- Finite element analysis is a sophisticated technology based on the principle of discretization and numerical approximation to solve scientific and engineering problems. In this methodology any structure under consideration is discretized into small geometric shapes and the material properties are analyzed over these small elements. This method scores over the general strength of material methods in the way that in this technique complex beam elements with differential cross sectional geometry can be analyzed quite easily. The purpose of this project is to study the simple approach of analyzing the torsional vibration in a branched geared system. The overall mass matrix, damping matrix, stiffness matrix, and load vector of the direct-transmitted system are obtained with the conventional finite element method (FEM) by assembling the elemental property matrices of all the shaft elements contained in the torsional system. Then equations of motion of the whole vibrating system are defined. Solution of the equations of motion gives the dynamic responses and solution of the associated eigenvalue equation provides the natural frequencies and the mode shapes of the system. The influence of the shaft mass on the natural frequencies of a torsional system is also studied. And thence the results obtained using Finite Element Method are compared with those obtained by the numerical method as devised by 'Holzer' or 'Transfer Matrix'.
Introduction
1.1 Finite Element Method
Finite element analysis is based on the principle of discretization and numerical approximation to solve scientific and engineering problems. In this method, a complex region defining a continuum is discretized into simple geometric shapes called the finite elements. The material properties and the governing relationships are considered over these elements and are expressed in terms of unknown elements at the corners. An assembly process duly considering the loading and constraints results in a set of equations. Solution of these equations gives the approximate behaviour of the continuum. The application of this method ranges from deformation and stress analysis of automotives, aircrafts, buildings, bridge structures to field analysis of other flow problems. With the advent of new computer technologies and CAD systems complex problems can be modeled with relative ease. Several alternative configurations can be tested on a computer before the first prototype is built. All these above suggests that we need to keep pace with these developments by understanding the basic theory, modeling techniques and computational aspects of finite element analysis.
1.1.1 Historical Background
The term finite element was first coined and used by Clough in 1960. Basic idea of Finite Element Method originated from advances in the air craft structural analysis. In early 1960s engineers used this method for approximate solutions of problems in stress analysis, fluid flow, heat transfer and other areas. By late 1950s, the key concepts of stiffness matrix and element assembly existed essentially in the form used today. A book by Argyris in 1955 on energy theorems and matrix methods laid a foundation for further development in finite element analysis was published by Zienkiwiz and Chung in 1967. NASA issued request for proposals for the development of the finite element software NASTRAN in 1965. In late 1960s and 1970s finite element analysis was applied to non-linear problems and layer deformations. The method was provided with a rigorous mathematical foundation in 1973 with the publication of Strang and Fix's 'An Analysis of The Finite Element Method' has since been generalized into a branch of applied mathematics for numerical modeling of physical systems in a wide variety of engineering disciplines, e.g., electromagnetism and fluid dynamics. Today the advent of mainframe computational techniques and powerful microcomputers has made this method within the practical applicability of industries and engineers.
1.1.2 Application
A variety of specializations under the umbrella of the mechanical engineering discipline (such as aeronautical, biomechanical, and automotive industries) commonly use integrated FEM in design and development of their products. Several modern FEM packages include specific components such as thermal, electromagnetic, fluid, and structural working environments. In a structural simulation, FEM helps tremendously in producing stiffness and strength visualizations and also in minimizing weight, materials, and costs.
FEM allows detailed visualization of where structures bend or twist, and indicates the distribution of stresses and displacements. FEM software provides a wide range of simulation options for controlling the complexity of both modeling and analysis of a system. Similarly, the desired level of accuracy required and associated computational time requirements can be managed simultaneously to address most engineering applications. FEM allows entire designs to be constructed, refined, and optimized before the design is manufactured.
This powerful design tool has significantly improved both the standard of engineering designs and the methodology of the design process in many industrial applications. The introduction of FEM has substantially decreased the time to take products from concept to the production line. It is primarily through improved initial prototype designs using FEM that testing and development have been accelerated. In summary, benefits of FEM include increased accuracy, enhanced design and better insight into critical design parameters, virtual prototyping, fewer hardware prototypes, a faster and less expensive design cycle, increased productivity, and increased revenue.
