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This chapter introduces multi-degree-of-freedom systems by deriving the undamped and damped equations of motion of constrained and unconstrained systems having more than one degree of freedom, and solving for the response to initial conditions. The solutions introduce modal coordinates that allow for the decoupling of the equations of motion. This chapter includes derivations and thorough discussion of the eigenvalue problem, including the damping conditions that lead to classical normal modes or nonclassical complex modes. This includes a comprehensive discussion of left and right eigenvectors and the transformations needed to decouple the equations of motion with complex modes. Mode shape orthogonality for both classical and complex modes as well as the rigorous treatment of rigid body modes in classically and nonclassically damped systems are covered. Derivation of damping matrices that yield classical normal modes is also covered. This chapter concludes with a thorough discussion and proof of the stationary and convexity of Rayleigh's Quotient and provides the reasons why structures in unforced vibration can only vibrate at discrete frequencies and in the associated mode shapes. This chapter concludes with references and solved problems that reinforce the material discussed in this chapter.

The vibration of multi-degree-of-freedom systems excited by external forces or base motion is thoroughly covered in this chapter. The equations of motion of systems excited by harmonic forces or by short transients such as step and impulse forces are discussed first, followed by a thorough discussion of base excitation. Beating and frequency sweep effects are also covered. This is followed by the derivation and solution of the equations of motion for nondeterministic excitation that can only be described statistically. Both frequency and time domain approaches are thoroughly covered. This chapter also covers the use of truncated modal coordinates and how to improve the accuracy of the solutions by use of mode acceleration and residual flexibility concepts. This chapter deals extensively with nonclassical complex modes, and in particular with rotating systems with gyroscopic effects. The derivations and discussion include whirl and energy dissipation in systems with gyroscopic moments. This chapter also has discussion on systems where the behavior of the system itself causes excitation, such as aerodynamic and pogo instabilities. This chapter concludes with references and solved problems that reinforce the material discussed in this chapter.

Experimental structural dynamics is a critical element of developing accurate models, acquiring performance data, and demonstrating that systems can sustain operational environments. This chapter provides an in-depth discussion of mode survey testing, including procedures used on complex structural dynamic systems such as launch vehicles and spacecraft. The discussion includes test facility requirements, preparation of the test article, test apparatus and test article instrumentation, conduct of the test, and test success criteria. This chapter includes discussion on accelerometers and their placement, as well as excitation apparatus such as electrodynamic shakers and impact hammers. This chapter addresses in detail direct measurement test procedures, as well as identification of modal parameters from frequency response functions. Each procedure is developed from fundamental principles. This chapter also deals with modal contamination and when is it appropriate to analytically orthogonalize empirical mode shapes; three orthogonalization procedures are discussed in detail. This chapter concludes with references and solved problems that reinforce the material discussed in this chapter.

Modern structural dynamic models are generally developed by discretizing the real-world continuous systems and applying Newton’s laws to the resulting concentrated mass points, or formulating the system’s strain and kinetic energies, adding the work done by the damping and external forces, and applying Lagrange’s equations. For classical systems such as strings, rods, beams, membranes, plates, and fluids in simple rigid tanks, it is possible to develop and solve the equations of motion of the continuous system. In this chapter, the equations of motion are derived and solved for elements where the restoring force is due to the tension in the element; strings and membranes fall into this category. In addition, elements where the restoring forces are due to the elastic properties of the system are addressed; rods, beams, and plates fall into this category. There is extensive discussion on vibration of spinning circular disks, and how to reduce vibration caused by gas jets. The equations of motion of fluids in rectangular and cylindrical rigid tanks are also derived and solved, and the sloshing behavior of the fluids is studied in detail. This chapter provides the explanations of why circular plates fixed at the center and fluids in cylindrical tanks yield responses that appear to rotate about the longitudinal axis. This chapter concludes with references and solved problems that reinforce the material discussed in this chapter.

