Abstract
A method for demonstrating a new scientific discovery made by the inventor about the nonlinear instability of vehicles, like aircrafts, automobiles and ocean vehicles. Said method includes a model and a three-gimbaled framework that permits said model to respond to inertial moments about the axes of which the moments of inertias are the smallest and the largest, wherein said model has restoring and damping capabilities along these two axes. Said method also comprises how to use a variable motor or a crank for controlling said model rotational motions about the intermediate principal axis of inertia with closed form formulas for the external driven frequencies and amplitudes to be used to excite the nonlinear instabilities of said model. Said model could be an aircraft, an automobile, a ship, or even a rectangular block.
Claims
1. A method of simulating three-degrees of freedom vehicle nonlinear rotational instability comprising: a. attaching a vehicle model to a three-gimbaled framework, wherein said vehicle model includes one top piece and one bottom piece, wherein said top and bottom pieces are identical and symmetric at least about the vertical plane which contains their longitudinal centerlines when said top and bottom pieces are in horizontal position, respectively, a two-axle-combination shaft providing two orthogonal axles about which said vehicle model rotates, wherein said top and bottom pieces are rigidly connected at two points, wherein said two points are located along a direction perpendicular to a two-axle plane which said two orthogonal axles form, wherein said two orthogonal axles coincide with the smallest and the largest principal axes of said vehicle model, respectively; b. a plurality of restoring and damping mechanisms in the directions of the smallest and the largest principal axes of inertia of said vehicle model, wherein said restoring and damping mechanisms comprise springs, dampers, gears, gear racks, posts, and guide brackets; c. a rotatable frame to provide controllable rotations about the intermediate principal axis of inertia of said vehicle model, wherein said rotatable frame is rotatably coupled to supporting elements and a motor driving system; d. a plurality of identical loads slidable on said vehicle model for adjustments of moments of inertias for said vehicle model; e. wherein said rotatable frame is capable of rotating in oscillation fashions with a range of amplitudes and a range of frequencies to demonstrate the nonlinear instability of rotations about the intermediate principal of axis of said vehicle model, wherein said amplitudes and frequencies are controlled precisely by said motor driving system, wherein said amplitudes cover a range given by wherein .sub.10 and .sub.30 is the natural circular frequencies about the smallest and the largest principal axes of inertias of said vehicle model, respectively, b.sub.1 and b.sub.3 are the damping coefficients about the smallest and the largest principal axes of inertia of said vehicle model, respectively, I.sub.x, I.sub.y, I.sub.z are the moments of inertias of said vehicle model about the principal axes X,Y,Z, respectively, and with I.sub.x<I.sub.y<I.sub.z, wherein said frequencies cover a range given by
{0.1|.sub.10.sub.30|, 2(.sub.10+.sub.30)}, wherein two dangerous frequencies are controlled by said motor driving system in terms of motor RPM given, respectively, by
.sub.motor=30|.sub.10.sub.30|(RPM)
.sub.motor=30(.sub.10+.sub.30)/(RPM), wherein .sub.motor is revolutions of motor shaft per minute, wherein said oscillation fashions are given by
=(R.sub.1/R.sub.2)sin(.sub.motort/30)(rad), wherein is a rotational motion of said rotatable frame, R.sub.1 is an effective length of a crank of said motor driving system, R.sub.2 is a radius of a gear of said motor driving system, demonstrating the first frequency case of said two dangerous frequencies by, fixing said motor RPM precisely at 30|.sub.10.sub.30|/ then changing said amplitude when said amplitude exceeds a threshold given by said rotating in oscillation about the intermediate principal of axis becoming unstable, demonstrating the second frequency case of said two dangerous frequencies by fixing said motor RPM precisely at 30(.sub.10+.sub.30)/ then changing said amplitude when said amplitude exceeds a threshold given by said rotating in oscillation about the intermediate principal of axis becoming unstable.
2. A method of simulating three-degrees of freedom vehicle nonlinear rotational instability comprising: attaching a three-gimbaled framework to a vehicle model, wherein said vehicle model includes one top piece and one bottom piece, wherein said top and bottom pieces are identical and symmetric at least about the vertical plane which contains their longitudinal centerlines when said top and bottom pieces are in horizontal position, respectively, a two-axle-combination shaft providing two orthogonal axles about which said vehicle model rotates, wherein said top and bottom pieces are rigidly connected at two points, wherein said two points are located along a direction perpendicular to a two-axle plane which said two orthogonal axles form, wherein said two orthogonal axles coincide with the smallest and the largest principal axes of said vehicle model, respectively; a plurality of restoring and damping mechanisms in the directions of the smallest and the largest principal axes of inertia of said vehicle model, wherein said restoring and damping mechanisms comprise springs, dampers, gears, gear racks, posts, and guide brackets; a rotatable frame to provide controllable rotations about the intermediate principal axis of inertia of said vehicle model, wherein said rotatable frame is rotatably coupled to supporting elements and a crank; a plurality of identical loads slidable on said vehicle model for adjustments of moments of inertias for said vehicle model; wherein said rotatable frame is capable of rotating in oscillation fashions with a range of amplitudes and a range of frequencies to demonstrate the nonlinear instability of rotations about the intermediate principal of axis of said vehicle model, wherein said amplitudes and frequencies are controlled by said crank, wherein said amplitudes cover a range given by wherein .sub.10 and .sub.30 is the natural circular frequencies about the smallest and the largest principal axes of inertias of said vehicle model, respectively, b.sub.1 and b.sub.3 are the damping coefficients about the smallest and the largest principal axes of inertia of said vehicle model, respectively, I.sub.x,I.sub.y,I.sub.z are the moments of inertias of said vehicle model about the principal axes X,Y,Z, respectively, and with I.sub.x<I.sub.y<I.sub.z, wherein said frequencies cover a range given by
{0.1|.sub.10.sub.30|,2(.sub.10+.sub.30)}, demonstrating the first frequency case of said two dangerous frequencies by fixing said frequency at |.sub.10.sub.30| then changing said amplitude when said amplitude exceeds a threshold given by said rotating in oscillation about the intermediate principal of axis becoming unstable, demonstrating the second frequency case of said two dangerous frequencies by fixing said frequency at (.sub.10+.sub.30) then changing said amplitude when said amplitude exceeds a threshold given by said rotating in oscillation about the intermediate principal of axis becoming unstable.
