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A semi-active flutter control scheme for a two-dimensional wing
Journal of Sound and Vibration (1995) 184(1), 1–7
A SEMI-ACTIVE FLUTTER CONTROL SCHEME FOR A TWO-DIMENSIONAL WING Z.-C. Y, L.-C. Z J.-S. J Department of Aircraft Engineering, Northwestern Polytechnic University, Xi’an, People’s Republic of China (Received 23 November 1992, and in final form 30 September 1993) Both theoretical and experimental investigations are performed for a scheme for ‘‘flutter taming’’—semi-active control of structural non-linear flutter. For a two-dimensional non-linear flutter system, a digital simulation method is used to verify the principle of semi-active flutter control and to study the response characteristics of the closed loop flutter system. Simulation results show that by adjusting automatically the non-linear stiffness parameter of the flutter system, the amplitude of the flutter response can be suppressed. In accordance with the theoretical analysis, a wind tunnel test model for semi-active flutter control is designed. A micromotor–slide block system serves as the parameter control executive element with the monitoring of response signal and the controlling of micromotor performed by a microcomputer. Wind tunnel tests confirm that the non-linear flutter can be controlled effectively by this technique. 7 1995 Academic Press Limited
1. INTRODUCTION
In recent years, a semi-active control technique has been proposed, in which active and passive control principles are combined, that may find applications in flutter control. The well-known ‘‘decoupler pylon’’ invented by Reed et al. [1] may be considered as a successful application of a semi-active control technique. In reference [2], Morino proposed a tentative idea of ‘‘flutter taming’’, which means reduction of the limit cycle flutter amplitude by applying feedback control. He demonstrated this idea by theoretical analysis using a multi-scale asymptotic method. If the limit cycle amplitude is not excessively large, instantaneous failure will not occur in the wing structure; instead, only a fatigue problem arises. It is evident that from the viewpoint of fatigue, a smaller amplitude results in less accumulation of damage. Hence, in the circumstance that flutter is inevitable for a structural configuration, the aforementioned ‘‘flutter taming’’ concept would appear to be attractive. In the study of structural non-linear flutter [3], we found that the non-linear stiffness has direct effect on the limit cycle flutter amplitude. Therefore there naturally emerges a scheme of reducing the flutter amplitude by active control of some stiffness factors. In this paper, an analytical model of this semi-active flutter control scheme is first presented and digital simulation is performed to study the principle of this scheme. Then a flutter control wind tunnel test model is designed and tested. The results show that the flutter can be effectively controlled by this technique. 2. ANALYTICAL MODEL AND DIGITAL SIMULATIONS
The equation of motion of the flutter semi-active control system may be written as (a list of nomenclature is given in the Appendix) [M]{q¨ } + [K]{q} = {Q} + {F}.
(1)
1 0022–460X/95/260001 + 07 $12.00/0
7 1995 Academic Press Limited
.-. .
2
Figure 1. A sketch of the semi-active control flutter system.
It is noted that here the control force {F} is different from that of the active flutter control scheme. It is not the aerodynamic force produced by an aerodynamic control surface, but it may be considered as an additional force resulting from adjusting the structural parameter by an actuator. If the parameter to be adjusted is the structural stiffness, then {F} is the additional elastic restoring force. Let {F} = −{E(q)}.
(2)
[M]{q¨ } + [K]{q} + {E(q)} = {Q}.
