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A corotational finite element method combined with floating frame method and a numerical procedure is proposed to investigate large steady-state deformation and infinitesimal-free vibrationaround the steady-state deformation of a rotating-inclined Euler beam at constant angular velocity. The element nodal forces are derived using the consistent second-order linearization of the nonlinear beam theory, the d'Alembert principle, and the virtual work principle in a current inertia element coordinates, which is coincident with a rotating element coordinate system constructed at the current configuration of the beam element. The governing equations for linear vibration are obtained by the first-order Taylor series expansion of the equation of motion at the position of steady-state deformation. Numerical examples are studied to demonstrate the accuracy and efficiency of the proposed method and to investigate the steady-state deformation and natural frequency of the rotating beam with different inclined angle, angular velocities, radius of the hub, and slenderness ratios.

Rotating beams are often used as a simple model for propellers, turbine blades, and satellite booms. Rotating beam differs from a nonrotating beam in having additional centrifugal force and Coriolis effects on its dynamics. The vibration analysis of rotating beams has been extensively studied [

It is well known that the spinning elastic bodies sustain a steady-state deformation (time-independent deformation) induced by constant rotation [^{’}Alembert principle and the virtual work principle. This formulation and numerical procedure were proven to be very effective by numerical examples studied in [

A rotating inclined beam, (a) top view, (b) side view.

The objective of this study is to present a corotational finite element method combined with floating frame method and a numerical procedure for large steady-state deformation and free vibration analysis of a rotating-inclined beam at constant angular velocity. The nodal coordinates, displacements and rotations, absolute velocities, absolute accelerations, and the equations of motion of the system are defined in terms of an inertia global coordinate system which is coincident with a rotating global coordinate system rigidly tied to the rotating hub, while the total deformations in the beam element are measured in an inertia element coordinate system which is coincident with a rotating element coordinate system constructed at the current configuration of the beam element. The rotating element coordinates rotate about the hub axis at the angular speed of the hub. The inertia nodal forces and deformation nodal forces of the beam element are systematically derived by the virtual work principle, the d^{’}Alembert principle, and consistent second-order linearization of the fully geometrically nonlinear beam theory [

The infinitesimal-free vibrations of rotating beam are measured from the position of the corresponding steady-state deformation. The governing equations for linear vibration of rotating beam are obtained by the first-order Taylor series expansion of the equation of motion at the position of steady-state deformation.

Dimensionless numerical examples are studied to demonstrate the accuracy and efficiency of the proposed method and to investigate the effect of inclination angle and slenderness ratio on the steady-state deformation and the natural frequency for rotating inclined Euler beams at different angular speeds.

Consider an inclined uniform Euler beam of length

Here only axial and lagwise bending vibrations are considered. It is well known that the beam sustains a steady-state deformations (time-independent deformation displacements) induced by constant rotation [

The following assumptions are made in derivation of the beam element behavior.

The beam is prismatic and slender, and the Euler-Bernoulli hypothesis is valid.

The unit extension of the centroid axis of the beam element is uniform.

The deformation displacements and rotations of the beam element are small.

The strains of the beam element are small.

In conjunction with the corotational formulation and rotating frame method, the third assumption can always be satisfied if the element size is properly chosen. Thus, only the terms up to the second order of deformation parameters and their spatial derivatives are retained in element position vector, strain, and deformation nodal forces by consistent second-order linearization in this study.

In order to describe the system, we define three sets of right-handed rectangular Cartesian coordinate systems.

A rotating global set of coordinates,

Element coordinates;

Coordinate systems.

In this study, the direction of the axis of the rotating hub is parallel to the

Let

Kinematics of Euler beam.