3. A method of simulating three-degrees of freedom rigid body nonlinear rotational instability comprising: a. a homogeneous rectangular block model attached to a three-gimbaled framework, wherein a shaft aligns with either the smallest or the largest principal axis of inertia of said block model, wherein said three-gimbaled framework comprises an inner gimbal frame and an outer gimbal frame providing rotational freedoms for said block model; a plurality of restoring and damping assemblies in the directions of the smallest and the largest principal axes of inertia of said block model, wherein said restoring and damping assemblies comprise machined springs with a plurality of cylindrical legs, rotational dampers, and bearings, wherein said cylindrical legs are slidable smoothly in a plurality of holes in said gimbal frames; b. said outer gimbal frame is rotatably coupled to supporting elements and a motor driving system to provide controllable rotations about the intermediate principal axis of inertia of said block model; c. said outer gimbal frame is capable of rotating in oscillation fashions with a range of amplitudes and a range of frequencies to demonstrate the nonlinear instability of rotations about the intermediate principal of axis of said block model, wherein said amplitudes and frequencies are controlled precisely by said motor driving system, wherein said amplitudes cover a range given by wherein .sub.10 and .sub.30 is the natural circular frequencies about the smallest and the largest principal axes of inertias of said block model, respectively, b.sub.1 and b.sub.3 are the damping coefficients about the smallest and the largest principal axes of inertia of said block model, respectively, I.sub.x,I.sub.y,I.sub.z are the moments of inertias of said block model about the principal axes X,Y,Z, respectively, and with I.sub.x<I.sub.y<I.sub.z, wherein said frequencies cover a range given by
{0.1|.sub.10.sub.30|,2(.sub.10+.sub.30)}, wherein two dangerous frequencies are controlled by said motor driving system in terms of motor RPM given, respectively, by
.sub.motor=30|.sub.10.sub.30|/(RPM)
.sub.motor=30(.sub.10+.sub.30)/(RPM), wherein .sub.motor is revolutions of motor shaft per minute, wherein said oscillation fashions are given by
=(R.sub.1/R.sub.2)sin(.sub.motort/30)(rad), wherein is a rotational motion of said outer gimbal frame, R.sub.1 is an effective length of a crank of said motor driving system, R.sub.2 is a radius of a gear of said motor driving system, demonstrating the first frequency case of said two dangerous frequencies by fixing said motor RPM precisely at 30|.sub.10.sub.30|/ then changing said amplitude when said amplitude exceeds a threshold given by said rotating in oscillation about the intermediate principal of axis becoming unstable, demonstrating the second frequency case of said two dangerous frequencies by fixing said motor RPM precisely at 30(.sub.10+.sub.30)/ then changing said amplitude when said amplitude exceeds a threshold given by said rotating in oscillation about the intermediate principal of axis becoming unstable.
4. A method of simulating three-degrees of freedom rigid body nonlinear rotational instability comprising: a homogeneous rectangular block model attached to a three-gimbaled framework, wherein a shaft aligns with either the smallest or the largest principal axis of inertia of said block model, wherein said three-gimbaled framework comprises an inner gimbal frame and an outer gimbal frame providing rotational freedoms for said block model; a plurality of restoring and damping assemblies in the directions of the smallest and the largest principal axes of inertia of said block model, wherein said restoring and damping assemblies comprise machined springs with a plurality of cylindrical legs, rotational dampers, and bearings, wherein said cylindrical legs are slidable smoothly in a plurality of holes in said gimbal frames; said outer gimbal frame is rotatably coupled to supporting elements and a crank to provide controllable rotations about the intermediate principal axis of inertia of said block model; said outer gimbal frame is capable of rotating in oscillation fashions with a range of amplitudes and a range of frequencies to demonstrate the nonlinear instability of rotations about the intermediate principal of axis of said block model, wherein said amplitudes and frequencies are controlled by said crank, wherein said amplitudes cover a range given by wherein .sub.10 and .sub.30 is the natural circular frequencies about the smallest and the largest principal axes of inertias of said block model, respectively, b.sub.1 and b.sub.3 are the damping coefficients about the smallest and the largest principal axes of inertia of said block model, respectively, I.sub.x,I.sub.y,I.sub.z are the moments of inertias of said block model about the principal axes X,Y,Z, respectively, and with I.sub.x<I.sub.y<I.sub.z, wherein said frequencies cover a range given by
{0.1|.sub.10.sub.30|,2(.sub.10+.sub.30)}, demonstrating the first frequency case of said two dangerous frequencies by fixing said frequency at |.sub.10.sub.30| then changing said amplitude when said amplitude exceeds a threshold given by said rotating in oscillation about the intermediate principal of axis becoming unstable, demonstrating the second frequency case of said two dangerous frequencies by fixing said frequency at (.sub.10+.sub.30) then changing said amplitude when said amplitude exceeds a threshold given by said rotating in oscillation about the intermediate principal of axis becoming unstable.
