Resource Letter CM-1 on the Teaching of Angular Momentum and Rigid Body Motion

1965 ◽  
Vol 33 (11) ◽  
pp. 879-887
Author(s):  
John I. Shonle
Keyword(s):  
Angular Momentum ◽  
Rigid Body ◽  
Rigid Body Motion ◽  
Body Motion

A Nonclassical Vibration Analysis of a Multiple Rotating Disk and Spindle Assembly

1997 ◽  
Vol 64 (1) ◽  
pp. 165-174 ◽  
Author(s):  
I. Y. Shen ◽  
C.-P. R. Ku
Keyword(s):  
Angular Momentum ◽  
Rigid Body ◽  
Vibration Analysis ◽  
Natural Frequencies ◽  
Spindle Assembly ◽  
Rotating Disk ◽  
Rigid Body Motion ◽  
Body Motion ◽  
Vibration Modes ◽  
Nodal Diameter

This paper studies natural frequencies and mode shapes of a spinning disk/spindle assembly consisting of multiple elastic circular plates mounted on a rigid spindle that undergoes infinitesimal rigid-body translation and rotation. Through use of Lagrangian mechanics, linearized equations of motion are derived in terms of Euler angles, rigid-body translation, and elastic vibration modes of each disk. Compared with a single rotating disk whose spindle is fixed in space, the free vibration of multiple disks with rigid-body motion is significantly different in the following ways. First of all, lateral translation of the spindle, rigid-body rotation (or rocking) of the spindle, and one-nodal diameter modes of each disk are coupled together. When all the disks (say N disks) are identical, the coupled disk/spindle vibration splits into N − 1 groups of “balanced modes” and a group of “unbalanced modes.” For each group of the balanced modes, two adjacent disks vibrate entirely out of phase, while other disks undergo no deformation. Because the out-of-phase vibration does not change the angular momentum, the natural frequencies of the balanced modes are identical to those of the one-nodal-diameter modes of each disk. For the group of the unbalanced modes, all disks undergo the same out-of-plane vibration resulting in a change of angular momentum and a steady precession of the spindle. As a result, the frequencies of the unbalanced modes are significantly lower than those of one-nodal-diameter modes of each disk. Secondly, axial translation of the spindle and the axisymmetric modes of each disk are couple together. Similarly, the coupled motion split into N − 1 groups of “balanced modes” and one group of “unbalanced modes,” where the frequencies of the balanced and unbalanced modes are identical to and smaller than those of the axisymmetric modes of each disk, respectively. Thirdly, the rigid-body motion of the spindle does not affect disk vibration modes with two or more nodal diameters. Response of those modes can be determined through the classical vibration analysis of rotating disks. Moreover, vibration response of the disk/spindle assembly from a ground-based observer is derived. Finally, a calibrated experiment is conducted to validate the theoretical predictions.

A planar reconfigurable linear rigid-body motion linkage with two operation modes

2014 ◽  
Vol 228 (16) ◽  
pp. 2985-2991 ◽  
Author(s):  
Guangbo Hao ◽  
Xianwen Kong ◽  
Xiuyun He
Keyword(s):  
Rigid Body ◽  
Single Mode ◽  
Rigid Body Motion ◽  
Body Motion ◽  
Reference Guide ◽  
Revolute Joints ◽  
Operation Modes ◽  
Straight Line ◽  
Motion Stage ◽  
Motion Mode

A planar reconfigurable linear (also rectilinear) rigid-body motion linkage (RLRBML) with two operation modes, that is, linear rigid-body motion mode and lockup mode, is presented using only R (revolute) joints. The RLRBML does not require disassembly and external intervention to implement multi-task requirements. It is created via combining a Robert’s linkage and a double parallelogram linkage (with equal lengths of rocker links) arranged in parallel, which can convert a limited circular motion to a linear rigid-body motion without any reference guide way. This linear rigid-body motion is achieved since the double parallelogram linkage can guarantee the translation of the motion stage, and Robert’s linkage ensures the approximate straight line motion of its pivot joint connecting to the double parallelogram linkage. This novel RLRBML is under the linear rigid-body motion mode if the four rocker links in the double parallelogram linkage are not parallel. The motion stage is in the lockup mode if all of the four rocker links in the double parallelogram linkage are kept parallel in a tilted position (but the inner/outer two rocker links are still parallel). In the lockup mode, the motion stage of the RLRBML is prohibited from moving even under power off, but the double parallelogram linkage is still moveable for its own rotation application. It is noted that further RLRBMLs can be obtained from the above RLRBML by replacing Robert’s linkage with any other straight line motion linkage (such as Watt’s linkage). Additionally, a compact RLRBML and two single-mode linear rigid-body motion linkages are presented.

