Martin W. Trethewey

Finite element analysis of elastic beams subjected to moving dynamic loads

Abstract A method for the dynamic analysis of elastic beams subjected to dynamic loads induced by the arbitrary movement of a spring-mass-damper system is presented. The governing equations for the interaction between the beam and the moving dynamic system are derived, based on a finite element formulation. This set of equations is a system of second order differential equations with time dependent coefficients. The governing equations are solved with a Runge-Kutta integration scheme to obtain the dynamic response for both the support beam and the moving system. The method is capable of handling any time dependent dynamic system motion profile with complex boundary conditions in a computationally efficient fashion. Comparison of results with several simplified test conditions previously reported shows excellent agreement. The analysis is applied to a high-speed machining operation to demonstrate the unique capabilities and characteristics of the method.

Discretization considerations in moving load finite element beam models

Abstract This paper investigates continuum discretization for finite element models analyzing a moving load on an elastic beam. Moving load analysis is shown to require accurate evaluation of beam deformations over the entire length of an element, and not only the nodes. Model accuracy is shown to be related to element interpolation which in turn directly affects three aspects of the moving load finite element model; (1) calculation of equivalent nodal reactions for the moving load; (2) calculation of transmitted forces from a moving sprung mass, and; (3) system responses for a moving load initially positioned within the support beam span. The analysis indicates that the model accuracy can be maintained at an acceptable level provided that, in general, the number of elements used to discretize the support structure continuum is at least two to eight times greater than the number used in static analysis.

A model to predict minimum required clamp pre-loads in light of fixture–workpiece compliance

Abstract The determination of minimum required clamp pre-loads is an important process in the design of machining fixtures. This paper presents a linear, clamp pre-load (LCPL) model that can be applied to fixture–workpiece systems whose compliance is load invariant. The model considers the static deformation of the fixture–workpiece system in response to the clamping process and the machining process. Sources of compliance throughout a fixture–workpiece system are considered. The model computes the minimum required pre-loads necessary to prevent workpiece slip at the fixture–workpiece joints throughout the machining process. This paper also describes an experimental study that was used to characterize the accuracy of the LCPL model with regard to the application of a ramping external load to a fixture–workpiece system. Over the contact conditions tested, the LCPL model was observed to overestimate the minimum required clamp pre-loads by an average of 7%. This experimental study also revealed the sensitivity of the computed pre-loads to the relative compliance of the fixture elements as well as the coefficient of friction.

Using torsional vibration analysis as a synergistic method for crack detection in rotating equipment

A non-intrusive torsional vibration method for monitoring and tracking small changes in crack growth of reactor coolant pump shafts is presented in this paper. This method resolves and tracks characteristic changes in the natural torsional vibration frequencies that are associated with shaft crack propagation. The focus of this effort is to develop and apply the torsional vibration shaft cracking monitoring technique on a Westinghouse 93A reactor coolant pump. While this technique is being applied to reactor coolant pumps, it is generally applicable to many types of rotating equipment, including centrifugal charging pumps, condensate and feed water pumps, and may be used to detect and track changes in blade natural frequencies in gas or steam turbines. A laboratory scale rotor test bed was developed to investigate shaft cracking detection techniques under controlled conditions. The test bed provides a mechanism to evaluate sensing technologies and algorithm development. For accurate knowledge of the crack characteristics (crack depth and front), a sample shaft was seeded with a crack that was propagated using a three-point bending process. Following each crack growth step, the specimen was evaluated using ultrasonic inspection techniques for crack characterization. After inspection, the shaft was inserted in the rotor test bed for analysis and to track changes in shaft torsional vibration features. The torsional vibration measurement method has demonstrated the ability to reliably detect changes in the first natural shaft frequency in the range of 0.1 to 0.2 Hz. This technique shows the potential to enable online structural health diagnostics and ultimately the prevention of shaft or even possibly blade failure due to crack growth.

Finite element analysis of an elastic beam structure subjected to a moving distributed mass train

This paper investigates the finite element analysis of an elastic beam structure subjected to a moving distributed load. The equations of motion are derived for a moving distributed mass train, of some mass per unit length, which is assumed to always remain in contact with the beam support structure. The finite element formulation is verified for several classes of moving load problems (i.e. a point force, a point mass, and a continuous load). Results for the proposed formulation are shown to be in excellent agreement with those available from the literature for specialised cases (i.e. a simply supported beam subjected to a moving point force, moving point mass, and moving continuous force train traveling at a constant velocity). A parametric evaluation of elastic beams subjected to a moving distributed mass train is performed. The effects of velocity, mass and train length on the beams dynamic response are evaluated with respect to the beam stiffness and mass. Impact factors at the beam center are shown to increase with the velocity and mass of the moving mass train. The impact factors are also more sensitive to velocity than the mass distribution.