Using the approximation

Here, the lateral deflection of the centroid axis,

Making use of assumptions

The axial displacements of the centroid axis may be determined from the lateral deflections and the unit extension of the centroid axis using (

From (

Substituting (

From (

The absolute velocity and acceleration vectors of point

Note that the current element coordinates constructed at the current configuration of the beam element rotate about the hub axis at the angular velocity of the hub. Thus, the centripetal acceleration and Coriolis acceleration corresponding to the inertia forces of the rotating beam are unique. For nonrotating beam,

Let

The element nodal force vector is obtained from the d^{’}Alembert principle and the virtual work principle in the current inertia element coordinates. The virtual work principle requires that

If the element size is chosen to be sufficiently small, the values of the deformation parameters of the deformed element defined in the current element coordinate system may always be much smaller than unity. Thus the higher-order terms of deformation parameters in the element internal nodal forces may be neglected. However, in order to include the nonlinear coupling among the bending and stretching deformations, the terms up to the second order of deformation parameters and their spatial derivatives are retained in element deformation nodal forces by consistent second-order linearization of

From (

From (

Substituting (

The element matrices considered are element tangent stiffness matrix

Using the direct stiffness method, the element tangent stiffness matrix may be assembled by the following submatrices:

The element mass matrix may be assembled by the following submatrices:

The element centripetal stiffness matrix may be assembled by the following submatrices:

The element gyroscopic matrix may be assembled by the following submatrices:

For convenience, the dimensionless variables defined in Table

Dimensionless variables.

Variables | Dimensionless variables | |
---|---|---|

Coordinates | ||

Time | ||

Length of beam element | ||

Area moment of inertia | ||

Radius of hub | ||

Displacements | ||

spatial derivatives of displacement | ||

Time derivatives of displacement | ||

Force and moment | ||

Angular velocity | ||

Natural frequency |

The dimensionless nonlinear equations of motion for a rotating beam with constant angular velocity may be expressed by

For the steady-state deformations,

Here, an incremental-iterative method based on the Newton-Raphson method is employed for the solution of nonlinear dimensionless steady-state equilibrium equations at different dimensionless rotation speed

Substituting (

We will seek a solution of (

Substituting (

Equation (

To verify the accuracy of the present method and to investigate the steady deformation and the natural frequencies of rotating-inclined beams with different inclination angle

For simplicity, only the uniform beam with rectangular cross section is considered here. The maximum steady-state axial strain

To investigate the effect of the lateral deflection on the steady-state deformation and the natural frequency of rotating Euler beams, here cases with and without considering the lateral deflection are considered. The corresponding elements are referred to as EA element and EB element, respectively. For EA element, all terms in (

The example first considered is the rotating-inclined beams with dimensionless radius of the hub

Comparison of results for different cases (

EA10 | 0 | 0 | 0 | .174788 | 1.05957 | 1.57241 | 2.82495 | 4.75610 | 5.19546 | 8.00214 | ||

EA50 | 0 | 0 | 0 | .174787 | 1.05953 | 1.57086 | 2.82431 | 4.71413 | 5.19120 | 7.86206 | ||

0 | EA100 | 0 | 0 | 0 | .174787 | 1.05953 | 1.57081 | 2.82431 | 4.71283 | 5.19119 | 7.85600 | |

[ | 0 | 0 | 0 | .17479 | 1.05953 | 1.57080 | 2.82431 | 4.71239 | 5.19119 | — | ||

0° | [ | 0 | 0 | 0 | .17580 | 1.10172 | 1.57080 | 3.08486 | 4.71239 | 6.04510 | — | |

EA10 | 6.93309 | 0 | 0 | .198616 | 1.08756 | 1.57615 | 2.85333 | 4.75729 | 5.22384 | 8.02928 | ||

EA50 | 7.15492 | 0 | 0 | .198514 | 1.08726 | 1.57616 | 2.85243 | 4.71534 | 5.21931 | 7.86274 | ||

0.06 | EA100 | 7.18210 | 0 | 0 | .198511 | 1.08726 | 1.57457 | 2.85242 | 4.71403 | 5.21930 | 7.85669 | |

[ | 7.20000 | 0 | 0 | .19862 | 1.08760 | 1.57455 | 2.85276 | 4.71360 | 5.21962 | — | ||