Computation of Nonlinear Response of Hydraulically Actuated Structures

1997 ◽  
Author(s):  
T. D. Burton ◽  
C. P. Baker ◽  
J. Y. Lew
Keyword(s):  
Rigid Body ◽  
Flexible Structures ◽  
Rigid Body Motion ◽  
Body Motion ◽  
Flexible Beam ◽  
Test Bed ◽  
Ode Models ◽  
Hydraulically Actuated ◽  
Pressure Dynamics ◽  
Low Order

Abstract The maneuvering and motion control of large flexible structures are often performed hydraulically. The pressure dynamics of the hydraulic subsystem and the rigid body and vibrational dynamics of the structure are fully coupled. The hydraulic subsystem pressure dynamics are strongly nonlinear, with the servovalve opening x(t) providing a parametric excitation. The rigid body and/or flexible body motions may be nonlinear as well. In order to obtain accurate ODE models of the pressure dynamics, hydraulic fluid compressibility must generally be taken into account, and this results in system ODE models which can be very stiff (even if a low order Galerkin-vibration model is used). In addition, the dependence of the pressure derivatives on the square root of pressure results in a “faster than exponential” behavior as certain limiting pressure values are approached, and this may cause further problems in the numerics, including instability. The purpose of this paper is to present an efficient strategy for numerical simulation of the response of this type of system. The main results are the following: 1) If the system has no rigid body modes and is thus “self-centered,” that is, there exists an inherent stiffening effect which tends to push the motion to a stable static equilibrium, then linearized models of the pressure dynamics work well, even for relatively large pressure excursions. This result, enabling linear system theory to be used, appears of value for design and optimization work; 2) If the system possesses a rigid body mode and is thus “non-centered,” i.e., there is no stiffness element restraining rigid body motion, then typically linearization does not work. We have, however discovered an artifice which can be introduced into the ODE model to alleviate the stiffness/instability problems; 3) in some situations an incompressible model can be used effectively to simulate quasi-steady pressure fluctuations (with care!). In addition to the aforementioned simulation aspects, we will present comparisons of the theoretical behavior with experimental histories of pressures, rigid body motion, and vibrational motion measured for the Battelle dynamics/controls test bed system: a hydraulically actuated system consisting of a long flexible beam with end mass, mounted on a hub which is rotated hydraulically. The low order ODE models predict most aspects of behavior accurately.

Structures of the Phosphazenes [ClC(NPCl3)2]+PCl6 − and [CH3C(NPCl3)2]+SbCl6 − at 90 K

1997 ◽  
Vol 53 (6) ◽  
pp. 953-960 ◽  
Author(s):  
F. Belaj
Keyword(s):  
Rigid Body ◽  
Motion Analysis ◽  
Rigid Body Motion ◽  
Body Motion ◽  
Torsion Angles ◽  
Ionic Compounds ◽  
Intramolecular Motions

The asymmetric units of both ionic compounds [N-(chloroformimidoyl)phosphorimidic trichloridato]trichlorophosphorus hexachlorophosphate, [ClC(NPCl3)2]+PCl^{-}_{6} (1), and [N-(acetimidoyl)phosphorimidic trichloridato]trichlorophosphorus hexachloroantimonate, [CH3C(NPCl3)2]+SbCl^{-}_{6} (2), contain two formula units with the atoms located on general positions. All the cations show cis–trans conformations with respect to their X—C—N—P torsion angles [X = Cl for (1), C for (2)], but quite different conformations with respect to their C—N—P—Cl torsion angles. Therefore, the two NPCl3 groups of a cation are inequivalent, even though they are equivalent in solution. The very flexible C—N—P angles ranging from 120.6 (3) to 140.9 (3)° can be attributed to the intramolecular Cl...Cl and Cl...N contacts. A widening of the C—N—P angles correlates with a shortening of the P—N distances. The rigid-body motion analysis shows that the non-rigid intramolecular motions in the cations cannot be explained by allowance for intramolecular torsion of the three rigid subunits about specific bonds.

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