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 are we here? Why do I have a job? 2 • From “Characteristics of Space Shuttle Main Engine Failures”, H. Cikanek, AIAA Joint Propulsion Conference, 1987, San Diego, CA Failure of Lox Inlet Splitter to Nozzle Blows Engine – “During development and Out of Santa Susanna Test Stand 1985 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.” 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, Testing, and Operation of Rocket Engines: – How Rocket Engines Work – Turbomachinery – Rocket Nozzles – Rocket Engine Loads – System Hardware and Propellant Feedlines 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 of Structural Dynamics –Natural Frequency of SDOF System 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 Solution is Eigenvalue l = w2   w= Rad sec m m Define w ≡ Natural Frequency 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 k • W = Excitation Frequency • p = Harmonic Excitation Force m p=F cosWt o • w = System Natural Frequency = k lb c m in sec eq. of motion: mu  cu  ku  F cosWt • z = critical damping ratio = c  c o c 2 km critical   steady state response part of the solution is u (t)) U cos Wt  ss F define static response U to force F using F  kU U  o st o o st st k and define the "Complex Frequency Response" Dynamic Response U at frequency W H(W)= U(W)  H(W)U Static Response U st st 1 Long Derivation H(W)  W 1 r2 2 2zr2 where we define the Frequency Ratio r  w Resonance is defined when W w, ie, r = 1. 1   At r =1, H W   Quality Factor Q 2z Frequency Response Example P=2 lb * cos(Wt) , m=1 lbsec2 in c=0.6 lb  s e c , k=9lb in in k 9 w   3 m 1 |U| c 0.6 z   0.1 2 km 2 (9)(1) F 2 U  o   0.222 static k 9 At resonance, |U|=Q U = static r 1 |U | (.2222) 1.111 2z Demo: Joe- Bob the 2.8 Bungee Jumper For example, at W  2.8, r= W   . 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 Discrete DOF Systems 8 Solutions for Undamped, Free Vibration of MDOF Systems with N dof's. M uKu{0}     Assume m solutions (  eigenvectors modes) of form {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  F  .. N M where 1 2 3 M  ] ]   ] ]   ] ]    M F   C F   K F   P(t) Generalized (or Modal) Force - math dot product of each mode with  ]   ]   ]   ]T   excitation force vector `I   `z   `l   F P(t) . \ m\ m\ - means response directly proportional to similarity of spatial shape of each mode with spatial shape of the force (Orthogonality).