A method for identifying friction of a bearing-equipped trailing edge flap drive mechanism

Abstract

The application provides a bearing-equipped trailing edge flap drive mechanism friction force identification method, which comprises the following steps: applying a series of improved triangular waves with different speeds to a driver and performing closed-loop control, then performing zero-phase digital filtering on the collected flap angle signals to suppress the interference of random noise on the signals, adjusting the derivation step to suppress the chattering phenomenon of the speed to obtain the flap angle speed, then intercepting the uniform speed rising section and the uniform speed falling section, then specifying a series of angle speeds, identifying the friction model of the relationship between the friction force and the speed, then identifying the relationship between the identified parameters and the flap angle, and finally obtaining the friction force of the flap drive mechanism with respect to the flap angle and the flap angle speed.

Description

Technical Field

[0001] The present application belongs to the technical field of active control of trailing edge flap rotors, and in particular relates to a friction force identification method for a trailing edge flap drive mechanism with a bearing. Background Art

[0002] Rotor-induced vibration and noise are the primary sources of vibration and noise during helicopter flight, severely impacting the service life of instruments and meters, as well as passenger comfort. Active trailing-edge flap control (ACF), a technology for actively controlling helicopter vibration and noise developed over the past decade, has become a research hotspot. ACF rotors use actuators to drive additional flaps at the trailing edge of the blades to deflect according to a prescribed pattern, thereby altering the rotor's aerodynamic force distribution. This reduces alternating loads on the hub and suppresses disc noise, achieving vibration or noise reduction.

[0003] The trailing edge flap drive mechanism is typically located at the outer end of the blade, integrated into the box cavity formed by the blade airfoil. The flap driver is typically a piezoelectric actuator installed inside the blade, with one end fixed and the other end connected to the trailing edge flap via a drive rod. The trailing edge flap is fixed to the blade via a bearing or flexible mechanism. During operation, the trailing edge flap is driven by voltage to flap at a given frequency (single or multiple frequencies). In operation, the flap drive mechanism must withstand the huge centrifugal force caused by the rotation of the blade, and the flap flapping motion must also overcome complex aerodynamic loads.

[0004] To achieve flapping motion, spherical plain bearings, thrust bearings, ball bearings, and self-lubricating bushing bearings are often used in the design of the connection between the drive link and the blades. When the flaps are operated with a sinusoidal excitation voltage, bearing friction often causes the sinusoidal flapping waveform to flatten at its peak during low-frequency flapping motion, introducing undesirable frequency components such as double and triple frequencies.

[0005] It is difficult to eliminate this phenomenon through control laws such as PID. Therefore, it is necessary to identify the friction in the flap drive mechanism. On the one hand, it can guide the design of the flap drive mechanism and minimize the influence of friction on the flap flapping motion. On the other hand, in the servo closed-loop control of the flap, an identification-based friction model can be introduced to compensate for the friction, reduce or eliminate the flat-top phenomenon at the peak of the sine wave, and improve the control accuracy.

[0006] Unlike conventional motor control based on speed control, friction in motor speed control is generally only related to speed. The motor load may also experience random changes, but generally only in torque, while the normal force between the stator and mover of the rotating components does not change much. In the deflection motion of the trailing edge flap, the planar motion of the actuator is converted into deflection motion through the drive link and the flap's lever arm, which also acts as an amplification. The flap's load generally varies with the angle, causing the normal force on the friction surfaces such as the bearings at both ends of the drive link to change.

[0007] Therefore, the friction force of the trailing edge flap drive mechanism with bearings is not only a function of the flap flapping speed, but also a function of the angle. In addition, due to the opposite direction of the friction force in the flap ascending section (positive speed) and the descending section (negative speed), the flap driving force will suddenly change, so that the friction model parameters of the positive speed and negative speed will also be very different.

[0008] In addition, unlike the continuous rotation of loads such as motors, the flapping motion of the flap is a reciprocating motion, making it difficult to implement conventional friction identification based on constant speed tracking in the flap flapping motion. Summary of the Invention

[0009] Purpose of the invention: To propose a friction force identification method for a trailing edge flap drive mechanism with a bearing, aiming to provide an identification method for the friction force that varies with angle and angular velocity under reciprocating motion.

