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DOE OSTI2022

Cyber-Physical Dynamic System (CPDS) Modeling for Frequency Regulation and AGC Services of Distributed Energy Resources

The substantial integration of renewable energy brings significant challenges to balance the system in real time because of the variability and intermittency of renewable power. For the reliable system operation, the frequency regulation service is used to stabilize the system frequency through automatically balancing the system generation and load. On one hand, with the substantially increasing deployment of renewable energy in electricity system, the requirement of frequency regulation (FR) services increases significantly. On the other hand, the current main resource of FR services, the controllable conventional generation, is continuously decreasing in the system generation mix. This means that in the future high renewable penetration power system, additional and alternative reliable FR services providers such as distributed energy storage (DES) resources should be explored. Although the capability of utility-scale energy storage to provide FR services has been demonstrated, the integrated control and dynamic modeling of distributed energy resource (DER) providing frequency regulation grid services has been rarely explored. There are several challenges to adopt DERs to provide reliable grid services as illustrated in FERC 755. First, the distributed installation of DES requires a comprehensive cyber-physical dynamic system (CPDS) modeling to fully consider the impacts of the communication latency variability on its real time FR provision capability. Unlike the conventional generators, there are two-layer communication between DERs and system operators which increase the communication delay. Second, unlike the conventional generators whose dynamic models have been comprehensively studied, the difference among individual DER components' power-dynamic characteristic brings challenges in the accurate dynamic AGC modeling of its power-frequency relationship. Third, the temporal dependent state of charge uncertainty of DERs challenges DERs' power and frequency regulation capacities scheduling in the look-ahead generation scheduling. Therefore, the aggregator should optimize the frequency regulation provision from individual DER in real time. To overcome these challenges, this project proposes a cyber-physical dynamic system (CPDS) model to handle the uncertainty of DERs two-layer communication latency and power dynamics. The variability of DERs' communication delay and dynamic constraints will be comprehensively modeled. The DERs' AGC model with communication delay is designed to validate DESs' frequency regulation services. Like current performance-based frequency regulation evaluation, the delivery of DERs' frequency regulation will be assessed through post-analysis of the actual AGC response with respect to the AGC control signal from system operators. Consequently, the reliability improvement with DERs providing reliable frequency regulation services can be evaluated from a comprehensive perspective considering all the dynamics of communication and power.

24 POWER TRANSMISSION AND DISTRIBUTION↗

DOE OSTI2013

Heating of ions by high frequency electromagnetic waves in magnetized plasmas

The heating of ions by high frequency electrostatic waves in magnetically confined plasmas has been a paradigm for studying nonlinear wave-particle interactions. The frequency of the waves is assumed to be much higher than the ion cyclotron frequency and the waves are taken to propagate across the magnetic field. In fusion type plasmas, electrostatic waves, like the lower hybrid wave, cannot access the core of the plasma. That is a domain for high harmonic fast waves or electron cyclotron waves—these are primarily electromagnetic waves. Previous studies on heating of ions by two or more electrostatic waves are extended to two electromagnetic waves that propagate directly across the confining magnetic field. While the ratio of the frequency of each wave to the ion cyclotron frequency is large, the frequency difference is assumed to be near the ion cyclotron frequency. The nonlinear wave-particle interaction is studied analytically using a two time-scale canonical perturbation theory. The theory elucidates the effects of various parameters on the gain in energy by the ions—parameters such as the amplitudes and polarizations of the waves, the ratio of the wave frequencies to the cyclotron frequency, the difference in the frequency of the two waves, and the wave numbers associated with the waves. For example, the ratio of the phase velocity of the envelope formed by the two waves to the phase velocity of the carrier wave is important for energization of ions. For a positive ratio, the energy range is much larger than for a negative ratio. So waves like the lower hybrid waves will impart very little energy to ions. The theoretical results are found to be in good agreement with numerical simulations of the exact dynamical equations. The analytical results are used to construct mapping equations, simplifying the derivation of the motion of ions, which are, subsequently, used to follow the evolution of an ion distribution function. The heating of ions can then be properly quantified in terms of the wave parameters and can be conveniently used to find ideal conditions needed to heat ions by high frequency electromagnetic waves.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

DOE OSTI2019

Absolute frequency measurements for emission transitions of molecular iodine in the range of 1053 – 1068 nm