LAS | 7.20000 | 0 | 0 | — | — | — | — | — | — | — | ||

EA10 | 1.72680 | 1.93098 | 5.47630 | .181049 | 1.06661 | 1.57335 | 2.83206 | 4.75639 | 5.20256 | 8.00889 | ||

5° | 0.03 | EA50 | 1.78195 | 1.93546 | 5.47699 | .181021 | 1.06651 | 1.57180 | 2.83136 | 4.71443 | 5.19823 | 7.86221 |

EA100 | 1.78870 | 1.93560 | 5.47701 | .181020 | 1.06651 | 1.57175 | 2.83136 | 4.71312 | 5.19822 | 7.85616 | ||

LAS | 1.79486 | 2.03794 | 5.88301 | — | — | — | — | — | — | — | ||

EA10 | .173298 | 1.29008 | 3.72294 | .175410 | 1.06028 | 1.57252 | 2.82567 | 4.75613 | 5.19619 | 8.00281 | ||

30° | 0.01 | EA50 | .178615 | 1.29224 | 3.72299 | .175407 | 1.06024 | 1.57097 | 2.82503 | 4.71416 | 5.19191 | 7.86207 |

EA100 | .179264 | 1.29231 | 3.72300 | .175407 | 1.06024 | 1.57092 | 2.82503 | 4.71285 | 5.19190 | 7.85601 | ||

LAS | .179904 | 1.29904 | 3.75000 | — | — | — | — | — | — | — | ||

EA10 | .0500345 | 2.59364 | 7.49504 | .174836 | 1.05978 | 1.57253 | 2.82520 | 4.75612 | 5.19573 | 8.00229 | ||

90° | 0.01 | EA50 | .0500384 | 2.59784 | 7.49506 | .174835 | 1.05974 | 1.57098 | 2.82456 | 4.71415 | 5.19145 | 7.86205 |

EA100 | .0500216 | 2.59797 | 7.49507 | .174835 | 1.05974 | 1.57093 | 2.82456 | 4.71284 | 5.19144 | 7.85599 | ||

LAS | .0500000 | 2.59807 | 7.50000 | — | — | — | — | — | — | — |

Comparison of results for different cases (

EA10 | 0 | 0 | 0 | .351601 | .220349 | .617105 | .121008 | .200340 | .300117 | .421052 | ||

EA50 | 0 | 0 | 0 | .351601 | .220341 | .616949 | .120893 | .199838 | .298509 | .416903 | ||

0 | EA100 | 0 | 0 | 0 | .351601 | .220341 | .616948 | .120893 | .199837 | .298506 | .416896 | |

[ | 0 | 0 | 0 | .352 | .2203 | .6169 | .12089 | .19984 | .29851 | — | ||

0° | [ | 0 | 0 | 0 | .3516 | .22034 | .616972 | .120902 | — | — | — | |

EA10 | 6.93309 | 0 | 0 | 9.00457 | 2.50186 | 4.13423 | .591446 | .784725 | .992927 | 1.21760 | ||

EA50 | 7.15492 | 0 | 0 | 8.96239 | 2.47424 | 4.06068 | .580524 | .771309 | .976120 | 1.19365 | ||

0.06 | EA100 | 7.18210 | 0 | 0 | 8.96152 | 2.47312 | 4.05756 | .580088 | .770833 | .975634 | 1.19316 | |

[ | 7.20000 | 0 | 0 | 8.952 | 2.4708 | 4.0536 | .57955 | .77017 | .97486 | — | ||

