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SDEE: Lectures 1 and 2

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Alessandro Palmeri

This document provides an introduction and overview of a module on Structural Dynamics and Earthquake Engineering. It discusses motivations for studying structural dynamics, including examples of structural failures like the Tacoma Narrows Bridge and Millennium Bridge that collapsed due to resonant vibrations. It outlines the aims and content of the module, which will cover equations of motion for single-degree-of-freedom and multi-degree-of-freedom oscillators, modal analysis, damped structures, and seismic analysis including response spectra. The document concludes with the assessment structure and schedule for the module.

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Structural Dynamics & Earthquake Engineering Dr Alessandro Palmeri Motivations for this module Introduction to the module Equations of motion for SDoF oscillators Structural Dynamics & Earthquake Engineering Lecture #1: Introduction + Equations of Motion for SDoF Oscillators Dr Alessandro Palmeri Architecture, Building & Civil Engineering @ Loughborough University Tuesday, 3rd October 2017
Structural Dynamics & Earthquake Engineering Dr Alessandro Palmeri Motivations for this module Introduction to the module Equations of motion for SDoF oscillators Why bother about Structural Dynamics? http://www.youtube.com/watch?v= XggxeuFDaDU http://en.wikipedia.org/wiki/ Tacoma_Narrows_Bridge_(1940) The 1940 Tacoma Narrows Bridge It was a steel suspension bridge in the US state of Washington. Construction began in 1938, with the opening on 1st July 1940 From the time the deck was built, it began to move vertically in windy conditions (construction workers nicknamed the bridge Galloping Gertie). The motion was observed even when the bridge opened to the public. Several measures to stop the motion were ineffective, and the bridge’s main span finally collapsed under 64 km/h wind conditions the morning of 7th
Structural Dynamics & Earthquake Engineering Dr Alessandro Palmeri Motivations for this module Introduction to the module Equations of motion for SDoF oscillators Why bother about Structural Dynamics? http://www.youtube.com/watch?v= eAXVa__XWZ8 http://en.wikipedia.org/wiki/ Millennium_Bridge,_London The Millennium Bridge It is an iconic steel suspension bridge for pedestrians crossing the River Thames in London. Construction began in 1998, with the opening on 10th June 2000. Londoners nicknamed the bridge the Wobbly Bridge after participants in a charity walk to open the bridge felt an unexpected and uncomfortable swaying motion. The bridge was then closed for almost two years while modifications were made to eliminate the wobble entirely. It was reopened in 2002.
Structural Dynamics & Earthquake Engineering Dr Alessandro Palmeri Motivations for this module Introduction to the module Equations of motion for SDoF oscillators Why bother about Structural Dynamics? http: //dx.doi.org/10.1063/1.2963929 http://www.enea.it/en/en/ research-development/ new-technologies/ materials-technologies Shaking Table Tests The video shows the last, destructive shaking table test conducted in 2007 on a 1:2 scale model of a masonry building resembling a special type (the so-called Tipo Misto Messinese), which is proper to the reconstruction of the city of Messina after the 1783 Calabria earthquake. The model, incorporating a novel timber-concrete composite slab, has been tested on the main shaking table available at the ENEA Research Centre Casaccia in Rome, both before and after the reinforcement with FRP materials.
Structural Dynamics & Earthquake Engineering Dr Alessandro Palmeri Motivations for this module Introduction to the module Equations of motion for SDoF oscillators Why bother about Structural Dynamics?
Structural Dynamics & Earthquake Engineering Dr Alessandro Palmeri Motivations for this module Introduction to the module Equations of motion for SDoF oscillators Aim of the Module This module is aimed at: developing knowledge and understanding of vibrational problems in structural engineering and to provide the basic analytical and numerical tools to assess the dynamic response of structures with special emphasis on the seismic analysis and design to Eurocode 8
Structural Dynamics & Earthquake Engineering Dr Alessandro Palmeri Motivations for this module Introduction to the module Equations of motion for SDoF oscillators Module Description: Parts 1 and 2 Dynamics of single-degree-of-freedom (SDoF) oscillatiors Equations of motion for SDoF oscillators Modelling and evaluation of damping in SDoF oscillators Dynamic response of SDoF oscillators in both time and frequency domains Dynamics of multi-degree-of-freedom (MDoF) structures Equations of motion for MDoF structures Modal analysis and mode superposition method Classically and non-classically damped MDoF structures Dynamic response of MDoF structures in both time and frequency domains
Structural Dynamics & Earthquake Engineering Dr Alessandro Palmeri Motivations for this module Introduction to the module Equations of motion for SDoF oscillators Module Description: Part 3 Earthquake engineering Engineering seismology (basics) and seismic action on structures Elastic response spectrum and ductility-dependent design spectra Lateral force method Response spectrum method Design options to improve the seismic performance of existing and new buildings A look into the (near) future: performance-based earthquake engineering (PBEE)
Structural Dynamics & Earthquake Engineering Dr Alessandro Palmeri Motivations for this module Introduction to the module Equations of motion for SDoF oscillators Assessment of Module 2-hour examination (65% of the module) Section A (50% of the exam marks) on Parts 1 and 2 Solving 2 out of 3 questions (each one weighting 25%) Section B (50% of the exam marks) on Part 3 Solving 1 out of 2 questions (including some discursive reasonings Coursework comprising a group written assignment based on Part 1 (35% of the module) Issued in Week 3 (Tuesday 17th October 2017) Handed in Week 9 (Monday 27th November 2016) Returned in Week 12 (Tuesday 9th January 2018)
Structural Dynamics & Earthquake Engineering Dr Alessandro Palmeri Motivations for this module Introduction to the module Equations of motion for SDoF oscillators Schedule of the Module Lecture #1 Lecture #2 Tutorial Week 1 Introduction Equation of motion for SDoF oscillators Part 1 (Tue) Fourier analysis for structural dynamics applications Frequency response function for SDoF oscillators — Week 2 State-space formulation for SDoF oscillators Energy dissipation devices for vibration control Part 1 (Tue) Equations of motion for undamped MDoF structures Modal analysis — Week 3? Classically and non-classically damped MDoF structures Newmark- method Part 2 (Tue) State-space formulation for MDoF structures Coursework brief Part 2 (Fri) 4th -order Runge-Kutta method — — Week 7 Basics of engineering seismology Strong motion characteristics Part 2 Week 8 Elastic response spectrum Ductility µ, behaviour factor q and design spectra Part 3 Week 9† Lateral force method Part 3 Week 10 Response spectrum method Part 3 Week 11 Performance-based earthquake engineering Part 3 Christmas break Week 12 ‡ Revision — ? Coursework brief is issued in Week 3 † Group report is handed in Week 9 – On Monday 27th November 2017 ‡ Feedback is given in Week 12 (before the Exams)
Structural Dynamics & Earthquake Engineering Dr Alessandro Palmeri Motivations for this module Introduction to the module Equations of motion for SDoF oscillators Timetable of the module Extra sessions in weeks 1 to 3, to support the development of the coursework No lectures and tutorials in weeks 4 to 6 Weekly bookable one-hour surgery sessions throughout the term
Structural Dynamics & Earthquake Engineering Dr Alessandro Palmeri Motivations for this module Introduction to the module Equations of motion for SDoF oscillators Changes introduced this year 1 Slightly simplified the coursework, without changing the eight (still 35%) 2 REVIEW Lecture Capture 3 One-hour surgery session every week Starting from Week 2 Typically, Wednesday 11:30am to 12:30pm, in my office (RT.0.15) Typically, 10 minutes per student/group (requires pre-booking) Can be rearranged in some weeks, e.g. Friday 3:00 to 4:00pm in Week 4
Structural Dynamics & Earthquake Engineering Dr Alessandro Palmeri Motivations for this module Introduction to the module Equations of motion for SDoF oscillators Questions?