[0010] The present application provides a method for identifying friction force of a trailing edge flap drive mechanism with a bearing, the method comprising:

[0011] According to the angle range and deflection frequency of the flap deflection according to the sine law, the angular velocity range of the theoretical flap deflection is calculated;

[0012] According to the theoretical angular velocity range of flap deflection, an improved triangular wave is used to excite the flap drive mechanism, and PI closed-loop control is adopted to obtain a series of flap angle response signals and PI control output signals under positive and negative flap deflection speeds;

[0013] selecting a cutoff frequency based on the flap angle response signal;

[0014] Using a zero-phase digital low-pass filter to filter the flap angle response signal according to the cutoff frequency to obtain a filtered flap angle response signal;

[0015] selecting a derivation step size, and using the derivation step size to derive the filtered flap angle response signal to obtain an actual angular velocity of flap deflection;

[0016] intercepting the uniform ascending segment and descending segment signals in the filtered response signal and their corresponding PI control output signals according to the actual angular velocity of the flap deflection;

[0017] Fitting the signals of the uniform speed ascending segment and descending segment with the PI control output signal to obtain a relationship between the signals of the uniform speed ascending segment and descending segment and the PI control output signal;

[0018] Based on the relationship between the uniform speed ascending segment and descending segment signals and the PI control output signal, the stiffness and friction force of the flap elastic load are obtained.

[0019] Preferably, the method further comprises:

[0020] selecting a friction model based on a curve of a change in the friction force versus the angular velocity of the actual flap deflection;

[0021] Based on the friction model, the relationship between the friction force and the angular velocity of the actual flap deflection is identified.

[0022] Preferably, the method further comprises:

[0023] Based on the relationship between the friction force and the actual angular velocity of the flap deflection, the relationship between the friction force and the actual flap deflection angle and the actual angular velocity of the flap deflection is identified.

[0024] Preferably, the calculating of the theoretical angular velocity range of flap deflection according to the angular range and deflection frequency of the flap deflected according to the sinusoidal law comprises:

[0025] Obtain the deflection frequency of the flap deflected according to the sinusoidal law, the flap angle offset and the amplitude of the flap sinusoidal motion;

[0026] Calculating a flap circular frequency according to a deflection frequency of the flap deflected according to a sinusoidal law;

[0027] Calculating a theoretical flap deflection angle based on the flap circular frequency, the flap angle offset, and the amplitude of the flap sinusoidal motion;

[0028] Based on the theoretical flap deflection angle, an angular velocity range of the theoretical flap deflection is obtained.

[0029] Preferably, the improved triangular wave is characterized in that the peaks and troughs are modified to transition to sine waves, and the connection with the rising and falling sections of the triangular wave is smooth and the first-order derivative is smooth.

[0030] Preferably, the improved triangle wave formula is as follows:

[0031] u=A sin +A sin*cos(2πft-π),t≤T sin

[0032] u=V*(tT sin +T1),T sin <t≤T2-T sin

[0033] u=A tri -A sin +A sin *cos(2πf(t-T2)),T2-T sin <t≤T2+T sin

[0034] u=A tri -V*(t-T2-T sin +T1),T2+T sin <t≤2T2-T sin

[0035] u=A sin +A sin *cos(2πf(t-2T2)-π),T2+T sin <t≤2T2-T sin

[0036] Among them, u represents the command signal, A tri Indicates the amplitude of the triangle wave, A sin Indicates the amplitude of the sine wave at the peak and trough of the improved triangle wave, V represents the speed of the triangle wave ramp section, T sin It represents half of the duration of the sine wave at the peak or trough, T1 represents the time difference when u of the original triangle wave and the improved triangle wave are equal in the uniform rising section, and T2 represents half of the period of the improved three-level wave.

[0037] Preferably, selecting the cutoff frequency based on the response signal includes:

[0038] Select 0-phase digital low-pass filter;

[0039] The response signal is filtered using the 0-phase digital low-pass filter, and the cutoff frequency is selected by comparing the signals before and after filtering to determine whether the amplitude is attenuated.

[0040] Preferably, the expression for the relationship between the uniform speed ascending segment and descending segment signals and the PI control output signal is:

[0041] f=k flap θ flap -u PI

[0042] Where f represents friction, kflap represents the stiffness of the flap drive mechanism's elastic load related to the flap angle, u PI Represents the PI control output signal, θ flap Represents the flap angle response signal during the uniform ascent and descent phases.

