Dynamics of and waves Department of Mechanical and Civil Engineering Ecole Centrale Paris, France September 16, 2008
Outline 1 2 Pre-stressed Bernoulli-Euler in moving frames Kinematical assumption Resultant force and moment 3 and torsion waves waves Free vibrating modes 4 Beam Finite Elements 5
Vibrations of a rotating beam Circular cross-section: Does it rotate or not? Rectangular cross-section: Is it the same? What about a square cross-section?
Outline 1 2 Pre-stressed Bernoulli-Euler in moving frames Kinematical assumption Resultant force and moment 3 and torsion waves waves Free vibrating modes 4 Beam Finite Elements 5
The Bernoulli-Euler beam Kinematical assumptions The mean fiber: x s ds o = 0 S o Rigid cross sections: u(x, t) = u o (x 3, t) + θ(x 3, t) x s normal to the mean fiber: θ = θ 3 e 3 + e 3 u o
The Bernoulli-Euler beam Gradient and strains The gradient and the small strain tensors: D x (u) = (u o3e 3 + θ x s ) e 3 + Θ On the cross sections: ɛ = (u o3e 3 + θ x s ) s e 3 D x (u)e 3 = u o + θ x s ɛe 3 = (u o3 u os x s )e 3 + 1 2 θ 3e 3 x s
The Bernoulli-Euler beam rate of momentum and angular momentum of a cross-section d dt S Velocity and acceleration: v(x, t) = v o + u o + Ωu o + θ x s + Ω(θ x s ) γ(x, t) = γ o (x, t) + ü o + 2Ω u o + Ω 2 u o + θ x s + 2Ω( θ x s ) + Ω 2 (θ x s ) Rates of momentum and angular momentum: γ t d {}}{ ρvds = ργds = m l (γ dt oo + ü o + 2Ω u o + Ω 2 u o ) S S (x s + θ x s ) (ρv)ds = j θ = J m ( θ) J 3 θ3 e 3 Ω=0
Outline 1 2 Pre-stressed Bernoulli-Euler in moving frames Kinematical assumption Resultant force and moment 3 and torsion waves waves Free vibrating modes 4 Beam Finite Elements 5
The Bernoulli-Euler beam Resultant force and moment The total Cauchy stress tensor σ t The resultant force: Q t = σ t nds S t The moment with respect to the current mean fiber x 3 e 3 + u o : M t = (x s + θ x s ) (σ t n)ds S t
The Bernoulli-Euler beam Balances of momentum and angular momentum Balance of momentum: ( ) Q(x 3 + dx 3 ) Q(x 3 ) + f ext ds + t ext dl dx 3 = m l γ m dx 3 S S M(x 3 + dx 3 ) M(x 3 ) + (ndx 3 ) Q + c l dx 3 = j θ dx 3 Resultant and moment : Q t + f t = m l γ t M t + n Q t + c t = j θ
Outline 1 2 Pre-stressed Bernoulli-Euler in moving frames Kinematical assumption Resultant force and moment 3 and torsion waves waves Free vibrating modes 4 Beam Finite Elements 5
The Bernoulli-Euler beam Linearization around a compression-tension state Initial stress state: σ ref = σ ref e 3 satisfying: Sσ ref + f ref = m l γ ref Linear elasticity around the initial stress state: σ = S σ ref = C e ɛ linearized resultant force and moment: Q = Q t Q ref = u osn ref + Q s + SEu o3e 3 M = M t = EJ(e 3 u s) + µθ 3J(e 3 ) linearized resultant and moment equation: M + e 3 (Q N ref u os) + c l = j θ Q + f l = m l γ = m l (γ t γ ref e 3 )
Outline 1 2 Pre-stressed Bernoulli-Euler in moving frames Kinematical assumption Resultant force and moment 3 and torsion waves waves Free vibrating modes 4 Beam Finite Elements 5
The Bernoulli-Euler beam in a Galilean frame Traction- (when ω = ω 3 e 3 ): Torsion (when ω = ω 3 e 3 ): (SEu o3) + f l3 = m l ü o3 µ(j 3 θ 3) + c l3 = J m3 θ3 along one of the principal axis of the section e α in a Galilean frame : (EIu ) + (N ref u ) + c l + f l = I m ü }{{ +m } l ü 0 with I = I(e α ) = J(e 3 e α ) e 3.
