NMBD Dynamic Vibration Absorber

Derived from appendix T080194, and applied to the Non-Magnetic Blade-spring Damper (NMBD) for the Quad UIM internal blade-spring mode

coupled two-mass oscillator derivation

following J.C. Snowdon, Vibration and Shock in Damped Mechanical Systems, John Wiley & Sons, cr 1968, section 4.2, pp. 80-83.

derivation

The derivation below corrects Snowdon's equations. Snowdon's formulation for the amplitude of the transfer function from ground to primary mass motion is correct, but his real and imaginary terms are incorrect.

eqn1[t_]:=(Subscript[M, 1]) (Subscript[x, 2])''[t]-(Subscript[k, 1]) (Subscript[G, 1])[ω]((Subscript[x, 1])[t]-(Subscript[x, 2])[t])+(Subscript[k, 2]) (Subscript[G, 2])[ω]((Subscript[x, 2])[t]-(Subscript[x, 3])[t]);

eqn3=LaplaceTransform[eqn1[t],t,s]/.{(Subscript[x, 1])'[0]-> 0,(Subscript[x, 1])[0]-> 0,LaplaceTransform[(Subscript[x, 1])[t],t,s]-> (OverBar[(Subscript[x, 1])])[s],(Subscript[x, 2])'[0]-> 0,(Subscript[x, 2])[0]-> 0,LaplaceTransform[(Subscript[x, 2])[t],t,s]-> (OverBar[(Subscript[x, 2])])[s],(Subscript[x, 3])'[0]-> 0,(Subscript[x, 3])[0]-> 0,LaplaceTransform[(Subscript[x, 3])[t],t,s]-> (OverBar[(Subscript[x, 3])])[s]}

eqn2[t_]:=(Subscript[M, 2]) (Subscript[x, 3])''[t]-(Subscript[k, 2]) (Subscript[G, 2])[ω]((Subscript[x, 2])[t]-(Subscript[x, 3])[t]);

eqn4=LaplaceTransform[eqn2[t],t,s]/.{(Subscript[x, 1])'[0]-> 0,(Subscript[x, 1])[0]-> ζ,LaplaceTransform[(Subscript[x, 1])[t],t,s]-> (OverBar[(Subscript[x, 1])])[s],(Subscript[x, 2])'[0]-> 0,(Subscript[x, 2])[0]-> 0,LaplaceTransform[(Subscript[x, 2])[t],t,s]-> (OverBar[(Subscript[x, 2])])[s],(Subscript[x, 3])'[0]-> 0,(Subscript[x, 3])[0]-> 0,LaplaceTransform[(Subscript[x, 3])[t],t,s]-> (OverBar[(Subscript[x, 3])])[s]}

soln=Solve[{eqn3==0,eqn4==0},{(OverBar[(Subscript[x, 2])])[s],(OverBar[(Subscript[x, 3])])[s]}]

T[ω_]:=Simplify[(OverBar[(Subscript[x, 2])])[s]/(OverBar[(Subscript[x, 1])])[s]/.soln/.s->I ω]

Snowdon's equation 4.7:

T[ω]

Using the following substitutions/defintions
μ=((Subscript[M, 1])/((Subscript[M, 1])+(Subscript[M, 2])));
n=((Subscript[ω, a])/(Subscript[ω, 0]));
(Subscript[ω, 0])=(Sqrt[(((Subscript[k, 1]) (Subscript[Y, 1])[(Subscript[ω, 00])])/((Subscript[M, 1])+(Subscript[M, 2])))]);
(Subscript[ω, a])=(Sqrt[(((Subscript[k, 2]) (Subscript[Y, 2])[(Subscript[ω, 00])])/(Subscript[M, 2]))]);
Ω=(ω/(Subscript[ω, 0]));
and thefollowing notation for the complex modulus:

(Subscript[G, 1])[ω_]:=(Subscript[Y, 1])(1+I (Subscript[δ, 1]));
(Subscript[G, 2])[ω_]:=(Subscript[Y, 2])(1+I (Subscript[δ, 2]));

The expression for the transfer function is as follows:

T[ω]

Transfer function,: Real Numerator Term

(Subscript[R, N])=FullSimplify[Re[Expand[T[ω][[1,{1,2,3,4}]]]],{(Subscript[k, 1]),(Subscript[k, 2]),(Subscript[Y, 1]),(Subscript[Y, 2]),(Subscript[δ, 1]),(Subscript[δ, 2]),μ,w0,wa,Ω,ω,(Subscript[M, 1]),(Subscript[M, 2])}∈Reals]

c=-n^2/((Subscript[k, 1]) (Subscript[k, 2]) (Subscript[Y, 1]) (Subscript[Y, 2]))

Simplify[c (Subscript[R, N])]

Simplify[Expand[c (Subscript[R, N])]//.ω^2 n^2 (Subscript[M, 2])/((Subscript[k, 2]) (Subscript[Y, 2]))->Ω^2 ((Subscript[Y, a])/(Subscript[Y, 2]))]

This is the same as Snowdon's Equation 4.13 for (Subscript[I, N])

Transfer function,: Imaginary Numerator Term

(Subscript[I, N])=FullSimplify[Im[Expand[T[ω][[1,{1,2,3,4}]]]],{(Subscript[k, 1]),(Subscript[k, 2]),(Subscript[Y, 1]),(Subscript[Y, 2]),(Subscript[δ, 1]),(Subscript[δ, 2]),μ,w0,wa,Ω,ω,(Subscript[M, 1]),(Subscript[M, 2])}∈Reals]

Simplify[c (Subscript[I, N])]

Simplify[Expand[c (Subscript[I, N])]//.ω^2 n^2 (Subscript[M, 2])/((Subscript[k, 2]) (Subscript[Y, 2]))->Ω^2 ((Subscript[Y, a])/(Subscript[Y, 2]))]

This is the negative of Snowdon's Equation 4.12 for (Subscript[R, N])

Transfer function,: Real Denominator Term

(Subscript[R, D])=FullSimplify[Re[1/T[ω][[1,5]]],{(Subscript[k, 1]),(Subscript[k, 2]),(Subscript[Y, 1]),(Subscript[Y, 2]),(Subscript[δ, 1]),(Subscript[δ, 2]),μ,w0,wa,Ω,ω,(Subscript[M, 1]),(Subscript[M, 2])}∈Reals]

Simplify[c (Subscript[R, D])]

FullSimplify[Expand[Expand[c (Subscript[R, D])//.{ω^2 ((Subscript[M, 1])+(Subscript[M, 2]))->((Subscript[k, 1]) (Subscript[Y, 1])) Ω^2 ((Subscript[Y, 0])/(Subscript[Y, 1])),ω^2 n^2 (Subscript[M, 2])/((Subscript[k, 2]) (Subscript[Y, 2]))->Ω^2 ((Subscript[Y, a])/(Subscript[Y, 2]))}]]//.{ω^2 ((Subscript[M, 1])+(Subscript[M, 2]))->((Subscript[k, 1]) (Subscript[Y, 1])) Ω^2 ((Subscript[Y, 0])/(Subscript[Y, 1])),ω^2 n^2 (Subscript[M, 2])/((Subscript[k, 2]) (Subscript[Y, 2]))->Ω^2 ((Subscript[Y, a])/(Subscript[Y, 2]))}]

This is the Snowdon's Equation 4.15 for (Subscript[I, D])

Transfer function,: Imaginary Denominator Term

(Subscript[I, D])=FullSimplify[Im[1/T[ω][[1,5]]],{(Subscript[k, 1]),(Subscript[k, 2]),(Subscript[Y, 1]),(Subscript[Y, 2]),(Subscript[δ, 1]),(Subscript[δ, 2]),μ,w0,wa,Ω,ω,(Subscript[M, 1]),(Subscript[M, 2])}∈Reals]

