CN117290960B - Resistance rudder wing flutter analysis method considering unsteady aerodynamic force correction - Google Patents
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Abstract
The invention discloses a resistance rudder wing flutter analysis method considering unsteady aerodynamic force correction, belonging to the field of resistance rudder flutter analysis of an all-wing aircraft; the method comprises the following steps: aiming at a target resistance rudder wing, obtaining unsteady aerodynamic forces of the wing under different modal motions by a computational fluid dynamics method, further obtaining a high-precision aerodynamic coefficient matrix at a flutter point, and carrying out equal proportion correction on a series of low-precision aerodynamic coefficient matrices obtained by an engineering method. Calculating flutter of the corrected aerodynamic coefficient matrix based on a frequency domain flutter solving method of the matched Mach number; if the calculated results of the two times of flutter do not meet the convergence condition, the flutter speed and the frequency are substituted into the unsteady aerodynamic solution for iteration until convergence, and the corrected flutter speed under different cracking angles is obtained. The method considers the influence of the cracking angle of the resistance rudders on the flow field, improves the accuracy of flutter analysis of the wing of the resistance rudders, and simultaneously considers the calculation efficiency.
Description
Technical Field
The invention belongs to the field of drag rudder flutter analysis of flying wing type aircraft, and particularly relates to a drag rudder wing flutter analysis method considering unsteady aerodynamic force correction.
Background
The split resistance rudder (simply referred to as resistance rudder) is one of the main heading control devices of the existing flying-wing aircraft, and can be used as an efficient yaw device to improve the performance of the flying-wing layout aircraft in terms of yaw stability and control. However, in a large cracking angle state, complicated flow field states such as airflow separation or vortex flow and the like can be caused, and the traditional solving method cannot accurately analyze the flutter characteristics.
The current research methods aiming at similar problems mainly comprise the following steps: a fluid-solid coupling method, an engineering correction method and a wind tunnel test method. The problematic and borrowed parts are as follows:
First, a large number of numerical simulation calculations greatly reduce the calculation efficiency, increase the calculation cost, and are disadvantageous and popularized.
Secondly, when analyzing the special aeroelasticity problem brought by other complex flow fields, the used engineering correction method or the method by means of wind tunnel test can be popularized to the flutter analysis of the resistance rudder wing.
Therefore, it is necessary to provide a method for analyzing flutter of a resistance rudder wing, which can achieve both calculation efficiency and accuracy.
Disclosure of Invention
In order to solve the problem that the influence of the cracking rudder on the flow field cannot be considered in the engineering aerodynamic force calculation method, the invention provides the resistance rudder wing flutter analysis method which considers the unsteady aerodynamic force correction, and a series of low-precision data obtained in the engineering method can be corrected only by a small amount of high-precision numerical simulation calculation data, so that the flutter calculation accuracy of the resistance rudder wings with different cracking angles can be improved.
The method for analyzing the flutter of the resistance rudder wing considering the unsteady aerodynamic force correction comprises the following specific steps:
Step one, aiming at a target resistance rudder wing, carrying out modal analysis on the wing by a finite element analysis method to obtain a finite element model of the wing;
and secondly, acquiring initial reduction frequency values corresponding to each flutter point under the Mach number determined by the finite element model by adopting an engineering aerodynamic force calculation method, and obtaining low-precision aerodynamic force coefficients at each frequency by corresponding to table lookup, wherein each reduction frequency corresponds to an AIC matrix.
The initial reduction frequency value is artificially given;
Step three, aiming at a flutter point A under the current working condition, calculating the flutter speed and frequency of the resistance rudder wing at the flutter point A based on a frequency domain flutter solving method; and the main mode affecting the vibration is obtained.
The specific process is as follows:
first, the equation for solving the chatter vibration by the p-K method is as follows:
V represents the flight speed of the wing; m and K are mass and stiffness matrices generated by the structural finite element method; ρ represents the atmospheric density; q represents generalized coordinates; p is an operator of dimension 1, p=g+ik; where g=γk, γ represents the attenuation rate of vibration; q (ik) is the AIC matrix.
Then, under the given atmospheric density rho and flying speed V, solving the two vibration modes obtained by the above formula, respectively comparing the two vibration modes with the obtained reduced frequency k value, and selecting a k value which is closer to the obtained reduced frequency k value to recheck the aerodynamic matrix coefficient, so that iteration is repeated until satisfaction, at the moment, the first branch mode of free vibration of the wing in the airflow is sought, and other main modes can be found out similarly.
