Haddow s Experiment:

schematic drawing of Haddow's experimental set-up moving piston non-contacting motion sensor beams of spring steel m position varies to adjust frequencies blocks of solid steel m shaker Haddow s Experiment: terra firma Theoretical and Experimental study of Modal Interaction in a Two-Degree-of-Freedom Structure J. of Sound and Vibration 984, 97(3) p 45-473 experimental verification of the saturation phenomenon part ll

w same element when the beam is deflected dx an element in the undeflected beam x, segment of the undeflected beam w w + dx x the axial force is nearly zero, so we assume that the original length of the element, dx, does not change during the motion dx dx expanded view w dx x dv = dx dl " " indicates that projection of the deflected beam is always shorter than that of the undeflected beam dx = dl + dx dl = dx x x w w w w dl = dx dl = + dx x x dl dv w dx dl = dv = dx x V x= l w = x x = 0 dx the horizontal displacement of the free end toward the fixed end

recall the free-vibration modes: φ φ ( ωi ) ( ωi ) k m l 3 3 i ( x) = x x 0 x l k m l ( EI ) ( ω ) i ( EI ) ( ω 3 ) i ( ωi ) 3 3 3 ( k ωi m ) l ( k ωi m ) l ( k ωi m ) l k m l k m l k m l 3 3 4 i ( x) = l x + l x + l 3l EI EI 0 x l x now we describe the shapes of the deflected beams in terms of the free-vibration modes (, ) (, ) φ φ () () w x t φ x x u t = w x t φ x x u t () t the u are arbitrary functions of time to be determined i () () w x, t w x, t φ x φ x u t φ x φ x u t = = = x w x, t w x, t x φ x φ x u t φ x φ x u t () () V i l i w = x i dx = ( φ i) φφ ( φ ) x = 0 l i x = 0 u + uu + u dx = C u + C uu + C u i i i i i i3 u li li li V C C C3 = uu where Ci ( i) dx, Ci ( i i) dx, Ci3 ( i) dx, V C C C = φ φ φ φ 3 = x= 0 = x= 0 x= 0 u

the velocity of the shaker head (moving piston) is described by () W t = Fcos( Ωt) S the kinetic energy of m is given by the kinetic energy of m is given by ( S ) T = m W + W + V T = m W S + W V + W + V () t () t where W ( t) w x, t i i x= l () t () t W Φ Φ u where ij φij ( x) W = Φ Φ Φ u i x= l j u uu V C C C3 V C C C3 = uu uu uu V C C C = + 3 C V C C 3 u uu

( V i) note: contains fourth-order terms in the u, which in turn leads to third-order terms in EoM; ( V i ) so we neglect in the expressions for kinetic energy ( S ) ( S S ) T = m W + W = m W + W W + W i = m F cos Ω t + Fcos Ωt( Φ u +Φ u ) + ( Φ ) u + ΦΦ uu + ( Φ ) u + hot... = S + + + = S + S + S + + T m W W V W V m W W W W W V WV W WV = m F cos Ω t + Fcos Ωt( Φ u +Φu ) + ( Φ ) u + Φ Φ uu + ( Φ ) u ( F t)( C uu C uu C uu C u u ) ( u u )( C uu C uu C uu C u u ) 4 cos Ω + + + 4 Φ +Φ + + + 3 3 + Φ + Φ Φ + Φ + Φ +Φ u uu u u ( )( ) u Cuu ++ Cuu + Cuu + C3uu + hot...

the expressions for potential energy l l w U = ( EI) ( ) dx = EI φ u + φ u dx x= 0 x x= 0 ( EI ) ( φ ) ( u φφuu φ u dx EI K u Kuu K3u ) x = 0 l = + + = + + l w x = 0 x l U = EI dx = EI φ u + φ u dx = EI K u + K u u + K u x = 0 3 u K K K3 U = U+ U = uu K K K 3 u L = T + U following Lagrange's procedure, we obtain equations with the following form:

following Lagrange's procedure, we obtain equations with the following form: u m m u k k u m m u k k u b b b3 d d d3 + + uu + b b b uu uu 3 d d d + 3 u uu uu e e u g + F Ω t = F Ωt sin cos e e u g u uu Mu +Ku +B uu + D uu + uu + FsinΩ teu = FGcos Ωt u uu u uu - - - - - u + ( MKu ) + ( MB) uu + ( MD) uu + uu + cos Ω t( MEu ) = ( MG) cos Ωt u uu u uu u μ 0 u ω 0 u X X X3 Y Y Y3 + + u 0 μ uu uu uu u + 0 ω u X X X + + 3 Y Y Y 3 u uu Z Z u G + Fsin Ω t F cos t Z Z = Ω u G

CASE I: Ω near ( + μ ) ( σ ) = ( μ ) ( σ ) ( σ ) ( β ) the equations to eliminate troublesome terms can be reduced to i DA A 4A A exp i T 0 i DA + A 4A exp i T Fexp i T = 0 A = a exp i i i i ω modulation equations: a + μa aasinγ = 0 aβ + aacos γ = 0 a + μ a + a sinγ Fsinγ = 0 a β + μ a + a cosγ + Fcosγ = 0 γ = σt β + β γ = σ T β ω = ω + εσ Ω= ω + εσ steady-state solutions: ) a = 0, a = σ F + μ ) σ σ+ σ μμ σ+ σ σ+ σ = ± μσ + μ = μ + a F a γ = σt β+ β and γ = σt β are constant β = σt γ β = σt+ σt γ γ