Chapter 7 deals rigorously with the variability that one encounters in structural dynamics analyses. This chapter first covers samples and probability density functions, stationary and ergodic vibration, probability density functions of time histories, and properties of probability density functions, including Normal, Rayleigh, Gamma, Gumbel, Lognormal, and Bivariate Normal. This chapter addresses how to verify that a given data ensemble has a particular distribution, and how to deal with small sample sizes. There is extensive discussion on statistics of time history envelope and phase functions, and statistics of peaks and largest peak. Monte Carlo analysis and order statistics are also covered in detail. Tables are included that provide the required enclosure factors for upper tolerance limits for Normal and Rayleigh distributed small sample sets. In addition, order statistics tables are provided that give the optimum sample size and the number of kept values for a desired enclosure and confidence limit. This chapter also includes extensive discussion on the statistical combination of vibration responses, even when the contributors have different probability density functions. This chapter concludes with references and solved problems that reinforce the material discussed in this chapter.

Computation of responses of large complex systems that require multidisciplinary approaches is rigorously covered in this chapter. This chapter addresses when and how to include gravity effects, the behavior of structures vibrating in fluids, the response of systems to families of forcing functions, and to forcing functions solely described by statistical properties. Equations of motion, including initial conditions required to account for nonzero force values at the start of the solution, are developed for systems such as launch vehicles. Atmospheric-flight equations of motion, including aeroelasticity, are rigorously derived. These are then used to derive equations of motion required to compute atmospheric turbulence/gust responses. In addition, the equations for the buffet response of complex systems, both frequency and time domain, developed in Volume I, are thoroughly covered. The equations for the static-aeroelastic response of system in atmospheric flight are also derived and solved for both trimmed and nontrimmed flight conditions. In each load analysis covered in this chapter, the statistical treatment of response values is consistent with the material presented in Chapter 7 , Probability and Statistics. The last section addresses computing the response of structures to earthquake ground motion. This chapter concludes with references and solved problems that reinforce the material discussed in this chapter.

The efficient computation of responses via response recovery transformation matrices (RRTMs), and if specifically loads and stresses, via load transformation matrices (LTMs), is addressed in this chapter. The family of RRTMs covered in this chapter also includes acceleration and displacement/rattle space response recovery matrices. This chapter includes the derivation of response recovery matrices, and their inclusion/use with structural dynamic models developed using the Component Mode Synthesis and Component Mode Substitution procedures discussed in Chapter 2 . This chapter also includes the derivation of the full mode acceleration approach for computing loads with models developed with truncated modal coordinates. Residual flexibility implementation of response recovery matrices is also covered. This chapter concludes with references and solved problems that reinforce the material discussed in this chapter.

This chapter provides an introduction to the finite element method as used to develop structural dynamic models. The proper way to discretize a continuous system to develop stiffness, mass, and damping matrices is presented. To provide an introduction to the development of finite elements, beam and bar finite elements are developed using both force–displacement and interpolation functions; this includes derivation of lumped and consistent mass matrices. The proper application of external forces, including discrete and distributed forces with fixed end moments and shears, is covered, and the use of interpolation functions to model distributed forces is included. Coordinate transformations based on direction cosines and Euler angles are derived, and the reduction in the size of finite element models is covered. In addition, there is extensive discussion on comparing analytical models to mode survey test data, and a process for improving the agreement between data and analytical model is provided. This chapter concludes with references and solved problems that reinforce the material discussed in this chapter.

Analytical derivation of damping is extremely difficult and in most cases not possible. As a result, damping properties are typically specified on a mode-by-mode basis because of the availability of test data that can be used either directly, or can be used as a guide for similar systems for which data are not yet available. Occasionally there is need to develop from the modal damping properties damping matrices that correspond to the physical coordinate sets in which the finite element model mass and stiffness matrices were developed. Proportional damping, in which combinations of mass and stiffness matrices are used to develop the physical coordinate damping matrix, has been used extensively. It is the purpose of this work to discuss the appropriateness of including a mass proportional term. It is concluded that damping formulations that include a scaled mass term are fundamentally flawed and should not be used.