1.1.3 Fundamental Concepts
It is very difficult to make the alegebraic equations for the entire domain
Divide the domain into a number of small, simple elements
A field quantity is interpolated by a polynomial over an element
Adjacent elements share the DOF at connecting nodes
1.1.3 FEM Advantages
Can readily handle complex geometry
The heart and power of FEM
Can handle a wide variety of engineering problems
Solid and Fluid Mechanics
Dynamics
Heat Problems
Electrostatic Problems
Can handle complex structures
Indeterminate structures can be solved
Can handle complex loading
Nodal load (point load)
Element load (pressure, thermal, inertial forces)
Time or frequency dependent loading
1.2 Torsional Vibration
1.2.1 Vibration
Vibration refers to mechanical oscillations about an equilibrium point. The oscillations may be periodic such as the motion of a pendulum or random such as the movement of a tire on a gravel road. Vibration is occasionally "desirable". For example the motion of a tuning fork, the reed in a woodwind instrument or harmonica, or the cone of a loudspeaker is desirable vibration, necessary for the correct functioning of the various devices. More often, vibration is undesirable, wasting energy and creating unwanted sound - noise. For example, the vibrational motions of engines, electric motors, or any mechanical device in operation are typically unwanted. Such vibrations can be caused by imbalances in the rotating parts, uneven friction, the meshing of gear teeth, etc. Careful designs usually minimize unwanted vibrations.
Free vibration occurs when a mechanical system is set off with an initial input and then allowed to vibrate freely. Examples of this type of vibration are pulling a child back on a swing and then letting go or hitting a tuning fork and letting it ring. The mechanical system will then vibrate at one or more of its "natural frequencies" and damp down to zero.
Forced vibration is when an alternating force or motion is applied to a mechanical system. Examples of this type of vibration include a shaking washing machine due to an imbalance, transportation vibration (caused by truck engine, springs, road, etc), or the vibration of a building during an earthquake. In forced vibration the frequency of the vibration is the frequency of the force or motion applied, with order of magnitude being dependent on the actual mechanical system.
1.2.2 Torsional Vibration
Torsional vibration is angular vibration of an object-commonly a shaft along its axis of rotation. Torsional vibration is often a concern in power transmission systems using rotating shafts or couplings where it can cause failures if not controlled. In ideal power transmission systems using rotating parts the torques applied or reacted are "smooth" leading to constant speeds. In reality this is not the case. The torques generated may not be smooth (e.g., internal combustion engines) or the component being driven may not react to the torque smoothly (e.g., reciprocating compressors). Also, the components transmitting the torque can generate non-smooth or alternating torques (e.g., worn gears, misaligned shafts). Because the components in power transmission systems are not infinitely stiff these alternating torques cause vibration along the axis of rotation.
Torsional vibrations may result in shafts from following forcings:
Inertia forces of reciprocating mechanisms (such as pistons in Internal Combustion engines)
Impulsive loads occurring during a normal machine cycle (e.g. during operations of a punch press)
Shock loads applied to electrical machineries (such as a generator line fault followed by fault removal and automatic closure)
Torques related to gear tooth meshing frequencies, turbine blade passing frequencies, etc.
For machines having massive rotors and flexible shafts (where system natural frequencies of torsional vibrations may be close to, or within, the source frequency range during normal operation) torsional vibrations constitute a potential design problem area.
In such cases designers should ensure the accurate prediction of machine torsional frequencies and frequencies of any of the torsional load fluctuations should not coincide with torsional natural frequencies.
Hence, determination of torsional natural frequencies of a dynamic system is very important.
1.3 Geared Systems
A gear is a rotating machine part having cut teeth, or cogs, which mesh with another toothed part in order to transmit torque. Two or more gears working in tandem are called a transmission and can produce a mechanical advantage through a gear ratio and thus may be considered a simple machine. Geared devices can change the speed, magnitude, and direction of a power source. The most common situation is for a gear to mesh with another gear, however a gear can also mesh a non-rotating toothed part, called a rack, thereby producing translation instead of rotation. The gears in a transmission are analogous to the wheels in a pulley. An advantage of gears is that the teeth of a gear prevent slipping. When two gears of unequal number of teeth are combined a mechanical advantage is produced, with both the rotational speeds and the torques of the two gears differing in a simple relationship. In transmissions which offer multiple gear ratios, such as bicycles and cars, the term gear, as in first gear, refers to a gear ratio rather than an actual physical gear. The term is used to describe similar devices even when gear ratio is continuous rather than discrete, or when the device does not actually contain any gears, as in a continuously variable transmission.