Results of high-precision frequency measurements are presented for certain components of the {sup 127}I{sub 2} hyperfine structure, which correspond to emission transitions of the B – X band in the range of 1053 – 1068 nm. The frequencies are measured by using a femtosecond optical frequency synthesiser based on a Ti:sapphire laser. The hyperfine structure is resolved by the method of three-level laser spectroscopy. The frequency-doubled cw Nd : YAG laser is used as an excitation radiation source, and the probe radiation in the range of 1050 – 1070 nm is generated by an external-cavity diode laser. The frequencies of both lasers are simultaneously locked to those of two adjacent hyperfine structure components of iodine lines, which have a common upper level. Correspondingly, the frequency of the Nd : YAG laser is locked to the frequency of the absorption transition component, and the diode laser frequency – to the frequency of the emission transition component. The radiation frequency of the diode laser ‘locked’ in this way is measured. Frequencies of 18 hyperfine structure components for six emission lines corresponding to the bands (32 – 54) and (32 – 53) are measured. The relative uncertainty of the measurement is 8 × 10{sup –10}. (paper)

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

DOE OSTI2018

Narrowing Frequency Probability Density Function for Achieving Minimized Uncertainties in Power Systems Operation – a Stochastic Distribution Control Perspective

With the increased penetration of renewables and other distributed energy resources (DERs), the frequency response in power systems exhibits stochastic nature, where the standard deterministic swing equation should be either represented or transferred into an Ito stochastic differential equation whose solution should be the non-stationary transient behavior of the probability density function (PDF) of the frequency. This transfers the frequency quality control for power systems into a stochastic distribution control problem, where the shape of the frequency distribution, namely the frequency PDF, needs to be directly controlled. In this context, this requires the development of control strategies via controlling the power balance so that the PDF of the frequency is made as narrow as possible whilst centered at its required mean value simple because such a sharp distribution shape means less uncertainties or randomness for the frequency response. In this paper, the formulation of the stochastic swing equation will be made first taking into account of DERs. This will then be followed by the development of stochastic distribution control model that links the power sources with the PDF of the frequency using Fokker Planck Kolmogorov (FPK) equations. A generic constrained optimization problem will be formulated where the cost function is composed of a functional distance between the actual and the desired PDFs of the frequency. A feasible solution using B-spine Neural Networks based stochastic distribution control model will be described. Using the obtained stochastic distribution control model, a set of control algorithms will be described that uses available power sources to shape the PDF of the frequency or to minimize the randomness of the frequency via minimized entropy approach. Future directions will be discussed in the later part of the paper

Swing equation, Power Systems, stchastic distribut↗

DOE OSTI2023

Impact of Transportation Electrification on the System's Dynamic Frequency Response

Transportation electrification is an integral component of the energy decarbonization transition. This paper investigates the impact of distributed energy resources (DERs), including distributed photovoltaics (DPV) and electric vehicles (EVs), in the primary frequency response of the power grid. Increasing DER adoption poses challenges to maintaining grid frequency stability. However, DERs' ability to provide fast frequency regulation services-primary frequency response (PFR) and secondary frequency response (SFR)-can be exploited to recover the frequency after an N-1 contingency event in the system. This paper also investigates the importance of a droop control strategy through dynamic models of DPV and EV to provide the primary frequency regulation services following the contingency event. A dynamic EV model, based on the PVD1 model Western Electricity Coordinating Council (WECC) introduced, has been used for the simulation. Further, DERs' primary frequency response is studied for five different cases of DER penetration levels after the system is exposed to the generator trip. Additionally, different frequency regulation capacities of EVs are analyzed. The studies show that an increment in DERs capacity providing effective PFR can improve the system frequency nadir and stabilize the frequency faster after the generation trip contingency.

distributed energy resources↗

DOE OSTI2021

Optimized allocation method of the VSC-MTDC system for frequency regulation reserves considering the cost

In recent years, the interconnection of asynchronous power grids through the VSC-MTDC system has been proposed and extensively studied in light of the potential benefits of economical bulk power exchanges and frequency regulation reserves sharing. This paper proposed an optimized allocation method for sharing frequency regulation reserves among the interconnected power systems and the corresponding frequency regulation control of the VSC-MTDC system under emergency frequency deviation events. Firstly, the frequency regulation reserve classification is proposed. In the classification, the available frequency response capacity reserves of each interconnection are divided into commercial reserves and regular reserves. While the commercial reserves are procured through long-term contracts, the regular reserves are purchased based on market prices of frequency regulation services. Secondly, based on the proposed frequency regulation reserve classification, a novel frequency regulation control is then introduced for the VSC-MTDC system. This control method could minimize the costs of the disturbed power grid for the needed frequency response supports from the other power grids. Simulation verifications are performed on a modified IEEE 39 bus system and a highly reduced power system model representing the North American grids. The simulation verification indicates that the developed frequency regulation control significantly reduced ancillary service costs of the disturbed power grid.

