Equations of motion for a rotor blade, including gravity, pitch action and rotor speed variations

Wind Energy ◽  
2007 ◽  
Vol 10 (3) ◽  
pp. 209-230 ◽  
Author(s):  
B. S. Kallesøe
Keyword(s):  
Rotor Blade ◽  
Equations Of Motion ◽  
Rotor Speed

FREE VIBRATION ANALYSIS OF TRIPLY COUPLED PRE-TWISTED ROTOR BLADES BY THE DIFFERENTIAL QUADRATURE METHOD

2011 ◽  
Vol 11 (01) ◽  
pp. 127-147 ◽  
Author(s):  
O. SEPAHI ◽  
M. R. FOROUZAN ◽  
P. MALEKZADEH
Keyword(s):  
Rotor Blade ◽  
Differential Quadrature Method ◽  
Equations Of Motion ◽  
Free Vibration Analysis ◽  
Differential Quadrature ◽  
Rotor Blades ◽  
Dynamic Responses ◽  
Quadrature Method ◽  
Method Of Solution ◽  
Combined Flap

Key parameters on the dynamic characteristics of triply coupled pretwisted rotor blades are investigated. The issues of concern include the combined flap-wise bending, chord-wise bending, and torsion vibrations of the pretwisted rotor blade, considering the centrifugal force and Coriolis effects. The governing differential equations of motion presented by Houbolt and Brooks for the rotor blade are used as the basis of study, which contain many factors previously ignored. The differential quadrature method is adopted as the method of solution for its ease in implementation, accuracy, and fast convergence. The dynamic responses of the rotor blade are obtained for different cases of coupling and geometries, which agree well with existing results. The dynamic responses of the rotor blades are plotted against parameters such as angular velocity, pretwisting angle, and hub radius in proper curves and discussed in details.

Movement of Deposited Water on Turbomachinery Rotor Blade Surfaces

2006 ◽  
Author(s):  
John Williams ◽  
John B. Young
Keyword(s):  
Rotor Blade ◽  
Water Movement ◽  
Equations Of Motion ◽  
Surface Friction ◽  
Blade Surface ◽  
Inertia Effects ◽  
Aero Engine ◽  
Blade Tip ◽  
Fan Blade ◽  
Bypass Ratio

A theoretical approach for calculating the movement of liquid water following deposition onto a turbomachine rotor blade is described. Such a situation can occur during operation of an aero-engine in rain. The equation of motion of the deposited water is developed on an arbitrarily oriented plane triangular surface facet. By dividing the blade surface into a large number of facets and calculating the water trajectory over each one crossed in turn, the overall trajectory can be constructed. Apart from the centrifugal and Coriolis inertia effects, the forces acting on the water arise from the blade surface friction, and the aerodynamic shear and pressure gradient. Non-dimensionalisation of the equations of motion provides considerable insight and a detailed study of water flow on a flat rotating plate set at different stagger angles demonstrates the paramount importance of blade surface friction. The extreme cases of low and high blade friction are examined and it is concluded that the latter (which allows considerable mathematical generalisation) is the most likely in practice. It is also shown that the aerodynamic shear force, but not the pressure force, may influence the water motion. Calculations of water movement on a low-speed compressor blade and the fan blade of a high bypass ratio aero-engine suggest that in low rotational speed situations most of the deposited water is centrifuged rapidly to the blade tip region.

scholarly journals Multidisciplinary Aerodynamic Design of a Rotor Blade for an Optimum Rotor Speed Helicopter

2017 ◽  
Vol 7 (6) ◽  
pp. 639 ◽  
Author(s):  
Jiayi Xie ◽  
Zhifeng Xie ◽  
Ming Zhou ◽  
Jun Qiu
Keyword(s):  
Rotor Blade ◽  
Aerodynamic Design ◽  
Rotor Speed

Performance of Quiet Helicopter

2020 ◽  
Vol 2020 (1) ◽  
pp. 1-17
Author(s):  
Jarosław Stanisławski
Keyword(s):  
Equations Of Motion ◽  
Rotor Blades ◽  
Main Rotor ◽  
Rotor Speed ◽  
Flexible Rotor ◽  
Slowing Down ◽  
Helicopter Flight ◽  
Operational Envelope ◽  
The Galerkin Method ◽  
Lower Limits

AbstractNoise generated by helicopters is one of the main problems associated with the operation of rotorcrafts. Requirements for reduction of helicopter noise were reflected in the regulations introducing lower limits of acceptable rotorcraft noise. A significant source of noise generated by helicopters are the main rotor and tail rotor blades. Radical noise reduction can be obtained by slowing down the blade tips speed of main and tail rotors. Reducing the rotational speed of the blades may decrease rotor thrust and diminish helicopter performance. The problem can be solved by attaching more blades to main rotor. The paper presents results of calculation regarding improvement of the helicopter performance which can be achieved for reduced rotor speed but with increased number of rotor blades. The calculations were performed for data of hypothetical light helicopter. Results of simulation include rotor loads and blade deformations in chosen flight conditions. Equations of motion of flexible rotor blades were solved using the Galerkin method which takes into account selected eigen modes of the blades. The simulation analyzes can help to determine the performance and loads of a quiet helicopter with reduced rotor speed within the operational envelope of helicopter flight states.