Description
BRIEF DESCRIPTION OF THE DRAWINGS
(1) The following descriptions of drawings of the preferred embodiments are merely exemplary in nature and are not intended to limit the scope of the invention, its application, or uses in any way.
(2) FIG. 1 is a perspective view of a prior related art of a demonstrator for showing the roll inertial coupling.
(3) FIG. 2 is a perspective view of the apparatus as a demonstrator related to aircrafts in accordance with the first preferred embodiment.
(4) FIG. 3a is perspective view of the aircraft model and the roll and yaw axles without the supporting frames. This aircraft model has the pitch moment of inertia as the intermediate between the roll and yaw inertias. FIG. 3b is a perspective view of the aircraft model assembly. FIG. 3c is the perspective views of the roll and yaw axle assembly.
(5) FIG. 4 is a side view of the aircraft model in FIG. 3a.
(6) FIG. 5 is a back view of the aircraft model in FIG. 3a.
(7) FIG. 6 is a top view of the aircraft model in FIG. 3a.
(8) FIG. 7 is a zooming-in perspective view of the restoring and damping assemblies of the aircraft model in FIG. 3a.
(9) FIG. 8 is a zooming-in side view of the aircraft model in FIG. 4 without parts 109a, 109c, 110a, 110c, and 111a.
(10) FIG. 9a is a perspective view of the spring and damper assembly as one unit and FIG. 9b is a perspective view of the spring and the damper to be used separately.
(11) FIG. 10 is a sectional view of the cutting plane A-A in FIG. 6.
(12) FIG. 11 is a perspective view of the roll restoring and damping assembly 105 of the aircraft model in FIG. 2.
(13) FIG. 12a is an assembly view of the wing and the balance load in FIG. 3a.
(14) FIG. 12b is a zooming-in view of the Dovetail groove sliding connector in FIG. 12a.
(15) FIG. 13 is a perspective view of the apparatus with the motor-driving assembly for precision control of the external exciting moments.
(16) FIG. 14a is a perspective zooming-in view of the motor-driving assembly of FIG. 13. FIG. 14b is a view of FIG. 14a without the gear rack 151 and the slotted link 153. FIG. 14c is a side view of FIG. 14a with the aircraft model pitch down at 45.
(17) FIG. 15 is a cranking arm with the sliding pin.
(18) FIG. 16a is a perspective view of the apparatus with a bus model as a demonstrator related to automobiles for the case in which the pitch moment of inertia of an automobile is the intermediate between the other two inertias. FIG. 16b is the zooming-in view of the bus model without the supporting frames and without the top adjustable weight 270a shown in FIG. 16a.
(19) FIG. 17 is another way of mounting the model to the base in order to simulate the case in which the yaw moment of inertia of an automobile (like some loading conditions of big trucks) is the intermediate between the other two inertias.
(20) FIG. 18 is a perspective view of the apparatus with a ship model as a demonstrator related to ocean vehicles.
(21) FIG. 19 is a top view of the ship model of FIG. 18 without the supporting frames.
(22) FIG. 20 is a zooming-in perspective view of the ship model of FIG. 18 without the supporting frame and with a shell section cut out amidships.
(23) FIG. 21 is a perspective view of the apparatus with an aircraft model which has the roll moment of inertia as the intermediate between the other two inertias.
(24) FIG. 22 is the zooming-in view of FIG. 21 without the supporting frames.
(25) FIG. 23 is a perspective view of an apparatus with a rigid body model of rectangular block.
(26) FIG. 24 is a perspective view of the individual elements of the restoring and damping assembly.
(27) FIG. 25 is a side view of one of the roll restoring and damping assembly.
(28) FIG. 26 is a side view of one of the yaw restoring and damping assembly.
