Publications

Comparison of pure sinusoidal vibration to random vibration or combinations of the two is an important and useful subject for dynamic testing. The objective of this chapter is to succinctly document the technical background for converting a sine-sweep test specification into an equivalent random vibration test specification. The information can also be used in reverse, i.e., to compare a random vibe spec with a sine-sweep, although that is less common in practice. Because of inherent assumptions involved in such conversions, it is always preferable to test to original specifications and conduct this conversion when other options are impractical. This chapter outlines the theoretical premise and relevant equations. An example of implementation with hypothetical but realistic data is provided that captures the conversion of a sinusoid to an equivalent ASD. The example also demonstrates how to account for the rate of sine-sweep to the duration of the random vibration. A significant content of this chapter is the discussion on the statistical distribution of peaks in a narrow-band random signal and the consequences of that on the damage imparted to a structure. Numerical simulations were carried out to capture the effect of various combinations of narrow-band random and pure sinusoid superimposed on each other. The consequences of this are captured to provide guidance on accuracy and conservatism.

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Imposing a boundary condition on a structure can significantly alter its dynamic properties. However, sometimes the specifics of the new boundary conditions are not known. When the effects of a boundary condition are uncertain or there is not enough information, engineers need to excite the complex structure to obtain these modified properties. In order to experimentally obtain the new properties, engineers need multiple experiments and many outputs for interpolation in order to sufficiently represent the entire structure. The researchers attached a stinger to a cantilever beam, acting as a new transverse restraint of unknown properties. This paper presents a conversion expression that predicts the dynamic behavior of any point in the system with the new boundary condition. This expression relies only on one impact hammer experiment with one output and the model of the stinger-free cantilever beam, referred to as the simple structure. Researchers estimated the Transfer Function (TRF) of the beam and compared it with an experimentally measured TRF to validate the method. The mean absolute error of the estimated TRF compared to the experimental TRF is 1.99 dB. This demonstrates the use of the proposed method for estimating unmeasured TRFs in a system with an uncertain boundary condition using a single input, single output (SISO) test and a model of the simple structure.

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Multi-Input-Multi-Output (MIMO) vibration testing is considered more representative of the true loading environment (flight or wind induced vibration) where the inputs are not through a single point. The derivation of N inputs for testing typically involves matching the response at M locations (outputs). This involves inversion of a N × M Transfer Functions (TRF) matrix corresponding to the N input and M output locations. The matrix inversion is affected by both mathematical and physical parameters (ill-conditioned matrix, structural modes, signal noise). Tikhonov regularization is commonly used in inverting an ill-conditioned N × M matrix. A low value of the Tikhonov regularization parameter minimizes the distortion of the original equations while a higher value can minimize error. In practice this introduces an interesting dilemma where obtaining realistic input loads and maintaining accuracy of output are often pitted against each other. A study was conducted using data synthesized from a simply-supported plate structure with known vibration modes with added noise at outputs. The objective of the study was to understand how noise or errors in the output and the Transfer function affect the input. This leads to a more judicious choice of the Tikhonov parameter that can achieve a balance between reducing input loads while preserving desired accuracy of output vibration.

Interpretation of field data from shock tests and subsequent assessment of product safety margins via laboratory testing are based on the shock response spectra (SRS). The SRS capture how a single degree of freedom (SDOF) structure responds to the shock at differing frequencies and, therefore, no longer contain the duration or other temporal parameters pertaining to the shock. A single duration can often be included in the technical specification or in the recreation of acceleration vs. time history from the specified SRS; however, there is little basis for that beyond technical judgment. The loss of such temporal information can result in the recreated SRS being the same while its effect on a system or component can be different. This paper attempts to quantify this deficiency as well as propose a simple method of capturing damping from shock waves that can allow the original waveform to be more accurately reconstructed from the SRS. In this study the decay rate associated with various frequencies that comprise the overall shock was varied. This variation in the decay rate leads to a variation in the acceleration vs. time history, which can be correlated to a “Damage Index” that captures the fatigue damage imparted to the object under shock. Several waveforms that have the same SRS but varying rates of decay for either high- or low-frequency components of the shock were investigated. The resulting variation in stress cycles and Damage Index is discussed in the context of the lognormal distribution of fatigue failure data. It is proposed that, along with the SRS, the decay rate is also captured to minimize the discrepancy between field data and representative laboratory tests.

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