Literature Survey
2.1 Rotating machines
Rotating machines like steam turbines, compressors, generators most probably develop excessive dynamic stresses, when they run at speeds near their natural frequencies in torsional vibration. Continuous operation of the machinery under such conditions can lead to premature fatigue failure of system components. One of the major obstacles in the measurement and subsequent detection of torsional vibration in a machine is that torsional oscillations cannot be detected without special equipment. However, prediction of torsional natural frequencies of a system and consequent design changes that avoid the torsional natural frequencies from occurring in the operating speed range of a machine is necessary. Many a time, torsional vibration produces stress reversals causing metal fatigue and gear tooth impact forces. All rigid bodies such as flywheel, inner and outer parts of a flexible coupling, a turning disk, can be considered as rigid disks, whose mass moments of inertia can be found out easily. Flexible couplings and thin shafts, whose polar mass moments of inertia are small, can be considered as massless shafts, whose torsional stiffness can be determined easily. Where the shaft diameter is large and its polar mass moment of inertia cannot be neglected as in the case of steam turbine or generator shaft, we can either consider a large number of sections and lump the inertia of each section as a rigid as a rigid disk while retaining a large number of stations and lump the inertia of each station as a rigid disk while keeping the elasticity of the shaft as in massless torsional shafts, or alternatively we can consider a distributed inertia and stiffness of the shaft between sections, where the shaft diameter is large and that its inertia cannot be ignored. Hence the system reduces to either several rigid disks connected by massless elastic shafts or distributed mass and elastic shafts. If one part of the system is coupled to another part through gears, the system interias and stiffnesses should be reduced to one reference speed. [1]
So we see that, the torsional characteristics of a system greatly depend on the stiffness and inertia in the train. While some properties of the system can be changed, generally the system inertia cannot be altered as required. Considering the case of a pump, whose inertia properties are depend upon the dimensions of its impellers, shafts, driving motors etc. Now, a change in geometries and overall sizes to effect torsional characteristics may faint the consideration of factors like the pump hydraulics and lateral vibrations. Besides, the selection of the driver, which is primarily based on the power and load requirements, can hardly be fixed based on torsional characteristics. The typical engineering objectives of torsional vibration analysis are listed below [2]:
1. Predicting the torsional natural frequencies of the system.
2. Calculating the effect of the natural frequencies and vibration amplitudes of changing one or more design parameters (for e.g. "sensitivity analysis").
3. Determining vibration amplitudes and peak torque under steady-state torsional excitation.
4. Computing the dynamic torque and gear tooth loads under transient conditions (for e.g., during machine startup).
2.2 Gear Dynamics
Gears are comprehensively used for power transmission in many engineering machines like vehicles and industrial devices. The dynamic behavior of gears has a great influence on noise and vibration of the system that gears drive. Hence, design of gear systems affects the performance of such machines significantly. Many papers have been published in the past on the effect of dynamics of gears on the response of the system [3-5]. It seems that most of the works are on vibrations of gears caused by backlash, alteration of tooth profile and eccentricities. The backlash detection and its influence in geared shafts has been investigated by N Sarkar et al [6]. Ambili and A Fregolent [7] found out the modal parameters of spur gear system using Harmonic Balance Method. Studying of design of compact spur gears including the effects of tooth stress and dynamic response was carried by PH Lin et al [4]. Non-linear behavior of gear system with backlash and varying stiffness was studied by S. Nastivas & S. Theodassiadas [8]. The study of modal analysis of compliant multi-body geared systems done by H. Vinayak and R. Singh [9]. The effect of time-variant meshing stiffness and non-uniform gear speed on dynamic performance using Finite Elements was analysed by Y. Wang et al [10]. A gear system can be seen as a system of rotors interacting with one another dynamically. Therefore, linear as well as gyroscopic whirling phenomena are expected to exist in addition to vibrations caused by tooth, bearings and shaft interactions. The rotor effect is considered negligible in the gear systems with bearing supports in both ends because design can be made to minimize lateral deflections; however can be a significant effect in overhung gears such as seen in hypoid gear systems. A dynamic model for geared multi-body system containing gear, bar and shaft was proposed and a new gear element, particularly developed based on a Finite element theory by Yong Wang et al. [11]
2.3 Methods of finding Natural Frequencies
The various methods are:
Dunkerly's Equation
Gives good results if damping is negligible and the frequencies of the harmonics are much higher than the fundamentals.
Rayleigh Method
Here, the dynamic mode shape or modal vector is assumed to estimate the natural frequency.
Holzer's Method
This method assumes a trial frequency. A solution is found when the assumed frequency satisfies the constraints of the problem using a systematic tabulation of frequency equations.
Transfer Matrix
Here, the concept of state vector and transfer matrices are applied to the technique of Holzer's Method.
Finite Element Method
An eigenvalue equation is formed using elemental stiffness matrices (using displacement or direct stiffness method) and mass matrices (using Lagrange's Equations.
For this project the Finite Element Method is used mainly to determine the natural frequencies and mode shapes and then the results hence obtained were compared using those calculated using Holzer's Method. So now, the Holzer's Method and FEM are explained in detail.
2.4 Holzer's Method
Holzer method is basicaly a systematic tabulation of the frequency equations of the system. The method has general applications, spanning systems with rectilinear and angular motions, damped or undamped, un-branched or branched. Here, a trial frequency is assumed. When the assumed frequency satisfies the constraints
