S. Neogy

Stability Analysis of a Flexible Spinning and Precessing Rotor with Non-symmetric Shaft

The present work deals with stability analysis of a spinning non-symmetric shaft with a non-central disk mounted on a rotating (precessing) base, where the spin axis and the precession axis intersect at right angles. The nutation speed is zero and the spin and precession speeds are considered to be uniform. The motion of the rotor is such that it undergoes small elastic deformation superposed on rigid body rotation. The shaft-disk system is assumed to be axially and torsionally stiff. A four-degree- of-freedom model is considered for the stability analysis. A non-symmetric shaft (e.g., shaft with rectangular or elliptic cross-section, shaft with a keyway, cracked shaft etc.) of a rotor has dissimilar stiffness in two perpendicular transverse planes. The governing equations for such a rotor are expressed in the precessing but non-spinning frame. Since the governing equations of motion are found to have periodic stiffness terms, a variant of Hill’s method is adopted for stability analysis. The stability borderlines are constructed with respect to the spin speed and precession speed.

Vibration Control of a Structure and a Rotor Using One-sided Magnetic Actuator and a Digital Proportional-derivative Control

The present work aims at vibration control of a structure/rotor using a one-sided magnetic actuator. For such an actuator, the actuator force is a nonlinear function of current through the coil of the actuator and the gap between the actuator and the structure/rotor. Though the actuator introduces sufficient nonlinearity to the system, it has the advantage that, for applying force in a direction, this actuator needs to be placed in only one side of the structure/rotor. For vibration control of the structure/rotor, a simple digital proportional derivative (PD) control technique is suggested here. It is considered that the displacement signal from the sensor is sampled at specific instants of time and a series of discrete values of displacement is made available to the digital controller. The controller processes the above values of displacement to compute the values of control current that produces the values of appropriate control force at the corresponding instants. The input current to the actuator is kept constant between two consecutive sampling instants. The response of the controlled system (the structure/rotor along with the digital controller) is computed using standard time-marching algorithms (with time steps much smaller than the sampling interval). The effect of the sampling interval in the response pattern is observed for a single-degree-of-freedom (SDOF) system, a finite element model of a beam and a four degree-of-freedom model of a rotor.

Unbalance Response Analysis of a Spinning Rotor Mounted on a Precessing Platform

The present work deals with analysis of a uniformly spinning shaft with a non-central disc mounted on a rotating (precessing) base, where the spin axis and the precession axis intersect at right angle. The motion of the rotor is such that it undergoes small elastic deformation superposed on rigid body rotation about a point. It is assumed that the shaft is axially and torsionally stiff and the disc has four degrees of freedom. Due to unbalance excitation, somewhat like a rotor on an orthotropic support, this rotor has also been found to undergo backwardwhirl.

Flexible spinning and precessing rotor–stability analysis based on different analytical and finite element models

This study analyzes the elastic vibration of a simultaneously spinning and precessing cantilevered rotor for its stability margin and whirl frequency. The governing equations suggest that the stability is largely governed by two counteracting effects – the centrifugal stiffening and the precession softening. The concentrated mass and inertia of the disc as well as the distributed mass of the shaft contribute to both of these effects. A finite element formulation shows that along with the standard matrices for conventional rotor dynamic analysis, two completely new ones are obtained to account for the effect of precession. Two- and four-degrees-of-freedom models indicate that the rotor is always stable irrespective of its precession speed. But, interestingly, results from the converged finite element model show that the rotor will be unstable beyond a moderately high value of precession speed. The reason for this can be attributed to the shape of deformation of the rotor during its motion. This shape is only approximate in two- and four-degrees-of-freedom models. The Campbell diagrams computed using the four-degrees-of-freedom model and the finite element model are compared and presented.

Simulating Rotation about a Point

The present paper deals with a student friendly but general treatment of rotation about a point. The conventional treatment followed in Dynamics of machines only deals with steady state precession. Engineering Mechanics courses formulate general theory but due to presence of nonlinear differential equations, avoid their solution. Such an approach may produce less impact on the students. The authors propose to use readily available differential equation solvers and graphics facilities to numerically solve the equations and show the animated motion. Such an approach is expected to boost the interest of the students.

Demonstration of Different Beam Models by Simple Experiment

Experimental validation of various beam models is presented. The validation could form part of a course in strength of materials, mechanical vibration and finite element analysis of structures. Natural frequency is used as the key parameter. Such a presentation will also acquaint students with the various instruments used in a dynamics laboratory. However, it suggests that the finite element method, as well as other analytical or numerical tools, merely model the original structure and so reminds students of the importance of experiments.

A building block-like approach for kinematic analysis of plane mechanisms

Powerful packages based on multibody dynamics can solve virtually any dynamic system. But to the user they are black boxes. Kinematic analysis of plane mechanisms is vital to mechanical engineering. This analysis hardly requires such versatile tools. Further blind usage of these powerful tools does not permit the user to develop insight into the mechanisms. On the other hand, packages based on simple but modular approach are ideally suited for the purpose. The aim of the preset work is to rigorously define the independently solvable modules and to develop a program using these modules, on MATLAB platform, which can solve a large variety of plane mechanisms.

ADAMS as an aid for teaching undergraduate dynamics

The objective of the present work is to use multi body dynamics package (like MSC Adams) for motivating undergraduate students in a Dynamics course by means of simulations—a step which lies between traditional board work and actual experimentation. It also deepens his understanding of mechanics by allowing him to compare ADAMS solution with manual solution and solution obtained through small user developed program segments. The student learns to appreciate why the manual solutions are specialized cases of general solutions and how and why trajectories are determined. In the process it discusses the logical structure of Dynamics as implemented in these packages. Such an approach would also introduce them to such powerful tools at an early stage. It provides detailed solution of three problems using Adams and brief solutions to eight more. It is expected once these examples are mastered, most of the problems in undergraduate dynamics can be tackled using Adams.

Detection of Higher Harmonics in a Slider Crank Mechanism of Variable Obliquity Ratio and its Use in the Determination of Displacement Transmissibility

The present paper deals with the design and manufacture of a slider crank mechanism with a variable obliquity ratio. It uses a double eccentric to serve this purpose. The slider crank mechanism can be operated at various speeds and different obliquity ratios. An accelerometer attached to the slider helps obtain its acceleration. The presence of higher harmonics is detected using fast Fourier transforms. The experimentally obtained values are compared with standard theoretical results. Further, a cantilever can be fixed to the slider under displacement excitation. Accelerations measured at the root and tip of the cantilever are used to calculate displacement transmissibility. The experimentally obtained values are compared with those obtained using finite elements. It is expected that such an approach will boost the interest of the students as it bridges theory with experimental work, which is so vital for engineering education.

A Combination of a Piezoelectric Vibrator and a Permanent Magnet for Non-Contact Positive and Negative Damping of a Beam

An attempt has been made to develop a non-contact actuator consisting of a piezoelectric vibrator and a permanent magnet. In a simple control system the actuator dissipates energy and works as a damper. The actuator can also work as a negative damper, by supplying energy to the system and making it unstable. The demonstration of a negatively damped unstable system makes this experiment interesting for students.