DETAILED DESCRIPTION OF PREFERRED EMBODIMENTS
(29) Referring to the drawings, more particularly to FIG. 2, preferred the apparatus comprises a flat round base 101, an upside-down U type supporting frame 103 (the outer frame) rigidly mounted on the base 101, a crank 102 rigidly connected to another rectangular frame 104 (the inner frame) runs through a bearing (not shown in the figure) which is mounted in the outer frame 103 so that the crank can smoothly rotate the inner frame 104, an aircraft model 106 comprising two symmetric (top and bottom) parts about XOY plane is also symmetric (left and right) about XOZ plane as shown in FIG. 3a so that the principal inertial axes of the model are along the roll, pitch, and yaw axes of the model. The top and bottom parts are kept at certain distance and connected with each other at the two points located at the tips of wings 109a, 109b, 109c, and 109d, and the yaw axles in order to create one rigid piece of aircraft model. The significance of the model being two parts connected only at the two points and the yaw axles is that the empty space between the two parts provides space for a long roll axle and for the model to yaw up to almost 90. The center of gravity (COG) of the model is located at Point O as shown in FIG. 3a and FIG. 10. The COG coincides with the intersection point of roll, pitch, and yaw axes. Therefore, the roll, pitch, and yaw axes coincide with the X, Y and Z axes of the coordinate system as shown in FIG. 3a. A long roll axle 108 supporting the model 106 is in turn supported by two bearings (not shown) at the two ends of 108 and the two bearings are mounted in the two opposite sides of the frame 104. Yaw restoring and damping assemblies 107a and 107b are mounted on the top part of the model, and yaw restoring and damping 107c and 107d (not shown) on the bottom part of the model as shown in FIG. 3a. A roll restoring and damping assembly 105 in FIG. 2 and FIG. 11 is mounted on one side of the frame 104. In this embodiment, the pitch moment of inertia is the intermediate between the roll and yaw inertias, which represents the case of the most commercial aircrafts, for example, Boeing 737.
(30) As shown in FIG. 3a, the aircraft model 106 has a top fuselage 112a and a bottom fuselage 112b, wings 109a and 109b are rigidly connected to top fuselage and wings 109c and 109d to the bottom fuselage, spacer 111a connecting the top wing 109a and the bottom wing 109c, spacer 111b connecting the top wing 109b and the bottom wing 109d, the spacers are to keep the top and bottom fuselage with wings apart at enough distance to allow the model 106 to freely rotate up to almost 90 about the yaw axis (Z axis), adjustable weights 110a, 110b, 110c, and 110d sliding along two grooves which are perpendicular to each other and located on the wings and the weights, respectively, these weights could be positioned almost anywhere on the wings to vary the moment of inertias and to tune the COG of the model to coincide with the origin of the roll, pitch, and yaw axes, the weights 110a, 110b, 110c, and 110d are identical, the roll axle 108 is along the center line of the model and connected to the yaw axles 115a and 115b at the COG by a cross connector 113 as shown in FIG. 3c. As shown in FIG. 5 two yaw restoring and damping assemblies 107a and 107b are mounted on the top fuselage while two other yaw restoring and damping assemblies 107c and 107d are mounted on the bottom fuselage, the four spring and damping assemblies 107a, 107b, 107c, and 107d are identical. As shown in FIG. 3a, an axle collar 131 and a gear 125 located on each very ends of the roll axle 108 keep the axle fixed along the X direction when the model is moving. The standalone aircraft model 106 is shown in FIG. 3b. The roll and yaw axle assembly 132 is shown in FIG. 3c.
(31) A zooming-in perspective view of the restoring and damping assemblies 107a and 107b on the top fuselage 112a is given in FIG. 7. A side view of the spring and damping assemblies 107a and 107c is given in FIG. 8. The assembly 107a comprises spring and damper combinations 119a1 and 119a2, a gear rack 117a, a right angle bracket 118a mounted on the top fuselage to serve as a guide for the linear motion of the gear rack 117a, spring posts 120a1 and 120a4 are mounted on the top fuselage while spring posts 120a2 and 120a3 are mounted on the two ends of the gear rack 117a. As shown in FIG. 5, the assembly 107a is located on one side of a gear 116a and the yaw axle 115a while assembly 107b is symmetrically located on the other side of the gear 116a and the yaw axle 115a. As shown in FIG. 7, similarly, the assembly 107b (also shown in FIG. 5) comprises spring and damper combinations 119b1 and 119b2, a gear rack 117b, a right angle bracket 118b mounted on the top fuselage to serve as a guide for the linear motion of the gear rack 117b. Spring posts 120b1 and 120b4 are mounted on the top fuselage while spring posts 120b2 and 120b3 are mounted on the two ends of gear rack 117b. Similar arrangements of the restoring and damping assemblies 107c (as shown in FIG. 8) and 107d (as shown in FIG. 5) are located on the bottom fuselage 112b as shown in FIG. 5. The gear 116a is mounted on the yaw axle 115a which goes through a ball bearing 114a as shown in FIG. 7 and FIG. 10. The FIG. 10 shows a section view along a plane which is cut at A-A of FIG. 6. As shown in FIG. 10, the ball bearing 114a sitting in a hole on the fuselage 112a is suspended on the cross connector 113, the gear 116a is fixed on the yaw axle 115a by a screw, the yaw axle 115a is fitted through the ball bearing 114a so that the aircraft model has the freedom to rotate about the yaw axle 115a. The yaw axle 115a, the gear 116a and the roll axle 108, however, do not have the degree of freedom in yaw. The aircraft model suspended on the roll axle 108 by the cross connector 113 as shown in FIG. 8 and FIG. 10 has three rotational degrees of freedom (roll, pitch, and yaw). The roll axle 108 supported at the ends by two bearings, one of which is ball bearing 126 as shown in FIG. 11. The two ball bearings are sitting in holes on the two opposite sides of frame 104 and only one side is shown in FIG. 11. The model together with the frame 104 and the roll axle 108 has the freedom to rotate about the pitch axis and all three of them may be driven by the crank as shown in FIG. 2. Therefore the aircraft model has unlimited freedom in roll and pitch, but only has a freedom to yaw up to almost 90 since the model can only yaw up to the points where the spacers 111a and 111b in FIG. 3a touch the roll axle 108. Since the apparatus is designed to demonstrate the onset of the nonlinear instability which happens when the model is only pitching with certain frequencies and amplitudes, and then roll and yaw would grow from zero to large degrees without any help of external roll and yaw moments, the near 90 freedom of yaw of the model is large enough for this purpose. Because any three-gimbal framework, including the current invention, has the so-called gimbal-lock problem that is when one of the rotational axes rotates and coincides with another axis then one rotational freedom is lost, the yaw motion at about 90 of the model is not accurate anyway to simulate the free rotations of aircrafts. However, the nonlinear instability onset is accurately simulated because the model does not have gimbal-lock problem when roll and yaw are zero. To simulate a real aircraft dynamics in infinite freedoms in roll, pitch, and yaw, the inventor also invented a gimbal-lock-free flight simulator of which a patent will be applied separately.