(3)
Then equation (1) becomes
Therefore, it can be seen that E{(q)} is a non-linear stiffness term in the equation. For a fundamental, but elementary study, it is convenient to consider a two-dimensional wing. The sketch is shown in Figure 1. The governing equations are mh + Sa a¨ + Kh h = Qh + Fh (h, a),
Sa h + Ia a¨ + Ka a = Qa + Fa (h, a),
(4)
where h is the plunging displacement, a is the pitching displacement, the dot above these characters means d/dt, t is the time, m, Sa and Ia are the mass, static moment and moment of inertia per unit span respectively, Kh is the plunging stiffness coefficient, Ka is the pitching stiffness coefficient, and Fh (h, a) Fa (h, a) are the control forces in plunging and pitching respectively. The unsteady aerodynamic force and moment from Theodorsen theory are expressed as Qh = −prb 2G − 2prVbC(k)G, Qa = prb 2[abG − 0·5Vba˙ − 0·125b 2a¨ ] + 2prVb 2(0·5 + a)C(k)G,
(5)
where G = [Va + h + (0·5 − a)ba˙ ]. By transforming the governing differential equations (4) into the Laplace domain and using a non-dimensional Laplace argument, s¯ = (b/V)s, the non-dimensional governing equations in the Laplace domain are obtained as (m + 1)s¯ 2H + (ma xa − a)s¯ 2a¯ + s¯ a¯ + m(bvh /V)2H = −2C(s¯ )A(s¯ ) + Fh (H , a¯ ), (ma xa − a)s¯ 2H + (ma ra2 + a 2 + 0·125)s¯ 2a¯ + (0·5 − a)s¯ a¯ + ma (bva /V)2a¯ = (1 + 2a)C(s¯ )A(s¯ ) + Fa (H , a¯ ),
(6)
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3
Figure 2. The simulation block diagram of the flutter control system.
where A(s¯ ) = a¯ + s¯ H + (0·5 − a)s¯ a¯ , ra2 = Ia /ma b 2,
m = m/prb 2,
vh = zkh /mh ,
xa = Sa /ma b,
va = zka /Ia .
H = h/b and H , a¯ are the Laplace transforms of H and a respectively; C(s¯ ) is the approximate Laplace transform of C(k): C(s¯ ) = (1 + 1·774s¯ )(1 + 10·61s¯ )/(1 + 2·745s¯ )(1 + 13·51s¯ ). After some manipulations, the final form of the governing equations for digital simulation is s¯ 2H = C1 H + C2 s¯ a¯ + C3 a¯ + C4 C(s¯ ) + C5 Fh (H , a¯ ), s¯ 2a¯ = D1 H + D2 s¯ a¯ + D3 a¯ + D4 C(s¯ ) + D5 Fa (H , a¯ ),
(7)
where Ci (i = 1, 2, . . . , 5) and Di (i = 1, 2, . . . , 5) are composed of the model physical parameters. From equations (7) the simulation block diagram of the semi-active flutter control system can be constructed and drawn as shown in Figure 2. The system has four integral elements, two linear time lag elements, and two feedback control elements, so the calculation can be performed by the digital simulation method introduced in reference [3]. In accordance with the experimental model, the control scheme is designed to be Fh (h, a) = 0,
Fa (h, a) = 0
3 Fa (h, a) = nKa (a − zna s)
for a E as , for a q as ,
(8)
where as is the control threshold value and n is a control parameter. Note that the control force exists only in the pitching degree of freedom. In fact, the flutter system turns out to be non-linear with respect to the non-linear pitching stiffness because of this control force. From equation (8), it can be seen that the characteristics of the non-linear stiffness and the control force are all determined by the value of parameter n. Thus the limit cycle amplitude of the flutter can be suppressed by adjusting n, and the goal of flutter alleviation is achieved.
4
.-. . 3. WIND TUNNEL TEST
3.1. The wind tunnel test model is a two-dimensional wing flutter system. The wooden wing has a NACA 0015 profile. There is a balancing weight cabin in the middle section of the wing, so that its inertia parameters can be adjusted. The details of the model construction are shown in Figure 3. The model wing is supported by two brackets which can slide along a vertical column, and are restrained by a helical spring which provides plunging stiffness for the wing. Each end axle of the wing is fitted on a ball bearing on the bracket, and wing pitching freedom is allowed. A vertical leaf spring with one end fixed on the wing left end axle is the element intended to provide pitching stiffness for the wing. The free end of the leaf spring is limited by a fork, which can move up and down along a vertical guiding rail welded to the bracket. The fork allows the leaf spring to move freely only in a gap, the width of which can be adjusted through a screw. The movement of the fork is driven by a micromotor/guidescrew set which is installed in the bracket. When the fork moves upward, the effective length of the leaf spring is increased, and hence the pitching stiffness of the wing is reduced. On the contrary, downward movement of the fork leads to an increase of pitching stiffness. Thus the micromotor acts as the actuator for adjusting pitching stiffness in the feedback control circuit. This is equivalent to changing n in equations (8). 3.2. The diagram of the experimental set-up is shown in Figure 4. The signal of pitching freedom is picked up by a sensor and then is put into a pulse transforming circuit. The
Figure 3. The wind tunnel test model. 1, Wooden wing; 2, leaf spring; 3, axle; 4, guiding rail; 5, fork; 6, bracket; 7, micromotor; 8, helical spring; 9, guiding column.