Finally, under the condition of unchanged flying height, the flying speed V is properly increased; repeating the steps, so as to find out the frequency and attenuation rate of two modes of the wing at the incremental speed;
The speed V is increased step by step to obtain a series of corresponding modal frequencies and attenuation rates, and a V-gamma graph and a V-omega graph are drawn; the corresponding speed is the flutter speed V F at the flutter point A, wherein the point intersecting with the abscissa in the V-gamma diagram;
Fourthly, at the flutter point A, the wing makes reciprocating sinusoidal motion according to the flutter frequency, and the incoming flow speed is the flutter speed; calculating an unsteady aerodynamic coefficient under the main modal movement by adopting a computational fluid dynamics method to further obtain a high-precision aerodynamic coefficient;
The specific formula is as follows:
step five, replacing the low-precision aerodynamic coefficient at the flutter point A with the high-precision aerodynamic coefficient, obtaining a correction coefficient S of the high-precision low-precision matrix at the flutter point A, and calculating corrected data under each reduction frequency by using the correction coefficient S;
the correction coefficient S formula is
Wherein AIC h(k0) represents high-precision data at dither point a, AIC l(k0) represents low-precision data at dither point a, AIC l(kn) represents low-precision data at the remaining reduction frequency, and AIC c(kn) represents corrected data at the remaining reduction frequency.
Step six, multiplying the low-precision aerodynamic coefficient matrix under each reduced frequency with the respective corrected data points to obtain the respective corresponding high-precision aerodynamic coefficient matrix, thereby obtaining corrected series AIC matrices under all reduced frequencies;
And step seven, returning to the step three, recalculating the flutter speed and the frequency after the first iteration by using the corrected series AIC matrix, judging whether the relative difference between the flutter speed and the frequency before the iteration is less than 5%, and if so, meeting the convergence condition to obtain the final corrected flutter speed and frequency. Otherwise, substituting the first iteration result into a new round of unsteady aerodynamic force calculation, and carrying out second correction until the adjacent two iteration results reach the convergence standard, thereby obtaining a final correction result.
Compared with the prior art, the invention has the beneficial effects that:
1. The method for analyzing the flutter of the resistance rudder wing by considering the unsteady aerodynamic force correction can consider the influence of the cracking angle of the resistance rudder wing on the wing surface pressure, and the change rule of the flutter speed increased along with the cracking angle is basically consistent with the wind tunnel test result, so that the accuracy of the engineering calculation method is greatly improved.
2. Compared with a fluid-solid coupling method, the method for analyzing the flutter of the resistance rudder wing by considering unsteady aerodynamic force correction does not need to establish an excessively fine finite element model, the vibration mode and the frequency of the main flutter mode are consistent with those of a test, the structural characteristics of the target wing can be reflected, and the method has higher calculation efficiency.
Drawings
FIG. 1 is a flow chart of a method for analyzing flutter of a resistance rudder wing taking into account unsteady aerodynamic force correction;
Figure 2 is a schematic view of an embodiment model of a drag rudder wing as used in an embodiment of the present invention.
Fig. 3 is a schematic view of a finite element structure of an embodiment of a resistance rudder wing according to an embodiment of the present invention.
FIG. 4 is a graph of interpolated points and interpolated airfoil deformations as employed in an embodiment of the present invention.
FIG. 5 is a step diagram of deriving a high-precision AIC matrix from modal aerodynamic results as employed by an embodiment of the present invention.
FIG. 6 is a graph comparing AIC data before and after correction as employed in an embodiment of the present invention.
Fig. 7 is a flow chart of frequency domain dither solution based on matched mach numbers, as employed in an embodiment of the present invention.
