Flapping motion
To examine the flapping motion more fully we assume, unless otherwise stated, that the flapping hinge is on the axis of rotation. This simplifies the considerations without hiding anything of significance. Referring to Fig 4.12, the flapping takes place under conditions of dynamic equilibrium, about the hinge, between the aerodynamic lift (the exciting function), the centrifugal force (the ‘spring’ or restraining force) and the blade inertia (the damping). In other words, the once-per-cycle oscillatory motion is that of a dynamic system in resonance. The flapping moment equation is seen to be
We shall return to this equation later. The centrifugal force is by far the largest force acting on the blade and provides an essential stability to the flapping motion. The degree of stability is highest in the hover condition (where the flapping angle is constant) and decreases as the advance ratio increases. Bramwell’s consideration of the flapping equation (p. 153 et seq.) leads in effect to the conclusion that the motion is dynamically stable for all realistic values of m. Maximum flapping velocities occur where the resultant air velocity is at its highest and lowest, that is at 90° and 270° azimuth. Maximum displacements occur 90° later, that is at 180° (upward) and 0° (downward). These displacements mean that the plane of rotation of the blade tips,the tip-path plane(TPP),is tilted backwards relative to the plane normal to the rotor shaft, the shaft normal plane (SNP). In hover the blades cone upwards at a constant angle a0, say, to the shaft normal plane. The coning angle is that at which the blade weight is supported by the aerodynamic lift. Its existence has an additional effect on the orientation of the TPP during rotation in forward flight. Figure 4.13 shows that because of the coning angle, the flight velocity V has a lift-increasing effect on a blade at 180° (the forward blade) and a lift-decreasing effect on a blade at 0° (the rearward blade). This asymmetry in lift is, we see, at 90° to the side-to-side asymmetry discussed earlier: its effect is to tilt the TPP laterally and since the point of lowest tilt follows 90° behind the point of lowest lift, the TPP is tilted downwards to the right, that is on the advancing side. The coning and disc tilt angles are normally no more than a few degrees. Since in any steady state of the rotor the flapping motion is periodic, the flapping angle can be expressed in the form of a Fourier series:
To examine the flapping motion more fully we assume, unless otherwise stated, that the flapping hinge is on the axis of rotation. This simplifies the considerations without hiding anything of significance. Referring to Fig 4.12, the flapping takes place under conditions of dynamic equilibrium, about the hinge, between the aerodynamic lift (the exciting function), the centrifugal force (the ‘spring’ or restraining force) and the blade inertia (the damping). In other words, the once-per-cycle oscillatory motion is that of a dynamic system in resonance. The flapping moment equation is seen to be
We shall return to this equation later. The centrifugal force is by far the largest force acting on the blade and provides an essential stability to the flapping motion. The degree of stability is highest in the hover condition (where the flapping angle is constant) and decreases as the advance ratio increases. Bramwell’s consideration of the flapping equation (p. 153 et seq.) leads in effect to the conclusion that the motion is dynamically stable for all realistic values of m. Maximum flapping velocities occur where the resultant air velocity is at its highest and lowest, that is at 90° and 270° azimuth. Maximum displacements occur 90° later, that is at 180° (upward) and 0° (downward). These displacements mean that the plane of rotation of the blade tips,the tip-path plane(TPP),is tilted backwards relative to the plane normal to the rotor shaft, the shaft normal plane (SNP). In hover the blades cone upwards at a constant angle a0, say, to the shaft normal plane. The coning angle is that at which the blade weight is supported by the aerodynamic lift. Its existence has an additional effect on the orientation of the TPP during rotation in forward flight. Figure 4.13 shows that because of the coning angle, the flight velocity V has a lift-increasing effect on a blade at 180° (the forward blade) and a lift-decreasing effect on a blade at 0° (the rearward blade). This asymmetry in lift is, we see, at 90° to the side-to-side asymmetry discussed earlier: its effect is to tilt the TPP laterally and since the point of lowest tilt follows 90° behind the point of lowest lift, the TPP is tilted downwards to the right, that is on the advancing side. The coning and disc tilt angles are normally no more than a few degrees. Since in any steady state of the rotor the flapping motion is periodic, the flapping angle can be expressed in the form of a Fourier series:
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