LAS | 7.20000 | 0 | 0 | — | — | — | — | — | — | — | ||

EA10 | 1.73113 | 3.88303 | .0835171 | 4.54714 | 1.27448 | 2.17658 | .323098 | .443024 | .577262 | .726698 | ||

5° | 0.03 | EA50 | 1.78396 | 6.00526 | .0838194 | 4.53348 | 1.26220 | 2.15028 | .319777 | .439167 | .572464 | .719965 |

EA100 | 1.78936 | 6.20203 | .0838218 | 4.53320 | 1.26179 | 2.14942 | .319678 | .439068 | .572368 | .719873 | ||

LAS | 1.79486 | 101.897 | 14.70753 | — | — | — | — | — | — | — | ||

EA10 | .117174 | 8.73588 | .429688 | 1.29068 | .405580 | .836390 | .143462 | .221484 | .319586 | .439110 | ||

30° | 0.008 | EA50 | .114341 | 9.36150 | .429979 | 1.28848 | .404156 | .836065 | .143631 | .221433 | .318254 | .434653 |

EA100 | .113410 | 9.38784 | .429986 | 1.28840 | .404108 | .836056 | .143643 | .221458 | .318289 | .434691 | ||

LAS | .115138 | 41.5692 | 6.00000 | — | — | — | — | — | — | — | ||

EA10 | 00632587 | 8.11012 | .747138 | .561367 | .232168 | .566051 | .113317 | .190889 | .289637 | .409722 | ||

90° | 0.003 | EA50 | 00388224 | 8.15298 | .747250 | .560585 | .232182 | .566299 | .113204 | .190326 | .287895 | .405359 |

EA100 | .00351740 | 8.15396 | .747254 | .560558 | .232181 | .566306 | .113202 | .190322 | .287886 | .405342 | ||

LAS | .00450000 | 11.6913 | 1.68750 | — | — | — | — | — | — | — |

To investigate the effect of the lateral deflection on the steady-state deformation and the natural frequency of rotating-inclined beams, the cases with and without considering the lateral deflection are studied for

Dimensionless frequencies for rotating beam with different inclination angle (

EA | EB | EA | EA | EA | EB | [ | EA | EB | [ | |

0° | 7.61582 | 7.61579 | 0 | 0 | .105565 | .105427 | .105 | .411754 | .410792 | .418 |

10° | 7.53275 | 7.53893 | .021843 | .119537 | .105513 | .104869 | .105 | .411356 | .410001 | .417 |

20° | 7.28556 | 7.31066 | .043509 | .236923 | .105359 | .103195 | .103 | .410160 | .407642 | .414 |

30° | 6.88032 | 6.93792 | .064820 | .350059 | .105101 | .100399 | .100 | .408162 | .403758 | .410 |

40° | 6.32700 | 6.43205 | .085602 | .456935 | .104737 | .0964721 | .096 | .405357 | .398421 | .405 |

50° | 5.63927 | 5.80840 | .105681 | .555686 | .104264 | .0913941 | .091 | .401742 | .391733 | .398 |

60° | 4.83425 | 5.08596 | .124890 | .644628 | .103679 | .0851262 | .085 | .397314 | .383830 | .390 |

70° | 3.93214 | 4.28663 | .143064 | .722301 | .102976 | .0775919 | .077 | .392079 | .374876 | .381 |

80° | 2.95575 | 3.43472 | .160043 | .787503 | .102151 | .0686418 | .068 | .386048 | .365073 | .371 |

90° | 1.93010 | 2.55611 | .175673 | .839324 | .101193 | .0579597 | .057 | .379245 | .354659 | .361 |

To investigate the effect of angular speed on the steady-state deformation and natural frequency of rotating beams with different slenderness ratios and inclination angles, the following cases are considered: slenderness ratio

Dimensionless frequencies for rotating beam with different inclination angle (

0 | 0 | 0 | 0 | .900168 | .559057 | 1.54325 | 1.57086 ( | 2.96396 | 4.71413 | |

.010 | 1.48999 | 0 | 0 | .909817 | .560283 | 1.54452 | 1.57097 ( | 2.96528 | 4.71415 | |

.020 | 5.96060 | 0 | 0 | .938126 | .563945 | 1.54830 | 1.57130 ( | 2.96926 | 4.71421 | |

0° | .030 | 13.4138 | 0 | 0 | .983359 | .569996 | 1.55455 | 1.57190 ( | 2.97586 | 4.71432 |

.040 | 23.8528 | 0 | 0 | 1.04313 | .578359 | 1.56302 | 1.57296 ( | 2.98509 | 4.71449 | |