FIG. 8 is an iterative flow chart employed by an embodiment of the present invention;
FIG. 9 is a diagram of the final correction result according to the embodiment of the present invention.
| 1-Wing | 2-Spar | 3-Steering engine |
| 4-Wing rib | 5-Finite element model spar | 6-Finite element model rotating shaft |
| 7-Finite element model control surface area | 8-Finite element model spar section | |
| 10-Control surface | 20-Finite element control surface wing rib | 30-Finite element airfoil rib |
| 101-Fourth wing box upper and lower control surfaces | 102-Upper and lower control surfaces of fifth wing box | 201-First control surface rib |
| 202-Second control surface wing rib | 203-Third control surface wing rib | 301 First airfoil rib |
| 302-Second airfoil rib | 303-Third airfoil rib | 304-Fourth airfoil rib |
| 305-Fifth airfoil rib | 306-Sixth airfoil rib | 307-Seventh airfoil rib |
Detailed Description
The present invention is further described in detail below with reference to the drawings and examples for the purpose of facilitating understanding and practicing the present invention by those of ordinary skill in the art. It is apparent that the described embodiments are only some embodiments of the present invention, not all embodiments, and that all other embodiments obtained by persons having ordinary skill in the art without making creative efforts based on the embodiments in the present invention should fall within the protection scope of the present invention.
According to the method for analyzing the flutter of the resistance rudder wing by considering the unsteady aerodynamic force correction, the high-efficiency flutter of the resistance rudder wing is analyzed by correcting the low-precision aerodynamic force influence coefficient matrix (AIC matrix for short) obtained by an engineering aerodynamic force calculation method through the high-precision aerodynamic force influence coefficient matrix obtained by calculating the hydrodynamic force;
The method comprises the following steps: the method comprises the steps of carrying out modal analysis on a resistance rudder wing through a finite element analysis method, approximately representing structural dynamics characteristics of a target wing through a modal superposition method, analyzing main modes, flutter speeds and frequencies of flutter through an engineering aerodynamic force calculation method such as a dipole grid method, and simultaneously obtaining a low-precision aerodynamic force influence coefficient matrix under the main flutter modes to serve as a correction basis.
Obtaining unsteady aerodynamic force under different modal motions of the target wing through a computational fluid dynamics (computational fluid dynamic, CFD) method, deducing to obtain a high-precision aerodynamic force influence coefficient matrix at a flutter point, replacing a low-precision aerodynamic force influence coefficient matrix at the frequency, carrying out same-proportion correction on the aerodynamic force influence coefficient matrix at other frequencies according to conversion proportion at the flutter point to obtain a corrected aerodynamic force influence coefficient matrix, and calculating to obtain the flutter speed and frequency after the first iteration based on a frequency domain flutter solving method of matching Mach numbers.
If the vibration speed and the frequency do not meet the convergence condition compared with the vibration speed and the frequency under the condition of no correction, substituting the first iteration result into a new round of unsteady aerodynamic force calculation, and carrying out second correction in the same way until the adjacent two iteration results reach the convergence standard, thereby obtaining the final correction result.
As shown in fig. 1, the specific steps are as follows:
Step one, aiming at a target resistance rudder wing, carrying out modal analysis on the wing by a finite element analysis method to obtain a finite element model of the wing;
Obtaining initial reduction frequency values corresponding to each flutter point under the Mach number determined by the finite element model by adopting an engineering aerodynamic calculation method, obtaining low-precision aerodynamic coefficients at each frequency by corresponding to table lookup, and obtaining a AIC matrix corresponding to each reduction frequency by restoring transient motion (Laplacian domain) introduced in an unsteady aerodynamic formula into simple harmonic motion (frequency domain) in engineering software.
The initial reduction frequency value is artificially given;
step three, aiming at a flutter point A under the current working condition, calculating the flutter speed and frequency of a resistance rudder wing at the flutter point A based on a frequency domain flutter solving method matched with Mach numbers; and the main mode affecting the vibration is obtained.
The specific process is as follows:
first, the flutter is solved by the p-K method, and in ZAERO software the equation is as follows:
V represents the flight speed of the wing; m and K are mass and stiffness matrices generated by the structural finite element method; ρ represents the atmospheric density; q represents generalized coordinates; p is an operator of dimension 1, p=g+ik; where g=γk, γ represents the attenuation rate of vibration; q (ik) is the AIC matrix.
Then, under the condition of given atmospheric density rho and flying speed V, under the condition of a certain reduced frequency k value, the aerodynamic coefficient in a corresponding table is searched with Mach number Ma, namely, two vibration roots can be obtained by solving the above method, the two vibration roots are respectively compared with the obtained reduced frequency k value, and the aerodynamic matrix coefficient is rechecked by selecting a closer k value, so that iteration is repeated until satisfaction, at the moment, the first branch mode of free vibration of the wing in the airflow is sought, and other main modes can be found out similarly.
