Fundamentals of Fundamentals of Structural Vibration Speaker: - PDF document

Fundamentals of Fundamentals of Structural Vibration Speaker: Speaker: Prof. FUNG Tat Ching Date & Time: Wed 20 August 2014, 1:30 - 5:30 pm Venue: CEE Seminar Room D (N1-B4C-09B) School of Civil and Environmental Engineering N Nanyang Technological University T h l i l U i it 1 Topics in Fundamentals of Structural Vibration (1.5 hrs) � SDoF Systems � MDoF Systems � Dynamic Equilibrium y q � Mode Shapes p � Natural Freq/Period � Modal decomposition � Damping ratio Damping ratio � Modal responses Modal responses � Phase lag � Disp Resp Factor � Disp Resp Factor � Response Spectrum 2

Course Outline for CV6103 Lecture Course Content Chapters in (21 hours) Textbook 1 1 Single-Degree-of-Freedom Systems Si l D f F d S t Equations of motion 1.2 – 1.6 Free vibration 2.1 – 2.2 2 Single-Degree-of-Freedom Systems 3.1 – 3.4 Response to harmonic and periodic excitations 3.12 – 3.13 3 Single-Degree-of-Freedom Systems g g y Response to arbitrary, step and pulse excitations 4.1 – 4.11 4 Multi-Degree-of-Freedom Systems Equations of motion Equations of motion 9.1 – 9.2 9.1 9.2 Natural vibration frequencies and modes 10.1 – 10.7 5 Multi-Degree-of-Freedom Systems Free vibration response Free vibration response 10 8 – 10 15 10.8 – 10.15 6 Multi-Degree-of-Freedom Systems Forced vibration response 12.1 – 12.7 7 7 S Systems with Generalized Degrees of Freedom t ith G li d D f F d Generalized coordinates and their applications 14.3, 17.1 3 Textbook and References ● Main Text ● Chopra, A. K., Dynamics of Structures: Theory Ch A K D i f St t Th and Applications to Earthquake Engineering, Prentice Hall 4 rd Edition 2011 Prentice Hall, 4 rd Edition, 2011. ● References ● References ● Clough, R. W., and Penzien, J., Dynamics of Structures, Dynamics of Structures, McGraw-Hill, 1993. ● Meirovitch, L. Fundamentals of , Vibrations, McGraw-Hill, 2001. 4

Why Is There A Need To Do Dynamic Analysis? ● Static analysis ● External Load = Internal Force ● External Load Internal Force � Magnitude of loading & stiffness ● Dynamic analysis Dynamic analysis ● External Load ≠ Internal Force � Magnitude of loading & stiffness f & ff � Frequency characteristics of loading, and the dynamic properties of structures (mass stiffness damping) properties of structures (mass, stiffness, damping) 5 Examples of SDOF Systems � Water tank � � Mass concentrated at one location Mass concentrated at one location � Supports assumed to be massless � � Can be modeled as a SDOF system C b d l d SDOF t � Pendulum � Rod is assumed to be massless � Only allowed to rotate about hinge � � Can be modelled as a SDOF system Can be modelled as a SDOF system 6

Equation of Motion Damping force Restoring force & & m u f f p + + = D S External force External force Newton’s Second Law of Motion: p f f m u & & − − = S D Inertia force D’Alembert’s Principle (Dynamic equilibrium): (Dynamic equilibrium): p f f f 0 with f m u & & − − − = = S D I I 7 Single-degree-of-freedom (SDoF) Systems u(t) c Typical m representation: p ( t ) p ( ) (mass) (mass) Mass-spring-damper system k Assumptions: � External force p ( t ) � External force p ( t ) � Linear elastic restoring force � Restoring force ku d d u � Li � Linear viscous damping i d i c d � Damping force t 2 d u & & & m u c u k ku p ( t ( t ) ) � I � Inertia force ti f m + + = 2 d t 8

Undamped Free Vibration & & & m u c u ku p ( t ) ● Equation of Motion: + + = ● with c = 0, p ( t ) = 0 k 2 & & m & & u u u u 0 0 m u u + ku + ku 0 0 or + + ω ω = ω ω n = = n m & = & ● Initial conditions: u u ( 0 ), u u ( 0 ) = ● Exact Solution: u u & ( ( 0 0 ) ) u ( t ) u ( 0 ) cos t sin t = ω + ω n n ω n ( ) See Page 46 in Chopra u ( t ) u cos t or = ω − θ max n 2 2 ⎛ ⎛ ⎞ ⎞ u & ( 0 ) u & ( 0 ) 2 ( ) ⎜ ⎟ tan 1 u u ( 0 ) − θ = = + ⎜ ⎟ 9 max u ( 0 ) ω ⎝ ω ⎠ n n Free Vibration of a System without y Damping u max u max u max 10

Periods of Vibration of Common Structures Common Structures Period 20-story moment resisting frame 1.9 sec 10-story moment resisting frame 1.1 sec 1-story moment resisting frame 0.15 sec 20-story braced frame 1.3 sec 10-story braced frame 0.8 sec 1-story braced frame 0.1 sec Gravity dam 0.2 sec Suspension bridge 20 sec 11 Viscously Damped Free Vibration m u & & c u & ku 0 Equation of Motion: + + = c & k u & & u u 0 0 Divided by m : + + = m m c c c c (reasons will be clear later) ζ = = Let 2 m c ω n cr zeta k 2 & & ζω & u 2 u u 0 ω n = Hence + + ω = n n m as before c 2 m 2 km Note: = ω = c cr n � critical damping ratio 12