[0043] Beneficial technical effects of this application:

[0044] 1. Under the condition that the flap flapping motion angle range is limited, the improved triangular wave can be used to increase the angle range of the flap angle uniform change section under closed-loop control to improve the recognition accuracy;

[0045] 2. When only the flap angle can be measured but the angular velocity cannot be directly measured, the random noise of the signal can be suppressed by means of zero-phase digital low-pass filtering and derivative step size setting to improve the speed identification accuracy;

[0046] 3. A friction force identification method related to angle and angular velocity is proposed. Conventional friction identification of rotating mechanisms is only related to angular velocity. BRIEF DESCRIPTION OF THE DRAWINGS

[0047] Figure 1 This is a flow chart for identifying friction force in a trailing edge flap drive mechanism with bearings;

[0048] Figure 2 This is a comparison diagram between the improved triangle wave signal and the conventional triangle wave signal;

[0049] Figure 3 To improve the schematic diagram of each time node in the triangle wave signal;

[0050] Figure 4 Schematic diagram of the trailing edge flap drive mechanism;

[0051] Among them: 1, diamond-shaped piezoelectric actuator; 2, driving connecting rod; 3, trailing edge flap; 4, flap rotation axis. DETAILED DESCRIPTION

[0052] The present application provides a friction force identification method for a trailing edge flap drive mechanism with a bearing, aiming to provide an identification method for the friction force that varies with angle and angular velocity in a reciprocating motion situation.

[0053] This invention proposes a friction force identification method for a trailing-edge flap drive mechanism with a bearing. This method applies a series of modified triangular waves at different speeds to the driver and performs closed-loop control. The collected flap angle signal is then subjected to zero-phase digital filtering to suppress random noise interference. The derivative step size is adjusted to suppress velocity chattering to obtain the flap angular velocity. A uniform ascending segment and a uniform descending segment are then intercepted. A series of angular velocities are then specified. A friction model is then identified for the relationship between friction force and velocity. The relationship between the identified parameters and the flap angle is then identified. Ultimately, the friction force in the flap drive mechanism related to the flap angle and flap angular velocity is determined. The specific steps are as follows:

[0054] Step 1: Calculate the angular velocity range of the flap deflection according to the angular range and deflection frequency of the flap deflection according to the sinusoidal law;

[0055] θ=θ0+θ amp sin(ωt)

[0056]

[0057] In the above formula, θ represents the flap angle, represents the flap angular velocity, ω represents the flap circular frequency, θ0 represents the flap angle offset, θ amp Represents the amplitude of the sinusoidal motion of the flap.

[0058] Step 2: Based on the angular velocity range obtained in step 1, an improved triangular wave is used to excite the flap drive mechanism, and PI closed-loop control is used to obtain a series of responses under positive and negative flap deflection speeds.

[0059] The improved triangular wave is characterized by the transition from peaks to troughs to sinusoidal waves, and the smooth connection between the rising and falling sections of the triangular wave and the smooth first-order derivative. This can avoid or reduce the oscillation phenomenon of the closed-loop controller at the peaks and troughs, and increase the angular range of the uniform speed control section. The improved triangular wave formula is as follows:

[0060] u=u amp +u amp *cos(2πft-π),t≤T sin

[0061] u=V*(tT sin +T1),T sin <t≤T2-T sin

[0062] u=Au amp +u amp *cos(2πf(t-T2)),T2-T sin <t≤T2+T sin

[0063] u=AV*(t-T2-T sin +T1),T2+T sin <t≤2T2-T sin

[0064] u=u amp +u amp *cos(2πf(t-2T2)-π),T2+T sin <t≤2T2-T sin

[0065] In the above formula, A represents the amplitude of the triangle wave, u amp The amplitude of the sine wave at the peak and trough of the improved triangle wave, V represents the speed of the triangle wave ramp section, T sin It represents half of the duration of the sine wave at the peak or trough, T1 represents the time from the minimum angle of the uniform rising section in the improved triangular wave to the 0 angle, and T2 represents half of the period of the improved tertiary wave.

[0066] Step 3: Filter the signal using a zero-phase digital filter. By comparing the signals before and after filtering to see if the amplitude has attenuated, select the appropriate low-pass filter cutoff frequency (to suppress noise interference in the signal without causing phase delay).