Finally, under the condition of unchanged flying height, changing the flying speed V (namely, making proper increase on the basis of the original speed V); repeating the steps, so as to find out the frequency and attenuation rate of two modes of the wing at the incremental speed;
The speed V is increased step by step to obtain a series of corresponding modal frequencies and attenuation rates, and a V-gamma graph and a V-omega graph are drawn; the corresponding speed is the flutter speed V F at the flutter point A, wherein the point intersecting with the abscissa in the V-gamma diagram;
Calculating the reduction frequency as using the dither speed V F and the dither frequency omega
Wherein L represents a half chord length;
Fourthly, at the flutter point A, the wing makes reciprocating sinusoidal motion according to the flutter frequency, and the incoming flow speed is the flutter speed; calculating an unsteady aerodynamic coefficient under the main modal movement by adopting a computational fluid dynamics method to further obtain a high-precision aerodynamic coefficient;
The specific formula is as follows:
step five, replacing the low-precision aerodynamic coefficient at the flutter point A with the high-precision aerodynamic coefficient, obtaining a correction coefficient S of the high-precision low-precision matrix at the flutter point A, and calculating corrected data under each reduction frequency by using the correction coefficient S;
the correction coefficient S formula is
Wherein AIC h(k0) represents high-precision data at dither point a, AIC l(k0) represents low-precision data at dither point a, AIC l(kn) represents low-precision data at the remaining reduction frequency, and AIC c(kn) represents corrected data at the remaining reduction frequency.
Step six, multiplying the low-precision aerodynamic coefficient matrix under each reduced frequency with the respective corrected data points to obtain the respective corresponding high-precision aerodynamic coefficient matrix, thereby obtaining corrected series AIC matrices under all reduced frequencies;
And step seven, returning to the step three, recalculating the flutter speed and the frequency after the first iteration by using the corrected series AIC matrix, judging whether the relative difference between the flutter speed and the frequency before the iteration is less than 5%, and if so, meeting the convergence condition to obtain the final corrected flutter speed and frequency. Otherwise, substituting the first iteration result into a new round of unsteady aerodynamic force calculation, and carrying out second correction until the adjacent two iteration results reach the convergence standard, thereby obtaining a final correction result.
Examples:
As shown in fig. 2, the model appearance of the drag rudder wing in this embodiment is schematically shown, the wing 1 is a long straight wing with a span length of 1.2m and a chord length of 0.4m, the wing adopts a NACA0015 wing, the main spar 2 is located at a chord length 0.3 times from the front edge, and the rudder axis is located at a chord length 0.7 times from the front edge. The wing section consists of six wing boxes, wherein the rear end of the fourth wing box is a cracking control surface 101, the rear end of the fifth wing box is a cracking control surface 102, and the cracking angle is controlled through a rotating shaft at the chord length of 0.7 chord length from the front edge of the wing.
Finite element analysis is carried out on a model of the resistance rudder wing, and a finite element model of the target wing is obtained, as shown in fig. 3, and is composed of a main beam 5, a wing rib 30 for replacing the strength of a wing box and a rotating shaft 6, wherein a control surface area 7 and a cross section of the main beam are indicated as 8. Analyzing the main vibration modes and frequencies of the finite element model by adopting engineering aerodynamic force calculation methods such as a dipole grid method and the like, wherein the main vibration modes are a first order and a third order, and are bending vibration; the two-order flutter frequency is also basically consistent with the wind tunnel test model. As shown in table 1:
TABLE 1
| Mode order | Simulation frequency/Hz | Experimental frequency/Hz |
| 1 | 1.337 | 1.383 |
| 3 | 11.901 | 11.855 |
Then, a series of low-precision AIC matrixes at different Mach numbers and different reduction frequencies are obtained through an engineering aerodynamic force calculation method, and the mode shape data and the low-precision AIC matrixes are used as the basis of a subsequent correction method.