Structural Dynamics & Earthquake Engineering Dr Alessandro Palmeri Motivations for this module Introduction to the module Equations of motion for SDoF oscillators Newton’s Second Law of Motion Sir Isaac Newton (25 Dec 1642 – 20 Mar 1726) The acceleration ! a of a body is parallel and directly proportional to the net force ! F acting on the body, is in the direction of the net force, and is inversely proportional to the mass m of the body, i.e. ! F = m ! a (1)
Structural Dynamics & Earthquake Engineering Dr Alessandro Palmeri Motivations for this module Introduction to the module Equations of motion for SDoF oscillators Newton’s Second Law of Motionr What are the implications of Newton’s Second Law of Motion from the point of view of a structural engineer? Structures are not rigid, therefore they always experience deformations when subjected to loads The acceleration a is the rate of variation of the velocity v, which in turn is the rate of variation of the displacement u, that is: a(t) = d dt v(t) = d2 dt2 u(t) (2) It follows that, if the displacements vary rapidely, the inertial forces (proportional to the masses) have to be included in the dynamic equilibrium of the structure
Structural Dynamics & Earthquake Engineering Dr Alessandro Palmeri Motivations for this module Introduction to the module Equations of motion for SDoF oscillators Equation of Motion: Unforced Undamped SDoF Oscillator u(t) k m The figure on the left-hand side shows a typical graphical representation of an unforced and undamped Single- Degree-of-Freedom (SDoF) oscillator, which consists of: A rigid mass, ideally moving on frictionless wheels, whose value m is usually expressed in kg (or Mg= 103 kg= 1 ton) An elastic spring, whose stiffness k is usually expressed in N/m (or kN/m)
Structural Dynamics & Earthquake Engineering Dr Alessandro Palmeri Motivations for this module Introduction to the module Equations of motion for SDoF oscillators Equation of Motion: Unforced Undamped SDoF Oscillator When the oscillator is displaced (u(t) is positive if directed to the right), then an elastic force fk(t) = k u(t) arises in the spring at the generic time t, and is applied to the left on the mass According to Newton’s Second Law of Motion, the acceleration a(t) is given by the net force acting on the mass m at time t, divided by the mass itself, i.e. a(t) = fk(t)/m, and is directed to the left as well The acceleration a(t) is therefore negative, and the inertial force fm(t) = m a(t) felt by the oscillator is directed to the left in order to reconcile the sign conventions
Structural Dynamics & Earthquake Engineering Dr Alessandro Palmeri Motivations for this module Introduction to the module Equations of motion for SDoF oscillators Equation of Motion: Unforced Undamped SDoF Oscillator The equation of motion for this unforced and undamped SDoF oscillator reads: fm(t) + fk(t) = 0 ) m a(t) + k u(t) = 0 ) m ¨u(t) + k u(t) = 0 where the overdot means derivation with respect to time t: v(t) = d dt u(t) = ˙u(t); a(t) = d dt v(t) = ¨u(t)
Structural Dynamics & Earthquake Engineering Dr Alessandro Palmeri Motivations for this module Introduction to the module Equations of motion for SDoF oscillators Equation of Motion: Unforced Undamped SDoF Oscillator The free (i.e. unforced) vibration of an undamped SDoF oscillator is ruled by a 2nd-order homogeneous linear ordinary differential equation (ODE) with constant coefficients: m ¨u(t) + k u(t) = 0 (3) This equation can be rewritten as: ¨u(t) + !2 0 u(t) = 0 (4) where !0 is the so-called undamped natural circular frequency of vibration for the SDoF oscillator. This new quantity is expressed in rad/s and is given by: !0 = r k m (5)
Structural Dynamics & Earthquake Engineering Dr Alessandro Palmeri Motivations for this module Introduction to the module Equations of motion for SDoF oscillators Equation of Motion: Unforced Undamped SDoF Oscillator Similarly, the undamped natural period of vibration (measured in s) is: T0 = 2 ⇡ !0 (6) and the undamped natural frequency of vibration (measured in Hz= 1/s): ⌫0 = 1 T0 = !0 2 ⇡ (7) The adjective natural is used because such quantities only depend on the natural mechanical properties of the oscillator, i.e. the mass m and the stiffness k
Structural Dynamics & Earthquake Engineering Dr Alessandro Palmeri Motivations for this module Introduction to the module Equations of motion for SDoF oscillators Equation of Motion: Unforced Undamped SDoF Oscillator Golden Gate Bridge San Francisco, California The fundamental natural periods of vibration of this iconic suspension bridge, having the main span of 1,300 m, are: 18.2 s for the lateral movement 10.9 s for the vertical movement 4.43 s for the torsional movement
Structural Dynamics & Earthquake Engineering Dr Alessandro Palmeri Motivations for this module Introduction to the module Equations of motion for SDoF oscillators Equation of Motion: Unforced Undamped SDoF Oscillator We have then to solve this homogenous differential equation: ¨u(t) + !2 0 u(t) = 0 (4) and the solution can be expressed as: u(t) = C1 cos(!0 t) + C2 sin(!0 t) (8) where C1 and C2 are two arbitrary integration constants, whose values are usually determined by assigning the initial condition in terms of displacement u(0) = u0 and velocity v(0) = ˙u(0) = v0 at the initial time instant t = 0
Structural Dynamics & Earthquake Engineering Dr Alessandro Palmeri Motivations for this module Introduction to the module Equations of motion for SDoF oscillators Equation of Motion: Unforced Undamped SDoF Oscillator The first initial condition leads to: u(0) = C1 cos(!0 ⇥ 0) + C2 sin(!0 ⇥ 0) = u0 ) C1 ⇥ 1 + C2 ⇥ 0 = u0 ) C1 = u0 (9) By differentiating the displacement function u(t), given by Eq. (8), and taking into account the identity above, one gets: ˙u(t) = d dt u(t) = u0 !0 sin(!0 t) + C2 !0 cos(!0 t) We can then apply the second initial condition: ˙u(0) = u0 !0 sin(!0 ⇥ 0) + C2 !0 cos(!0 ⇥ 0) = v0 ) u0 ⇥ 0 + C2 !0 ⇥ 1 = v0 ) C2 = v0 !0 (10)
Structural Dynamics & Earthquake Engineering Dr Alessandro Palmeri Motivations for this module Introduction to the module Equations of motion for SDoF oscillators Equation of Motion: Unforced Undamped SDoF Oscillator Substituting the values of constants C1 and C2, given by Eqs. (9) and (10), the mathematical expressions of displacement and velocity of a SDoF experiencing free vibration can be obtained: u(t) = u0 cos(!0 t) + v0 !0 sin(!0 t) (11) ˙u(t) = v0 cos(!0 t) u0 !0 sin(!0 t) (12)
Structural Dynamics & Earthquake Engineering Dr Alessandro Palmeri Motivations for this module Introduction to the module Equations of motion for SDoF oscillators Equation of Motion: Unforced Undamped SDoF Oscillator By using the trigonometric identities, one can prove that: u(t) = ⇢0 cos(!0 t + '0) (13) where ⇢0 is the amplitude of the motion and '0 is the phase at the initial time instant t = 0: ⇢0 = q C2 1 + C2 2 = s u2 0 + ✓ v0 !0 ◆2 (14) tan('0) = C2 C1 = v0/!0 u0 = v0 !0 u0 (15)
