Method for determining an unbalance of a shaft-elastic rotor with reference to the outward deflection

Abstract

In a method for determining an equivalent modal unbalance for the first bending characteristic form of a shaft-elastic rotor, which unbalance is to be compensated for, a rotor model is created describing the geometric shape and material properties of the shaft-elastic rotor. The magnitude of compliance of the rotor model is calculated at a measurement point and at the center of gravity of the rotor at an assumed speed. The shaft-elastic rotor is received in a rotatable bearing and accelerated to the assumed speed which is below its first critical speed. Subsequently, the magnitude of outward deflection at the measurement point of the shaft-elastic rotor rotating at the assumed speed can be measured. The equivalent modal unbalance for the first bending characteristic form of the shaft-elastic rotor, which unbalance is to be compensated for, can be calculated from the magnitudes of the calculated compliance and the measured outward deflection.

Claims

1. A method for determining an equivalent modal unbalance for the first bending characteristic form of a shaft-elastic rotor, which unbalance is to be compensated for, comprising the steps of: creating a simple rotor model which describes only the geometric shape and the material properties of the shaft-elastic rotor; calculating the magnitude of the static compliance of the model of the rotor at at least one point of measurement and at the center of gravity of the rotor at at least one assumed speed; receiving the shaft-elastic rotor in a rotatable bearing and accelerating the rotor to the assumed speed which is below the first critical speed thereof; measuring the magnitude of the outward deflection at the point of measurement of the shaft-elastic rotor rotating at the assumed speed; calculating the equivalent modal unbalance for the first bending characteristic form of the shaft-elastic rotor, which unbalance is to be compensated for, from the magnitudes of the calculated compliance and the measured outward deflection, without taking rotor-dynamic effects into account.

2. The method according to claim 1, wherein the rotor model is created according to the finite element method.

3. The method according to claim 1, wherein the vector quantity of the outward deflection of the shaft-elastic rotor is measured by means of a contactless displacement transducer which detects a point on the outer circumference of the rotor.

4. The method according to claim 1, wherein a bearing of the shaft-elastic rotor is taken into account when creating the rotor model.

5. The method according to claim 1, wherein a radial runout error of the rotor is compensated for at the point of measurement.

6. The method according to claim 1, wherein the rotor is accelerated to a speed which corresponds to a maximum of 50% of its first critical speed.

7. The method according to claim 1, wherein the rotor is accelerated to a speed which corresponds to a maximum of 30% of its first critical speed.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

(1) The invention will be explained in more detail with reference to embodiments of the invention, which are illustrated in the drawings, in which:

(2) FIG. 1 shows a rotor which is to be compensated for, and

(3) FIG. 2 shows a comparison of different approximation methods for calculating an outward deflection of a rotor.

DETAILED DESCRIPTION OF THE DRAWINGS

(4) FIG. 1 shows a rotor which is to be compensated for. A shaft-elastic rotor 1 is rotatably mounted in bearings 2. The rotor 1 has an unbalance which is to be compensated for. The rotor 1 is accelerated by means of a corresponding drive to a previously defined speed which is preferably below the first critical speed of the rotor 1.

(5) An outward deflection of the rotor 1 is measured at the speed at at least one point. The vector quantity of the outward deflection of the shaft-elastic rotor 1 can be measured by means of a contactless displacement transducer 3 which detects a point on the outer circumference of the rotor 1. The displacement transducer 3 detects a radial deflection of the rotor 1 at the point of measurement. Examples of displacement transducers 3 are capacitive or inductive displacement transducers (eddy current sensors) or laser triangulation sensors. The measurement data from the displacement transducer 3 can be forwarded to an evaluation unit.

(6) In addition, a rotor model is formed. This can also be calculated by the evaluation unit. This is a simple numerical model, for an open tube or a solid shaft, for example, that does not take rotor-dynamic effects into account. However, more complex geometries can also be used, such as a rotor comprising end plates and pins. Material properties, such as the modulus of elasticity and density, and geometric data, such as length, wall thickness and/or diameter, are detected from the rotor 1.