(32) When the model yaws, the yaw axles 115a and 115b (shown in FIG. 10) are not yawing. The gear racks 117a, 117b (in FIG. 7), 117c (in FIG. 8), and 117d (not shown) are capable to slide linearly along the tracks guided by 118a (in FIG. 7), 118b (in FIG. 7), 118c (in FIG. 8), and 118d (not shown), respectively. When the model yaws, the gear 116a (in FIG. 7) will drive the gear rack 117a and 117b to slide, and in a similar way, the gear 116b (not shown in FIG. 7) will drive the gear rack 117c and 117d (not shown in FIG. 7) to slide. When these gear racks slide, the spring and damper combinations 119a1, 119a2, 119b1, 119b2 (in FIG. 7), 119c1, 119c2 (in FIG. 8), 119d1 (not shown), and 119d2 (not shown) will extended or compressed from the original setup. Therefore, the restoring and damping functions in yaw can be achieved. To keep the model symmetric about XOY and XOZ planes, these springs and dampers are identical within one model set. However, the set of springs and dampers can be changed (all eight at a time) to vary the yaw natural frequency and damping of the model. As an example, the smallest level of yaw damping that the apparatus can achieve is the case without any dampers installed for yaw motion, meaning the spring and the damper are separated as shown in FIG. 9b and only the springs are installed. In this case the yaw damping is due to the structural fictions of the apparatus.
(33) For roll motion, a similar restoring and damping assembly 105 (shown in FIG. 2 and FIG. 11) is installed at least on one side of the frame 104. These springs and dampers can be changed accordingly to vary the roll natural frequency and damping of the model. The assembly 105 comprises a gear rack 123, a track guide 124, a gear 125, two spring and damper combinations 122-1 and 122-2, four spring posts 121-1, 121-2, 121-3 and 121-4 as shown in FIG. 11. The gear 125 is mounted rigidly on the roll axle 108 as shown in FIG. 11. When the model rolls, it drives the roll axle 108 and the gear 125 to rotate with it. The gear 125 in turn drives the gear rack 123 to move linearly along the track guided by the bracket 124. In such way, the restoring and damping in roll is achieved. A ball bearing sitting in a hole in the frame 104 to provide a smooth rotation for the roll axle 108 is 126 as shown in FIG. 11 and another ball bearing (not shown) is located on the other end of 108. Again, the smallest roll damping can be achieved by uninstalling the two roll dampers. In this case, the damping is due to the structural frictions of the apparatus.
(34) The positions of the half cylinder weights can be adjusted on the surface of the wings. For example, as shown in FIG. 12a, there are a Dovetail groove 128 on the wing 109a along the wingspan direction and a Dovetail groove 129 along the centerline of the half cylinder weight 110a which is connected with the wing 109a by a connector 127 which has Dovetail shapes on the top and bottom orthogonally as shown in FIG. 12b. The connector 127 is capable to slide along the groove 128 while the weight 110a is capable to slide on the connector 127 along the groove 129 and to be locked in position by a screw 130 on the wing 109a. An arrangement of the adjustable weight 110b on wing 109b is symmetric to that on wing 109a about the XOZ plane as shown in FIG. 3a and FIG. 6. Arrangements of the adjustable weights 110c on wing 109c and 110d on wing 109d are symmetric to that on wings 109a and 109b about XOY plane, respectively. The four adjustable weights 110a, 110b, 110c, and 110d are so positioned that the model is symmetric about XOY and XOZ planes and the center of gravity (COG) of the model is tuned to coincide with the intersection point O of roll, pitch, and yaw axes as shown in FIG. 3a.