Type of Motion ζ : damping ratio c cr : critical damping coefficient Three scenarios ζ < 1, i.e. c < c cr ⇒ under-damped (oscillating) � ζ ζ = 1, i.e. c = c cr ⇒ critically damped � ζ > 1, i.e. c > c cr ⇒ over-damped � 13 Typical Damping Ratio Structure ζ Welded steel frame e ded stee a e 0.010 0 0 0 Bolted steel frame 0.020 Uncracked prestressed concrete 0.015 Uncracked reinforced concrete 0.020 Cracked reinforced concrete Cracked reinforced concrete 0 035 0.035 Glued plywood shear wall p y 0.100 Nailed plywood shear wall 0.150 Damaged steel structure 0.050 Damaged concrete structure 0.075 Structure with added damping 0.250 14

Effects of Damping in Free Vibration ⎛ ⎛ ⎞ ⎞ & u u ( ( 0 0 ) ) u u ( ( 0 0 ) ) + + ζω ζω t t ⎜ ⎜ ⎟ ⎟ − ζω ζω n u ( ( t ) ) e u ( ( 0 0 ) ) cos t sin i t = ω + ω n ⎜ ⎟ D D ⎝ ω ⎠ D 2 1 ζ 1 ζ ω ω = = ω ω − D n T T n T = D ρ = 2 1 ζ 15 u − max Decay of Motion One way to measure damping is from rate of decay from free vibration decay from free vibration T D T D ( ) t − ζω u ( t ) e u cos t = ω − θ n max D (exactly) ( y) Since peaks are separated by T D , ⎛ ⎞ ζω t − u e 2 πζ ζ n ⎜ ⎜ ⎟ ⎟ ( ( ) ) i i exp exp T T exp exp = = ζω ζω = ⎜ ⎟ n D ( ) t T u − ζω + e 2 n D 1 − ζ ⎝ ⎠ 16 i 1 +

Type of Excitations Steady-state ● Harmonic / Periodic Excitations Responses ● Commonly encountered in engineering ● Commonly encountered in engineering � Unbalanced rotating machinery � Wave loading � Wave loading ● Basic components in more general periodic excitations excitations � Fourier series representation ● More General Excitations More General Excitations T Transient i t Responses ● Step/Ramp Forces ● Pulses Excitations 17 Equation of Motion m u & & c u & ku p ( t ) + + = Resonance ● Linear ● ⇒ u ( t ) can be replaced ● ⇒ u ( t ) can be replaced by u ( t ) + any free vibration responses p / 2 ω ω > n ● For example, Rapidly Slowly loaded loaded ● c = 0, u ( t ) = u 0 sin ω t c = 0 u ( t ) = u sin ω t loaded ● (- m ω 2 + k ) u 0 sin ω t = p ( t ) ● ⇒ p ( t ) = p 0 sin ω t p p 1 k k u 0 0 = ω n = 0 2 m k ( ) 1 / − ω ω 18 n

Excitation p p sin t sin t ω ω 0 ω 0 . 2 = ω n u u ( ( 0 0 ) ) 0 0 = = Response p ω u & ( 0 ) n 0 = k k p 1 0 u p ( t ) sin t = ω 2 k k ( ( ) ) 1 1 − ω / / ω ω ω n 19 Undamped Resonant Systems ● For resonance, ω = ω n Derivation: See Page 70 & 72 in Chopra 1 p ( ( ) ) 0 u u ( ( t t ) ) t t cos cos t t sin sin t t = = − ω ω ω ω − ω ω n n n 2 k & u ( 0 ) 0 , u ( 0 ) 0 = = ● Response grows indefinitely p g y ● Becomes infinite after infinite duration 20

Harmonic Vibration of Viscous Damping Equation of Motion Equation of Motion k Sinusoidal force ω n = & & & m m u u c c u u ku ku p p sin sin t t m m + + + + = ω ω 0 0 Excitation frequency Amplitude of force Amplitude of force Particular Solution S u p ( t ) C sin t D cos t Derivation: See = ω + ω p Page 73 in Chopra Page 73 in Chopra 2 p 1 ( / ) − ω ω 0 n C = 2 2 2 2 2 2 k k [ [ 1 1 ( ( / / ) ) ] ] [ [ 2 2 / / ] ] − ω ω + + ζ ζω ω n n p 2 / − ζω ω 0 n D D = = 2 2 2 k [ 1 ( / ) ] [ 2 / ] − ω ω + ζω ω n n 21 Steady-State Solution The particular solution can also be written as: ( ( ) ) u u p ( ( t t ) ) u u sin sin t t = = ω − ω φ φ 0 D tan 1 − 2 2 − u u C C D D tan where where = + + φ φ = 0 0 C Using previously derived results for C and D , p 1 0 u u = = max max 0 0 k k [ [ ] ] 2 2 2 2 ( ) [ ( ) ] 1 / 2 / − ω ω + ζ ω ω n n ( ) 2 / ζ ω ω 1 n tan − φ = 2 ( ( ) ) 1 1 / / − ω ω ω ω n 22