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- 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 #2: Equations of Motion for SDoF Oscillators Dr Alessandro Palmeri Civil and Building Engineering @ Loughborough University Tuesday, 2nd February 2016
- 2. 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)
- 3. 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
- 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: 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)
- 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 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
- 6. 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)
- 7. 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) + ω2 0 u(t) = 0 (4) where ω0 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: ω0 = k m (5)
- 8. 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 π ω0 (6) and the undamped natural frequency of vibration (measured in Hz= 1/s): ν0 = 1 T0 = ω0 2 π (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
- 9. 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
- 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 We have then to solve this homogenous differential equation: ¨u(t) + ω2 0 u(t) = 0 (4) and the solution can be expressed as: u(t) = C1 cos(ω0 t) + C2 sin(ω0 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
- 11. 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(ω0 × 0) + C2 sin(ω0 × 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 ω0 sin(ω0 t) + C2 ω0 cos(ω0 t) We can then apply the second initial condition: ˙u(0) = −u0 ω0 sin(ω0 × 0) + C2 ω0 cos(ω0 × 0) = v0 ∴ u0 × 0 + C2 ω0 × 1 = v0 ⇒ C2 = v0 ω0 (10)
- 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 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(ω0 t) + v0 ω0 sin(ω0 t) (11) ˙u(t) = v0 cos(ω0 t) − u0 ω0 sin(ω0 t) (12)
- 13. 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) = ρ0 cos(ω0 t + ϕ0) (13) where ρ0 is the amplitude of the motion and ϕ0 is the phase at the initial time instant t = 0: ρ0 = C2 1 + C2 2 = u2 0 + v0 ω0 2 (14) tan(ϕ0) = C2 C1 = v0/ω0 u0 = v0 ω0 u0 (15)
- 14. 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
- 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 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)
- 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 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 ζ0 ω0 ˙u(t) + ω2 0 u(t) = 0 (18) where ζ0 is the so-called equivalent viscous damping ratio for the SDoF oscillator, given by: ζ0 = c 2 m ω0 = c ccr (19) in which ccr is the so-called critical damping coefﬁcient: ccr = 2 m ω0 = 2 √ k m = 2 k ω0 (20)
- 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 Damped SDoF Oscillator One can mathematically prove that, assuming initial conditions u(t) = u0 > 0 and v0 = 0: If c = ccr (or ζ0 = 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 ζ0 > 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 ζ0 < 1), then the system is said to be underdamped, and the mass oscillates about the equilibrium position with a progressively decreasing amplitude
- 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 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
- 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 Damped SDoF Oscillator The vast majority of civil engineering structures are underdamped, and the range of values for the viscous damping ratio ζ0 is between 0.01 and 0.07, ζ0 = 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−ζ0 ω0 t C1 cos(ω0 t) + C2 sin(ω0 t) (21) In this case, the displacement function u(t) is a harmonic wave of circular frequency ω0 and exponentially decaying amplitude. Again, the integration constants C1 and C2 have to be determined by imposing the initial conditions
- 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 For the free vibration of an underdamped SDoF system, the time history of the deﬂection takes the expression: u(t) = e−ζ0 ω0 t u0 cos(ω0 t) + v0 + ζ0 ω0 u0 ω0 sin(ω0 t) (22) or, equivalently: u(t) = ρ0 e−ζ0 ω0 t cos(ω0 t + ϕ0) (23) in which: ρ0 = C 2 1 + C 2 2 = u2 0 + v0 + ζ0 ω0 u0 ω0 2 (24) tan(ϕ0) = C2 C1 = v0 + ζ0 ω0 u0 ω0 u0 (25)
- 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 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
- 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 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
- 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: 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(ωf t), where F0 and ωf 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)
- 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: 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(ωf t) (26) and in terms of accelerations is: ¨u(t) + 2 ζ0 ω0 ˙u(t) + ω2 0 u(t) = F0 m sin(ωf t) (27)
- 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: 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−ζ0 ω0 t C1 cos(ω0 t) + C2 sin(ω0 t) (29)
- 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: Forced SDoF Oscillator For a sinusoidal load, F(t) = F0 sin(ωf t), the particular integral takes the expression: up(t) = ρp sin(ωf + ϕp) (30) That is, up(t) is again a sinusoidal function with the same circular frequency ωf as the input, while amplitude ρp and phase angle ϕp are given by: ρp = F0 k 1 (1 − β2) 2 + (2 ζ0 β)2 (31) tan(ϕp) = 2 ζ0 β 1 − β2 (32) in which β = ωf/ω0
- 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 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 (β = ωf/ω0 = 0.2; ω0 = 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) ζ0 = 0; b) ζ0 = 0.05
- 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: Forced SDoF Oscillator Alternatively, the particular integral for a sinusoidal excitation can be expressed as: up(t) = ust D(β) sin(ωf + ϕp) (33) where: ust = F0 k is the static displacement due to the force F0 D(β) is the so-called dynamic ampliﬁcation factor, which depends on the frequency ratio β = ωf/ω0: D(β) = ρp ust = 1 (1 − β2) 2 + (2 ζ0 β)2 (34)
- 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 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 ϕp (b) as functions of the frequency ratio β for different values of the viscous damping ratio ζ0
- 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: Forced SDoF Oscillator For β → 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 β → 1 and ζ0 = 0 (undamped case), the system experience an unbounded resonance, meaning that D → +∞ and (theoretically) the amplitude of the dynamic response keeps increasing with time If β = 1 and 0 ≤ ζ0 < 0.5, then D > 1, i.e. the amplitude of the steady-state dynamic response ρp is larger than the static displacement ust; moreover, the less ζ0, the larger is D If β > 1.41, then D < 1, i.e. (independently of ζ0) the amplitude of the steady-state dynamic response ρp is less than the static displacement ust For β → +∞, 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 ζ0 ≥ 0.707, then D < 1, meaning that (independently of the frequency ratio β) the amplitude of the steady-state dynamic response is less than the static displacement
- 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: 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 (ζ0 = 0.20) SDoF oscillator subjected to a sinusoidal force F(t) = F0 sin(ω0 t), with ωf = 2 π, and for four different values of the frequency ratio β = ωf/ω0

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