MIT - 16.20 Fall, 2002 Unit 19 General Dynamic Considerations Reference: Elements of Vibration Analysis , Meirovitch , McGraw-Hill, 1975. Paul A. Lagace, Ph.D. Professor of Aeronautics & Astronautics and Engineering Systems Paul A. Lagace ? 2001 MIT - 16.20 Fall, 2002 VI. (Introduction to) Structural Dynamics Paul A. Lagace ? 2001 Unit 19 - 2 MIT - 16.20 Fall, 2002 Thus far have considered only static response. However, things also move, this includes structures. Can actually identify three “categories ” of response: A. (Quasi) - Static [ “ quasi ” because the load must first be applied] B. Dynamic C. Wave Propagation What is the key consideration in determining which regime one is in? --> the frequency of the forcing function Example : Mass on a Spring Figure 19.1 Representation of mass on a spring Paul A. Lagace ? 2001 Unit 19 - 3 qt MIT - 16.20 Fall, 2002 A) Push very slowly Figure 19.2 Representation of force increasing slowly with time t = time The response is basically determined by: F = k q Ft ? () = () ≈ F k k Figure 19.3 Deflection response versus time for mass in spring with loads slowly increasing with time (F/k) at any point Paul A. Lagace ? 2001 Unit 19 - 4 MIT - 16.20 Fall, 2002 B) Push with an oscillating magnitude Figure 19.4 Representation of force with oscillating magnitude The response also oscillates Figure 19.5 Representation of oscillating response Paul A. Lagace ? 2001 Unit 19 - 5 MIT - 16.20 Fall, 2002 C) Whack mass with a hammer ? Force is basically a unit impulse Figure 19.6 Representation of unit impulse force Force has very high frequencies Response is (structural) waves in spring with no global deflection Paul A. Lagace ? 2001 Unit 19 - 6 MIT - 16.20 Fall, 2002 --> Represent this as Figure 19.7 Representation of regions of structural response versus frequency of forcing function (Quasi) - Static Dynamics Wave Propagation Static What determines division points between regimes? --> borderline between quasi-static and dynamic is related to natural frequency of structure . Depends on: ? structural stiffness ? structural “characteristic length ” --> gives natural frequency of structure --> borderline between dynamic and waves is related to speed of waves (sound) in material . Depends on: ? modulus ? density speed = E ρ Paul A. Lagace ? 2001 Unit 19 - 7 MIT - 16.20 Fall, 2002 --> These are not well-defined borderlines ? depends on specifics of configuration ? actually transition regions, not borders ? interactions between behaviors So illustration is: Figure 19.8 Representation of regions of structural response versus frequency of forcing function (Structural) Wave (Quasi) - Static Dynamics Propagation Static f(natural f(speed of frequency of waves in structure) material) = region of transition Paul A. Lagace ? 2001 Unit 19 - 8 MIT - 16.20 Fall, 2002 Statics -- Unified and 16.20 to date Waves -- Unified (Structural) Dynamics -- 16.221 (graduate course). Look at what we must include/add to our static considerations Consider the simplest ones … The Spring-Mass System Are probably used to seeing it as: Figure 19.9 General representation of spring-mass system Paul A. Lagace ? 2001 Unit 19 - 9 MIT - 16.20 Fall, 2002 For easier relation to the structural configuration (which will later be made), draw this as a rolling cart of mass attached to a wall by a spring: Figure 19.10 Alternate representation of spring-mass system [Force/length] k ? The mass is subjected to some force which is a function of time ? The position of the mass is defined by the parameter q ? Both F and q are defined positive in the positive x-direction Static equation: F = kq ? What must be added in the dynamic case? Inertial load(s) = - mass x acceleration In this case: inertial load = ? mq ˙˙ Paul A. Lagace ? 2001 Unit 19 - 1 0 q MIT - 16.20 Fall, 2002 where: d ? () = dt (derivative with respect to time) Drawing the free body diagram for this configuration: Figure 19.11 Free body diagram for spring-mass system ∑ F = 0 ? F ? k q ? m ˙˙ = 0 ? mq ˙˙ + k q = F t () Basic spring-mass system (no damping) This is a 2nd order Ordinary Differential Equation in time . When the Ordinary/Partial Differential Equation is in space , need Boundary Conditions. Now that the Differential Equation is in time, need Initial Conditions. Paul A. Lagace ? 