24 POWER TRANSMISSION AND DISTRIBUTION↗

NASA NTRSDate not supplied

In-Orbit Selection of Cryocooler Drive Frequencies for XRISM/Resolve

The XRISM/Resolve instrument cooling system uses adiabatic demagnetization refrigerators (ADRs) to cool the detectors to 50 mK and two types of cryocoolers to reduce heat load on the He tank, Joule-Thomson and Stirling cryocoolers. Resolve was designed with tunable cryocooler frequencies so that any mechanical interference and its effect on performance could be minimized. Measurements at different cryocooler drive frequencies were used to quantify the interference and narrow the acceptable range of frequency options. For Resolve the choice of cryocooler drive frequencies dramatically influences the instrument performance. Poor choices have been shown to degrade ADR cooling power and degrade the spectroscopic performance of the instrument. However, good choices have been shown to be free of undesirable features, allowing the instrument to satisfy its performance requirements with significant margin. Thus, the choice of drive frequencies is a critical factor for overall instrument performance. The drive frequencies of the Joule-Thomson cooler operating near 52 Hz, and the Stirling coolers operating near 15 Hz were adjusted in three distinct scan sequences to efficiently narrow the candidate frequency options. The frequency choices were based on stability of the 50 mK control thermometer, changes in the ADR demagnetization rate, and changes in detector noise. Scans across the drive frequency range of each cryocooler were performed multiple times during ground testing. Surprisingly, the results were not repeatable across test campaigns and did not conform to any discernable trend, but they were repeatable when remeasured within a single cool-down cycle. The conclusion was that the good/bad drive frequencies changed during Dewar warmup/cooldown, and possibly after mechanical vibration tests. Therefore, the settings could not be finalized before launch, and scans during instrument commissioning were necessary. This paper describes the cryocooler frequency scan measurements and compares the results during instrument commissioning to those performed during ground tests.

XRISM↗

DOE OSTI2023

Impact of Transportation Electrification on the System's Dynamic Frequency Response: Preprint

Transportation electrification is an integral component of the energy decarbonization transition. This paper investigates the impact of distributed energy resources (DERs), including distributed photovoltaics (DPV) and electric vehicles (EVs), in the primary frequency response of the power grid. Increasing DER adoption poses challenges to maintaining grid frequency stability. However, DERs' ability to provide fast frequency regulation services-primary frequency response (PFR) and secondary frequency response (SFR)-can be exploited to recover the frequency after an N-1 contingency event in the system. This paper also investigates the importance of a droop control strategy through dynamic models of DPV and EV to provide the primary frequency regulation services following the contingency event. A dynamic EV model, based on the PVDl model Western Electricity Coordinating Council (WECC) introduced, has been used for the simulation. Further, DERs' primary frequency response is studied for five different cases of DER penetration levels after the system is exposed to the generator trip. Additionally, different frequency regulation capacities of EVs are analyzed. The studies show that an increment in DERs capacity providing effective PFR can improve the system frequency nadir and stabilize the frequency faster after the generation trip contingency.

ADVANCED PROPULSION SYSTEMS↗

DOE OSTI2016

High-energy sources at low radio frequency: the Murchison Widefield Array view of Fermi blazars