CASE I: Ω near ω steady-state solutions: ) Ai = ai exp i i = T = T + T ( β ) β σ γ β ( σ σ γ γ ) ( γ γ ) 0 u= Aexp ( iωt0) + cc = aexp ( iβ) exp ( iωt0) + cc = aexp ( 0 ) i ωt + β + cc = aexp i ωt0 + ( σt+ σt γ γ ) + cc = aexp i 0 ( ) ω + εσ T + σ T + γ γ + cc = aexp i ω εσ T0 ( γ γ ) cc aexp i T0 ( γ γ ) cc + + + = Ω + + = a cos ΩT + u = Aexp ( iωt0) + cc = aexp ( iβ) exp ( iωt0) + cc = aexp ( 0 ) i ωt + β + cc = aexp i ( ωt 0 σt γ) cc aexp{ i ( ω εσ) T0 γ } cc aexp i ( T0 γ) cc + + = + + = Ω + = a cos ΩT γ 0 w ( x, t) φ φ acos Ωt ( γ + γ ) w ( x, t ) φ φ acos( Ωt γ ) continued on the next slide

CASE I: Ω near ω (continued) steady-state solutions: ) (, ) φ φ 0 ( x, t) φ φ a cos( Ωt γ ) w x t F w μ + σ comparisons of the theoretical (asymptotic) solutions and experimental results follow the stability of the steady-state solutions are obtained in the famliar way

modal amplitudes as functions of the amplitude of the excitation (constant frequency of the excitation, Ω ω ) Haddow s experimental and theoretical results (taken from his paper) supercritical instability subcritical instability note the sub- and super-critical instabilities, which depend on the detuning parameters and are predicted by theory, do appear in the experimental results the unstable responses predicted by the theory do not appear in the experimental results, but are indicated (guessed) for one case saturation and jump phenomena predicted by theory do appear a = 0, a = σ F + μ guessed, not observed a σ σ+ σ μμ = σ + σ ± F μσ + μ a σ + σ = μ + from Haddow

modal amplitudes as functions of the frequency of the excitation (constant amplitude of the excitation F, and Ω = ω + εσ ) modal amplitudes σ σ modal amplitudes a local minimum appears where there is perfect tuning in sharp contrast with the response of a linear system jumps appear here also: increasing frequency, decreasing frequency Ω if the amplitude of the excitation is small enough, the amplitude of the first mode is zero and the solution essentially is the solution of the linear problem

summary of the modal amplitudes as functions of both amplitude and frequency of the excitation for Ω = ω + εσ

results of the stability study for Ω near ω when the combination of amplitude, F, and frequency, σ, of the excitation lies in: Region I, the steady-state response always corresponds to the nonlinear solution, ) above Region II, the steady-state response always corresponds to the linear solution, ) above Region III, the steady-state response can correspond to either, depending on the initial conditions

CASE II: Ω near ω ( μ ) ( σ ) ( σ ) ( + μ ) ( σ ) = ( β ) the equations to eliminate troublesome terms can be reduced to i DA + A 4A A exp i T Fexp i T = 0 i DA A 4A exp i T 0 A = a exp i i i i modulation equations: a + μa aasinγ Fsinγ = 0 aβ + aacos γ+ Fcosγ = 0 a + μ a + a sinγ = 0 a β + μ a + a cosγ = 0 γ = σt β + β γ = σ T β ω = ω + εσ Ω= ω + εσ steady-state solutions: a + μμ σ σ σ a μ σ σ σ μ a F + + + = 0 cubic in a a 6 4 a = note: as, 0 σ a μ + ( σ σ) F and a the 'linear' solution σ + μ γ = σt β + β and γ = σ T β are constant β = σ T γ β = σ T σt γ + γ

CASE II: Ω near ω steady-state solutions: Ai = ai exp ( iβi) β = σt γ β = σt σt γ + γ u = Aexp ( iωt0) + cc = aexp ( iβ) exp ( iωt0) + cc = aexp i( T0 ) cc ω + β + = aexp i( ωt0 σt γ) cc aexp i ( ω εσ) T0 γ + + = { + } + cc = aexp i( ΩT0 γ) + cc = acos( Ωt γ) u= Aexp ( iωt0) + cc = aexp ( iβ) exp ( iωt0) + cc = aexp ( 0 ) i ω T + β + cc = aexp i( T0 T T ) cc aexp{ i ( ) T0 } cc ω + σ σ γ + γ + = ω ε σ εσ γ γ + + + = aexp{ i ( ω+ εσ) T0 γ + γ } + cc = aexp i ( T0 ) cc Ω γ + γ + = a cos ΩT γ + γ 0 ( Ω γ ) ( Ω γ + γ ) w x, t φ φ acos t w x, t φ φ acos T0

a comparison of modal amplitudes as functions of the amplitude of the excitation modal amplitudes theory a comparison of theoretical predictions and experimental observations for modal amplitudes as functions of the amplitude of the excitation a jump phenomenon can occur, depending on the detuning parameter, σ : increasing amplitude, decreasing amplitude modal amplitudes experiment

modal amplitudes modal amplitudes as functions of the frequency of the excitation: a comparison between theoretical predictions and experimantal observations note that the steady-state response is unstable at perfect tunning modal amplitudes again the amplitudes of the response have a local minimum near perfect tuning jump phenomena are possible

summary of the modal amplitudes as functions of both amplitude and frequency of the excitation for Ω = ω + εσ

when the combination of amplitude, F, and frequency, σ, of the excitation lies in: Region I, there is only one steady-state response and it is stable Region II, there is a continual exchange of energy between the modes Region III, there are three steady-state responses, with the middle-amplitude response being unstable