[0067] Step 4: Derivative the measured angle signal with a suitable derivation step length to obtain the angular velocity. (A suitable derivation step length can suppress the chattering phenomenon that may be caused by the first-order derivative of the angle signal, i.e., the angular velocity).

[0068] Step 5: According to the angular velocity obtained in step 4, intercept the uniform rising and falling segments of the triangular wave signal;

[0069] Step 6: For the uniform speed control segment signal intercepted in step 5, the relationship between the angle and the control output voltage (or the output force of the driver) is obtained by fitting. According to the formula f=k flap θ-u, calculate the stiffness k of the flap elastic load flap ;

[0070] Where f represents friction, k flap represents the stiffness of the elastic load related to the flap angle in the flap drive mechanism, and u represents the actual control input;

[0071] Step 7: For the uniform speed control segment signal intercepted in step 5, fit the relationship between the angle and the control output voltage (or driver output force) at different angular velocities. Keeping the angle the same, calculate the positive speed friction force and negative speed friction force corresponding to different angular velocities at this angle. According to the curve of friction force changing with angular velocity, select a suitable friction model, such as the friction model of Coulomb friction + viscous friction, and identify the relationship between friction force and speed at this angle.

[0072] The relationship between the excitation voltage applied to the flap drive mechanism and the flap angle can be expressed as follows:

[0073]

[0074] F load =f+k flap θ

[0075] In the above formula, J represents the flap moment of inertia, represents the flap angular acceleration, F load The load representing the flap deflection motion.

[0076] In order to identify the friction force of the flap drive mechanism related to the recommended angular velocity and angle, the triangular wave closed-loop control of step 2 is applied and the uniform speed section under closed-loop control is selected. At this time, the flap angular acceleration have:

[0077] u=f+k flap θ

[0078] When the angle remains unchanged, for the friction model composed of Coulomb friction + viscous friction, the friction force f can be expressed as:

[0079]

[0080] In the above formula, represents the friction model coefficient when the speed is positive, Indicates the friction model coefficient when the velocity is negative

[0081] Step 8: Based on the angular range of flap deflection, refer to step 7 to identify the relationship between friction force and angular velocity at different angles. On this basis, identify the relationship between parameters and angles in the friction model.

[0082] Within the flap angle range, by specifying different flap angles, it is obvious that it is easy to identify a series of friction model parameters. Then, the relationship between the parameters of this series of friction models and the flap angle is identified, and the following is obtained:

[0083]

[0084] In summary, the friction force of the flap drive mechanism related to the flap angle and flap angular velocity can be obtained through experimental identification.

[0085] In other embodiments of the present application, for a pair of 4 blades, the base frequency of the ACF rotor system is 6 Hz, the trailing edge flap drive mechanism (such as Figure 4 ) The flap deflection and drive rods in the propeller are connected by bearings. The maximum excitation frequency of the four-blade flap is generally 5 / rev, or 30Hz. If the flap deflection angle range is 0° to 6°, the friction force of the mechanism within its operating range is identified as follows:

[0086] Step 1: Calculate the flap circular frequency based on the maximum deflection frequency of the flap deflected according to the sinusoidal law. The formula is as follows:

[0087] ω=2πf

[0088] In the above formula, f represents the deflection frequency, ω represents the flap circular frequency, and π represents pi.

[0089] Step 2: Based on the angular range of the flap deflection according to the sinusoidal law and the deflection circular frequency, calculate the angular velocity range of the flap deflection. The formula is as follows:

[0090] θ=θ0+θ amp sin(ωt)

[0091]

[0092] In the above formula, θ represents the flap angle, represents the flap angular velocity, ω represents the flap circular frequency, θ0 represents the flap angle offset, θ amp represents the amplitude of the sinusoidal motion of the flap, and t represents the time.

[0093] Step 3: Based on the angular velocity range calculated in step 2, the flap drive mechanism is excited using an improved triangular wave at certain speed intervals. PI closed-loop control is used to obtain a series of flap angle response signals under positive and negative flap deflection speeds.

