The tool RD-01 computes the natural frequencies and unbalanced mass response of a Laval-Jeffcott rotor. The tool also gives the whirl direction at each rotation speed.

In addition, a tutorial on how to use the tool in computation is provided and a subject review on fundamental theories and useful formulas is presented.

**Consider a Jeffcott rotor as shown in the following figure. **

The input in the tool is

After clicking "Run", the results will be shown in another page. The Campbell diagram is

The X/Y vibration amplitude to unbalanced mass response and whirl direction are

The analytical results are also shown as follows.

Natural Frequencies

Translational Motion

Natural Frequency - BW: 93.70708108 Hz

Natural Frequency - FW: 93.70708108 Hz

Damped Natural Frequency - BW: 93.70708108 Hz

Damped Natural Frequency - FW: 93.70708108 Hz

Rotational Motion

Natural Frequency - BW: (-0.15077837)*Ω+(0.15915494)*(0.94736842*Ω^{2} + 328421.05263158)^{0.5} Hz

Natural Frequency - FW: (0.15077837)*Ω+(0.15915494)*(0.94736842*Ω^{2} + 328421.05263158)^{0.5} Hz

Note: Since the coupling between displacement and rotation is not considered in the model, rotational natural frequencies cannot be excited by unbalanced mass.

Unbalanced Mass Response

x(t) = x_{c} cos(Ωt) + x_{s} sin(Ωt)

y(t) = y_{c} cos(Ωt) + y_{s} sin(Ωt)

where

x_{c} = {(1*Ω^{2})*(346660.65788193-1*Ω^{2})}/{(346660.65788193-1*Ω^{2})^{2} + (0*Ω)^{2}}

x_{s} = {(0*Ω)*(346660.65788193-1*Ω^{2})}/{(346660.65788193-1*Ω^{2})^{2} + (0*Ω)^{2}}

y_{c} = {(-0*Ω)*(346660.65788193-1*Ω^{2})}/{(346660.65788193-1*Ω^{2})^{2} + (0*Ω)^{2}}

y_{s} = {(1*Ω^{2})*(346660.65788193-1*Ω^{2})}/{(346660.65788193-1*Ω^{2})^{2} + (0*Ω)^{2}}

(Unit of Ω: rad/s)

**Fig. 1** Laval-Jeffcott rotor

Laval-Jeffcott rotor is widely used to study and understand basic rotordynamics phenomena. As shown in Fig. 1, Laval-Jeffcott rotor consist of a single disk placed at the midspan of a flexible, massless and uniform shaft. The shaft is supported by two identical rigid bearings, which are infinitely stiff. The system is symmetric and the first two fundamental motions, i.e. translational and rotational motions, are decoupled and can be studied separately, as shown in Fig. 2.

(a) Translational motion

(b) Rotational motion

**Fig. 2** Two fundamental motions

**Translational Motion**

The translational motion at the disk center is described by two translational displacement (x,y), as shown in Fig. 3.

The governing equation of the translational motion is in the form of

where

m is the mass of the disk;

Ω is the shaft rotating speed;

c is the bearing damping coefficient;

k is the equivalent stiffness of the system, combining shaft stiffness k_{s} and bearing stiffness k_{b}, which is

with k_{s} = 48EI/L^{3}

**Rotational Motion**

The governing equation of the rotational motion is in the form of

where

I_{d} is the diametral Inertia;

I_{p} is the polar Inertia;

k_{r} = 12EI/L represents the angular restoring moment from the shaft.

Since the governing equations for translational and rotational motions are decoupled, they can be solved separately. Then the natural frequency and unbalanced mass response can be obtained.

For more explanation about natural frequencies, critical speeds, and unbalanced mass response of a rotating flexible shaft-disk system, please check the subject review in software RD-03.

Laval-Jeffcott model (RD-01) is a lumped-parameter model of rotors while the shaft-disk model (RD-03) is a distributed-parameter model.

1. D. Childs, Turbomachinery Rotordynamics: Phenomena, Modeling, & Analysis, 1993, John Wiley & Sons Inc, New York.

2. W.J. Chen and E.J. Gunter, Introduction to Dynamics of Rotor-Bearing Systems, 2007, Trafford Publishing.

3. M. Lalanne and G. Ferraris, Rotordynamics Prediction in Engineering, 1998, Wiley.