## 7. Prestressed Beams

The eigenfrequencies and eigenmodes of a helicopter rotor modelled with beam elements are calculated. The rotor consists of four blades connected to each other by springs. Each blade is modelled with a separate branch, thus illustrating the use of multiple branches. With the key word 'meshtype' the mesh consisting of one or four branches (one per blade), respectively, can be selected. In the latter case the way in which the various branches are joined (automatic or by specifying the connectivity explicitly) can be selected. Prestresses modelling forces induced by the rotation can be specified.

The geometry data as well as the material properties (all isotropic elastic) are listed in the tables below.

Table 10. Geometry

 length L 1.7 [m] area A 0.01 [m2] head width a 0.2 [m] shaft height h 0.1 [m] Iyy 2.8e-8 [m4] Izz 2.8e-8 [m4] Ip 2.*Iyy Sy 100.[m2] Sz 100.[m2]

Table 11. Material properties

 Part E-modulus ν density ρ [N/m2] [kg/m3] rotor 1.E10 0. 117.6 springs 1E6 0. 100.

The end point of the shaft is fully clamped. The mass matrix is consistent.

Table 12. Input options

 meshtype Type of mesh to be generated: meshtype=1 selects one branch, meshtype=2 selects four branches. jointype Branch join type: jointype=1 selects automatic join, jointype=2 selects manual join (meshtype=2 only). prestr Specifies the value of the pre-stress: Analysis is performed with prestr=1E7)or without prestress (prestr=0).

The solutions are independent of the mesh type or join type and vary only due to the prestress. The table below lists the ten lowest eigenfrequencies. Figure 69 shows the 1st and 4th eigenmode as well as the undeformed shape. Notice that the first 5 eigenfrequencies are virtually the same, as are the 6th, 7th and 8th eigenfrequencies and the 9th and 10th.

Table 13. Eigenfrequencies helicopter rotor

 Prestress 1st 2nd 3rd 4th 5th 0.0 2.987 2.987 2.987 2.987 2.987 1E7 44.178 44.189 44.189 44.189 44.189 Prestress 6th 7th 8th 9th 10th 0.0 3.1136 3.1138 3.197 18.643 18.643 1E7 44.3967 44.3972 44.590 130.31 130.35