A filtering method based on the empirical mode decomposition (EMD) is developed to remove spurious, quasi-periodic features in measured wind profiles. These oscillatory features are associated with the pendulum motion of the sonde responding to the balloon’s lateral self-induced motions. Since these oscillations are not indicative of the actual wind, current balloon sounding systems apply digital filters to remove these artifacts. Data from two GPS-based wind profiling systems, the National Weather Service Radiosonde Replacement System (RRS) and the Low-Resolution Automated Meteorological Profiling System (AMPS), will be examined. These profiling systems employ pre-defined low-pass digital filters that have prescribed frequency cutoffs. Since the frequencies of the oscillations vary both temporally and spatially, conservative filter cutoffs must be used to ensure that these artifacts are suppressed. On the other hand, the EMD is a data-driven approach that resolves a time series as a summation of its unique intrinsic mode functions (IMF). As such, the EMD-based filtering is able to adaptively identify and remove the IMFs associated with these oscillations. The filter’s characteristics will be examined using synthetic data that exhibits the random oscillations of a pendulum responding to fractional Brownian motions. The practical effectiveness of the EMD filtering will be demonstrated using the RRS and AMPS data.

Measurements of solid rocket motor head end pressure and structural acceleration recorded during flights of a heavy lift launch vehicle are used to investigate if an interaction between the motor internal flow and structural motion exists. These data reveal that a locking of frequency and phase occurs over a 34-s period towards the end of the motor burn. A feedback relationship involving the motor pressure oscillations and structural accelerations, therefore, exists for this launch vehicle. The observed interaction significantly increases the pressure oscillation amplitudes relative to those measured in ground tests at the same burn time. This finding highlights a limitation of motor stability analysis methodology, in which structural motion is traditionally neglected. It appears that the potential for coupling between the motor pressure oscillations and structural response should be accounted for in predictions of solid rocket motor stability. The observed interaction also has implications for the prediction of loads induced by solid rocket motor pressure oscillations. Under some conditions the use of forcing functions based solely on ground test pressure measurements may underpredict flight loads.

Parameters that characterize the perturbational pressure and flow at the inlet and outlet of a pump are established through pogo stability analyses for a launch vehicle. Ground tests of the launch vehicle's engine indicate the presence of unsteady pump cavitation, and some flights of the launch vehicle exhibit frequency, amplitude, and phase locking between axial structural acceleration and engine chamber pressure-a condition emblematic of propulsion-structure interaction, or pogo. Models developed for several missions of the subject launch vehicle are used to establish the ranges of the pump parameters that yield instability during the flight pogo occurrences and stability at other times. The resulting nominal values of normalized pump cavitation stiffness and mass flow gain for the launch vehicle's engine fall into the ranges 0.62-0.86 and 0.31-0.59, respectively. These ranges account for sensitivity with respect to dynamic pump gain and to structural damping for the axial modes of the coupled launch vehicle-payload system. It is shown that cavitation stiffness is generally the predominate pump parameter when the feedline hydraulic and axial structural modes are separated in frequency. However, if the frequencies of these modes are in close proximity, mass flow gain has a strong destablizing effect.

The aerodynamic e ow state on launch vehicle payload fairings and aircraft wings can change abruptly during transonic e ight if the angle of attack reaches a critical value. The nonlinear pressure variation associated with a e ow-state change induces transient structural responses that may converge to a limit-cycle oscillation (LCO). In this steady state, the work conducted during thee ow-statechangesbalances theenergy dissipation from structural damping.AnalysisofthistransonicLCOphenomenonisoftenconductedusingasemi-empirical,unsteadypressure variation in which the levels for the e ow states are determined from steady wind-tunnel test data. The presented theory addresses the condition in which the e ow-state changes occur near a quasi-steady, nonzero angle of attack. The analysis for the resulting asymmetric forcing function complements the authors’ existing derivations for a symmetric forcing function at zero angle of attack. Both the asymmetric and the symmetric analyses develop closed-form equations for the structural response frequency and amplitude. These expressions show that the solution space contains a subcritical Hopf bifurcation when the critical angle of attack equals the quasi-steady angle of attack. They also show that a saddle-node bifurcation, or fold, corresponds to the critical angle of attack beyond which LCO will not occur for a given quasi-steady angle of attack.