.050 | 37.2822 | 0 | 0 | 1.11488 | .588935 | 1.57113 | 1.57704 ( | 2.99691 | 4.71472 | |

.060 | 53.7078 | 0 | 0 | 1.19620 | .601604 | 1.57358 ( | 1.58936 | 3.01129 | 4.71502 | |

.005 | .371545 | .073310 | .412011 | .902582 | .559363 | 1.54357 | 1.57089 ( | 2.96429 | 4.71414 | |

.010 | 1.48623 | .289977 | 1.62160 | .909786 | .560280 | 1.54449 | 1.57100 ( | 2.96528 | 4.71415 | |

5° | .015 | 3.34417 | .640707 | 3.55360 | .921661 | .561805 | 1.54595 | 1.57126 ( | 2.96693 | 4.71416 |

.020 | 5.94559 | 1.11150 | 6.09535 | .938019 | .563932 | 1.54784 | 1.57177 ( | 2.96925 | 4.71416 | |

.025 | 9.29075 | 1.68547 | 9.11211 | .958617 | .566654 | 1.55003 | 1.57268 ( | 2.97222 | 4.71415 | |

.030 | 13.3800 | 2.34473 | 12.4630 | .983170 | .569962 | 1.55233 | 1.57416 ( | 2.97585 | 4.71413 | |

.002 | .054292 | .067510 | .379956 | .900509 | .559102 | 1.54330 | 1.57087 ( | 2.96401 | 4.71413 | |

.004 | .217174 | .269614 | 1.51631 | .901534 | .559237 | 1.54341 | 1.57090 ( | 2.96415 | 4.71413 | |

30° | .006 | .488657 | .605041 | 3.39858 | .903243 | .559461 | 1.54355 | 1.57103 ( | 2.96440 | 4.71412 |

.008 | .868763 | 1.07170 | 6.00953 | .905635 | .559775 | 1.54362 | 1.57132 ( | 2.96474 | 4.71409 | |

.010 | 1.35752 | 1.66673 | 9.32547 | .908710 | .560177 | 1.54351 | 1.57191 ( | 2.96518 | 4.71403 | |

.002 | .019999 | .135091 | .760426 | .900208 | .559076 | 1.54327 | 1.57087 ( | 2.96398 | 4.71413 | |

.004 | .080009 | .540362 | 3.04080 | .900333 | .559132 | 1.54324 | 1.57098 ( | 2.96404 | 4.71412 | |

90° | .006 | .180072 | 1.21579 | 6.83830 | .900554 | .559224 | 1.54295 | 1.57140 ( | 2.96415 | 4.71407 |

.008 | .320253 | 2.16132 | 12.1478 | .900889 | .559349 | 1.54211 | 1.57246 ( | 2.96431 | 4.71394 | |

.010 | .500631 | 3.37678 | 18.9613 | .901363 | .559506 | 1.54041 | 1.57447 ( | 2.96451 | 4.71365 |

Dimensionless frequencies for rotating beam with different inclination angle (

0 | 0 | 0 | 0 | .702550 | .437859 | 1.21530 | 1.57086 | 2.35176 | 3.82646 | |

.010 | 1.48998 | 0 | 0 | .714917 | .439441 | 1.21696 | 1.57096 | 2.35349 | 3.82822 | |

.020 | 5.96058 | 0 | 0 | .750712 | .444154 | 1.22192 | 1.57125 | 2.35868 | 3.83349 | |

0° | .030 | 13.4137 | 0 | 0 | .806600 | .451898 | 1.23014 | 1.57173 | 2.36731 | 3.84227 |

.040 | 23.8527 | 0 | 0 | .878442 | .462516 | 1.24155 | 1.57241 | 2.37932 | 3.85452 | |

.050 | 37.2820 | 0 | 0 | .962316 | .475813 | 1.25605 | 1.57328 | 2.39467 | 3.87021 | |

.060 | 53.7076 | 0 | 0 | 1.05500 | .491563 | 1.27352 | 1.57435 | 2.41329 | 3.88930 | |