An experimental evaluation of coefficients of static friction of common workpiece-fixture element pairs

Abstract Most machining fixtures utilize clamping forces and friction at fixture–workpiece joints to help prevent the workpiece from slipping out of the fixture during machining. The magnitudes of the clamping forces required are a direct function of the coefficients of static friction at the joints. Recently, analytical methods have been developed to predict minimum clamping forces. However, these methods require accurate estimates of the friction coefficients. One source of friction data are handbooks. However, these data are typically listed relative to the materials of the contacting elements and are otherwise completely generalized. This paper will illustrate that the coefficient of static friction for typical fixture–workpiece joints is not a simple function of the workpiece materials. Instead it is also a function of factors such as fixture element geometry, workpiece surface topography, clamping forces, the presence or absence of cutting fluids, and normal joint rigidity.

Blade and Shaft Crack Detection Using Torsional Vibration Measurements Part 1: Feasibility Studies

The primary goal of the development project was to demonstrate the feasibility of detecting changes in blade natural frequencies (such as those associated with a blade crack) on a turbine using non-contact, non-intrusive measurement methods. The approach was to set up a small experimental apparatus, develop a torsional vibration detection system, and maximize the dynamic range and the signal to noise ratio. The results of the testing and analysis clearly demonstrated the feasibility of using torsional vibration to detect the change in natural frequency of a blade due to a change in stiffness such as those associated with a blade crack.

Blade and Shaft Crack Detection Using Torsional Vibration Measurements Part 3: Field Application Demonstrations

The primary goal of the this paper is to summarize field demonstrations of the feasibility of detecting changes in blade and shaft natural frequencies (such as those associated with a blade or shaft crack) on operating machinery using non-contact, non- intrusive measurement methods. Laboratory demonstration of feasibility and some special signal processing issues were addressed in Parts 1 and 2 [1, 2]. Part 3 primarily addresses the results of application of this non-intrusive torsional vibration sensing to: a large wind tunnel fan; a jet engine high-pressure disk; a hydro station turbine; and to a large coal- fired power plant induced-draft (ID) fan motors. During the operation of rotating equipment, torsional natural frequencies are excited by turbulence, friction, and other random forces. Laboratory testing was conducted to affirm the potential of this method for diagnostics and prognostics of blade and shafting systems. Field installation at the NASA Ames National Full-Scale Aerodynamic Facility (NFAC) reaffirmed the ability to detect both shaft and blade modes. Installation on a high-pressure (HP) disk in a jet engine test cell at General Electric Aircraft Engines demonstrated that the fundamental mode of the turbine blades was clearly visible during operation. Field installation at a hydro power station demonstrated that the first few shaft natural frequencies were visible, and correlated well with finite element results. Finally, field installation on the ID fan motors also showed the first few shaft torsional modes. These field tests have resulted in high confidence in the feasibility of the application of this technique for diagnosing and tracking shaft and blade cracks in operating machinery.

Blade and Shaft Crack Detection Using Torsional Vibration Measurements Part 2: Resampling to Improve Effective Dynamic Range

The primary goal of the development project was to demonstrate the feasibility of detecting changes in blade bending natural frequencies (such as those associated with a blade crack) on a turbine using non-contact, non-intrusive measurement methods. The approach was to set up a small experimental apparatus, develop a torsional vibration detection system, and maximize the dynamic range and the signal to noise ratio. The results of the testing and analysis clearly demonstrated the feasibility of using torsional vibration to detect the change in natural frequency of a blade due to a change in stiffness such as those associated with a blade crack. However, it was found that harmonics of shaft operating speed, created as an unwanted artifact of the measurement method, resulted in spectral regions in which the effective dynamic range was inadequate to detect low-level torsional vibration associated with the natural frequencies. The loss of effective dynamic range was due to the “skirts” created by the sampling window. Application of order resampling, followed by frequency resampling, to the torsional vibration waveform increased the effective dynamic range and improved the ability to identify shaft torsional and blade bending natural frequencies.

A Non-Intrusive Technique f or On-Line Shaft Crack Detection a nd Tracking

A non-intrusive torsional vibration method for monitoring and tracking small changes in crack growth of shafts is presented in this paper. This method resolves and tracks characteristic changes in the natural torsional vibration frequencies that are associated with shaft crack propagation. While this technique is being applied to reactor coolant pumps (RCPs) it is generally applicable to any type of rotating equipment, including drivelines, and can be applied to detecting and tracking changes in blade natural frequencies in gas or steam turbines. This technique was first developed on a laboratory scale rotor test bed to investigate shaft cracking detection techniques under controlled conditions. The test bed provided a mechanism to evaluate sensing technologies and algorithm development. For accurate knowledge of the crack characteristics (crack depth and front), a shaft was seeded with a crack which was then propagated using a three-point bending process. Following each crack growth step, the specimen was evaluated using ultrasonic inspection techniques for crack characterization. After inspection, the shaft was inserted in the rotor test bed for analysis of shaft torsional vibration features. Following success in detecting and tracking crack growth on the test bed, this process was then take to a much bigger machine for verification. In the summer of 2004, the Applied Research Laboratory, along with other EPRI team members (Southern Co. and Jeumont Industrie), instrumented a 41% scale reactor coolant pump in Jeumont, France. On this platform, the team successfully detected and tracked a seeded cut through the shaft. The torsional vibration measurement method has demonstrated the ability to reliably detect changes in the first natural shaft frequency in the range of 0.1 to 0.2 Hz. This technique shows the potential to enable online structural health diagnostics and ultimately prevent shaft or even possibly blade failure due to crack growth