-
5
Figure 4. A diagram of the experimental set-up.
on-line monitoring of the feedback response signal is accomplished by a microcomputer system; the microprocessor determines whether the flutter occurs and then controls the operation of the micromotor. As a consequence, the fork is driven to move up or down. 3.3. The flutter control tests were performed in a closed circuit low speed wind tunnel with an open test section of 1 m diameter; the maximum air speed attainable is 60 m/s. The installation of the model in the wind tunnel is shown in Figure 5. The pitching and plunging stiffnesses are represented by the two natural frequencies of the primary linear system, which are obtained by vibration test as vh = 15·7 rad/s and va = 48·3 rad/s; other parameters are listed in Table 1. The linear critical flutter speed found by the wind tunnel test is 25 m/s, and the response of the open loop system diverges rapidly. When the control loop is closed, the response amplitude increases to as , the micromotor drives the fork down, and thus the pitching stiffness is automatically adjusted to suppress the response to a very small value which depends on the preset as . When the air speed is decreased, the motor then turns in reverse and drives the fork back to its original place. Some of the experimental and simulation results are compared in the form of time histories in Figure 6, which shows that the
N
Figure 5. A photograph of the model set-up.
.-. .
6
T 1 Parameters of the test model mh
ma
xa
ra2
a
b(m)
as (degrees)
58·4
98·1
0·31
0·36
0·3
0·15
2
semi-active control scheme is feasible. Both digital simulation and model test results in Figure 6 show that the flutter control process may be divided into three stages: i.e., flutter occurring, flutter controlling and full controlled stages. The stage of flutter occurring is related to the sampling time of the pulse transforming circuit and the time of the flutter controlling stage depends on the rotation speed of the motor.
4. CONCLUDING REMARKS
(1) The digital simulation method may be used as an effective technique to study the feasibility of the semi-active flutter control, and it can provide some elementary information for the design of the control element and control scheme. (2) Both simulation and experimental results show that, by using the semi-active control technique, flutter can be suppressed with a simple control scheme. (3) The semi-active flutter control scheme provides a new tool for the non-linear flutter suppression which often appears in the control surface flutter and wing/store flutter problems.
Figure 6. Control results at V = 30 m/s. (a) Digital simulation; (b) wind tunnel test.
ACKNOWLEDGMENTS This project was supported by the Science Foundation of Aeronautics of China and the Doctorate Foundation of China.
-
7
REFERENCES 1. W. H. R III, J. T. F, J. and H. L. R 1979 NASA-79-0791. Decoupler pylon: a simple, effective wing/store flutter suppressor. 2. L. M 1984 CCAD-TR-84-01. Flutter taming—a new tool for the aeroelastic designer. 3. Z. C. Y and L. C. Z 1988 Journal of Sound and Vibration 123, 1–13. Analysis of limit cycle flutter of an airfoil in incompressible flow.