Structural Dynamics & Earthquake Engineering Dr Alessandro Palmeri Motivations for this module Introduction to the module Equations of motion for SDoF oscillators Equation of Motion: Unforced Undamped SDoF Oscillator u(t) t m a) u0 ρ0 T0=2π/ω0 t b) u(t) T0=2π/ω0 u0 ρ0 u0>0 c) u(t) T0+ϕ0/ω0 u0 t ϕ0/ω0 u0<0 t d) ϕ0/ω0 u0<0 u(t) u0>0 ϕ0/ω0 T0 /2+ϕ0/ω0 Free undamped vibration of a SDoF oscillator for given initial conditions: a) u0 > 0 and ˙u0 = 0; b) u0 > 0 and ˙u0 > 0; c) u0 > 0 and ˙u0 < 0; d) u0 < 0 and ˙u0 > 0
Structural Dynamics & Earthquake Engineering Dr Alessandro Palmeri Motivations for this module Introduction to the module Equations of motion for SDoF oscillators Equation of Motion: Unforced Damped SDoF Oscillator The unrealistic feature of the solution presented in the previous slides is that the motion of the SDoF oscillator appears to continue indefinitely The reason is that the effects of energy dissipation phenomena are not considered We must include a damping force in the equation of motion, which acts in parallel with the elastic force fk (t) = k u(t), proportional to the displacement, and the inertial force fm(t) = m ¨u(t), proportional to the acceleration Damping mechanics are very difficult to model, and the simplest approach is to assume a viscous type of energy dissipation, so that the damping force is proportional to the velocity: fc(t) = c ˙u(t) (16) where c is the so-called viscous damping coefficient, usually expressed in N⇥s/m (or kN⇥s/m)
Structural Dynamics & Earthquake Engineering Dr Alessandro Palmeri Motivations for this module Introduction to the module Equations of motion for SDoF oscillators Equation of Motion: Unforced Damped SDoF Oscillator The equation of motion (with viscous damping) becomes: m ¨u(t) + c ˙u(t) + k u(t) = 0 (17) Normalisng all terms with respect to the mass m, one obtains: ¨u(t) + 2 ⇣0 !0 ˙u(t) + !2 0 u(t) = 0 (18) where ⇣0 is the so-called equivalent viscous damping ratio for the SDoF oscillator, given by: ⇣0 = c 2 m !0 = c ccr (19) in which ccr is the so-called critical damping coefficient: ccr = 2 m !0 = 2 p k m = 2 k !0 (20)
Structural Dynamics & Earthquake Engineering Dr Alessandro Palmeri Motivations for this module Introduction to the module Equations of motion for SDoF oscillators Equation of Motion: Unforced Damped SDoF Oscillator One can mathematically prove that, assuming initial conditions u(t) = u0 > 0 and v0 = 0: If c = ccr (or ⇣0 = 1), then the system is said to be critically damped, and the mass returns to the equilibrium configuration (u = 0) without oscillating If c > ccr (or ⇣0 > 1), then the system is said to be overdamped, and the mass returns to the equilibrium position at a slower rate, again without oscillating If c < ccr (or ⇣0 < 1), then the system is said to be underdamped, and the mass oscillates about the equilibrium position with a progressively decreasing amplitude
Structural Dynamics & Earthquake Engineering Dr Alessandro Palmeri Motivations for this module Introduction to the module Equations of motion for SDoF oscillators Equation of Motion: Unforced Damped SDoF Oscillator 0 1 2 3 4 1.0 0.5 0.0 0.5 1.0 T0 0 Ζ0 3 Ζ0 1 Ζ0 0.25 Ζ0 0.05 Ζ0 0
Structural Dynamics & Earthquake Engineering Dr Alessandro Palmeri Motivations for this module Introduction to the module Equations of motion for SDoF oscillators Equation of Motion: Unforced Damped SDoF Oscillator The vast majority of civil engineering structures are underdamped, and the range of values for the viscous damping ratio ⇣0 is between 0.01 and 0.07, ⇣0 = 0.05 being for instance the assumed value for reinforced concrete frames and steel frames with bolted joints The solution of Eq. (18) can be posed in the form: u(t) = e ⇣0 !0 t h C1 cos(!0 t) + C2 sin(!0 t) i (21) In this case, the displacement function u(t) is a harmonic wave of circular frequency !0 and exponentially decaying amplitude. Again, the integration constants C1 and C2 have to be determined by imposing the initial conditions
Structural Dynamics & Earthquake Engineering Dr Alessandro Palmeri Motivations for this module Introduction to the module Equations of motion for SDoF oscillators Equation of Motion: Unforced Damped SDoF Oscillator For the free vibration of an underdamped SDoF system, the time history of the deflection takes the expression: u(t) = e ⇣0 !0 t  u0 cos(!0 t) + v0 + ⇣0 !0 u0 !0 sin(!0 t) (22) or, equivalently: u(t) = ⇢0 e ⇣0 !0 t cos(!0 t + '0) (23) in which: ⇢0 = q C 2 1 + C 2 2 = s u2 0 + ✓ v0 + ⇣0 !0 u0 !0 ◆2 (24) tan('0) = C2 C1 = v0 + ⇣0 !0 u0 !0 u0 (25)
Structural Dynamics & Earthquake Engineering Dr Alessandro Palmeri Motivations for this module Introduction to the module Equations of motion for SDoF oscillators Equation of Motion: Unforced Damped SDoF Oscillator u0 t u(t) u0 > 0 ρ0e-ζ0ω0 t −ρ0e-ζ0ω0 t T0=2π/ω0 ϕ0 /ω0 Free vibration of an underdamped SDoF oscillator with initial conditions u0 > 0 and v0 = ˙u0 > 0
Structural Dynamics & Earthquake Engineering Dr Alessandro Palmeri Motivations for this module Introduction to the module Equations of motion for SDoF oscillators Equation of Motion: Unforced Damped SDoF Oscillator t u(t) 0 1 2 3 -50 -25 0 25 50 ζ0=0.05 ζ0=0.1 ζ0=0.2 ζ0=0.4 t u(t) 0 1 2 3 -50 -25 0 25 50 ζ0=0.05 t u(t) 0 1 2 3 -50 -25 0 25 50 ζ0=0.1 t u(t) 0 1 2 3 -50 -25 0 25 50 ζ0=0.2 t u(t) 0 1 2 3 -50 -25 0 25 50 ζ0=0.4
Structural Dynamics & Earthquake Engineering Dr Alessandro Palmeri Motivations for this module Introduction to the module Equations of motion for SDoF oscillators Equation of Motion: Forced SDoF Oscillator u(t) k c m F(t)=F0 sin(ωf t) a) u(t) ku(t) cu(t) . m b) mu(t) .. F(t)=F0 sin(ωf t) Let us now consider the forced vibration of a SDoF oscillator subjected to an harmonic input F(t) = F0 sin(!f t), where F0 and !f are amplitude and circular frequency of the force The figure above shows the sketch of the oscillator (a) and the forces acting on the mass m at a generic time instant t (b)
Structural Dynamics & Earthquake Engineering Dr Alessandro Palmeri Motivations for this module Introduction to the module Equations of motion for SDoF oscillators Equation of Motion: Forced SDoF Oscillator u(t) k c m F(t)=F0 sin(ωf t) a) u(t) ku(t) cu(t) . m b) mu(t) .. F(t)=F0 sin(ωf t) The equations of motion in terms of forces is: m ¨u(t) + c ˙u(t) + k u(t) = F0 sin(!f t) (26) and in terms of accelerations is: ¨u(t) + 2 ⇣0 !0 ˙u(t) + !2 0 u(t) = F0 m sin(!f t) (27)