2001 Unit 19 - 1 1 MIT - 16.20 Fall, 2002 2nd Order ? need 2 Initial Conditions Here: t = 0 q = 0 ˙ q = 0 @ some initial values given (may often be zero) Will look at how to solve this in the next unit. There is another consideration that generally occurs in real systems -- DAMPING . For the spring-mass system, this is represented by a dashpot with a constant c which produces a force in proportion to the velocity: Figure 19.12 Representation of spring-mass system with damping [Force/length] [Force/length/time] q ˙ q Paul A. Lagace ? 2001 Unit 19 - 1 2 MIT - 16.20 Fall, 2002 Here the free body diagram is: Figure 19.13 Free body diagram of spring-mass system with damping ∑ F = 0 ? mq ˙˙ + cq ˙ + k q = F t () Basic spring-mass system (with damping) From here on: neglect damping Can build on what has been done and go to a … Paul A. Lagace ? 2001 Unit 19 - 1 3 MIT - 16.20 Fall, 2002 Multi-Mass System For example, consider two masses linked by springs: Each mass has stiffness, (k i ) mass (m i ) and force (F i ) with associated deflection, q i Figure 19.14 Representation of multi-mass (and spring) system Consider the free body diagram for each mass: Paul A. Lagace ? 2001 Unit 19 - 1 4 q MIT - 16.20 Fall, 2002 ? Mass1 Figure 19.15 Face body diagram of Mass 1 in multi-mass system ∑ F = 0 yields: F 1 + k 2 ( q 2 ? q 1 ) ? k q ? m 11 = 0 11 ˙˙ Paul A. Lagace ? 2001 Unit 19 - 1 5 q MIT - 16.20 Fall, 2002 ? Mass 2 Figure 19.16 Free body diagram of Mass 2 in multi-mass system ∑ F = 0 yields: F 2 ? k 2 ( q 2 ? q 1 ) ? m 2 ˙˙ 2 = 0 Rearrange and unite these as (grouping terms): mq ˙˙ + ( k 1 + k 2 ) q 1 ? k 2 q 2 = F 1 11 mq 2 ? k 2 q 1 + kq 2 = F 2 2 ˙˙ 2 --> Two coupled Ordinary Differential Equations Paul A. Lagace ? 2001 Unit 19 - 1 6 q q MIT - 16.20 Fall, 2002 Write in matrix form: ? m 1 0 ? ? ˙˙ 1 ? ? ( k 1 + k 2 ) ? k 2 ? ? q 1 ? ? F 1 ? ? 0 m 2 ? ? ˙˙ 2 ? ? ? ? = ? ? ? ? ? ? + ? ? ? k 2 k 2 ? ? q 2 ? ? F 2 ? or: mq ˙˙ + kq = F ~~ ~~ ~ mass stiffness matrix matrix Note that the stiffness matrix is symmetric (as it has been in all other considerations) k ij = k ji This formulation can then be extended to 3, 4 ….n masses with m i = mass of unit i k i = stiffness of spring of unit i q i = displacement of unit i F i = force acting on unit i etc. Paul A. Lagace ? 2001 Unit 19 - 1 7 MIT - 16.20 Fall, 2002 Will next consider solutions to this equation. But first talk about why these considerations are important in structures. First issue -- what causes such response are: Dynamic Structural Loads Generic sources of dynamic loads: ? Wind (especially gusts) ? Impact ? Unsteady motion (inertial effects) ? Servo systems ? ? ? How are these manifested in particular types of structures? Paul A. Lagace ? 2001 Unit 19 - 1 8 MIT - 16.20 Fall, 2002 Aircraft ? Gust loads and turbulence flutter ( aeroelasticity is interaction of aerodynamic, elastic and inertial forces) ? Servo loads (and aero loads) on control surfaces Spacecraft Paul A. Lagace ? 2001 Unit 19 - 1 9 MIT - 16.20 Fall, 2002 Automobiles, Trains, etc. Civil Structures Earthquakes and Buildings What does this all result in? A response which is comprised of two parts: ? rigid-body motion ? elastic deformation and vibration of structure Paul A. Lagace ? 2001 Unit 19 - 2 0 MIT - 16.20 Fall, 2002 Note that: ? Peak dynamic deflections and stresses can be several times that of the static values ? Dynamic response can (quickly) lead to fatigue failure (Helicopter = a fatigue machine!) ? Discomfort for passengers (think of a car without springs) So there is a clear need to study structural dynamics Before dealing with the continuous structural system, first go back to the simple spring-mass case and learn: ? Solutions for spring-mass systems ? How to model a continuous system as a discrete spring-mass system then … ? Extend the concept to a continuous system Paul A. Lagace ? 2001 Unit 19 - 2 1