Structural Dynamics of Rocket Engines Andy Brown, Ph.D. NASA/Marshall Space Flight Center AIAA SciTech Forum San Diego, California ER41/Propulsion Structures & 3 January 2016 Dynamic Analysis Why do I have a job? Why are we here? 2 • From “Characteristics of Space Shuttle Main Engine Failures”, H. Cikanek, AIAA Joint Propulsion Conference, 1987, San Diego, CA – “During development and operation of the SSME, 27 ground test failures of sufficient severity to be termed “major incident” have occurred.” – “Most SSME failures were a result of design deficiencies stemming from inadequate definition of dynamic loads. High cycle fatigue was the most frequent mechanism leading to failure.” Failure of Lox Inlet Splitter to Nozzle Blows Engine Out of Santa Susanna Test Stand Agenda 3 • Introduction to NASA’s new SLS • Short Review of Basics of Structural Dynamics • The Critical Role of Structural Dynamics in the Design, Analysis, and Testing of Rocket Engines: – How Rocket Engines Work – Turbomachinery – Rocket Nozzles – Rocket Engine Loads – System Hardware and Propellant Feedlines • Will need to introduce various Structural Dynamics Analysis Methods throughout presentation – “Two Minute Tutorials”. Travelling To and Through Space Space Launch System (SLS) – America’s Heavy-lift Rocket  Provides initial lift capacity of 70 metric tons (t), evolving to 130 t  Carries the Orion Multi-Purpose Crew Vehicle (MPCV) and significant science payloads  Supports national and international missions beyond Earth’s orbit, such as near-Earth asteroids and Mars 70t 130t  Builds on the proven success of Saturn and Shuttle Solid Rocket Friction Stir Shell Buckling MPCV Stage Adapter Selective Laser RS-25 (SSME) Core Booster Test Welding for Core Structural Test Assembly Melting Engine Stage Engines in Stage Parts Inventory Test of RS25 Core Stage Engine for Space Launch System Basics 5 • Free Vibration, Undamped Single Degree of Freedom System u F  mu x m mu  Ku  0 1) Steady State, simplest, worth remembering: u(t)  Acos(wt) Assume solution u=u(t) is of form u(t)  Awsin(wt) u(t)  Aw2 cos(wt) Now plug these equalities into eq of motion: m(Aw2 coswt)  k(Acoswt)  0 Acoswt(k w2m)  0 For A coswt = 0, A has to = 0 , i.e., no response (“trivial solution”) Therefore, k w2m  0 k k w2   w= Rad sec m m Define l ≡ Eigenvalue = w2 ≡ Natural Frequency2 So, solution for u= u(t) is where A depends on the u(t)  A cos( k t) initial conditions m Response to Harmonic Excitation • W = Excitation Frequency k • p = Harmonic Excitation Force m p=F cosWt o k • w = System Natural Frequency = m c • z = critical damping ratio = c  c c 2 km critical eq. of motion: mu  cu  ku  F cosWt o Now, define static response U to force F using st o F F  kU  U  o o st st k then we can define the "Complex Frequency Response" Dynamic Response U H(W)= U(W)  H(W)U Static Response U st st 1 H(W)  W where we define the Frequency Ratio r  1 r2 2 2zr2 w Resonance is defined when W w, ie, r=1. 1   At r=1, H W   Quality Factor Q 2z Frequency Response Example Example: F=2; c=0.6; m=1; k=9 k w  3 m |U| c z  0.1 2 km F U  o  0.222 static k At resonance, |U|=Q U = static r 1 |U | (.2222) 1.111 2z Demo: Joe- Bob the Bungee Jumper For W  2.8, r= W  2. 8  . 9 3 3 3 , so w 3 1 1 H(W)    4.408 1 r22  2zr2 10.9333322  2*.1*0.933332 Modal Analysis of Multiple DOF Systems 8 Solutions for Undamped, Free Vibration of MDOF Systems with N dof's. M uKu{0}     Assume solution of form (m spatial solutions  eigenvectors=modes) {u} {} ei(wmtm) m m m=1,M, where M N Continuous Discrete MDOF w(x) U (x)  {} m m u u 61  u (t)  ei(w1t1) u 21 2m 2m 41 u 21  x 41 t (sec)  61 w(x) Other Spatial Solutions are other Mode Shapes 9 Clamped-Free Boundary Conditions Mode 1 at f hz 1 Mode 2 at f hz 2 Mode 3 at f hz 3 Now, if resonance, forced response required, need to know about Generalized Coordinates/Modal Superposition • Frequency and Transient Response Analysis uses Concept of Modal Superposition using Generalized (or Principal Coordinates).  ]   ]   ]    M u  C u  K u  P(t) • Mode Superposition Method – transforms to set of uncoupled, SDOF equations that we can solve using SDOF methods. • First obtain [F] . Then introduce coordinate transformation: mass M    ]  u  F  N M  ] ]   ] ]   ] ]    M F   C F   K F   P(t) Generalized (or Modal) Force - dot product of each mode with  ]   ]   ]   ]T   excitation force vector `I  C`   `   F P(t) . \ \ \ - means response directly proportional to similarity of spatial shape of each mode with spatial shape of the force (Orthogonality).