Structural Dynamics & Earthquake Engineering Dr Alessandro Palmeri Motivations for this module Introduction to the module Equations of motion for SDoF oscillators Equation of Motion: Forced SDoF Oscillator The general solution to this inhomogeneous 2nd-order ODE is given by: u(t) = uh(t) + up(t) (28) where: up(t) is the particular integral, i.e. any function satisfying Eq. (26) (or equivalently Eq. (27)), which in turn depends on the particular forcing function F(t) uh(t) is the general solution of the related homogeneous ODE (Eqs. (17) and (18)), which is given by the time history of free vibration of the SDoF oscillator with generic integration constants C1 and C2 (see Eq. (21)) uh(t) = e ⇣0 !0 t h C1 cos(!0 t) + C2 sin(!0 t) i (29)
Structural Dynamics & Earthquake Engineering Dr Alessandro Palmeri Motivations for this module Introduction to the module Equations of motion for SDoF oscillators Equation of Motion: Forced SDoF Oscillator For a sinusoidal load, F(t) = F0 sin(!f t), the particular integral takes the expression: up(t) = ⇢p sin(!f + 'p) (30) That is, up(t) is again a sinusoidal function with the same circular frequency !f as the input, while amplitude ⇢p and phase angle 'p are given by: ⇢p = F0 k 1 q (1 2) 2 + (2 ⇣0 )2 (31) tan('p) = 2 ⇣0 1 2 (32) in which = !f/!0
Structural Dynamics & Earthquake Engineering Dr Alessandro Palmeri Motivations for this module Introduction to the module Equations of motion for SDoF oscillators Equation of Motion: Forced SDoF Oscillator ζ 0 =0 0.0 2.5 5.0 7.5 10.0 -0.50 -0.25 0.00 0.25 0.50 a) t 0.0 2.5 5.0 7.5 10.0 -0.50 -0.25 0.00 0.25 0.50 b) t ζ 0 =0.05 Dynamic response u(t) of a SDoF oscillator (solid line) compared to the particular integral up(t) (dashed line) for a sinusoidal excitation ( = !f/!0 = 0.2; !0 = 7 rad/s; F0/m = 10 m/s2 ; u0 = 0; v0 = 1.5 m/s) and two different values of the viscous damping ratio: a) ⇣0 = 0; b) ⇣0 = 0.05
Structural Dynamics & Earthquake Engineering Dr Alessandro Palmeri Motivations for this module Introduction to the module Equations of motion for SDoF oscillators Equation of Motion: Forced SDoF Oscillator Alternatively, the particular integral for a sinusoidal excitation can be expressed as: up(t) = ust D( ) sin(!f + 'p) (33) where: ust = F0 k is the static displacement due to the force F0 D( ) is the so-called dynamic amplification factor, which depends on the frequency ratio = !f/!0: D( ) = ⇢p ust = 1 q (1 2) 2 + (2 ⇣0 )2 (34)
Structural Dynamics & Earthquake Engineering Dr Alessandro Palmeri Motivations for this module Introduction to the module Equations of motion for SDoF oscillators Equation of Motion: Forced SDoF Oscillator β D 0 1 2 3 0 1 2 3 4 5 6 7 8 a) ζ0=0.0 ζ0=0.1 ζ0=0.2 ζ0=0.3 ζ0=0.5 ζ0=0.7 ζ0=1.0 0 1 2 3 0 ϕp b) π/4 π/2 3/4 π π β ζ0=0.0 ζ0=0.1 ζ0=0.2 ζ0=0.3 ζ0=0.5 ζ0=0.7 ζ0=1.0 ζ0=0.0 Dynamic amplification factor D (a) and phase angle 'p (b) as functions of the frequency ratio for different values of the viscous damping ratio ⇣0
Structural Dynamics & Earthquake Engineering Dr Alessandro Palmeri Motivations for this module Introduction to the module Equations of motion for SDoF oscillators Equation of Motion: Forced SDoF Oscillator For ! 0, the static case is approached (i.e. D ! 1), as the force F(t) varies very slowly and therefore the mass m can follows it For ! 1 and ⇣0 = 0 (undamped case), the system experience an unbounded resonance, meaning that D ! +1 and (theoretically) the amplitude of the dynamic response keeps increasing with time If = 1 and 0  ⇣0 < 0.5, then D > 1, i.e. the amplitude of the steady-state dynamic response ⇢p is larger than the static displacement ust; moreover, the less ⇣0, the larger is D If > 1.41, then D < 1, i.e. (independently of ⇣0) the amplitude of the steady-state dynamic response ⇢p is less than the static displacement ust For ! +1, the dynamic amplification factor goes to zero (i.e. D ! 0), as the force F(t) varies so rapidely that the mass m cannot follows it If ⇣0 0.707, then D < 1, meaning that (independently of the frequency ratio ) the amplitude of the steady-state dynamic response is less than the static displacement
Structural Dynamics & Earthquake Engineering Dr Alessandro Palmeri Motivations for this module Introduction to the module Equations of motion for SDoF oscillators Equation of Motion: Forced SDoF Oscillator 0 1 2 3 -3 -2 -1 0 1 2 3 β=0.1 t a) β=0.5 t0 1 2 3 -3 -2 -1 0 1 2 3 b) β=1 t0 1 2 3 -3 -2 -1 0 1 2 3 c) β=2 t0 1 2 3 -3 -2 -1 0 1 2 3 d) Normalised particular integral up(t)/ust (solid line) compared to the normalised pseudo-static response F(t)/(k ust) (dashed line) for a damped (⇣0 = 0.20) SDoF oscillator subjected to a sinusoidal force F(t) = F0 sin(!0 t), with !f = 2 ⇡, and for four different values of the frequency ratio = !f/!0

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SDEE: Lectures 1 and 2

  • 1. Structural Dynamics & Earthquake Engineering Dr Alessandro Palmeri Motivations for this module Introduction to the module Equations of motion for SDoF oscillators Structural Dynamics & Earthquake Engineering Lecture #1: Introduction + Equations of Motion for SDoF Oscillators Dr Alessandro Palmeri Architecture, Building & Civil Engineering @ Loughborough University Tuesday, 3rd October 2017
  • 2. Structural Dynamics & Earthquake Engineering Dr Alessandro Palmeri Motivations for this module Introduction to the module Equations of motion for SDoF oscillators Why bother about Structural Dynamics? http://www.youtube.com/watch?v= XggxeuFDaDU http://en.wikipedia.org/wiki/ Tacoma_Narrows_Bridge_(1940) The 1940 Tacoma Narrows Bridge It was a steel suspension bridge in the US state of Washington. Construction began in 1938, with the opening on 1st July 1940 From the time the deck was built, it began to move vertically in windy conditions (construction workers nicknamed the bridge Galloping Gertie). The motion was observed even when the bridge opened to the public. Several measures to stop the motion were ineffective, and the bridge’s main span finally collapsed under 64 km/h wind conditions the morning of 7th
  • 3. Structural Dynamics & Earthquake Engineering Dr Alessandro Palmeri Motivations for this module Introduction to the module Equations of motion for SDoF oscillators Why bother about Structural Dynamics? http://www.youtube.com/watch?v= eAXVa__XWZ8 http://en.wikipedia.org/wiki/ Millennium_Bridge,_London The Millennium Bridge It is an iconic steel suspension bridge for pedestrians crossing the River Thames in London. Construction began in 1998, with the opening on 10th June 2000. Londoners nicknamed the bridge the Wobbly Bridge after participants in a charity walk to open the bridge felt an unexpected and uncomfortable swaying motion. The bridge was then closed for almost two years while modifications were made to eliminate the wobble entirely. It was reopened in 2002.