(35) A method for demonstrating the nonlinear instability is summarized below. The crank 102 (in FIG. 2) drives the model in a harmonic oscillation in pitch which represents a control from a pilot. Oscillation pitch motion is the necessary motion happens during takeoff and landing for aircrafts. The governing equations of roll(), pitch(), and yaw(), of the model 106 as shown in FIG. 3b are given as,
I.sub.x{umlaut over ()}+b.sub.1{dot over ()}+k.sub.1=(I.sub.yI.sub.z){dot over ()}{dot over ()}+M.sub.x,Math. 11
=A.sub.21 cos(.sub.21t+.sub.21),Math. 12
I.sub.z{umlaut over ()}+b.sub.3{dot over ()}+k.sub.3=(I.sub.xI.sub.y){dot over ()}{dot over ()},Math. 13
wherein, I.sub.x, I.sub.y, and I.sub.z are the moment of inertias of the model 106 about X, Y and Z axes, respectively, b.sub.1 and b.sub.3 are the damping coefficients for roll and yaw, respectively, k.sub.1 and k.sub.3 are the restoring coefficients for roll and yaw, respectively, M.sub.x is the roll moment acting on the model 106 by the roll and yaw axle assembly 132 (FIG. 3c); A.sub.21, .sub.21, and .sub.21 are the amplitude, frequency, and phase of the pitch motion driven by the crank 102, respectively.
(36) Since the mass distribution of the assembly 132 is close to the roll axle as shown in FIG. 3c, the roll moment of inertia of the assembly 132 is very small compared with that of the aircraft model 106. Therefore the effects of the moment M.sub.x is very small and may be neglected. So we assume M.sub.x=0. This assumption is 100% accurate when the model only has pitch motions. Therefore when the crank drives the model to rotate in pitch, the governing equations of the model 106 becomes Math. 14, Math. 12, and Math.13.
I.sub.x{umlaut over ()}+b.sub.1{dot over ()}+k.sub.1=(I.sub.yI.sub.z){dot over ()}{dot over ()}Math. 14
As we know the equations Math. 14, Math. 12, and Math.13 have the following solutions,
=0,Math. 15
=A.sub.21 cos(.sub.21t+.sub.21),Math. 12
=0.Math. 16
However, according to the theory in the inventor's book, it was found that the motions represented by Math. 15, Math. 12, and Math. 16 will become unstable, then roll and yaw motions will grow exponentially under the conditions Math. 9 and Math. 10 described above.
(37) To do the demonstration, the moments of inertias of the model 106 as shown in FIG. 3b may be measured before it is installed on the assembly 132. After the model 106 is installed as shown in FIG. 2, the roll and yaw natural frequencies are measured by free-rotating tests to be .sub.10 and .sub.30, respectively. For example, in a prototype of the apparatus the frequencies were tuned to be about .sub.10=2=6.28 and .sub.30==3.14. This means that the natural roll frequency of the prototype is 1 (1/s) and the natural yaw frequency of it is 0.5 (1/s). The damping coefficients of roll and yaw can be also measured by free-decay tests, respectively. The preferred condition for the damping is the case in which the dampers in roll and yaw directions are not installed. This is the case of the minimum damping situations and therefore two dangerous pitch amplitude thresholds given by Math. 9 and Math. 10 are minimized as well.
(38) When the crank drives the model in a very low frequency, .sub.21<<|.sub.10.sub.30) (a very long period), the aircraft model only shows pitch motions controlled by the crank, no roll and yaw motion are observed. For example, for the prototype this case means .sub.21<<3.14 and the driving period needs to be much longer than 2 seconds, say 4 seconds or more. When the driving frequency .sub.21 is increased and approaching to |.sub.10.sub.30| together with a pitch amplitude exceeding the threshold given by Math. 10, roll and yaw double resonances should happen. In this case, violent roll and yaw motion are observed. For example in the prototype case, the first dangerous frequency is .sub.21=|.sub.10.sub.30|=3.14 and the pitch amplitude threshold is found by testing to be about 90. When maintaining the same frequency at .sub.21=3.14, but decreasing the pitch amplitudes to be about 40 which is less than the threshold of 90, a disappearance of the roll and yaw resonances should be observed, and the only motion remains is the pitch motion, meaning that the pitch motion is stable. When the crank driving frequency continues increase to the second dangerous frequency .sub.21=.sub.10+.sub.30 together with pitch amplitude exceeding the threshold value given by Math. 9, roll and yaw resonances should happen again. Since |.sub.10.sub.30| is smaller than .sub.10+.sub.30, the amplitude threshold based on .sub.21=|.sub.10.sub.30| by Math. 10 is larger than that based on .sub.21=.sub.10+.sub.30 by Math. 9. [For example for the prototype case, |.sub.10.sub.30|=3.14 and .sub.10+.sub.30=3=9.42, so the pitch amplitude threshold based on .sub.10+.sub.30=9.42 was 3 times smaller than the pitch amplitude threshold (about 90) based on .sub.21=|.sub.10.sub.30|=3.14. Therefore the pitch amplitude threshold for the second frequency .sub.21=9.42 was about 30. When the externally excited pitch amplitude was exceeding about 30 and the externally exciting frequency was at .sub.21=9.42, the roll and yaw double resonances were observed again]. When maintaining the pitch amplitude given by Math. 9 but increasing the pitch frequency to larger than .sub.10+.sub.30, a disappearance of roll and yaw resonances should be observed again, and the only motion remains is pitch in this case. For the prototype case, the roll and yaw resonances were excited by the inertial moments on the right hand sides of Math. 14 and Math.13, respectively. These inertial moments are nonlinear terms and neglected in the current practice by the linearization approximation in the industry. These phenomena observed above are consistent with the predictions of the theory found and described in detail in the book written also by the inventor.