.005 | .371544 | .093761 | .674835 | .705653 | .438254 | 1.21571 | 1.57089 | 2.35219 | 3.82689 | |

.010 | 1.48623 | .368279 | 2.62905 | .714878 | .439437 | 1.21695 | 1.57096 | 2.35349 | 3.82821 | |

5° | .015 | 3.34418 | .804892 | 5.66932 | .729982 | .441402 | 1.21900 | 1.57111 | 2.35565 | 3.83040 |

.020 | 5.94560 | 1.37717 | 9.52433 | .750592 | .444136 | 1.22184 | 1.57134 | 2.35869 | 3.83347 | |

.025 | 9.29074 | 2.05594 | 13.8991 | .776240 | .447626 | 1.22547 | 1.57166 | 2.36258 | 3.83740 | |

.030 | 13.3799 | 2.81357 | 18.5209 | .806412 | .451854 | 1.22987 | 1.57208 | 2.36734 | 3.84221 | |

.002 | .054292 | .086522 | .624206 | .702990 | .437917 | 1.21536 | 1.57087 | 2.35182 | 3.82652 | |

.004 | .217177 | .345191 | 2.48734 | .704308 | .438091 | 1.21554 | 1.57088 | 2.35202 | 3.82671 | |

30° | .006 | .488671 | .773349 | 5.56121 | .706507 | .438381 | 1.21583 | 1.57093 | 2.35234 | 3.82704 |

.008 | .868803 | 1.36665 | 9.79986 | .709585 | .438785 | 1.21621 | 1.57103 | 2.35280 | 3.82749 | |

.010 | 1.35760 | 2.11921 | 15.1409 | .713543 | .439301 | 1.21666 | 1.57120 | 2.35340 | 3.82806 | |

.002 | .019999 | .173193 | 1.24980 | .702604 | .437883 | 1.21532 | 1.57087 | 2.35179 | 3.82648 | |

.004 | .080020 | .692767 | 4.99671 | .702772 | .437956 | 1.21539 | 1.57090 | 2.35188 | 3.82657 | |

90° | .006 | .180129 | 1.55866 | 11.2327 | .703078 | .438073 | 1.21545 | 1.57103 | 2.35204 | 3.82670 |

.008 | .320419 | 2.77065 | 19.9428 | .703561 | .438232 | 1.21544 | 1.57135 | 2.35228 | 3.82688 | |

.010 | .500997 | 4.32814 | 31.1020 | .704273 | .438426 | 1.21527 | 1.57200 | 2.35265 | 3.82709 |

Dimensionless frequencies for rotating beam with different inclination angle (

0 | .000000 | 0 | 0 | .351520 | .219989 | .614602 | 1.20047 | 1.57086 | 1.97619 | |

.010 | 1.48998 | 0 | 0 | .375696 | .223163 | .617963 | 1.20402 | 1.57096 | 1.97983 | |

.020 | 5.96058 | 0 | 0 | .439855 | .232418 | .627926 | 1.21458 | 1.57124 | 1.99072 | |

0° | .030 | 13.4137 | 0 | 0 | .528670 | .247050 | .644140 | 1.23194 | 1.57172 | 2.00872 |

.040 | 23.8527 | 0 | 0 | .630838 | .266138 | .666082 | 1.25578 | 1.57240 | 2.03361 | |

.050 | 37.2820 | 0 | 0 | .740095 | .288755 | .693127 | 1.28568 | 1.57327 | 2.06512 | |

.060 | 53.7076 | 0 | 0 | .853195 | .314093 | .724614 | 1.32116 | 1.57433 | 2.10288 | |

.005 | .371547 | .184139 | 2.62891 | .357708 | .220785 | .615441 | 1.20136 | 1.57089 | 1.97710 | |

.010 | 1.48624 | .688585 | 9.52242 | .375636 | .223154 | .617940 | 1.20397 | 1.57105 | 1.97985 | |

5° | .015 | 3.34416 | 1.40679 | 18.5127 | .403666 | .227045 | .622062 | 1.20826 | 1.57140 | 1.98446 |