  • 4. Structural Dynamics & Earthquake Engineering Dr Alessandro Palmeri Motivations for this module Introduction to the module Equations of motion for SDoF oscillators Why bother about Structural Dynamics? http: //dx.doi.org/10.1063/1.2963929 http://www.enea.it/en/en/ research-development/ new-technologies/ materials-technologies Shaking Table Tests The video shows the last, destructive shaking table test conducted in 2007 on a 1:2 scale model of a masonry building resembling a special type (the so-called Tipo Misto Messinese), which is proper to the reconstruction of the city of Messina after the 1783 Calabria earthquake. The model, incorporating a novel timber-concrete composite slab, has been tested on the main shaking table available at the ENEA Research Centre Casaccia in Rome, both before and after the reinforcement with FRP materials.
  • 5. Structural Dynamics & Earthquake Engineering Dr Alessandro Palmeri Motivations for this module Introduction to the module Equations of motion for SDoF oscillators Why bother about Structural Dynamics?
  • 6. Structural Dynamics & Earthquake Engineering Dr Alessandro Palmeri Motivations for this module Introduction to the module Equations of motion for SDoF oscillators Aim of the Module This module is aimed at: developing knowledge and understanding of vibrational problems in structural engineering and to provide the basic analytical and numerical tools to assess the dynamic response of structures with special emphasis on the seismic analysis and design to Eurocode 8
  • 7. Structural Dynamics & Earthquake Engineering Dr Alessandro Palmeri Motivations for this module Introduction to the module Equations of motion for SDoF oscillators Module Description: Parts 1 and 2 Dynamics of single-degree-of-freedom (SDoF) oscillatiors Equations of motion for SDoF oscillators Modelling and evaluation of damping in SDoF oscillators Dynamic response of SDoF oscillators in both time and frequency domains Dynamics of multi-degree-of-freedom (MDoF) structures Equations of motion for MDoF structures Modal analysis and mode superposition method Classically and non-classically damped MDoF structures Dynamic response of MDoF structures in both time and frequency domains
  • 8. Structural Dynamics & Earthquake Engineering Dr Alessandro Palmeri Motivations for this module Introduction to the module Equations of motion for SDoF oscillators Module Description: Part 3 Earthquake engineering Engineering seismology (basics) and seismic action on structures Elastic response spectrum and ductility-dependent design spectra Lateral force method Response spectrum method Design options to improve the seismic performance of existing and new buildings A look into the (near) future: performance-based earthquake engineering (PBEE)
  • 9. Structural Dynamics & Earthquake Engineering Dr Alessandro Palmeri Motivations for this module Introduction to the module Equations of motion for SDoF oscillators Assessment of Module 2-hour examination (65% of the module) Section A (50% of the exam marks) on Parts 1 and 2 Solving 2 out of 3 questions (each one weighting 25%) Section B (50% of the exam marks) on Part 3 Solving 1 out of 2 questions (including some discursive reasonings Coursework comprising a group written assignment based on Part 1 (35% of the module) Issued in Week 3 (Tuesday 17th October 2017) Handed in Week 9 (Monday 27th November 2016) Returned in Week 12 (Tuesday 9th January 2018)
  • 10. Structural Dynamics & Earthquake Engineering Dr Alessandro Palmeri Motivations for this module Introduction to the module Equations of motion for SDoF oscillators Schedule of the Module Lecture #1 Lecture #2 Tutorial Week 1 Introduction Equation of motion for SDoF oscillators Part 1 (Tue) Fourier analysis for structural dynamics applications Frequency response function for SDoF oscillators — Week 2 State-space formulation for SDoF oscillators Energy dissipation devices for vibration control Part 1 (Tue) Equations of motion for undamped MDoF structures Modal analysis — Week 3? Classically and non-classically damped MDoF structures Newmark- method Part 2 (Tue) State-space formulation for MDoF structures Coursework brief Part 2 (Fri) 4th -order Runge-Kutta method — — Week 7 Basics of engineering seismology Strong motion characteristics Part 2 Week 8 Elastic response spectrum Ductility µ, behaviour factor q and design spectra Part 3 Week 9† Lateral force method Part 3 Week 10 Response spectrum method Part 3 Week 11 Performance-based earthquake engineering Part 3 Christmas break Week 12 ‡ Revision — ? Coursework brief is issued in Week 3 † Group report is handed in Week 9 – On Monday 27th November 2017 ‡ Feedback is given in Week 12 (before the Exams)
  • 11. Structural Dynamics & Earthquake Engineering Dr Alessandro Palmeri Motivations for this module Introduction to the module Equations of motion for SDoF oscillators Timetable of the module Extra sessions in weeks 1 to 3, to support the development of the coursework No lectures and tutorials in weeks 4 to 6 Weekly bookable one-hour surgery sessions throughout the term
  • 12. Structural Dynamics & Earthquake Engineering Dr Alessandro Palmeri Motivations for this module Introduction to the module Equations of motion for SDoF oscillators Changes introduced this year 1 Slightly simplified the coursework, without changing the eight (still 35%) 2 REVIEW Lecture Capture 3 One-hour surgery session every week Starting from Week 2 Typically, Wednesday 11:30am to 12:30pm, in my office (RT.0.15) Typically, 10 minutes per student/group (requires pre-booking) Can be rearranged in some weeks, e.g. Friday 3:00 to 4:00pm in Week 4
  • 13. Structural Dynamics & Earthquake Engineering Dr Alessandro Palmeri Motivations for this module Introduction to the module Equations of motion for SDoF oscillators Questions?