[0094] The improved triangular wave is characterized by modifying the peaks and troughs of the triangular wave to transition to a sine wave, and the connections with the rising and falling sections of the triangular wave are smooth, and the first-order derivative is smooth. This can avoid or reduce the oscillation phenomenon of the closed-loop controller at the peaks and troughs, increase the angular range of the uniform speed control section, and prevent the piezoelectric driver from being damaged by the step response. The improved triangular wave formula is as follows:

[0095] u=A sin +A sin *cos(2πft-π),t≤T sin

[0096] u=V*(tT sin +T1),T sin <t≤T2-T sin

[0097] u=A tri -A sin +A sin *cos(2πf(t-T2)),T2-T sin <t≤T2+T sin

[0098] u=A tri -V*(t-T2-T sin +T1),T2+T sin <t≤2T2-T sin

[0099] u=A sin +A sin *cos(2πf(t-2T2)-π),T2+T sin <t≤2T2-T sin

[0100] In the above formula, u represents the command signal, A tri Indicates the amplitude of the triangle wave, A sin Indicates the amplitude of the sine wave at the peak and trough of the improved triangle wave, V represents the speed of the triangle wave ramp section, T sin It represents half of the duration of the sine wave at the peak or trough, T1 represents the time difference when the original triangle wave and the improved triangle wave are equal in the uniform rising section, and T2 represents half of the period of the improved three-level wave. Figure 2 and Figure 3 .

[0101] PI closed-loop control is based on basic PID control. The discrete expression of PI control is:

[0102]

[0103] e′(k)=SP(k)-PV(k)

[0104] In the above formula, u PI (k) represents the PI control output signal of the kth control step in discrete control, Kp represents the proportional gain, Ki represents the integral gain, SP(k) represents the value of the improved triangular wave command signal u at the kth control step, PV(k) represents the value of the flap angle signal θ fed back at the kth control step, e′(k) represents the difference between the triangular wave command signal and the feedback flap angle signal at the kth control step, that is, the control error, It represents the cumulative sum of errors from the 0th control step to the kth control step.

[0105] Step 4: Use a zero-phase digital low-pass filter to filter the flap angle response signals at positive and negative flap deflection speeds obtained in step 3. The cutoff frequency of the low-pass filter is selected by comparing the amplitude of the signal before and after filtering to see if there is significant attenuation.

[0106] Step 5: Derivative the filtered flap angle signals obtained in Step 4 using a suitable derivation step length to obtain a series of measured flap angular velocities. The derivation step length is selected based on whether the derived flap angular velocity signals exhibit buffeting.

[0107] Step 6: Based on the series of measured flap angular velocities obtained in step 5, intercept the PI control output signal u in step 3 corresponding to the uniform ascending and descending segments in the angular velocity signal. PI and flap angle response signal θ flap .

[0108] Step 7: PI control output signal u of a series of uniform speed control segments intercepted in step 6 PI The flap angle response signal is fitted according to the following formula to obtain a series of relationships between flap angles and control output voltage (or driver output force);

[0109] f=k flap θ flap -u PI

[0110] In the above formula, f represents friction, k flap Indicates the stiffness of the flap drive mechanism's elastic load related to the flap angle.

[0111] Step 8: Assume that the flap angle in the relationship between the series of flap angles and the control output voltage obtained in Step 7 is the same, thereby obtaining a series of friction force values ​​at different flap angular velocities. For the obtained series of friction force values ​​and flap angular velocities at different flap angular velocities, select an appropriate friction model, such as a friction model consisting of Coulomb friction + viscous friction, and identify the parameters of the friction model. The friction model is as follows:

[0112]

[0113] In the above formula, represents the friction model coefficient when the flap angular velocity is positive, represents the friction model coefficient when the flap angular velocity is negative, sgn is the sign function, Indicates the flap angular velocity calculated based on the measured flap angle.

[0114] Step 9: Repeat step 8, and let the flap angle in step 8 be taken at a certain step interval within the flap angle range obtained in step 2, and identify the friction model parameters between the friction force and the flap angular velocity at different flap angles. The relationship between this series of friction model parameters and the corresponding flap angles is then identified through polynomial fitting, and the following is obtained:

[0115]

[0116] In the above formula, a n 、a0 represents and θ flap The polynomial coefficients obtained by fitting between n , b0 represents and θ flap The polynomial coefficients obtained by fitting between n 、c0 represents and θ flap The polynomial coefficients obtained by fitting between n d0 represents and θ flap The polynomial coefficients obtained by fitting between .

[0117] In summary, the relationship between the friction force of the flap drive mechanism and the flap angle and flap angular velocity was identified.