APPENDIX: NOMENCLATURE [M] [K] {Q} {F} {q} C(k) k V n r v mh ma Ia Kh Ka ab b mh ma xa ra2
generalized mass matrix generalized stiffness matrix generalized aerodynamics generalized control force generalized co-ordinates Theodorsen function =vb/V, reduced frequency air speed control parameter air density frequency mass per unit span in plunging mass per unit span in pitching inertia moment per unit span about stiffness centre stiffness in plunging stiffness in pitching see Figure 1 semi-chord of the airfoil =mh /prb 2 =Ia /prb 3 =Sa /ma b =Ia /ma b 2
A SEMI-ACTIVE FLUTTER CONTROL SCHEME FOR A TWO-DIMENSIONAL WING Z.-C. Y, L.-C. Z J.-S. J Department of Aircraft Engineering, Northwestern Polytechnic University, Xi’an, People’s Republic of China (Received 23 November 1992, and in final form 30 September 1993) Both theoretical and experimental investigations are performed for a scheme for ‘‘flutter taming’’—semi-active control of structural non-linear flutter. For a two-dimensional non-linear flutter system, a digital simulation method is used to verify the principle of semi-active flutter control and to study the response characteristics of the closed loop flutter system. Simulation results show that by adjusting automatically the non-linear stiffness parameter of the flutter system, the amplitude of the flutter response can be suppressed. In accordance with the theoretical analysis, a wind tunnel test model for semi-active flutter control is designed. A micromotor–slide block system serves as the parameter control executive element with the monitoring of response signal and the controlling of micromotor performed by a microcomputer. Wind tunnel tests confirm that the non-linear flutter can be controlled effectively by this technique. 7 1995 Academic Press Limited
1. INTRODUCTION
In recent years, a semi-active control technique has been proposed, in which active and passive control principles are combined, that may find applications in flutter control. The well-known ‘‘decoupler pylon’’ invented by Reed et al. [1] may be considered as a successful application of a semi-active control technique. In reference [2], Morino proposed a tentative idea of ‘‘flutter taming’’, which means reduction of the limit cycle flutter amplitude by applying feedback control. He demonstrated this idea by theoretical analysis using a multi-scale asymptotic method. If the limit cycle amplitude is not excessively large, instantaneous failure will not occur in the wing structure; instead, only a fatigue problem arises. It is evident that from the viewpoint of fatigue, a smaller amplitude results in less accumulation of damage. Hence, in the circumstance that flutter is inevitable for a structural configuration, the aforementioned ‘‘flutter taming’’ concept would appear to be attractive. In the study of structural non-linear flutter [3], we found that the non-linear stiffness has direct effect on the limit cycle flutter amplitude. Therefore there naturally emerges a scheme of reducing the flutter amplitude by active control of some stiffness factors. In this paper, an analytical model of this semi-active flutter control scheme is first presented and digital simulation is performed to study the principle of this scheme. Then a flutter control wind tunnel test model is designed and tested. The results show that the flutter can be effectively controlled by this technique. 2. ANALYTICAL MODEL AND DIGITAL SIMULATIONS
The equation of motion of the flutter semi-active control system may be written as (a list of nomenclature is given in the Appendix) [M]{q¨ } + [K]{q} = {Q} + {F}.
(1)
1 0022–460X/95/260001 + 07 $12.00/0
7 1995 Academic Press Limited
.-. .
2
Figure 1. A sketch of the semi-active control flutter system.
It is noted that here the control force {F} is different from that of the active flutter control scheme. It is not the aerodynamic force produced by an aerodynamic control surface, but it may be considered as an additional force resulting from adjusting the structural parameter by an actuator. If the parameter to be adjusted is the structural stiffness, then {F} is the additional elastic restoring force. Let {F} = −{E(q)}.
(2)
[M]{q¨ } + [K]{q} + {E(q)} = {Q}.