  • 14. Structural Dynamics & Earthquake Engineering Dr Alessandro Palmeri Motivations for this module Introduction to the module Equations of motion for SDoF oscillators Newton’s Second Law of Motion Sir Isaac Newton (25 Dec 1642 – 20 Mar 1726) The acceleration ! a of a body is parallel and directly proportional to the net force ! F acting on the body, is in the direction of the net force, and is inversely proportional to the mass m of the body, i.e. ! F = m ! a (1)
  • 15. Structural Dynamics & Earthquake Engineering Dr Alessandro Palmeri Motivations for this module Introduction to the module Equations of motion for SDoF oscillators Newton’s Second Law of Motionr What are the implications of Newton’s Second Law of Motion from the point of view of a structural engineer? Structures are not rigid, therefore they always experience deformations when subjected to loads The acceleration a is the rate of variation of the velocity v, which in turn is the rate of variation of the displacement u, that is: a(t) = d dt v(t) = d2 dt2 u(t) (2) It follows that, if the displacements vary rapidely, the inertial forces (proportional to the masses) have to be included in the dynamic equilibrium of the structure
  • 16. Structural Dynamics & Earthquake Engineering Dr Alessandro Palmeri Motivations for this module Introduction to the module Equations of motion for SDoF oscillators Equation of Motion: Unforced Undamped SDoF Oscillator u(t) k m The figure on the left-hand side shows a typical graphical representation of an unforced and undamped Single- Degree-of-Freedom (SDoF) oscillator, which consists of: A rigid mass, ideally moving on frictionless wheels, whose value m is usually expressed in kg (or Mg= 103 kg= 1 ton) An elastic spring, whose stiffness k is usually expressed in N/m (or kN/m)
  • 17. Structural Dynamics & Earthquake Engineering Dr Alessandro Palmeri Motivations for this module Introduction to the module Equations of motion for SDoF oscillators Equation of Motion: Unforced Undamped SDoF Oscillator When the oscillator is displaced (u(t) is positive if directed to the right), then an elastic force fk(t) = k u(t) arises in the spring at the generic time t, and is applied to the left on the mass According to Newton’s Second Law of Motion, the acceleration a(t) is given by the net force acting on the mass m at time t, divided by the mass itself, i.e. a(t) = fk(t)/m, and is directed to the left as well The acceleration a(t) is therefore negative, and the inertial force fm(t) = m a(t) felt by the oscillator is directed to the left in order to reconcile the sign conventions
  • 18. Structural Dynamics & Earthquake Engineering Dr Alessandro Palmeri Motivations for this module Introduction to the module Equations of motion for SDoF oscillators Equation of Motion: Unforced Undamped SDoF Oscillator The equation of motion for this unforced and undamped SDoF oscillator reads: fm(t) + fk(t) = 0 ) m a(t) + k u(t) = 0 ) m ¨u(t) + k u(t) = 0 where the overdot means derivation with respect to time t: v(t) = d dt u(t) = ˙u(t); a(t) = d dt v(t) = ¨u(t)
  • 19. Structural Dynamics & Earthquake Engineering Dr Alessandro Palmeri Motivations for this module Introduction to the module Equations of motion for SDoF oscillators Equation of Motion: Unforced Undamped SDoF Oscillator The free (i.e. unforced) vibration of an undamped SDoF oscillator is ruled by a 2nd-order homogeneous linear ordinary differential equation (ODE) with constant coefficients: m ¨u(t) + k u(t) = 0 (3) This equation can be rewritten as: ¨u(t) + !2 0 u(t) = 0 (4) where !0 is the so-called undamped natural circular frequency of vibration for the SDoF oscillator. This new quantity is expressed in rad/s and is given by: !0 = r k m (5)
  • 20. Structural Dynamics & Earthquake Engineering Dr Alessandro Palmeri Motivations for this module Introduction to the module Equations of motion for SDoF oscillators Equation of Motion: Unforced Undamped SDoF Oscillator Similarly, the undamped natural period of vibration (measured in s) is: T0 = 2 ⇡ !0 (6) and the undamped natural frequency of vibration (measured in Hz= 1/s): ⌫0 = 1 T0 = !0 2 ⇡ (7) The adjective natural is used because such quantities only depend on the natural mechanical properties of the oscillator, i.e. the mass m and the stiffness k
  • 21. Structural Dynamics & Earthquake Engineering Dr Alessandro Palmeri Motivations for this module Introduction to the module Equations of motion for SDoF oscillators Equation of Motion: Unforced Undamped SDoF Oscillator Golden Gate Bridge San Francisco, California The fundamental natural periods of vibration of this iconic suspension bridge, having the main span of 1,300 m, are: 18.2 s for the lateral movement 10.9 s for the vertical movement 4.43 s for the torsional movement
  • 22. Structural Dynamics & Earthquake Engineering Dr Alessandro Palmeri Motivations for this module Introduction to the module Equations of motion for SDoF oscillators Equation of Motion: Unforced Undamped SDoF Oscillator We have then to solve this homogenous differential equation: ¨u(t) + !2 0 u(t) = 0 (4) and the solution can be expressed as: u(t) = C1 cos(!0 t) + C2 sin(!0 t) (8) where C1 and C2 are two arbitrary integration constants, whose values are usually determined by assigning the initial condition in terms of displacement u(0) = u0 and velocity v(0) = ˙u(0) = v0 at the initial time instant t = 0
  • 23. Structural Dynamics & Earthquake Engineering Dr Alessandro Palmeri Motivations for this module Introduction to the module Equations of motion for SDoF oscillators Equation of Motion: Unforced Undamped SDoF Oscillator The first initial condition leads to: u(0) = C1 cos(!0 ⇥ 0) + C2 sin(!0 ⇥ 0) = u0 ) C1 ⇥ 1 + C2 ⇥ 0 = u0 ) C1 = u0 (9) By differentiating the displacement function u(t), given by Eq. (8), and taking into account the identity above, one gets: ˙u(t) = d dt u(t) = u0 !0 sin(!0 t) + C2 !0 cos(!0 t) We can then apply the second initial condition: ˙u(0) = u0 !0 sin(!0 ⇥ 0) + C2 !0 cos(!0 ⇥ 0) = v0 ) u0 ⇥ 0 + C2 !0 ⇥ 1 = v0 ) C2 = v0 !0 (10)