(39) In another embodiment as shown in FIG. 13, FIG. 14a, FIG. 14b, and FIG. 14c, the crank 102 in FIG. 2 is replaced by a more precise driving system which comprises a variable speed motor 156, a crank arm 157, a slider pin 158, a slotted link 153, a gear 150, a gear rack 151, track guides 152a, 152b, and 152c, a stand 155 to support the motor 156, and a stand 154 to support the gear rack 151. The slotted link 153 is rigidly connected to the gear rack 151. The effective length of the crank link 153 is adjustable by sliding and locking the slider pin 158 along the Dovetail groove in 157 as shown in FIG. 15. The setup in FIG. 13, FIG. 14a, FIG. 14b, and FIG. 14c is for demonstration purpose only to show how the motor drives the pitch rotation of the aircraft model. For a real experiment, the motor RPM and the position of the slider pin need to be adjusted according to the goal of the experiment. These adjustments are to be done according to the following formulas, Math. 17 and Math. 18. The two dangerous frequencies in terms of the RPM of the motor are calculated as
.sub.motor=30|.sub.10.sub.30|/(RPM)Math. 17
.sub.motor=30(.sub.10+.sub.30)/(RMP).Math. 18
The pitch motion of the aircraft model is given as
=(R.sub.1/R.sub.2)sin(.sub.motort/30)(rad),Math. 19
wherein R.sub.1 is the effective length of the crank link 157 and R.sub.2 is the radius of the gear 150. With the precise frequency and amplitude control of the pitch motions by the motor driving system, the above demonstrations can be repeated in a more precision fashion.
(40) In yet another embodiment, the apparatus can be used to demonstrate the nonlinear instability of automobiles. In this case, the aircraft model discussed above is replaced by a bus model to represent an automobile as shown in FIG. 16a. The apparatus comprises a flat round base 201, an upside-down U type supporting frame 203 (the outer frame) rigidly mounted on base 201, a crank 202 rigidly connected to another rectangular frame 204 (the inner frame) runs through a bearing (not shown in the figure) which is sitting in a hole in the outer frame 203 so that the crank can smoothly rotate the inner frame 204, a bus model comprising two symmetric (top and bottom) part assemblies about XOY plane as shown in FIG. 16a is also both symmetric (left and right) about XOZ plane and symmetric (front and rear) about YOZ plane so that the principal inertial axes of the model are along the roll, pitch, and yaw axes of the model. The top and bottom parts of the bus model are only connected at the yaw axle and at the two points located at the middle of the bus by spacers 272a as shown in FIGS. 16b and 272b (not shown) which are very similar in size like 111a and 111b for the aircraft model above. The center of gravity (COG) of the bus model is located at Point O as shown in FIG. 16b which coincides with the intersection point of roll, pitch, and yaw axes. Therefore the roll, pitch, and yaw axes coincide with the X, Y and Z axes of the coordinate system as shown in FIG. 16a. A long roll axle 208 (shown in FIG. 16a and FIG. 16b) supporting the bus model is in turn supported by two bearings (not shown) at the two ends of it. The two bearings are sitting in holes in the two opposite sides of the frame 204. Yaw restoring and damping assemblies 207a and 207b are mounted on the top part as shown in FIG. 16b, two symmetric yaw restoring and damping assemblies 207c (not shown) and 207d (not shown) are mounted on the bottom part of the bus model. Rectangular plates 212a and 212b of the bus model serve as the top and bottom bases, respectively, in the same function as the top and bottom parts of the fuselage of the aircraft model above. Adjustable weight 210a, 210b, 210c, and 210d (not shown) are movable on the plates 212a and 212b along the Dovetail grooves, respectively. Square poles 271a-1, 271a-2, 271a-3, and 271a-4 as shown in FIG. 16b are used to adjust the Z position of weight 270a which is shown in FIG. 16a (not in FIG. 16b). Square poles 271b-1, 271b-2, and 271b-3 as shown in FIG. 16b, and 271b-4 (not shown) are used to adjust the Z position of weight 270b as shown in FIG. 16b. With this bus model installed in the apparatus, similar nonlinear instability phenomena as described above can be observed because automobiles share the same governing equations Math. 3, Math, 4 and Math.5 with aircrafts. Therefore they have the same symptom.
(41) In general for most sedans, the pitch moments of inertias are the intermediate between the other two inertias. However, for some loading conditions of some automobiles, for example big trucks, the intermediate moment of inertia may be in the yaw direction, not in pitch. In this case the automobile will show nonlinear yaw instability instead of pitch. In order to show this yaw instability, the bus model is installed in a different way as shown in FIG. 17 wherein the crank is driving the model in yaw direction instead of pitch.