.020 | 5.94540 | 2.23838 | 27.6550 | .439745 | .232381 | .627773 | 1.21419 | 1.57195 | 1.99093 | |

.025 | 9.29010 | 3.11970 | 35.8627 | .481914 | .239066 | .635040 | 1.22176 | 1.57261 | 1.99922 | |

.030 | 13.3786 | 4.01960 | 42.7784 | .528564 | .246988 | .643816 | 1.23096 | 1.57329 | 2.00929 | |

.002 | .054294 | .172596 | 2.48732 | .352403 | .220106 | .614725 | 1.20060 | 1.57087 | 1.97632 | |

.004 | .217197 | .683326 | 9.79956 | .355052 | .220456 | .615081 | 1.20096 | 1.57097 | 1.97675 | |

30° | .006 | .488756 | 1.51188 | 21.5060 | .359467 | .221033 | .615636 | 1.20145 | 1.57136 | 1.97753 |

.008 | .869003 | 2.62713 | 36.9471 | .365637 | .221832 | .616344 | 1.20197 | 1.57227 | 1.97873 | |

.010 | 1.35790 | 3.99065 | 55.3127 | .373528 | .222846 | .617163 | 1.20241 | 1.57395 | 1.98043 | |

.002 | .020005 | .346383 | 4.99669 | .351634 | .220038 | .614651 | 1.20052 | 1.57089 | 1.97625 | |

.004 | .080103 | 1.38532 | 19.9425 | .352038 | .220178 | .614748 | 1.20053 | 1.57127 | 1.97654 | |

90° | .006 | .180478 | 3.11497 | 44.6708 | .352915 | .220392 | .614736 | 1.20014 | 1.57288 | 1.97728 |

.008 | .321226 | 5.52831 | 78.7975 | .354553 | .220651 | .614373 | 1.19878 | 1.57704 | 1.97891 | |

.010 | .502064 | 8.60794 | 121.586 | .357326 | .220924 | .613368 | 1.19581 | 1.58523 | 1.98201 |

Dimensionless frequencies for rotating beam with different inclination angle (

0 | .000000 | 0 | 0 | .351601 | .220341 | .616949 | .120893 | .199838 | .298509 | |

.010 | 14.8998 | 0 | 0 | 1.32581 | .432338 | .886010 | .151855 | .233376 | .333736 | |

.020 | 59.6058 | 0 | 0 | 2.52842 | .766172 | 1.38410 | .216189 | .309541 | .419056 | |

0° | .030 | 134.137 | 0 | 0 | 3.73648 | 1.11402 | 1.92511 | .289167 | .400408 | .525942 |

.040 | 238.527 | 0 | 0 | 4.94495 | 1.46597 | 2.47903 | .364653 | .496007 | .640880 | |

.050 | 372.820 | 0 | 0 | 6.15267 | 1.81983 | 3.03912 | .441165 | .593428 | .759068 | |

.060 | 537.076 | 0 | 0 | 7.35907 | 2.17482 | 3.60301 | .518237 | .691707 | .878747 | |

.005 | 3.71469 | .765147 | 5.97092 | .741097 | .289190 | .695401 | .129392 | .208701 | .307598 | |

.010 | 14.8600 | 1.67568 | 7.33938 | 1.32573 | .432307 | .885745 | .151766 | .233190 | .333432 | |

5° | .015 | 33.4431 | 2.57966 | 7.79173 | 1.92549 | .595911 | 1.12500 | .182037 | .268266 | .371976 |

.020 | 59.4713 | 3.46977 | 8.02050 | 2.52834 | .766160 | 1.38402 | .216153 | .309441 | .418839 | |

.025 | 92.9513 | 4.34189 | 8.15957 | 3.13220 | .939307 | 1.65211 | .252155 | .353965 | .470793 | |

.030 | 133.889 | 5.19316 | 8.25360 | 3.73641 | 1.11401 | 1.92507 | .289150 | .400356 | .525817 | |

.002 | .542410 | 1.31449 | 16.2578 | .436039 | .231506 | .625674 | .121676 | .200527 | .299138 | |