Low-frequency radio arrays are opening a new window for the study of the sky, both to study new phenomena and to better characterize known source classes. Being flat-spectrum sources, blazars are so far poorly studied at low radio frequencies. In this paper, we characterize the spectral properties of the blazar population at low radio frequency, compare the radio and high-energy properties of the gamma-ray blazar population, and search for radio counterparts of unidentified gamma-ray sources. We cross-correlated the 6100 deg 2 Murchison Widefield Array Commissioning Survey catalogue with the Roma blazar catalogue, the third catalogue of active galactic nuclei detected by Fermi-LAT, and the unidentified members of the entire third catalogue of gamma-ray sources detected by Fermi-LAT. When available, we also added high-frequency radio data from the Australia Telescope 20 GHz catalogue. We find low-frequency counterparts for 186 out of 517 (36%) blazars, 79 out of 174 (45%) gamma-ray blazars, and 8 out of 73 (11%) gamma-ray blazar candidates. The mean low-frequency (120–180 MHz) blazar spectral index is (α low ) = 0.57 ± 0.02: blazar spectra are flatter than the rest of the population of low-frequency sources, but are steeper than at ~GHz frequencies. Low-frequency radio flux density and gamma-ray energy flux display a mildly significant and broadly scattered correlation. Ten unidentified gamma-ray sources have a (probably fortuitous) positional match with low radio frequency sources. Low-frequency radio astronomy provides important information about sources with a flat radio spectrum and high energy. However, the relatively low sensitivity of the present surveys still misses a significant fraction of these objects. Finally, upcoming deeper surveys, such as the GaLactic and Extragalactic All-Sky MWA (GLEAM) survey, will provide further insight into this population.

79 ASTRONOMY AND ASTROPHYSICS↗

DOE OSTI2014

Broadband electromagnetic power harvester from vibrations via frequency conversion by impact oscillations

In this paper, we propose an electromagnetic power harvester that uses a transformative multi-impact approach to achieve a wide bandwidth response from low frequency vibration sources through frequency-up conversion. The device consists of a pick-up coil, fixed at the free edge of a cantilever beam with high resonant frequency, and two cantilever beams with low excitation frequencies, each with an impact mass attached at its free edge. One of the two cantilevers is designed to resonate at 25 Hz, while the other resonates at 50 Hz within the range of ambient vibration frequency. When the device is subjected to a low frequency vibration, the two low-frequency cantilevers responded by vibrating at low frequencies, and thus their thick metallic masses made impacts with the high resonance frequency cantilever repeatedly at two locations. This has caused it along with the pick-up coil to oscillate, relative to the permanent magnet, with decaying amplitude at its resonance frequency, and results in a wide bandwidth response from 10 to 63 Hz at 2 g. A wide bandwidth response between 10–51 Hz and 10–58 Hz at acceleration values of 0.5 g and 2 g, respectively, were achieved by adjusting the impact cantilever frequencies closer to each other (25 Hz and 45 Hz). A maximum output power of 85 μW was achieved at 5 g at 30 Hz across a load resistor, 2.68 Ω.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗

DOE OSTI2016

Transition from Townsend to radio-frequency homogeneous dielectric barrier discharge in a roll-to-roll configuration

The aim of this paper is to better understand the transition from Townsend to radio-frequency homogeneous dielectric barrier discharge (DBD) at atmospheric pressure. The study is done in an Ar/NH{sub 3} Penning mixture for an electrode configuration adapted to roll-to-roll plasma surface treatment. The study was led in a frequency range running from 50 kHz up to 8.3 MHz leading to different DBD modes with a 1 mm gas gap: Glow (GDBD), Townsend (TDBD), and Radio-frequency (RF-DBD). In the frequency range between TDBD and RF-DBD, from 250 kHz to 2.3 MHz, additional discharges are observed outside the inter-electrode gas gap. Because each high voltage electrode are inside a dielectric barrel, these additional discharges occur on the side of the barrel where the gap is larger. They disappear when the RF-DBD mode is attained in the 1 mm inter-electrode gas gap, i.e., for frequencies equal or higher than 3 MHz. Fast imaging and optical emission spectroscopy show that the additional discharges are radio-frequency DBDs while the inter-electrode discharge is a TDBD. The RF-DBD discharge mode is attained when electrons drift becomes low enough compared to the voltage oscillation frequency to limit electron loss at the anode. To check that the additional discharges are due to a larger gas gap and a lower voltage amplitude, the TDBD/RF-DBD transition is investigated as a function of the gas gap and the applied voltage frequency and amplitude. Results show that the increase in the frequency at constant gas gap or in the gas gap at constant frequency allows to obtain RF-DBD instead of TDBD. At low frequency and large gap, the increase in the applied voltage allows RF-DBD/TDBD transition. As a consequence, an electrode configuration allowing different gap values is a solution to successively have different discharge modes with the same applied voltage.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