The vibration speed in the uncorrected condition was 34.23m/s, the vibration frequency was 5.58Hz, and the reduction frequency at the vibration point was at this time
The method comprises the following steps of realizing the solution of unsteady aerodynamic force under different modal motions by a computational fluid dynamics method, selecting the two-order flutter main modes for solving, respectively controlling the wings to deform in the two-order modes, then calculating aerodynamic force influence coefficients of the ith-order mode on the jth-order mode, and finally forming a2×2 high-precision AIC matrix at the reduced frequency k=0.2, wherein the detailed process is as follows:
Firstly, a user-defined function (User Defined Function, UDF) is adopted in Ansys Fluent to realize grid deformation, and a custom macro program is written to realize the modal movement of the wing. As shown in fig. 4, the mode shape data is interpolated on a fluid grid through a surface spline interpolation method, and the physical displacement Z n＝Φn -q of the wing is calculated;
Where Φ n represents a modal matrix, q=q 1 ×sin (2pi ωt) represents a generalized displacement, and the wing makes a simple harmonic motion with a flutter frequency ω and an amplitude q 1.
Next, according to the formulaAnd solving modal aerodynamic force, and selecting a Transition k-kl-omega (3 eqn) turbulence model in Fluent to solve, wherein the incoming flow speed is set to be the flutter speed, and the time step is 5 multiplied by 10 -4 s. The output result modal aerodynamic force Q ij is in a trigonometric function rule and is fitted through a trigonometric functionTo obtain the amplitude F ij and the phaseThe procedure shown in fig. 5 results in a high-precision AIC matrix at this reduced frequency.
A series of low-precision AIC matrices are modified by the high-precision AIC matrix at the dither points. Taking a 20deg cracking angle as an example, a 11 is taken as an illustration of the correction method, as shown in fig. 6, where only high-precision data of k=0.2 at the chatter point is known, the low-precision data is replaced with this value, and the data at the rest of the reduced frequency is obtained by multiplying the same coefficient, according to the formulaCalculating; a series of corrected AIC data at different reduction frequencies is obtained. As shown in fig. 7, the frequency domain dither method based on the matching mach number is used to solve the dither speed and frequency by programming the AIC data after correction.
The first round of corrected chatter velocity and frequency are shown in table 2 below:
TABLE 2
Wherein: the unit of speed is m/s and the unit of frequency is Hz
After the first iteration, the vibration frequency is found to be about thirty percent different from that before the iteration, the reduction frequency at the vibration point is also changed by about thirty percent, at the moment, the vibration point is relatively large in difference before and after the iteration, and the result is considered to have certain instability.
The iterative flow is shown in FIG. 8, and the convergence criteria are set as follows when not satisfiedAnd when the frequency and the speed of the vibration at the moment are substituted into the CFD for calculation, high-precision data of the next iteration are obtained until the vibration points of the two iterations meet the convergence condition, the vibration points are considered to be stable, the iteration is stopped, and the final correction result is output.
The final correction result of this example is shown in fig. 9, and the correction method can embody the change rule that the flutter speed increases along with the increase of the cracking angle, and is more consistent with the wind tunnel test result. The correction method can consider the influence of airfoil pressure change on the flutter speed, and is more accurate than an engineering surface element method.
Repeating the steps for other cracking angle states to obtain the flutter characteristics under different cracking angles.
According to the invention, by the correction method based on unsteady aerodynamic force, the influence of airfoil pressure on flutter can be considered, and the defect of an engineering surface element method on the resistance rudder wing problem is overcome.
Compared with the fluid-solid coupling method, the method has the advantages that the calculation result of the engineering surface element method is adopted, and only a small amount of high-precision data is needed for correction, so that the calculation efficiency is greatly improved, and the calculation cost is reduced.
In the description of the present specification, the descriptions of the terms "one embodiment/manner," "some embodiments/manner," "example," "specific example," or "some examples," etc., mean that a particular procedure or feature described in connection with the embodiment/manner or example is included in at least one embodiment/manner or example of the application. In this specification, the schematic representations of the above terms are not necessarily for the same embodiment/manner or example. Moreover, the specific processes or features described may be applicable to other target airfoils in any one or more embodiments/modes or examples. Furthermore, the various embodiments/modes or examples described in this specification and the features of the various embodiments/modes or examples can be combined and combined by persons skilled in the art without contradiction.
The above description is only illustrative of the preferred embodiments of the present invention and of the principles of the technology employed. It will be appreciated by persons skilled in the art that the scope of the invention referred to in the present invention is not limited to the specific combinations of the technical features described above, but also covers other technical features formed by any combination of the technical features described above or their equivalents without departing from the inventive concept described above. Such as the above-mentioned features and the technical features disclosed in the present invention (but not limited to) having similar functions are replaced with each other.