30° | .004 | 2.15709 | 3.45499 | 31.0603 | .628044 | .264068 | .655502 | .124160 | .202509 | .300711 |

.006 | 4.84203 | 5.61881 | 37.4726 | .851755 | .312455 | .712896 | .129772 | .207598 | .305211 | |

.008 | 8.61268 | 7.78502 | 40.6536 | 1.08530 | .369468 | .788930 | .138176 | .215896 | .313013 | |

.0005 | .012524 | .216407 | 3.11010 | .352489 | .220632 | .617142 | .120911 | .199852 | .298522 | |

.0010 | .050204 | .860794 | 12.1582 | .357434 | .221288 | .616084 | .120749 | .199640 | .298279 | |

.0015 | .112068 | 1.89528 | 25.7173 | .371925 | .221920 | .610342 | .119952 | .198654 | .297181 | |

90° | .0020 | .194123 | 3.21653 | 40.7205 | .400095 | .222722 | .599365 | .118425 | .196788 | .295121 |

.0025 | .290775 | 4.68879 | 54.0272 | .440751 | .224723 | .587411 | .116669 | .194633 | .292734 | |

.0030 | .398355 | 6.21301 | 64.2905 | .489485 | .228876 | .578807 | .115177 | .192740 | .290595 | |

.0035 | .516056 | 7.74385 | 71.7152 | .542488 | .235452 | .575106 | .114128 | .191302 | .288900 |

Figures

The steady-state deformation of rotating beam, (a) deformed configuration, (b) axial displacement, and (c) lateral displacement (

The steady-state deformation of rotating beam, (a) deformed configuration, (b) axial displacement, and (c) lateral displacement (

The steady-state deformation of rotating beam, (a) deformed configuration, (b) axial displacement, (c) lateral displacement (

Figures

The first six vibration mode shapes of a rotating beam (

The first six vibration mode shapes of a rotating beam (

The first six vibration mode shapes of a rotating beam (

The first six vibration mode shapes of a rotating beam (

The third and fourth natural frequencies verse the dimensionless angular velocity (

In this paper, a corotational finite element formulation combined with the rotating frame method and numerical procedure are proposed to derive the equations of motion for a rotating-inclined Euler beam at constant angular velocity. The element deformation and inertia nodal forces are systematically derived by the virtual work principle, the d^{’}Alembert principle, and consistent second-order linearization of the fully geometrically nonlinear beam theory in the current element coordinates. The equations of motion of the system are defined in terms of an inertia global coordinate system which is coincident with a rotating global coordinate system rigidly tied to the rotating hub, while the total strains in the beam element are measured in an inertia element coordinate system which is coincident with a rotating element coordinate system constructed at the current configuration of the beam element. The rotating element coordinates rotate about the hub axis at the angular speed of the hub. The steady-state deformation and the natural frequency of infinitesimal-free vibration measured from the position of the corresponding steady-state deformation are investigated for rotating-inclined Euler beams with different inclination angles, slenderness ratios, and angular speeds of the hub.

The results of dimensionless numerical examples demonstrate the accuracy and efficiency of the proposed method. The present results show that the geometrical nonlinearities that arise due to steady-state lateral and axial deformations should be considered for the natural frequencies of the inclined-rotating beams. Due to the effect of the centrifugal stiffening, the lower dimensionless natural frequencies of lateral vibration increase remarked with increase of the dimensionless angular speed for slender beam. The decrease of the centrifugal stiffening effect of the rotating inclined beam caused by the increase of the inclination angle is alleviated by the increase of lateral deflection induced by the lateral centrifugal force. Due to effect of the Coriolis force and centrifugal stiffening, frequency veering phenomenon is observed when inclination angle

Finally, it may be emphasized that, although the proposed method is only applied to the two dimensional rotating cantilever beams with inclination angle here, the present method can be easily extended to three dimensional rotating beams with precone and setting angle.

The research was sponsored by the National Science Council, ROC, Taiwan, under contract NSC 98-2221-E-009-099-MY2.