DOE OSTI2013

Spatial frequency spectrum of the x-ray scatter distribution in CBCT projections

Purpose: X-ray scatter is a source of significant image quality loss in cone-beam computed tomography (CBCT). The use of Monte Carlo (MC) simulations separating primary and scattered photons has allowed the structure and nature of the scatter distribution in CBCT to become better elucidated. This work seeks to quantify the structure and determine a suitable basis function for the scatter distribution by examining its spectral components using Fourier analysis.Methods: The scatter distribution projection data were simulated using a CBCT MC model based on the EGSnrc code. CBCT projection data, with separated primary and scatter signal, were generated for a 30.6 cm diameter water cylinder [single angle projection with varying axis-to-detector distance (ADD) and bowtie filters] and two anthropomorphic phantoms (head and pelvis, 360 projections sampled every 1°, with and without a compensator). The Fourier transform of the resulting scatter distributions was computed and analyzed both qualitatively and quantitatively. A novel metric called the scatter frequency width (SFW) is introduced to determine the scatter distribution's frequency content. The frequency content results are used to determine a set basis functions, consisting of low-frequency sine and cosine functions, to fit and denoise the scatter distribution generated from MC simulations using a reduced number of photons and projections. The signal recovery is implemented using Fourier filtering (low-pass Butterworth filter) and interpolation. Estimates of the scatter distribution are used to correct and reconstruct simulated projections.Results: The spatial and angular frequencies are contained within a maximum frequency of 0.1 cm{sup −1} and 7/(2π) rad{sup −1} for the imaging scenarios examined, with these values varying depending on the object and imaging setup (e.g., ADD and compensator). These data indicate spatial and angular sampling every 5 cm and π/7 rad (∼25°) can be used to properly capture the scatter distribution, with reduced sampling possible depending on the imaging scenario. Using a low-pass Butterworth filter, tuned with the SFW values, to denoise the scatter projection data generated from MC simulations using 10{sup 6} photons resulted in an error reduction of greater than 85% for the estimating scatter in single and multiple projections. Analysis showed that the use of a compensator helped reduce the error in estimating the scatter distribution from limited photon simulations by more than 37% when compared to the case without a compensator for the head and pelvis phantoms. Reconstructions of simulated head phantom projections corrected by the filtered and interpolated scatter estimates showed improvements in overall image quality.Conclusions: The spatial frequency content of the scatter distribution in CBCT is found to be contained within the low frequency domain. The frequency content is modulated both by object and imaging parameters (ADD and compensator). The low-frequency nature of the scatter distribution allows for a limited set of sine and cosine basis functions to be used to accurately represent the scatter signal in the presence of noise and reduced data sampling decreasing MC based scatter estimation time. Compensator induced modulation of the scatter distribution reduces the frequency content and improves the fitting results.

62 RADIOLOGY AND NUCLEAR MEDICINE↗

NASA NTRS1988

An analytical study of the effects of transverse shear deformation and anisotropy on natural vibration frequencies of laminated cylinders

Natural vibration frequencies of orthotropic and anisotropic simply supported right circular cylinders are predicted using a higher-order transverse-shear deformation theory. A comparison of natural vibration frequencies predicted by first-order transverse-shear deformation theory and the higher-order theory shows that an additional allowance for transverse shear deformation has a negligible effect on the lowest predicted natural vibration frequencies of laminated cylinders but significantly reduces the higher natural vibration frequencies. A parametric study of the effects of ply orientation on the natural vibration frequencies of laminated cylinders indicates that while stacking sequence affects natural vibration frequencies, cylinder geometry is more important in predicting transverse-shear deformation effects. Interaction curves for cylinders subjected to axial compressive loadings and low natural vibration frequencies indicate that transverse shearing effects are less important in predicting low natural vibration frequencies than in predicting axial compressive buckling loads. The effects of anisotropy are more important than the effects of transverse shear deformation for most strongly anisotropic laminated cylinders in predicting natural vibration frequencies. However, transverse-shear deformation effects are important in predicting high natural vibration frequencies of thick-walled laminated cylinders. Neglecting either anisotropic effects or transverse-shear deformation effects leads to non-conservative errors in predicted natural vibration frequencies.

