effect. For Λ>0, depending on the value of charge, only bounded motion or the infinite series of unbounded motion or both are realized. Since in this theory the charge of each particle appears in the form e1e2 in the determining equation, the static balance condition in 1+1 dimensions turns out to be identical with the condition in Newtonian theory. We generalize this condition to nonzero momenta, obtaining the first exact solution to the static balance problem that does not obey the Majumdar–Papapetrou condition.

Exactcharged two-body motion and the static balance condition inlineal gravity

We find an exact solution to the chargedtwo-body problem in (1+1)-dimensional lineal gravity whichprovides the first example of a relativistic system that generalizesthe Majumdar-Papapetrou condition for static balance.

Static balancing of spatial six-degree-of-freedom parallel mechanisms with revolute actuators

The static balancing of spatial 6-degree-of-freedom parallel mechanisms or manipulators with revolute actuators is studied in this article. Two static balancing methods, namely, using counterweights and using springs, are used. The first method leads to mechanisms with a stationary global center of mass while the second approach leads to mechanisms whose total potential energy (including the elastic potential energy stored in the springs as well as the gravitational potential energy) is constant. The position vector of the global center of mass and the total potential energy of the manipulator are first expressed as functions of the position and orientation of the platform. Then, conditions for static balancing are derived from the resulting expressions. Finally, examples are given to illustrate the design methodologies. © 2000 John Wiley & Sons, Inc.

Static balancing of spatial four-degree-of-freedom parallel mechanisms

The static balancing of four types of spatial four-degree-of-freedom parallel mechanisms or manipulators is discussed in this paper. Two static balancing methods, namely, using counterweights and using springs, are used. The first method leads to a mechanism with a stationary global center of mass and the second one leads to a mechanism with constant total potential energy (including the elastic potential energy stored in the springs). In both cases, the resulting mechanisms can be brought to static equilibrium in any configuration of their workspace with zero actuator forces or torques. In order to obtain the balancing conditions, the position vector of the global center of mass and the total potential energy of the mechanisms are first expressed as functions of the position and orientation of the platform and of the special fifth leg. Then, the conditions for static balancing are derived from the expressions obtained. Examples are given to illustrate the application of the results to the design of statically balanced spatial four-dof mechanisms or manipulators.

Static balancing of 3-DOF planar parallel mechanisms

The static balancing of planar 3-DOF parallel mechanisms is addressed in this paper. Static balancing is defined as the set of conditions on mechanism dimensional and inertial parameters which, when satisfied, ensure that the weight of the links does not produce any torque (or force) at the actuators for any configuration of the mechanism, under static conditions. For the mechanisms studied here, conditions for static balancing are obtained, and it is shown that balancing is generally possible, even when the dimensional parameters are imposed, which is a useful property since dimensional parameters are usually obtained from kinematic design or optimization. Then, the conditions for the static balancing of the same mechanisms are derived for designs in which elastic elements are included. Finally, examples of balanced mechanisms are given. A dynamic study is performed.

Synthesis and dynamics of statically balanced direct-drive manipulators with decoupled inertia tensors

This paper addresses conceptual design of direct-drive manipulators which have good promise for high speed, high precision manipulation. In the design methodology presented, the procedure begins by considering the kinematic aspects and ends by configuring manipulator structures with promising kinematic and dynamic characteristics. Based on the conceptual design considerations, two novel manipulator designs, a 2 and 3 DOF manipulator, are proposed and analyzed. Conditions for static balancing and decoupled invariant dynamics for the 2 DOF manipulator are derived. Design guidelines are also derived for static balance of the 3 DOF manipulator and for achieving decoupled and nearly invariant inertia tensor (two constant diagonal elements out of three). The two designs have the advantage over the currently known direct-drive manipulators of achieving both of the two desirable mechanical features, namely: static balancing and decoupled inertia tensor.

Feedback Balanced Code for Multilevel PCM Transmission

This paper describes a new code format called the feedback balanced code (FBC) for multilevel pulse code modulation (PCM) transmission, which is easily implemented and is capable of efficient utilization of transmission cable. The FBC is a type of multilevel code which has no dc wander and is obtained by the polarity control of the output pulse sequence using negative feedback. The principle of the FBC and its static balance condition are discussed, and spectral analysis using transition probability is described. FBCs are shown to have spectral nulls at dc and at frequency f <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">o</inf> . Suppression of the low-frequency component is also represented by a transfer function of the FBC converter by employing an appropriate assumption of linearization. The suppression characteristic is much improved by using a double integration feedback network. A higher order FBC is also discussed, which has spectral nulls at frequency <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">f_{o}/n</tex> and its integral multiples. The preceding theoretic approach was well verified through experiments.