Collect[Expand[c (Subscript[I, D])]//.{ω^2 (Subscript[M, 1])/((Subscript[k, 1]) (Subscript[Y, 1]))+ω^2 (Subscript[M, 2])/((Subscript[k, 1]) (Subscript[Y, 1]))->Ω^2 ((Subscript[Y, 0])/(Subscript[Y, 1])),ω^2 n^2 (Subscript[M, 2])/((Subscript[k, 2]) (Subscript[Y, 2]))->Ω^2 ((Subscript[Y, a])/(Subscript[Y, 2])),n^2ω^4 (Subscript[M, 1]) (Subscript[M, 2]) /((Subscript[k, 1]) (Subscript[k, 2]) (Subscript[Y, 1]) (Subscript[Y, 2]))-> μ  Ω^4 ((Subscript[Y, 0])/(Subscript[Y, 1]))((Subscript[Y, a])/(Subscript[Y, 2]))},n]/.{ω^2 (Subscript[M, 1])/((Subscript[k, 1]) (Subscript[Y, 1]))+ω^2 (Subscript[M, 2])/((Subscript[k, 1]) (Subscript[Y, 1]))->Ω^2 ((Subscript[Y, 0])/(Subscript[Y, 1]))}

This is the negative of Snowdon's Equation 4.14 for (Subscript[R, D])

coupled two-mass oscillator formulation

Same formulation as above but restated without subscripts.

general transmissibility equation

The UIM blade spring (D060237) has a mass of .367 kgm. I’ll assume ~1/2 of this mass is involved in the 1st resonance.
The mass of the NMBD involved in its 1st elastic mode is ~2 x the mass of the dumb-bells (D1400296) = 2 x 17 = 34 gm
See NMBD damping results in T1600046

.055/M1

n=wa/w0;
M1=.367/2;
M2=eps M1;
mu=M1/(M1+M2);

d1[w_]:= 1/720;
d2[w_]:= d;
Y1[w_]:= w0^2(M1+M2);
Y2[w_]:= wa^2 M2;

Variables:
d = loss factor
eps = damping mass to primary mass ratio, M2/M1
wa = natural frequency of the damping element by itself, rad/sec 
w0 = natural frequency of the primary mass without the damping element added, rad/sec

The total mass of the NMBD = 55 gm. However the mass of the dumb-bells involved in its 1st mode is ~34gm
So eps = .034/M1 = 0.185
wa was estimated in T1500237 to be ~120 Hz (indirectly by fitting to some data)
However using viton elastic stiffness, the cross-sectional area of the viton spring and the dumbbell mass, one gets wa = 344 Hz
From T1600046, w0 = 110 Hz and d1 = 1/720

IN[w_]:=(w/w0)^2 (Y2[wa]/Y2[w])-n^2 (1-d1[w] d2[w]);
RN[w_]:=-(w/w0)^2 (Y2[wa]/Y2[w]) d1[w]+n^2 (d1[w]+d2[w]);
ID[w_]:= -mu (w/w0)^4 (Y1[w0]/Y1[w]) (Y2[wa]/Y2[w])+(w/w0)^2 (n^2 (Y1[w0]/Y1[w])+(Y2[wa]/Y2[w]))-n^2 (1-d1[w] d2[w]);
RD[w_]:= -(w/w0)^2 (n^2 (Y1[w0]/Y1[w]) d2[w]+(Y2[wa]/Y2[w]) d1[w])+n^2 (d1[w]+d2[w]);
T[w_]:=(RN[w]+I IN[w])/(RD[w]+I ID[w]);