(7) If the rotor 1 has a complex geometry, the rotor model can be created, for example, according to the finite element method. However, other numerical methods can also be used.

(8) It has been found that the magnitude of the force which generates the outward deflection can be calculated from the measured outward deflection (magnitude and position) by means of the rotor model. When considered while ignoring the rotor-dynamic effects, this force consists of two components, the consideration being particularly advantageous in particular up to a speed of approximately 50% of the first critical speed. The first component corresponds to the force, due to the unbalances distributed axially in the rotor 1 at the selected measuring speed. The second component corresponds to the force due to additional unbalances caused by mass displacement (outward deflection) at the measuring speed. Since the second component is caused by the first component, it is sufficient to eliminate the first component in order to eliminate the total force. By means of the method, it is mathematically possible to determine the first component using the rotor model and the measured value of the outward deflection at a specific speed.

(9) The necessary force for the outward deflection of a rotor 1 at a speed in the elastic range can, in a static consideration which ignores the rotor-dynamic effects (in particular to 50% of the first critical speed), be approximately represented as a function of two force components.
F=U.sub.U*.sup.2+U.sub.B*.sup.2

(10) U.sub.U is in this case the (distributed) initial unbalance of the rotor 1. U.sub.B is the unbalance component that is produced as a result of the outward deflection and the accompanying mass displacement. This can be represented by the rotor mass m.sub.W and the outward deflection of the centre of gravity x.sub.S, or the compliance of the centre of gravity h.sub.S and the outward deflection force F.
U.sub.B=m.sub.W*x.sub.S=m.sub.W*F*h.sub.S

(11) The total force can be expressed from the compliance h.sub.W of the rotor at the point of measurement (calculated from the rotor model) and the outward deflection x.sub.W of the rotor 1, measured by means of the displacement transducer 3, as follows:
F=x.sub.W/h.sub.W

(12) The desired value U.sub.U can therefore be represented by the following quantities:

(13) $U_U=\left(1-h_S*m_W*{}^2\right)*\frac{x_W}{h_W}*\frac{1}{{}^2}$

(14) By measuring outward deflection x.sub.W at the known speed by means of the displacement transducer 3 and entering the rotor parameters (geometry, material properties) into a rotor model and subsequently calculating the compliances of the rotor 1 at the point of measurement h.sub.W and the compliance of the centre of gravity h.sub.S of the rotor, an unbalance acting in an equivalent manner to the distributed unbalance can be calculated, which is the cause of the outward deflection of the rotor. The method for determining this equivalent unbalance for the first bending characteristic form of a shaft-elastic rotor 1 is divided in one embodiment into simple steps such as: creating the rotor model including bearing (analytic for simple geometries or as an FE model for more complex geometries); calculating the compliances of the rotor 1 at the point of measurement and the compliance of the centre of gravity; measuring the outward deflection of the rotor at at least one speed (compensation of the radial runout error of the shaft at the point of measurement is optionally necessary for this purpose); calculating the equivalent unbalance that generates a rotor outward deflection of this kind using the equation described above, for example. The calculated unbalance can be compensated for at at least one point of the rotor 1 by setting a balancing weight to reduce the outward deflection.

(15) FIG. 2 shows a comparison of different approximation methods for calculating an outward deflection of a rotor. Different approximation methods for calculating an outward deflection of a rotor were compared, namely a simple static model (only taking into account the unbalance without any influence of the unbalance-induced outward deflection), an embodiment of the method according to the invention and a complex rotor-dynamic calculation. For this purpose, an unbalance was artificially generated on the rotor and the outward deflection at different speeds was calculated using the above approximation methods. It can be seen that the approximation method that only takes the unbalance forces into account deviates by more than 10% from the rotor-dynamic complex calculation at only approximately 20% of the critical speed. The preferred method, which takes into account the mass displacement due to outward deflection (iteratively calculated), differs in the calculated example even at 50% of the critical speed by less than 5% from the rotor-dynamic calculation. This shows that the measurement of the outward deflection can provide information about the unbalance occurring on the rotor.