Campbell Diagram

In rotating machinery, the resonance frequencies often depend on the shaft speed. One reason for this is the so called gyroscopic effects. A plot with the different resonance frequencies as a function of shaft speed is of the called a Campbell diagram. In this example, Campbell diagram is drawn by using Mathematica.

The equation of free motion for the Clamped beam with a very thin disk will be:

(({{m, 0, 0, 0}, {0, m, 0, 0}, {0, 0, (Subscript[J, d]), 0}, {0, 0, 0, (Subscript[J, d])}}))(({{({{({{(Overscript[x, ..])}, {(Overscript[y, ..])}})}, {(Subscript[(Overscript[θ, ..]), y])}})}, {(Subscript[(Overscript[θ, ..]), x])}}))+(({{0, 0, 0, 0}, {0, 0, 0, 0}, {0, 0, 0, -(Subscript[J, p])}, {0, 0, (Subscript[J, p]), 0}}))(({{({{({{(OverDot[x])}, {(OverDot[y])}})}, {(Subscript[(OverDot[θ]), y])}})}, {(Subscript[(OverDot[θ]), x])}}))+(({{(Subscript[k, ff]), 0, (Subscript[k, fm]), 0}, {0, (Subscript[k, ff]), 0, -(Subscript[k, fm])}, {(Subscript[k, fm]), 0, (Subscript[k, mm]), 0}, {0, -(Subscript[k, fm]), 0, (Subscript[k, mm])}}))(({{({{({{x}, {y}})}, {(Subscript[θ, y])}})}, {(Subscript[θ, x])}}))=(({{({{({{0}, {0}})}, {0}})}, {0}}))

With elementary beam theory for a console beam and numerical values chosen for this example, the result is:

kff=12*Emod*it/l^3;
kfm=-6*Emod*it/l^2;
kmm=4*Emod*it/l;

Also,

m=100;
dmass=1;
l=1;
jd=m*dmass^2/4/4;
jp=jd*2;
dshaft=0.05;
Emod=2.1*10^11;
it=Pi*dshaft^4/64;

Due to the symmetrical conditions the disk motion will be circular and for nontrivial solutions the corresponding determinant needs to be zero,the result is:

f[Ω_,ω_]:=Ω+((kff-m*ω^2)*(kmm-jd*ω^2)-kfm^2)/(jp*ω (kff-m*ω^2));

For a non-rotation shaft the solution to the above equation gives

NSolve[f[0,ω],ω]

{{ω->-217.4373},{ω->217.43735},{ω->-41.0568},{ω->41.0568}}

And for instance if the rotational speed is 40 rad/s :

NSolve[f[40,ω],ω]

For each rotational speed, the result is four different eigenfrequencies. Use Mathematica to solve and plot them:

ωout[Ω_,ω_]:=NSolve[f[Ω,ω],ω]
ωout2[Ω_,ω_]:=Sort[{ωout[Ω,ω][[1,1,2]],ωout[Ω,ω][[2,1,2]],ωout[Ω,ω][[3,1,2]],ωout[Ω,ω][[4,1,2]]}]
sol1[Ω_,ω_]:=-ωout2[Ω,ω][[1]]
sol2[Ω_,ω_]:=-ωout2[Ω,ω][[2]]
sol3[Ω_,ω_]:=ωout2[Ω,ω][[3]]
sol4[Ω_,ω_]:=ωout2[Ω,ω][[4]]
line[Ω_]:=Ω

Plot[{sol1[Ω,ω],sol2[Ω,ω],sol3[Ω,ω],sol4[Ω,ω],line[Ω]},{Ω,0,200},AxesLabel->{Ω[rad/s],ω[rad/s]},PlotStyle->{Directive[Blue,Dotted],Directive[Red,Dotted],Red,Blue,Black}, ImageSize->Large]

The red line is the first forward whirl resonance frequency, and the blue line is the second forward resonance frequency. The dotted lines are the backward whirl frequency. The black line corresponds to an unbalance excitation. An unbalance will hit the resonance when the black line and the red line intersect. The following figure zooms in on this intersection.

Plot[{sol2[Ω,ω],sol3[Ω,ω],line[Ω]},{Ω,0,60},AxesLabel->{Ω[rad/s],ω[rad/s]},PlotStyle->{Directive[Red,Dotted],Red,Black,Green}, ImageSize->Large]

The resonance for the non-rotating shaft would be 41.1 rad/s, but due to gyroscopic effects the resonance occurs at 47.2 rad/s. The resonance can be calculated with:

NSolve[f[Ω,Ω]==0,Ω][[4,1,2]]

The backward whirl is the intersection between the black line and the dotted red line . In this case 36.3 rad/s :

NSolve[f[-Ω,Ω]==0,Ω][[4,1,2]]

The corresponding backward whirling frequency at 47.2 rad/s is also needed . In this case it becomes 34.9 rad/s :

NSolve[f[-47.2,ω]==0,ω][[4,1,2]]

The next step is to relate these theoretical values with Wolfram SystemModeler models . There are two models in the library : ForwardWhirling and BackwardWhirling . The first one is excited via the unbalance and the second one is run up to a certain speed and then the applied static force is released.

Finally, run the shaft at 47, 2 rad/s and evaluate the resonance frequencies . This is done with model ClampedGyroscopicFrequencyExcitation . The response becomes.

The FFT is shown below, values 5, 5 and 7, 5 Hz, i . e . 34.6 and 47.2 rad/s . Theoretical it was shown above that the backward whirl is 34, 9 rad/s at Ω = 47.2 rad/s .