(3)
Then equation (1) becomes
Therefore, it can be seen that E{(q)} is a non-linear stiffness term in the equation. For a fundamental, but elementary study, it is convenient to consider a two-dimensional wing. The sketch is shown in Figure 1. The governing equations are mh + Sa a¨ + Kh h = Qh + Fh (h, a),
Sa h + Ia a¨ + Ka a = Qa + Fa (h, a),
(4)
where h is the plunging displacement, a is the pitching displacement, the dot above these characters means d/dt, t is the time, m, Sa and Ia are the mass, static moment and moment of inertia per unit span respectively, Kh is the plunging stiffness coefficient, Ka is the pitching stiffness coefficient, and Fh (h, a) Fa (h, a) are the control forces in plunging and pitching respectively. The unsteady aerodynamic force and moment from Theodorsen theory are expressed as Qh = −prb 2G − 2prVbC(k)G, Qa = prb 2[abG − 0·5Vba˙ − 0·125b 2a¨ ] + 2prVb 2(0·5 + a)C(k)G,
(5)
where G = [Va + h + (0·5 − a)ba˙ ]. By transforming the governing differential equations (4) into the Laplace domain and using a non-dimensional Laplace argument, s¯ = (b/V)s, the non-dimensional governing equations in the Laplace domain are obtained as (m + 1)s¯ 2H + (ma xa − a)s¯ 2a¯ + s¯ a¯ + m(bvh /V)2H = −2C(s¯ )A(s¯ ) + Fh (H , a¯ ), (ma xa − a)s¯ 2H + (ma ra2 + a 2 + 0·125)s¯ 2a¯ + (0·5 − a)s¯ a¯ + ma (bva /V)2a¯ = (1 + 2a)C(s¯ )A(s¯ ) + Fa (H , a¯ ),
(6)
-
3
Figure 2. The simulation block diagram of the flutter control system.
where A(s¯ ) = a¯ + s¯ H + (0·5 − a)s¯ a¯ , ra2 = Ia /ma b 2,
m = m/prb 2,
vh = zkh /mh ,
xa = Sa /ma b,
va = zka /Ia .
H = h/b and H , a¯ are the Laplace transforms of H and a respectively; C(s¯ ) is the approximate Laplace transform of C(k): C(s¯ ) = (1 + 1·774s¯ )(1 + 10·61s¯ )/(1 + 2·745s¯ )(1 + 13·51s¯ ). After some manipulations, the final form of the governing equations for digital simulation is s¯ 2H = C1 H + C2 s¯ a¯ + C3 a¯ + C4 C(s¯ ) + C5 Fh (H , a¯ ), s¯ 2a¯ = D1 H + D2 s¯ a¯ + D3 a¯ + D4 C(s¯ ) + D5 Fa (H , a¯ ),
(7)
where Ci (i = 1, 2, . . . , 5) and Di (i = 1, 2, . . . , 5) are composed of the model physical parameters. From equations (7) the simulation block diagram of the semi-active flutter control system can be constructed and drawn as shown in Figure 2. The system has four integral elements, two linear time lag elements, and two feedback control elements, so the calculation can be performed by the digital simulation method introduced in reference [3]. In accordance with the experimental model, the control scheme is designed to be Fh (h, a) = 0,
Fa (h, a) = 0
3 Fa (h, a) = nKa (a − zna s)
for a E as , for a q as ,
(8)
where as is the control threshold value and n is a control parameter. Note that the control force exists only in the pitching degree of freedom. In fact, the flutter system turns out to be non-linear with respect to the non-linear pitching stiffness because of this control force. From equation (8), it can be seen that the characteristics of the non-linear stiffness and the control force are all determined by the value of parameter n. Thus the limit cycle amplitude of the flutter can be suppressed by adjusting n, and the goal of flutter alleviation is achieved.
4
.-. . 3. WIND TUNNEL TEST
3.1. The wind tunnel test model is a two-dimensional wing flutter system. The wooden wing has a NACA 0015 profile. There is a balancing weight cabin in the middle section of the wing, so that its inertia parameters can be adjusted. The details of the model construction are shown in Figure 3. The model wing is supported by two brackets which can slide along a vertical column, and are restrained by a helical spring which provides plunging stiffness for the wing. Each end axle of the wing is fitted on a ball bearing on the bracket, and wing pitching freedom is allowed. A vertical leaf spring with one end fixed on the wing left end axle is the element intended to provide pitching stiffness for the wing. The free end of the leaf spring is limited by a fork, which can move up and down along a vertical guiding rail welded to the bracket. The fork allows the leaf spring to move freely only in a gap, the width of which can be adjusted through a screw. The movement of the fork is driven by a micromotor/guidescrew set which is installed in the bracket. When the fork moves upward, the effective length of the leaf spring is increased, and hence the pitching stiffness of the wing is reduced. On the contrary, downward movement of the fork leads to an increase of pitching stiffness. Thus the micromotor acts as the actuator for adjusting pitching stiffness in the feedback control circuit. This is equivalent to changing n in equations (8). 3.2. The diagram of the experimental set-up is shown in Figure 4. The signal of pitching freedom is picked up by a sensor and then is put into a pulse transforming circuit. The
Figure 3. The wind tunnel test model. 1, Wooden wing; 2, leaf spring; 3, axle; 4, guiding rail; 5, fork; 6, bracket; 7, micromotor; 8, helical spring; 9, guiding column.