  • 24. Structural Dynamics & Earthquake Engineering Dr Alessandro Palmeri Motivations for this module Introduction to the module Equations of motion for SDoF oscillators Equation of Motion: Unforced Undamped SDoF Oscillator Substituting the values of constants C1 and C2, given by Eqs. (9) and (10), the mathematical expressions of displacement and velocity of a SDoF experiencing free vibration can be obtained: u(t) = u0 cos(!0 t) + v0 !0 sin(!0 t) (11) ˙u(t) = v0 cos(!0 t) u0 !0 sin(!0 t) (12)
  • 25. Structural Dynamics & Earthquake Engineering Dr Alessandro Palmeri Motivations for this module Introduction to the module Equations of motion for SDoF oscillators Equation of Motion: Unforced Undamped SDoF Oscillator By using the trigonometric identities, one can prove that: u(t) = ⇢0 cos(!0 t + '0) (13) where ⇢0 is the amplitude of the motion and '0 is the phase at the initial time instant t = 0: ⇢0 = q C2 1 + C2 2 = s u2 0 + ✓ v0 !0 ◆2 (14) tan('0) = C2 C1 = v0/!0 u0 = v0 !0 u0 (15)
  • 26. Structural Dynamics & Earthquake Engineering Dr Alessandro Palmeri Motivations for this module Introduction to the module Equations of motion for SDoF oscillators Equation of Motion: Unforced Undamped SDoF Oscillator u(t) t m a) u0 ρ0 T0=2π/ω0 t b) u(t) T0=2π/ω0 u0 ρ0 u0>0 c) u(t) T0+ϕ0/ω0 u0 t ϕ0/ω0 u0<0 t d) ϕ0/ω0 u0<0 u(t) u0>0 ϕ0/ω0 T0 /2+ϕ0/ω0 Free undamped vibration of a SDoF oscillator for given initial conditions: a) u0 > 0 and ˙u0 = 0; b) u0 > 0 and ˙u0 > 0; c) u0 > 0 and ˙u0 < 0; d) u0 < 0 and ˙u0 > 0
  • 27. Structural Dynamics & Earthquake Engineering Dr Alessandro Palmeri Motivations for this module Introduction to the module Equations of motion for SDoF oscillators Equation of Motion: Unforced Damped SDoF Oscillator The unrealistic feature of the solution presented in the previous slides is that the motion of the SDoF oscillator appears to continue indefinitely The reason is that the effects of energy dissipation phenomena are not considered We must include a damping force in the equation of motion, which acts in parallel with the elastic force fk (t) = k u(t), proportional to the displacement, and the inertial force fm(t) = m ¨u(t), proportional to the acceleration Damping mechanics are very difficult to model, and the simplest approach is to assume a viscous type of energy dissipation, so that the damping force is proportional to the velocity: fc(t) = c ˙u(t) (16) where c is the so-called viscous damping coefficient, usually expressed in N⇥s/m (or kN⇥s/m)
  • 28. Structural Dynamics & Earthquake Engineering Dr Alessandro Palmeri Motivations for this module Introduction to the module Equations of motion for SDoF oscillators Equation of Motion: Unforced Damped SDoF Oscillator The equation of motion (with viscous damping) becomes: m ¨u(t) + c ˙u(t) + k u(t) = 0 (17) Normalisng all terms with respect to the mass m, one obtains: ¨u(t) + 2 ⇣0 !0 ˙u(t) + !2 0 u(t) = 0 (18) where ⇣0 is the so-called equivalent viscous damping ratio for the SDoF oscillator, given by: ⇣0 = c 2 m !0 = c ccr (19) in which ccr is the so-called critical damping coefficient: ccr = 2 m !0 = 2 p k m = 2 k !0 (20)
  • 29. Structural Dynamics & Earthquake Engineering Dr Alessandro Palmeri Motivations for this module Introduction to the module Equations of motion for SDoF oscillators Equation of Motion: Unforced Damped SDoF Oscillator One can mathematically prove that, assuming initial conditions u(t) = u0 > 0 and v0 = 0: If c = ccr (or ⇣0 = 1), then the system is said to be critically damped, and the mass returns to the equilibrium configuration (u = 0) without oscillating If c > ccr (or ⇣0 > 1), then the system is said to be overdamped, and the mass returns to the equilibrium position at a slower rate, again without oscillating If c < ccr (or ⇣0 < 1), then the system is said to be underdamped, and the mass oscillates about the equilibrium position with a progressively decreasing amplitude
  • 30. Structural Dynamics & Earthquake Engineering Dr Alessandro Palmeri Motivations for this module Introduction to the module Equations of motion for SDoF oscillators Equation of Motion: Unforced Damped SDoF Oscillator 0 1 2 3 4 1.0 0.5 0.0 0.5 1.0 T0 0 Ζ0 3 Ζ0 1 Ζ0 0.25 Ζ0 0.05 Ζ0 0
  • 31. Structural Dynamics & Earthquake Engineering Dr Alessandro Palmeri Motivations for this module Introduction to the module Equations of motion for SDoF oscillators Equation of Motion: Unforced Damped SDoF Oscillator The vast majority of civil engineering structures are underdamped, and the range of values for the viscous damping ratio ⇣0 is between 0.01 and 0.07, ⇣0 = 0.05 being for instance the assumed value for reinforced concrete frames and steel frames with bolted joints The solution of Eq. (18) can be posed in the form: u(t) = e ⇣0 !0 t h C1 cos(!0 t) + C2 sin(!0 t) i (21) In this case, the displacement function u(t) is a harmonic wave of circular frequency !0 and exponentially decaying amplitude. Again, the integration constants C1 and C2 have to be determined by imposing the initial conditions
  • 32. Structural Dynamics & Earthquake Engineering Dr Alessandro Palmeri Motivations for this module Introduction to the module Equations of motion for SDoF oscillators Equation of Motion: Unforced Damped SDoF Oscillator For the free vibration of an underdamped SDoF system, the time history of the deflection takes the expression: u(t) = e ⇣0 !0 t  u0 cos(!0 t) + v0 + ⇣0 !0 u0 !0 sin(!0 t) (22) or, equivalently: u(t) = ⇢0 e ⇣0 !0 t cos(!0 t + '0) (23) in which: ⇢0 = q C 2 1 + C 2 2 = s u2 0 + ✓ v0 + ⇣0 !0 u0 !0 ◆2 (24) tan('0) = C2 C1 = v0 + ⇣0 !0 u0 !0 u0 (25)
  • 33. Structural Dynamics & Earthquake Engineering Dr Alessandro Palmeri Motivations for this module Introduction to the module Equations of motion for SDoF oscillators Equation of Motion: Unforced Damped SDoF Oscillator u0 t u(t) u0 > 0 ρ0e-ζ0ω0 t −ρ0e-ζ0ω0 t T0=2π/ω0 ϕ0 /ω0 Free vibration of an underdamped SDoF oscillator with initial conditions u0 > 0 and v0 = ˙u0 > 0