Claims (3)
1. The method for analyzing the flutter of the resistance rudder wing by considering the unsteady aerodynamic force correction is characterized by comprising the following specific steps of:
Step one, aiming at a target resistance rudder wing, carrying out modal analysis on the wing by a finite element analysis method to obtain a finite element model of the wing;
Step two, acquiring initial reduction frequency values corresponding to each flutter point under the Mach number determined by the finite element model by adopting an engineering aerodynamic force calculation method, and obtaining low-precision aerodynamic force coefficients at each frequency by corresponding to table lookup, wherein each reduction frequency corresponds to an AIC matrix;
step three, aiming at a flutter point A under the current working condition, calculating the flutter speed and frequency of the resistance rudder wing at the flutter point A based on a frequency domain flutter solving method; simultaneously, the main mode affecting the flutter is obtained;
Different working conditions correspond to different cracking angles of the resistance control surface;
Fourthly, at the flutter point A, the wing makes reciprocating sinusoidal motion according to the flutter frequency, and the incoming flow speed is the flutter speed; calculating an unsteady aerodynamic coefficient under the main modal movement by adopting a computational fluid dynamics method to further obtain a high-precision aerodynamic coefficient;
The specific formula is as follows:
V represents the flight speed of the wing; ρ represents the atmospheric density; q (ik) is an AIC matrix; q represents generalized coordinates;
Step five, replacing the low-precision aerodynamic coefficient at the flutter point A with the high-precision aerodynamic coefficient, obtaining a correction coefficient S of the high-precision low-precision matrix at the flutter point A, and calculating corrected data under the rest reduction frequency by using the correction coefficient S;
the correction coefficient S formula is
Wherein AIC h(k0) represents high-precision data at dither point a, AIC l(k0) represents low-precision data at dither point a, AIC l(kn) represents low-precision data at the remaining reduction frequency, AIC c(kn) represents corrected data at the remaining reduction frequency;
Step six, multiplying the low-precision aerodynamic coefficient matrix under each reduced frequency with the respective corrected data points to obtain the respective corresponding high-precision aerodynamic coefficient matrix, thereby obtaining corrected series AIC matrices under all reduced frequencies;
Step seven, returning to the step three, recalculating the flutter speed and the frequency after the first iteration by using the corrected series AIC matrix, judging whether the relative difference between the flutter speed and the frequency before the iteration is less than 5%, and if so, meeting the convergence condition to obtain the final corrected flutter speed and frequency; otherwise, substituting the first iteration result into a new round of unsteady aerodynamic force calculation, and carrying out second correction until the adjacent two iteration results reach the convergence standard, thereby obtaining a final correction result.
2. The method for analyzing the flutter of the resistance rudder wing taking into consideration the unsteady aerodynamic force correction according to claim 1, wherein in the second step, the initial reduction frequency value is artificially given.
3. The method for analyzing the flutter of the resistance rudder wing taking into account unsteady aerodynamic force correction as recited in claim 1, wherein the three specific processes of the step are as follows:
first, the equation for solving the chatter vibration by the p-K method is as follows:
m and K are mass and stiffness matrices generated by the structural finite element method; p is an operator of dimension 1, p=g+ik; where g=γk, γ represents the attenuation rate of vibration; l represents a half chord length;
Then, under the condition of given atmospheric density rho and flying speed V, solving the two vibration modes obtained by the above formula, respectively comparing the two vibration modes with the obtained reduced frequency k value, and selecting a k value which is closer to the two vibration modes to recheck the aerodynamic matrix coefficient, so that iteration is repeated until satisfaction, at the moment, the first branch mode of free vibration of the wing in the airflow is sought, and other main modes can be found out by the same method;
finally, under the condition of unchanged flying height, the flying speed V is properly increased; repeating the steps, so as to find out the frequency and attenuation rate of two modes of the wing at the incremental speed;
the speed V is increased step by step to obtain a series of corresponding modal frequencies and attenuation rates, and a V-gamma graph and a V-omega graph are drawn; ω represents the chatter frequency; the point intersecting with the abscissa in the V-gamma diagram corresponds to the flutter speed V F at the flutter point A.
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