-
5
Figure 4. A diagram of the experimental set-up.
on-line monitoring of the feedback response signal is accomplished by a microcomputer system; the microprocessor determines whether the flutter occurs and then controls the operation of the micromotor. As a consequence, the fork is driven to move up or down. 3.3. The flutter control tests were performed in a closed circuit low speed wind tunnel with an open test section of 1 m diameter; the maximum air speed attainable is 60 m/s. The installation of the model in the wind tunnel is shown in Figure 5. The pitching and plunging stiffnesses are represented by the two natural frequencies of the primary linear system, which are obtained by vibration test as vh = 15·7 rad/s and va = 48·3 rad/s; other parameters are listed in Table 1. The linear critical flutter speed found by the wind tunnel test is 25 m/s, and the response of the open loop system diverges rapidly. When the control loop is closed, the response amplitude increases to as , the micromotor drives the fork down, and thus the pitching stiffness is automatically adjusted to suppress the response to a very small value which depends on the preset as . When the air speed is decreased, the motor then turns in reverse and drives the fork back to its original place. Some of the experimental and simulation results are compared in the form of time histories in Figure 6, which shows that the
N
Figure 5. A photograph of the model set-up.
.-. .
6
T 1 Parameters of the test model mh
ma
xa
ra2
a
b(m)
as (degrees)
58·4
98·1
0·31
0·36
0·3
0·15
2
semi-active control scheme is feasible. Both digital simulation and model test results in Figure 6 show that the flutter control process may be divided into three stages: i.e., flutter occurring, flutter controlling and full controlled stages. The stage of flutter occurring is related to the sampling time of the pulse transforming circuit and the time of the flutter controlling stage depends on the rotation speed of the motor.
4. CONCLUDING REMARKS
(1) The digital simulation method may be used as an effective technique to study the feasibility of the semi-active flutter control, and it can provide some elementary information for the design of the control element and control scheme. (2) Both simulation and experimental results show that, by using the semi-active control technique, flutter can be suppressed with a simple control scheme. (3) The semi-active flutter control scheme provides a new tool for the non-linear flutter suppression which often appears in the control surface flutter and wing/store flutter problems.
Figure 6. Control results at V = 30 m/s. (a) Digital simulation; (b) wind tunnel test.
ACKNOWLEDGMENTS This project was supported by the Science Foundation of Aeronautics of China and the Doctorate Foundation of China.
-
7
REFERENCES 1. W. H. R III, J. T. F, J. and H. L. R 1979 NASA-79-0791. Decoupler pylon: a simple, effective wing/store flutter suppressor. 2. L. M 1984 CCAD-TR-84-01. Flutter taming—a new tool for the aeroelastic designer. 3. Z. C. Y and L. C. Z 1988 Journal of Sound and Vibration 123, 1–13. Analysis of limit cycle flutter of an airfoil in incompressible flow.
APPENDIX: NOMENCLATURE [M] [K] {Q} {F} {q} C(k) k V n r v mh ma Ia Kh Ka ab b mh ma xa ra2
generalized mass matrix generalized stiffness matrix generalized aerodynamics generalized control force generalized co-ordinates Theodorsen function =vb/V, reduced frequency air speed control parameter air density frequency mass per unit span in plunging mass per unit span in pitching inertia moment per unit span about stiffness centre stiffness in plunging stiffness in pitching see Figure 1 semi-chord of the airfoil =mh /prb 2 =Ia /prb 3 =Sa /ma b =Ia /ma b 2