Sine on Random Analysis: Alternatives and Challenges

All frequency domain representations of environments include assumptions about the nature of the environment. Analysis methods which include these frequency domain loads also apply these assumptions. In combining loads, it is important that the analysis does not violate the assumptions used in defining the loads. This is the basic difficulty in combining sine with random or shock loads in any analysis. This paper addresses the analysis challenges and pitfalls of creating a combined loading environment. An equivalent transient approach is demonstrated on a generic payload designed for captive carry by a helicopter using loads from Mil-Std-810. INTRODUCTION AND BACKGROUND Requirements for a product being designed or delivered often include shock and vibration environments to which the product will be subjected. These environments are usually supplied as frequency domain definitions that are applicable to standardized tests, such as sine dwell/sweep, random vibration, or shock response. These frequency domain definitions may be the only available definition of the actual environment. Analysis methods have been developed and are in common use for each of these excitation types. All frequency domain representations of environments include assumptions about the nature of the environment; the real environments seldom exactly match those assumptions. For example, a helicopter environment includes sinusoidal excitations due to the rotor rotational speed and blade pass frequency which are accurately represented by discrete sine tones. Helicopter environments also include random vibration caused by engine noise and turbulence as well as shock loads caused by lifting or dropping payloads or firing weapons. The available descriptions of these environments have been prepared by separating accelerometer measurements from flight into components that match each of these types of loads in order to allow suppliers to design or test their equipment using standardized tests and analysis methods that are consistent with each of these types of frequency domain data representations. However, the real environments have all these types of loads often occurring simultaneously. The structural design or evaluation needs to consider some method of load combination. In the frequency domain, the load combination challenges include combining stresses or forces resulting from different sine tones for steady state frequency analysis or different modes if a shock response analysis is performed. In addition, the random results, which are only available as statistical averages, also need to be added in some way. In combining loads, it is important that the analysis does not violate the assumptions used in defining the loads or processing the flight data to get the frequency domain loads. This is the basic difficulty in combining sine with random or shock loads in any analysis. This paper suggests that in some cases, it is valuable to reconstruct a transient from the frequency domain definitions in order to assess some aspects of the structural performance under the combined loading, as opposed to relying solely on frequency domain analysis for each load component and arbitrary load combinations. Realistic loads are obtained from Mil-Std-810 F for a helicopter since this data is representative or the direct source of many shock and vibration requirements. Mil-Std-810 F provides the distinct sinusoidal tones from the rotor and blade pass frequencies and random vibration levels as well as shock from dropping and lifting payloads and gun fire. The methods that are described here have been applied in the design of actual payloads that included these shock and vibration requirements. The methods described are applicable to many different structures, but the results shown here are from a generic model constructed to avoid using any proprietary data but realistic enough to illustrate the methods. STANDARD METHODS In this section, a review and comments about each of the frequency domain methods are provided, followed by some load combination approaches. Sine Loads Sinusoidal loads, defined in Mil-Std-810, for many different helicopters, can be a big problem for the payload designers. Representative amplitude versus frequency is shown in Figure 1. Note that the specification provides only a single discrete frequency for each sine tone. In the discussion, the specification states that frequencies vary less than one percent. The analysis for a sine vibration is just the steady state or particular solution to the dynamic differential equations of motion for the structure. The response of each structural mode for sine tones below that frequency is at the same amplitude as the input and is greatly attenuated for sine tones above that mode frequency. At the mode frequency, the input sine tone causes a response which is greatly amplified with peak amplitude inversely proportional to the damping at that mode frequency. The response to a series of sine tones is a response at each input frequency with no response at other frequencies. Even structural mode frequencies will not show up in the sine response because these structural modes will contribute only at each input frequency. Figure 1. Blackhawk sine environment from Mil-Std_810 F. Thus designing for the sine environment requires understanding damping for the payload and knowing its modes to a high degree of accuracy. If the modes are separated a few percentage points from the sine tone, the response will be greatly reduced over that which would occur if the input were aligned in frequency with a mode. However, if a sine tone matches a natural frequency, the response could be higher than the vibration input amplitude by a factor of 50 for a typical structural damping of 1%. Thus, sinusoidal response analysis requires determination of the appropriate damping and the uncertainty in mode frequencies of the structure, but, in the case of the helicopter, there is little uncertainty in the frequency of the excitation. The best way to handle the structural frequency and damping uncertainties is to perform a modal test if hardware is available. Realistic damping and frequencies can be measured to allow a sine domain analysis to be completed with high accuracy. If the payload item is just being designed, testing a similar structure may be effective in identifying appropriate damping. One approach to handle the uncertainty in the structural mode frequencies is to move the sine tone closer to the closest mode frequency in order to identify the worst case condition. Another approach might be to broaden the input sine tone to a band width equal to the mode uncertainty, as shown in Figure 2, even though in actuality it is the structural mode not the input that carries the uncertainty. Figure 2. Broadened sine tones to account for uncertainties in the input. Random Random loads provide the statistics of the environment after the sine tones have been removed from the flight data. A representative random environment from Mil-Std-810F is shown in Figure. 3. The challenge with random is the interpretation of the results, since the response predictions provide only the statistics of the response – not a deterministic actual response. In performing a test to qualify equipment for this environment, a shake table is driven to some physical acceleration with random amplitude and phase which has the specified power spectral density (PSD). This motion is not unique and will be different every time the test is performed. For direct use of the specified PSD in a random analysis, it is assumed that the environment is a stationary random process with zero mean. The response computed is the PSD of the response, which might be stress, deflection, or acceleration in a component. In order to determine the structural integrity, the PSD needs to be interpreted in terms of the probability that the response will exceed some allowable, such as yield stress. Common practice is to use the three standard deviations or 3σ value of the response as the peak load. From statistics, the integral of the PSD is the mean square value of the function, and the square root of this integral is the root mean square (RMS), or one σ value. This definition of peak is based on the probability density function for a Gaussian random process. The selection of 3σ as peak is equivalent to acceptance of the statistical probability of exceeding the allowable value by 0.27%. This stress is only a statistical quantity, and there is no phasing information associated with the RMS value. Any combination of the predicted response from a random analysis with the predicted responses due to other types of loads needs to take into account the statistical meaning of the random analysis and the fact that the phasing of each stress is unknown. Therefore, one should consider multiple stress cases with stresses added with different phasing possibilities in each case. Shock Response Spectra (SRS) Some shock loads are defined by a time domain description, such as a half sine pulse with amplitude and pulse width. A shock caused by a drop from a helicopter is shown in Figure 3, which is a haversine pulse with a width of 10 msec, which could be directly used in a transient analysis. However, for many complex shocks, such as missile launch, the environment is represented by a shock response spectrum (SRS). This frequency domain analysis definition consists of the peak amplitude of response of a single degree-of-freedom system, tuned to each frequency on the frequency axis, when subjected to the environment. The analysis method that corresponds to this definition assumes that a real structure can be represented by a finite number of single degree-of-freedom systems, which are its modes. The analysis process just scales the response of each mode to the amplitude of the SRS. If there is more than one mode in the frequency range of interest, the peak responses for each mode need to be combined with the peak responses for all other modes in some way to get the response of the structure without knowledge of phasing information in each mode.