NMBD application

case 1: UIM blade spring, no damper

plt1=LogPlot[Abs[T[Ω w0]] /. {w0 -> 2 Pi 110, wa -> 2 Pi 344, d -> 0, eps -> 0}, {Ω, .1, 10}, PlotRange -> All, PlotPoints -> 10000, MaxRecursion -> 15,PlotStyle->{Black},Frame->True,FrameLabel->{"Normalized Excitation Frequency, Ω (Hz)","Transfer Function Amplitude, T"}]

FindMaximum[Abs[T[ω]]/.{w0->2 Pi 110,wa->2 Pi 344,d->0,eps->0}, {ω,10,500}]

case 2: NMBD, wa >> w0, 75 F (23.8 C)

plt2=LogPlot[Abs[T[Ω w0]] /. {w0 -> 2 Pi 110, wa -> 2 Pi 344, d -> .68, eps -> .185}, {Ω, .1, 10}, PlotRange -> All, PlotPoints -> 10000, MaxRecursion -> 15,PlotStyle->{Red},Frame->True,FrameLabel->{"Normalized Excitation Frequency, Ω (Hz)","Transfer Function Amplitude, T"}]

FindMaximum[Abs[T[ω]]/.{w0->2 Pi 110,wa->2 Pi 344,d->0.68,eps->0.185}, {ω,10,500}]

case 3: NMBD, wa >> w0, 65 F (18.3 C)

plt3=LogPlot[Abs[T[Ω w0]] /. {w0 -> 2 Pi 110, wa -> 2 Pi 344 1.119, d -> .9, eps -> .185}, {Ω, .1, 10}, PlotRange -> All, PlotPoints -> 10000, MaxRecursion -> 15,PlotStyle->{Blue},Frame->True,FrameLabel->{"Normalized Excitation Frequency, Ω (Hz)","Transfer Function Amplitude, T"}]

FindMaximum[Abs[T[ω]]/.{w0->2 Pi 110,wa->2 Pi 344 1.119,d->0.9,eps->0.185}, {ω,10,500}]

compare NMBD cases 1,2,3

Show[plt1,plt2,plt3, PlotRange->{{.5,1.5},All}]

case 4: NMBD, wa >> w0, 75 F (23.8 C), d & M2 set to match T1600046-v2 Qdamped

plt4=LogPlot[Abs[T[Ω w0]] /. {w0 -> 2 Pi 110, wa -> 2 Pi 344, d -> .6, eps -> .185/2}, {Ω, .1, 10}, PlotRange -> All, PlotPoints -> 10000, MaxRecursion -> 15,PlotStyle->{Red},Frame->True,FrameLabel->{"Normalized Excitation Frequency, Ω (Hz)","Transfer Function Amplitude, T"}]

FindMaximum[Abs[T[ω]]/.{w0->2 Pi 110,wa->2 Pi 344,d->0.6,eps->0.185/2}, {ω,10,500}]

case 5: NMBD, wa >> w0, 65 F (23.8 C), d & M2 set to match T1600046-v2 Qdamped

Using d & M2 that match T160046-v2 Qdamped at 75F, but then estimating effect of temperature of 65 F, by increasing d by 1.32 factor and increasing wa frequency by 1.119 factor, predicted from Viton-B complex modulus variation with temperature

plt5=LogPlot[Abs[T[Ω w0]] /. {w0 -> 2 Pi 110, wa -> 2 Pi 344 1.119, d -> .6 1.32, eps -> .185/2}, {Ω, .1, 10}, PlotRange -> All, PlotPoints -> 10000, MaxRecursion -> 15,PlotStyle->{Blue},Frame->True,FrameLabel->{"Normalized Excitation Frequency, Ω (Hz)","Transfer Function Amplitude, T"}]

FindMaximum[Abs[T[ω]]/.{w0->2 Pi 110,wa->2 Pi 344 1.119,d->0.6 1.32,eps->0.185/2}, {ω,10,500}]

compare NMBD cases 1,4,5

Show[plt1,plt4,plt5, PlotRange->{{.95,1.05},{1,7}}]

197/173 //N

If the NMBD were assembled at 75