Development of a Set of Equations for Incorporating Disk Flexibility Effects in Rotordynamical Analyses

1991 ◽  
Author(s):  
George T. Flowers ◽  
Stephen G. Ryan
Keyword(s):  
Equations Of Motion ◽  
Second Order ◽  
Rotor System ◽  
Rotor Speed ◽  
Disk System ◽  
Rigid Disk ◽  
First Order ◽  
Transverse Vibrations ◽  
Sample System

Rotordynamical equations that account for disk flexibility are developed. These equations employ free – free rotor modes to model the rotor system. Only transverse vibrations of the disks are considered, with the shaft/disk system considered to be torsionally rigid. Second order elastic foreshortening effects that couple with the rotor speed to produce first order terms in the equations of motion are included. The approach developed in this study is readily adaptable for usage in many of the codes that are current used in rotordynamical simulations. The equations are similar to those used in standard rigid disk analyses but with additional terms that include the effects of disk flexibility. An example case is presented to demonstrate the use of the equations and to show the influence of disk flexibility on the rotordynamical behavior of a sample system.

Resonance Reliability Analysis of Aeroengine Compressor Rotor Blade

2014 ◽  
Vol 472 ◽  
pp. 79-84
Author(s):  
Hai Feng Gao ◽  
Guang Chen Bai
Keyword(s):  
Vibration Frequency ◽  
Rotor Blade ◽  
Random Variables ◽  
Second Order ◽  
Rotor Blades ◽  
Rotor Speed ◽  
Blade Vibration ◽  
Compressor Rotor ◽  
Set Up ◽  
The Impact

To describe the frequency distribution of the rotor blades and improve the optimization, resonance reliability of the rotor blades was analyzed in this paper. Considering the variety of rand-om variables, we jointly used finite element method and response surface method. The Campbell diagram was set up to describe blade resonance by analyzing the compressor rotor blade vibration characteristics. For the second-order vibration failure of the rotor blade, we considered the impact of random variables with the rotor blade material, the blade dimension and the rotor speed. The pro-bability distribution and allowable reliability of the second-order vibration frequency was calculated, and the sensitivity of the random variables influencing vibration frequency was completed. The res-ults show that the resonance reliability with the confidence level 0.95 of the rotor blade are = 0.99753 with the excited order =4 and =0.99767 with the excited order =5,and basically ag-ree with the design requirements when the rotor speed =9916.2, and the factors mainly affe-cting the distribution of the second-order vibration frequency of the blades include elastic modulus, density and the rotor speed, with the sensitivity probabilities 35.09%,34.56% and 24.15% respecti-vely.

Movement of Deposited Water on Turbomachinery Rotor Blade Surfaces

2006 ◽  
Vol 129 (2) ◽  
pp. 394-403 ◽  
Author(s):  
John Williams ◽  
John B. Young
Keyword(s):  
Rotor Blade ◽  
Water Movement ◽  
Equations Of Motion ◽  
Surface Friction ◽  
Blade Surface ◽  
Inertia Effects ◽  
Mathematical Generalization ◽  
Aero Engine ◽  
Blade Tip ◽  
Fan Blade

A theoretical approach for calculating the movement of liquid water following deposition onto a turbomachine rotor blade is described. Such a situation can occur during operation of an aero-engine in rain. The equation of motion of the deposited water is developed on an arbitrarily oriented plane triangular surface facet. By dividing the blade surface into a large number of facets and calculating the water trajectory over each one crossed in turn, the overall trajectory can be constructed. Apart from the centrifugal and Coriolis inertia effects, the forces acting on the water arise from the blade surface friction, and the aerodynamic shear and pressure gradient. Nondimensionalization of the equations of motion provides considerable insight and a detailed study of water flow on a flat rotating plate set at different stagger angles demonstrates the paramount importance of blade surface friction. The extreme cases of low and high blade friction are examined and it is concluded that the latter (which allows considerable mathematical generalization) is the most likely in practice. It is also shown that the aerodynamic shear force, but not the pressure force, may influence the water motion. Calculations of water movement on a low-speed compressor blade and the fan blade of a high bypass ratio aero-engine suggest that in low rotational speed situations most of the deposited water is centrifuged rapidly to the blade tip region.

