美国麻省理工大学：《结构力学》（英文版） Unit 19 General Dynamic Considerations

MT-1620 al.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 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

MT-1620 al.2002 VI.(Introduction to Structural dynamics Paul A Lagace @2001 Unit 19-2

MIT - 16.20 Fall, 2002 VI. (Introduction to) Structural Dynamics Paul A. Lagace © 2001 Unit 19 - 2

MT-1620 al.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 F t多 Paul A Lagace @2001 Unit 19-3

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

MT-1620 al.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 F(t F q 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

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

MT-1620 al.2002 B)Push with an oscillating magnitude Figure 19.4 Representation of force with oscillating magnitude 七 The response also oscillates Figure 19.5 Representation of oscillating response Paul A Lagace @2001 Unit 19-5

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

MT-1620 al.2002 C)Whack mass with a hammer Force is basically a unit impulse Figure 19.6 Representation of unit impulse force F 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 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

MT-1620 al.2002 Represent this as Figure 19.7 Representation of regions of structural response versus frequency of forcing function Quasi )-Static Dynamics Wave Propagation functon 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 Paul A Lagace @2001 Unit 19-7

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

MT-1620 al.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 fnatural f(speed of frequency of waves in structure material 22424= region of transition Paul A Lagace @2001 Unit 19-8

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

MT-1620 al.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 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

MT-1620 al.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 Forcellength →Ft The mass is subjected to some force which is a function of time The position of the mass is defined by the parameter g Both F and g 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 19-10

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 = − m q˙˙ Paul A. Lagace © 2001 Unit 19 - 10