  • 34. Structural Dynamics & Earthquake Engineering Dr Alessandro Palmeri Motivations for this module Introduction to the module Equations of motion for SDoF oscillators Equation of Motion: Unforced Damped SDoF Oscillator t u(t) 0 1 2 3 -50 -25 0 25 50 ζ0=0.05 ζ0=0.1 ζ0=0.2 ζ0=0.4 t u(t) 0 1 2 3 -50 -25 0 25 50 ζ0=0.05 t u(t) 0 1 2 3 -50 -25 0 25 50 ζ0=0.1 t u(t) 0 1 2 3 -50 -25 0 25 50 ζ0=0.2 t u(t) 0 1 2 3 -50 -25 0 25 50 ζ0=0.4
  • 35. Structural Dynamics & Earthquake Engineering Dr Alessandro Palmeri Motivations for this module Introduction to the module Equations of motion for SDoF oscillators Equation of Motion: Forced SDoF Oscillator u(t) k c m F(t)=F0 sin(ωf t) a) u(t) ku(t) cu(t) . m b) mu(t) .. F(t)=F0 sin(ωf t) Let us now consider the forced vibration of a SDoF oscillator subjected to an harmonic input F(t) = F0 sin(!f t), where F0 and !f are amplitude and circular frequency of the force The figure above shows the sketch of the oscillator (a) and the forces acting on the mass m at a generic time instant t (b)
  • 36. Structural Dynamics & Earthquake Engineering Dr Alessandro Palmeri Motivations for this module Introduction to the module Equations of motion for SDoF oscillators Equation of Motion: Forced SDoF Oscillator u(t) k c m F(t)=F0 sin(ωf t) a) u(t) ku(t) cu(t) . m b) mu(t) .. F(t)=F0 sin(ωf t) The equations of motion in terms of forces is: m ¨u(t) + c ˙u(t) + k u(t) = F0 sin(!f t) (26) and in terms of accelerations is: ¨u(t) + 2 ⇣0 !0 ˙u(t) + !2 0 u(t) = F0 m sin(!f t) (27)
  • 37. Structural Dynamics & Earthquake Engineering Dr Alessandro Palmeri Motivations for this module Introduction to the module Equations of motion for SDoF oscillators Equation of Motion: Forced SDoF Oscillator The general solution to this inhomogeneous 2nd-order ODE is given by: u(t) = uh(t) + up(t) (28) where: up(t) is the particular integral, i.e. any function satisfying Eq. (26) (or equivalently Eq. (27)), which in turn depends on the particular forcing function F(t) uh(t) is the general solution of the related homogeneous ODE (Eqs. (17) and (18)), which is given by the time history of free vibration of the SDoF oscillator with generic integration constants C1 and C2 (see Eq. (21)) uh(t) = e ⇣0 !0 t h C1 cos(!0 t) + C2 sin(!0 t) i (29)
  • 38. Structural Dynamics & Earthquake Engineering Dr Alessandro Palmeri Motivations for this module Introduction to the module Equations of motion for SDoF oscillators Equation of Motion: Forced SDoF Oscillator For a sinusoidal load, F(t) = F0 sin(!f t), the particular integral takes the expression: up(t) = ⇢p sin(!f + 'p) (30) That is, up(t) is again a sinusoidal function with the same circular frequency !f as the input, while amplitude ⇢p and phase angle 'p are given by: ⇢p = F0 k 1 q (1 2) 2 + (2 ⇣0 )2 (31) tan('p) = 2 ⇣0 1 2 (32) in which = !f/!0
  • 39. Structural Dynamics & Earthquake Engineering Dr Alessandro Palmeri Motivations for this module Introduction to the module Equations of motion for SDoF oscillators Equation of Motion: Forced SDoF Oscillator ζ 0 =0 0.0 2.5 5.0 7.5 10.0 -0.50 -0.25 0.00 0.25 0.50 a) t 0.0 2.5 5.0 7.5 10.0 -0.50 -0.25 0.00 0.25 0.50 b) t ζ 0 =0.05 Dynamic response u(t) of a SDoF oscillator (solid line) compared to the particular integral up(t) (dashed line) for a sinusoidal excitation ( = !f/!0 = 0.2; !0 = 7 rad/s; F0/m = 10 m/s2 ; u0 = 0; v0 = 1.5 m/s) and two different values of the viscous damping ratio: a) ⇣0 = 0; b) ⇣0 = 0.05
  • 40. Structural Dynamics & Earthquake Engineering Dr Alessandro Palmeri Motivations for this module Introduction to the module Equations of motion for SDoF oscillators Equation of Motion: Forced SDoF Oscillator Alternatively, the particular integral for a sinusoidal excitation can be expressed as: up(t) = ust D( ) sin(!f + 'p) (33) where: ust = F0 k is the static displacement due to the force F0 D( ) is the so-called dynamic amplification factor, which depends on the frequency ratio = !f/!0: D( ) = ⇢p ust = 1 q (1 2) 2 + (2 ⇣0 )2 (34)
  • 41. Structural Dynamics & Earthquake Engineering Dr Alessandro Palmeri Motivations for this module Introduction to the module Equations of motion for SDoF oscillators Equation of Motion: Forced SDoF Oscillator β D 0 1 2 3 0 1 2 3 4 5 6 7 8 a) ζ0=0.0 ζ0=0.1 ζ0=0.2 ζ0=0.3 ζ0=0.5 ζ0=0.7 ζ0=1.0 0 1 2 3 0 ϕp b) π/4 π/2 3/4 π π β ζ0=0.0 ζ0=0.1 ζ0=0.2 ζ0=0.3 ζ0=0.5 ζ0=0.7 ζ0=1.0 ζ0=0.0 Dynamic amplification factor D (a) and phase angle 'p (b) as functions of the frequency ratio for different values of the viscous damping ratio ⇣0
  • 42. Structural Dynamics & Earthquake Engineering Dr Alessandro Palmeri Motivations for this module Introduction to the module Equations of motion for SDoF oscillators Equation of Motion: Forced SDoF Oscillator For ! 0, the static case is approached (i.e. D ! 1), as the force F(t) varies very slowly and therefore the mass m can follows it For ! 1 and ⇣0 = 0 (undamped case), the system experience an unbounded resonance, meaning that D ! +1 and (theoretically) the amplitude of the dynamic response keeps increasing with time If = 1 and 0  ⇣0 < 0.5, then D > 1, i.e. the amplitude of the steady-state dynamic response ⇢p is larger than the static displacement ust; moreover, the less ⇣0, the larger is D If > 1.41, then D < 1, i.e. (independently of ⇣0) the amplitude of the steady-state dynamic response ⇢p is less than the static displacement ust For ! +1, the dynamic amplification factor goes to zero (i.e. D ! 0), as the force F(t) varies so rapidely that the mass m cannot follows it If ⇣0 0.707, then D < 1, meaning that (independently of the frequency ratio ) the amplitude of the steady-state dynamic response is less than the static displacement
  • 43. Structural Dynamics & Earthquake Engineering Dr Alessandro Palmeri Motivations for this module Introduction to the module Equations of motion for SDoF oscillators Equation of Motion: Forced SDoF Oscillator 0 1 2 3 -3 -2 -1 0 1 2 3 β=0.1 t a) β=0.5 t0 1 2 3 -3 -2 -1 0 1 2 3 b) β=1 t0 1 2 3 -3 -2 -1 0 1 2 3 c) β=2 t0 1 2 3 -3 -2 -1 0 1 2 3 d) Normalised particular integral up(t)/ust (solid line) compared to the normalised pseudo-static response F(t)/(k ust) (dashed line) for a damped (⇣0 = 0.20) SDoF oscillator subjected to a sinusoidal force F(t) = F0 sin(!0 t), with !f = 2 ⇡, and for four different values of the frequency ratio = !f/!0