Our experiment (j θ e 3 0, c l = f ls = N ref = 0, and ω = ω 3 e 3 ): ( EI(u os) ) = m l (ü os + 2Ω u os + Ω 2 u os ) with a circular cross-section: R T IR = I and thus in the fixed frame: ( EI(R T u os) ) = m l d 2 dt 2 (RT u os ) should be the same for a square cross-section. with a rectangular cross-section this invariance is lost, but when ω 0
Outline 1 2 Pre-stressed Bernoulli-Euler in moving frames Kinematical assumption Resultant force and moment 3 and torsion waves waves Free vibrating modes 4 Beam Finite Elements 5
Wave in and torsion waves Harmonic free propagating solutions: u(x 3, t) = cos(kx 3 ± ωt + ϕ), c = ω k, c g = ω k (ω = ω 3 e 3, ρ = cst): (SEu o3) + f l3 = m l ü o3 c n = Torsion (ω = ω 3 e 3, ρ = cst): µj 3 θ 3+c l3 = J m3 θ3 c t = General solutions for non dispersive waves: u o3 (x 3, t) = g(x 3 ± ct) E ρ µj3 ρ=cst µ = J m3 ρ < c n
Outline 1 2 Pre-stressed Bernoulli-Euler in moving frames Kinematical assumption Resultant force and moment 3 and torsion waves waves Free vibrating modes 4 Beam Finite Elements 5
Wave in waves in a Galilean frame The equation (N ref = 0): (EIu ) +(N ref u ) = m l ü EIk 4 +k 2 N ref ω 2 m l = 0 Stability condition ω is a real number k 2 / ] N ref EI, 0[ Four stable waves : Nref 2 k r = ± + 4ω2 EIm l N ref 2EI Dispersive propagating waves: 2EIω c f = 2 Nref 2 + 4ω2 EIm l N ref k i = ±i k 2 r + N ref EI = ω 4 EI ω m l = ω 0 Nref m l
Outline 1 2 Pre-stressed Bernoulli-Euler in moving frames Kinematical assumption Resultant force and moment 3 and torsion waves waves Free vibrating modes 4 Beam Finite Elements 5
Free vibrating modes and torsion in a Galilean frame modes under clamped-free conditions : u o3 (0) = 0, u o3(l) = 0 modes and eigenfrequencies: ( ) (1 + 2n)πx3 û n (x 3 ) = sin 2L ω n = (1 + 2n)π E 2L ρ
Free vibrating modes modes of a simply supported beam in a Galilean frame General : û(x 3 )=A r cos(k r x 3 )+A sin(k r x 3 )+B r cosh(k i x 3 )+B sinh(k i x 3 ) Boundary conditions: sin(k r L) = 0, k r 0 Eigenmodes and eigenfrequencies : ( û n 1 (x 3 ) = sin nπ x ) 3 L EI ω n 1 = nπ m l L 4 (nπ)2 + N ref L 2 m l Euler buckling: N lim = π2 EI L 2 ( u lim = sin π x ) 3 L
Outline 1 2 Pre-stressed Bernoulli-Euler in moving frames Kinematical assumption Resultant force and moment 3 and torsion waves waves Free vibrating modes 4 Beam Finite Elements 5
ulation of the equation Galilean frame The equation: (EIu ) + (N ref u ) + f l = m l ü For all w(x 3 ) with square integrable second derivatives: K e(u,w) {}}{ L 0 EIu w dx 3 + K g(u,w) {}}{ L 0 N ref u w dx 3 + = M(ü,w) {}}{ L 0 L m l üwdx 3 f l wdx 3 0 } {{ } P ext(w)
Outline 1 2 Pre-stressed Bernoulli-Euler in moving frames Kinematical assumption Resultant force and moment 3 and torsion waves waves Free vibrating modes 4 Beam Finite Elements 5
0 0 L 0.2L 0 L The shape Galilean frame The shape : w E A1(x 3 ) = 1 3ζ 2 + 2ζ 3, w E A2(x 3 ) = x 3 (ζ 1) 2 1 0.2L w E B1(x 3 ) = 3ζ 2 2ζ 3, w E B2(x 3 ) = x 3 ζ(ζ 1) 0 0 L 1 0 0 L 0
Outline 1 2 Pre-stressed Bernoulli-Euler in moving frames Kinematical assumption Resultant force and moment 3 and torsion waves waves Free vibrating modes 4 Beam Finite Elements 5
Elastic stiffness Galilean frame The stiffness matrix of a beam: 12 6h E 12 6h E K e = EI 4h 2 h 3 E 6h E 2h 2 E 12 6h E E sym 4h 2 E The stiffness matrix of a column clamped on two floors: K e = 12EI ( ) 1 1 h 3 1 1 E The stiffness matrix of a column with an hinge on one floor: K e = 3EI ( ) 1 1 h 3 1 1 E With two hinges?
What s new? Dynamics of beam in a moving frame with initial normal stresses,, torsion and waves, free vibrating modes of, beam finite element. What s left? A general solution method, random loads.