Jegley, Dawn C.↗

NASA NTRS1989

An approximation function for frequency constrained structural optimization

The purpose is to examine a function for approximating natural frequency constraints during structural optimization. The nonlinearity of frequencies has posed a barrier to constructing approximations for frequency constraints of high enough quality to facilitate efficient solutions. A new function to represent frequency constraints, called the Rayleigh Quotient Approximation (RQA), is presented. Its ability to represent the actual frequency constraint results in stable convergence with effectively no move limits. The objective of the optimization problem is to minimize structural weight subject to some minimum (or maximum) allowable frequency and perhaps subject to other constraints such as stress, displacement, and gage size, as well. A reason for constraining natural frequencies during design might be to avoid potential resonant frequencies due to machinery or actuators on the structure. Another reason might be to satisy requirements of an aircraft or spacecraft's control law. Whatever the structure supports may be sensitive to a frequency band that must be avoided. Any of these situations or others may require the designer to insure the satisfaction of frequency constraints. A further motivation for considering accurate approximations of natural frequencies is that they are fundamental to dynamic response constraints.

Canfield, R. A.↗

NASA NTRS1989

An analytical study of the effects of transverse shear deformation and anisotropy on natural vibation frequencies of laminated cylinders

Jegley, Dawn C.↗

NASA NTRS1990

Time and frequency measuring metrological equipment in the USSR

The complex of the means of providing time and frequency traceability in the USSR includes the system of time and frequency standards of the National Time and Frequency Calibration Service, time and frequency transfer facilities and local time and frequency standards. Control on measurement correctness is performed by the All-Union State Standard calibration service. The hardware of most of the above-mentioned systems is provided by the instruments developed by our institute. A common scientific and technological approach allowed us to create a unified system of time-frequency equipment composed of widely used serial instruments, sets, automated systems and complexes. Primary frequency standards of different classes, time and frequency references and instruments are based on the unified system. CH1-70 hydrogen frequency standard and its CH1-70A, CH1-80 modifications are used in the group time and frequency standards. Measuring time and data processing techniques, and also instrumentation specifications on the results of 10-year operation are given. The existing system provides time-frequency measurements with 2 times 10(exp -13) plus or minus 1 times 10(exp -14) accuracy.

Uljanov, Adolph A.↗

NASA NTRS1992

Funnel-shaped, low-frequency equatorial waves

Funnel-shaped, low-frequency radiation, as observed in frequency time spectrograms, is frequently found at the earth's magnetic equator which extends from the proton-cyclotron frequency up to the lower hybrid frequency. Ray-tracing calculations can qualitatively reproduce the observed frequency-time characteristics of these emissions if the waves are propagating in the fast magnetosonic mode starting with wave normal angles of about 88 deg at the magnetic equator. The funnel-shaped emissions are consistent with generation by protons with a ring-type velocity space distribution. A ring-shaped region of positive slope in the velocity space density distribution of protons is observed near the Alfven velocity, indicating that the ring protons strongly interact with the waves. Ray-tracing calculations show that for similar equatorial wave normal angles lower-frequency fast magnetosonic waves are more closely confined to the magnetic equator than higher-frequency fast magnetosonic waves. For waves refracted back toward the equator at similar magnetic latitudes, the lower-frequency waves experience stronger damping in the vicinity of the equator than higher-frequency waves. Also, wave growth is restricted to higher frequencies at larger magnetic latitudes. Wave damping at the equator and wave growth off the equator favors equatorial wave normal angle distributions which lead to the funnel-shaped frequency time characteristic.

Boardsen, S. A.↗

NASA NTRS1994

Natural frequency of uniform and optimized tetrahedral truss platforms

Qualitative and quantitative estimates for the fundamental frequency of uniform and optimized tetrahedral truss platforms are determined. A semiempirical equation is developed for the frequency of free-free uniform trusses as a function of member material properties, truss dimensions, and parasitic (nonstructural) mass fraction Mp/Mt. Optimized trusses with frequencies approximately two times those of uniform trusses are determined by varying the cross-sectional areas of member groups. Trusses with 3 to 8 rings, no parasitic mass, and member areas up to 25 times the minimum area are optimized. Frequencies computed for ranges of both Mp/Mt and the ratio of maximum area to minimum area are normalized to the frequency of a uniform truss with no parasitic mass. The normalized frequency increases with the number of rings, and both frequency and the ratio of maximum area to minimum area decrease with increasing Mp/Mt. Frequency improvements that are achievable with a limited number of member areas are estimated for a 3-ring truss by using Taguchi methods. Joint stiffness knockdown effects are also considered. Comparison of optimized and baseline uniform truss frequencies indicates that tailoring can significantly increase structural frequency; maximum gains occur for trusses with low values of Mp/Mt. This study examines frequency trends for ranges of structural parameters and may be used as a preliminary design guide.

Wu, K. Chauncey↗