208 R.L. Mayes and D.W. Linehan k Subscript for mode number TA Subscript for the test article 21.1 Motivation and Application Sandia National Laboratories has proposed a fatigue-damage metric based on the cumulative dissipated energy in a linear superposition of single-degree-of-freedom (SDoF) modal models as part of a framework for predicting failure of components subjected to random vibration loading. Edwards [1] showed how this method can predict the total energy absorbed by the test article in a random vibration environment. In order to utilize the proposed framework, the effective mass, fixed base natural frequency and damping is required for the SDoF modal models. Usually effective mass is obtained from finite element model (FEM) calculations, but in many cases, a FEM may not exist. In addition, the FEM may not be verified. It is desirable to have an experimental method to calculate the fixed base natural frequency, damping and the effective mass of each of the lower modes of the component. If the test article is available, it can be mounted on a fixture and a free modal test performed to extract parameters that can be utilized to calculate effective mass as shown by Mayes et al. [2]. In their work they showed that effective mass could be measured for the first ten modes in one direction within about 4 % of the test article mass. Their work addressed a test article with a mass of 72 kg for modes from about 35 to 1,350 Hz. Here we wish to extract the effective mass for a circuit board with a mass of 42.41 g and modal frequencies from 130 to 2,300 Hz. In order to establish the uncertainty of the method for this class of test article, the effective mass experiment and calculation is applied to a “truth” structure which is a uniform nylon plate with the same length and width and almost the same weight as the circuit board. Both will be attached to an aluminum plate fixture with two posts to which the circuit board (or truth structure) is attached. One can see the circuit board as well as the truth structure attached to the fixture in Fig. 21.1. Since the truth test article is relatively simple, a FEM of the truth structure was generated to calculate effective mass. The “truth” effective mass in the out of plane direction is calculated, and the test effective mass will be compared to that to quantify uncertainty of the effective mass from the test approach. Finally, the test effective mass will be extracted for the real circuit board and the uncertainty is assumed to be the same as derived from the truth test. 21.2 Effective Mass Concept and History The effective mass offers a physical interpretation of a physical system with multiple modes of vibration being excited dynamically from a base, similar to testing that occurs for many systems. The concept was proposed in the early 1970s by Bamford [3] with others. For a base excited system, it is represented as attached to a massless base, which will be excited in only one direction with acceleration, Rx, with each mode represented by a single degree of freedom oscillator as shown in Fig. 21.2. The mass of each oscillator is valued so that it applies the same force to the base as the real system. The springs are scaled so that the mass vibrates at the appropriate modal frequency. In general, only the modes that have the significant Fig. 21.1 Circuit board and fixture (left), truth plate and fixture (right)
RkJQdWJsaXNoZXIy MTMzNzEzMQ==