Claims

1. A method for identifying friction force of a trailing edge flap drive mechanism with a bearing, characterized in that: The method comprises: According to the angle range and deflection frequency of the flap deflection according to the sine law, the angular velocity range of the theoretical flap deflection is calculated; According to the theoretical angular velocity range of flap deflection, an improved triangular wave is used to excite the flap drive mechanism, and PI closed-loop control is adopted to obtain a series of flap angle response signals and PI control output signals under positive and negative flap deflection speeds; selecting a cutoff frequency based on the flap angle response signal; Using a zero-phase digital low-pass filter to filter the flap angle response signal according to the cutoff frequency to obtain a filtered flap angle response signal; selecting a derivation step size, and using the derivation step size to derive the filtered flap angle response signal to obtain an actual angular velocity of flap deflection; intercepting the uniform ascending segment and descending segment signals in the filtered response signal and their corresponding PI control output signals according to the actual angular velocity of the flap deflection; Fitting the signals of the uniform speed ascending segment and descending segment with the PI control output signal to obtain a relationship between the signals of the uniform speed ascending segment and descending segment and the PI control output signal; Based on the relationship between the uniform speed ascending segment and descending segment signals and the PI control output signal, the stiffness and friction force of the flap elastic load are obtained.

2. The method according to claim 1, characterized in that The method further comprises: selecting a friction model based on a curve of a change in the friction force versus the angular velocity of the actual flap deflection; Based on the friction model, the relationship between the friction force and the angular velocity of the actual flap deflection is identified.

3. The method according to claim 2, characterized in that The method further comprises: Based on the relationship between the friction force and the actual angular velocity of the flap deflection, the relationship between the friction force and the actual flap deflection angle and the actual angular velocity of the flap deflection is identified.

4. The method according to claim 3, characterized in that The calculation of the theoretical angular velocity range of the flap deflection according to the angle range and deflection frequency of the flap deflection according to the sinusoidal law includes: Obtain the deflection frequency of the flap deflected according to the sinusoidal law, the flap angle offset and the amplitude of the flap sinusoidal motion; Calculating a flap circular frequency according to a deflection frequency of the flap deflected according to a sinusoidal law; Calculating a theoretical flap deflection angle based on the flap circular frequency, the flap angle offset, and the amplitude of the flap sinusoidal motion; Based on the theoretical flap deflection angle, an angular velocity range of the theoretical flap deflection is obtained.

5. The method according to claim 3, characterized in that The improved triangular wave is characterized in that the peaks and troughs are modified to transition to sine waves, and the connection between the rising and falling sections of the triangular wave is smooth and the first-order derivative is smooth.

6. The method according to claim 5, characterized in that The improved triangle wave formula is as follows: u=A sin +A sin *cos(2πft-π),t≤T sin u=V*(tT sin +T1),T sin <t≤T2-T sin u=A tri -A sin +A sin *cos(2πf(t-T2)),T2-T sin <t≤T2+T sin u=A tri -V*(t-T2-T sin +T1),T2+T sin <t≤2T2-T sin u=A sin +A sin *cos(2πf(t-2T2)-π),T2+T sin <t≤2T2-T sin Among them, u represents the command signal, A tri Indicates the amplitude of the triangle wave, A sin Indicates the amplitude of the sine wave at the peak and trough of the improved triangle wave, V represents the speed of the triangle wave ramp section, T sin It represents half of the duration of the sine wave at the peak or trough, T1 represents the time difference when u of the original triangle wave and the improved triangle wave are equal in the uniform rising section, and T2 represents half of the period of the improved three-level wave.

7. The method according to claim 3, characterized in that The step of selecting a cutoff frequency based on the response signal comprises: Select 0-phase digital low-pass filter; The response signal is filtered using the 0-phase digital low-pass filter, and the cutoff frequency is selected by comparing the signals before and after filtering to determine whether the amplitude is attenuated.

8. The method according to claim 3, characterized in that The expression for the relationship between the uniform speed ascending and descending segment signals and the PI control output signal is: f=k flap i flap -u PI Where f represents friction, k flap represents the stiffness of the flap drive mechanism's elastic load related to the flap angle, u PI Represents the PI control output signal, θ flap Represents the flap angle response signal during the uniform ascent and descent phases.

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