Launch vehicles with liquid rocket engines have feed lines through which propellants flow to the engine. To prevent feedback between structural responses and propellant pressure and flow oscillations, a compliant device called a pogo accumulator is typically installed in the propellant feed line. Even if a catastrophic interaction is thus averted, the fluid-induced structural responses may exceed those for important flight events such as liftoff and atmospheric buffeting. In that case, the fluid-induced excitation must be predicted in order to ensure adequate structural margins for the launch vehicle and space vehicle hardware. Venting of compliant gas in the pogo accumulator prior to engine shutdown is known to exacerbate the fluid-induced excitation. In particular, for the Atlas V launch vehicle, a 5–7 Hz fluid mode with large pressure gains at the aft end of the liquid oxygen feed line often excites structural modes just prior to engine cutoff. A methodology for the prediction of these structural responses is presented.Copyright © 2014 by ASME

A method based on Principal Component Analysis (PCA) was developed to generate synthetic winds that may be used for studies of aerodynamic loads and the dispersion of rocket trajectories. A notable feature of the analysis is that it is performed on winds separated in time by certain time deltas (30, 60, 120 minutes, etc.). This approach produces synthetic winds that preserve relevant statistics of the observed winds. These include the variance, spatial spectra, correlation between wind components and temporal decorrelation. This is different from methods for computing synthetic winds based on standard autoregressive Markov techniques, such as the Global Reference Atmosphere Model (GRAM). We have examined the generation of synthetic winds that may be used within the GRAM framework. The GRAM combines a deterministic monthly mean and a stochastic perturbation to give individual realizations. A difficulty with the present GRAM formulation for calculating temporal effects is that each realization is uncorrelated with any other. In addition, GRAM predicts that the north-south and east-west components of the wind are uncorrelated and that their power spectral density falls off less rapidly with decreasing wavelength than observed. Our approach for computing the stochastic part resolves all of these difficulties and may be used with the deterministic part of GRAM to produce more realistic synthetic winds.

In structural modeling of launch vehicles, liquid propellant is sometimes rigidly attached to feedline walls. This assumption precludes the interaction of structural modes with propellant pressure and flow. An analysis of fluid-structure interaction (FSI) for the Atlas V launch vehicle revealed that structural models with rigidly-attached propellant yield unconservative response predictions under some conditions. In particular, during the maximum acceleration time of flight, pressure oscillations acting at bends in the Atlas V liquid oxygen (LO2 ) feedline excite 15–20 Hz structural modes that have considerable gain on the feedline and at the spacecraft interface. The investigation also revealed that the venting of gas from the pogo accumulator is an excitation source and changes the dynamic characteristics of the hydraulic system. The FSI simulation produced during the investigation can be adapted to mission-specific conditions, such that responses and loads are conservatively predicted for any Atlas V flight.Copyright © 2010 by ASME

∗ Senior Engineering Specialist, Structural Dynamics Department, P.O. Box 92957-M4/911, Member AIAA † Director, Structural Dynamics Department, P.O. Box 92957-M4/911, Associate Fellow AIAA ABSTRACT A direct least square (DLS) formulation of the KMA method is presented. The KMA method belongs to a class of procedures that refines dynamic models using test-measured modes and structural connectivity. Viewed as a constrained minimization problem, most of these methods have applied the Lagrange multiplier method (LMM) in their development. It is shown that the DLS and LMM approaches result in linear systems of equations that are algebraically equivalent and therefore, have identical solutions. By virtue of this equivalence, the DLS formulation of the KMA method is shown to be equivalent to its original LMM formulation. The DLS versions of other model refinement methods that preserve structural connectivity are also discussed. In this respect, the DLS approach provides a rigorous framework for unifying these optimal update procedures and indicate how their solutions should compare. Numerical results are presented that illustrates the equivalence between the LMM and DLS formulations and also the conditions under which the various methods yield identical solutions. The results also indicate that because of improved numerical conditioning, the DLS approach yields more accurate computational solutions and generally requires less computer storage than methods which were developed using Lagrange multipliers.