The Dynamics of a Multigimbal Hooke's-Joint Gyroscope

1978 ◽  
Vol 20 (5) ◽  
pp. 255-262 ◽  
Author(s):  
J. S. Burdess ◽  
C. H. J. Fox
Keyword(s):  
Angular Displacement ◽  
Angular Rate ◽  
Equations Of Motion ◽  
Displacement Sensor ◽  
Rotor Speed ◽  
Free Response ◽  
Linearized Equations ◽  
Stiffness Characteristics ◽  
Parallel Assembly ◽  
Hooke’S Joint

The paper considers the dynamics of a gyroscope where the gyro rotor is supported and driven through a gimbal suspension, which takes the form of a parallel assembly of universal couplings. The linearized equations of motion are derived and the free response, and the response to a constant applied rate of turn, are determined. It is shown that the form of each response is greatly influenced by the geometry, inertia and stiffness characteristics of the suspension and the rotor speed. Tuning conditions are identified and the performance of the gyroscope as either an angular rate or an angular displacement sensor is evaluated.

Transient Simulation Method for Autorotation in Forward Flight

2013 ◽  
Vol 284-287 ◽  
pp. 1001-1006 ◽  
Author(s):  
Hak Yoon Kim
Keyword(s):  
Steady State ◽  
High Speed ◽  
Equations Of Motion ◽  
Simulation Method ◽  
Transient Simulation ◽  
Rotor Speed ◽  
Forward Flight ◽  
Data Set ◽  
Shaft Angle ◽  
Lift And Drag

This paper describes the Transient Simulation Method (TSM) which predicts the steady state and performance of autorotation in forward flight. Flapping and rotational equations of motion are integrated from an arbitrary initial rotor speed, and the steady state of autorotation is obtained as a periodic solution through a transient process. The induced velocity field update method and the average thrust, lift, and drag computations during the transition are described in detail. TSM is then applied to the model rotor to validate the feasibility. High speed autorotation is simulated using an aerodynamic data set that is analyzed by the two-dimensional compressible Navier-Stokes Solver. Rotor speed variation for increases in airspeed at low shaft angle is presented and discussed. When TSM is used with sophisticated aerodynamic data analyzed as functions of the blade angle of attack, the Reynolds number, and the Mach number, the autorotation range for the collective pitch, velocity, and shaft angle can be reasonably explored.

Vibrations of a Rotating Beam

1966 ◽  
Vol 11 (4) ◽  
pp. 17-24 ◽  
Author(s):  
Jay L. Lipeles
Keyword(s):  
Boundary Conditions ◽  
Coordinate System ◽  
Equations Of Motion ◽  
Cone Angle ◽  
Elastic Stiffness ◽  
Degree Of Freedom ◽  
Mode Shapes ◽  
Flexible Beam ◽  
Rotor Speed ◽  
Classical Formula

The object of this paper is to analyze the coupled flatwise (out of plane) edgewise (in plane) vibrations of a beam rotating about one if its ends. The hub is assumed motionless and the vibration is considered to occur about a large deflected position. That is, coning and lagging angles are allowed to be large. The equations of motion are derived by a combination of techniques. The kinetic energy of the system is expressed in terms of coordinates lying in the beam. This is done by making use of four coordinate transformations that relate the beam coordinate system to a fixed coordinate system. The inertial load distribution is obtained by application of the first two terms of Lagrange's equation. These loads are used to compute the bending moment distribution which is substituted into Euler's beam bending equation. The equations of motion are solved by assuming a solution in the form of a linear combination of orthogonal modes. These equations are multiplied by the jth mode shape and integrated over the beam length. There are four terms resulting; the mass and elastic stiffness terms form a diagonal array and the Coriolis' and centrifugal spring terms form a full array. These equations may be solved by easily available matrix techniques. The modes chosen for the solution are the normal modes of the nonrotating beam. The advantage of this choice is that each of the modes already satisfies the problem boundary conditions. Since the non‐rotating modes are a good approximation to the rotating modes the series converges rapidly and can be cut off after a few terms. Several sample problems are worked out. First, the beam is assumed rigid and free to flap. The classical formula for flapping frequency is verified with the addition that the terms due to large cone and lag angles are included. Second, the same problem is done except that instead of the flapping degree of freedom the lagging degree of freedom is analyzed. The classical formula for lagging is also verified for the zero cone angle. When the cone angle is large this degree of freedom becomes statically unstable. Third, the above problem is redone for the coupled lagging — flapping degrees of freedom. Fourth, a flexible beam is assumed with zero cone and lag angles. Mode shapes and frequencies are computed as a function of rotor speed. It is shown that as rotor speed increases the beam mode shapes and frequencies approach those of a chain. That is, the elastic stiffness becomes negligible relative to the centrifugal stiffness. The advantages of the formulation developed in this paper (in addition to allowing consideration of large coning and lagging angles) are: 1) that the terms that involve rotor speed (the centrifugal spring and the Coriolis coupling) have that parameter as a factor multiplying the whole matrix so that if frequencies and modes are required over a range of rotor speeds the centrifugal and Coriolis' terms need only be calculated once; 2) at large rotor speeds the Myklestad analysis has difficulty converging but in this procedure, because the non‐rotating modes already satisfy the boundary conditions, there is no difficulty in convergence.

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