(42) In still yet another embodiment, the apparatus can be used to demonstrate nonlinear yaw instability of ocean going vessels, for example, containership. A ship model 306 with the yaw moment of inertia being the intermediate is installed on the apparatus as shown in FIG. 18. In general, a ship pitch moment of inertial is the intermediate one, but in some loading conditions, a ship can be loaded so that the yaw moment of inertial becomes the intermediate, this is especially true for containerships. This apparatus comprises a base 301, a crank 302 going through an outer frame 303 to be able to rotate an inner frame 304, a similar roll restoring and damping assembly 305 is mounted on the inner frame 304. A top view of the ship model is given in FIG. 19. As shown in FIG. 19, the ship model comprises two parts 312a (left) and 312b (right). The two-parts arrangement is similar as all models discussed above. The two parts are connected only at the pitch axle and at the two points along the yaw axle by the spacers 382a and 382b (not shown). The two parts 312a and 312b each have big cut-out amidships to make a room for the pitch restoring and damping assemblies as shown in FIG. 20. The ship model again is symmetric about XOY and XOZ planes. A long roll axle 308 connected with two pitch axles by a cross connector 313 shown in FIG. 19, similar as that shown in FIG. 3c, supports the model at center of the gravity. Adjustable weights 380a-1, 380a-2, 380b-1, and 380b-2 are movable along Dovetail grooves 381a-1, 381a-2, 381b-1, and 381b-2, respectively on the horizontal plane while adjustable weights 310b-1 and 310b-2 are movable in a similar way in a vertical plane as shown in FIG. 20.
(43) The roll and pitch restoring, and damping coefficients of the ship model are adjustable to match a real ship data measured by free-decay tests so that these coefficients include the added mass effects. Therefore the roll and pitch natural frequencies of the ship model are the same as that of the ship it is simulating, respectively. To demonstrate the nonlinear yaw instability in following and quartering seas of which the roll and pitch damping coefficients are the minimum, a preferred setup is the case without any damper installed, which represents the minimum damping effect the apparatus can achieve. Since the yaw moment of inertial is the intermediate one, the yaw amplitude thresholds and the critical frequencies for nonlinear instability are given as
(44)
wherein .sub.10 and .sub.20 are the roll and pitch natural frequencies of a ship including added mass effects, respectively; b.sub.1 and b.sub.2 are the damping coefficients in roll and pitch of a ship including added mass effects, respectively; I.sub.x,I.sub.y,I.sub.z are the moment of inertias in roll, pitch, and yaw of a ship in the air, respectively. The nonlinear instability theory in the inventor's book says that the yaw motion is stable until the yaw amplitude reaches the threshold values A.sub.YTH and at the critical frequencies .sub.31 given in Math. 20 or Math. 21. Therefore a similar demonstration like that for the aircraft model discussed above can be performed using this apparatus.
(45) In further yet another embodiment, the apparatus can be used to demonstrate the nonlinear roll instability for some type of aircrafts of which the roll moment of inertia is the intermediate between the pitch and yaw inertias, for example, B52 aircraft with a longer wingspan than the length which makes the roll moment of inertia exceed the pitch moment of inertia. In this case, the original aircraft model in FIG. 2 is replaced by another aircraft model with longer wingspan and more adjustable weights on the wings as shown in FIG. 21. The similar apparatus comprises an aircraft model 406, a base 401, an outer frame 403, a crank 402, an inner frame 404, a pitch restoring and damping assembly 405 and a long pitch axle 408. Again the model is symmetric about XOY and XOZ planes and the center of gravity is adjusted to locate at the origin O of the coordinate system as shown in FIG. 22. The model comprises a top part 412a and a bottom part 412b. Again 412a and 412b are connected only at the yaw axle and at the two points along the roll axis by spacers 411a and 411b as shown in FIG. 22. Adjustable weights 410a, 410e, 410b, 410f, 410c, 410g, 410d, and 410h are moveable along the Dovetail grooves on wings 409a, 409b, 409c, and 409d, and along the Dovetail grooves on the weights, respectively, by connecting to similar connectors (not shown) like 127 in FIG. 12b. These adjustable weights are used to adjust the moment of inertias of the aircraft model. Similar yaw restoring and damping assemblies are mounted on the top and bottom parts of the model, for example, assemblies 407a and 407b on the top part.
(46) Since the roll moment of inertia is the intermediate, the roll amplitude thresholds and the critical frequencies are given as
(47)
wherein .sub.20 and .sub.30 are the pitch and yaw natural frequencies of the aircraft model, respectively; b.sub.2 and b.sub.3 are the damping coefficients in pitch and yaw of the model, respectively; I.sub.x,I.sub.y,I.sub.z are the moment of inertias in roll, pitch, and yaw of the model, respectively. The nonlinear instability theory in the inventor's book shows that the roll motion is stable until the roll amplitude reaches the threshold values A.sub.RTH and at the critical frequencies .sub.11 given in Math. 22 or Math. 23. Therefore a similar demonstration like that for the aircraft model in FIG. 2 discussed above can be performed using this apparatus.
(48) In further yet another embodiment, a general-case apparatus with a three-gimbaled framework is illustrated in FIG. 23. This apparatus comprises a base 501; a crank 502; stanchions 503a and 503b; an outer ring 504; an inner ring 505; a rectangular block 506; roll restoring and damping assemblies 507a and 507b; yaw restoring and damping assemblies 508a and 508b. The rectangular block has three different moments of inertias such as the roll moment of inertia to be the smallest, the pitch moment of inertia the intermediate, and the yaw moment of inertia the largest. The block is symmetrically mounted in the apparatus with the principal roll, pitch, and yaw axes of the block to be aligned with the roll, pitch, and yaw axles, respectively as shown in FIG. 23. A roll axle is rigidly connected with the block and aligns with the roll axis as shown in FIG. 23. The roll axle is supported at the two ends by two bearings (not shown in FIG. 23) which are sitting in the holes on the inner ring 505 such that the block is smoothly rotate about the roll axle. Another two bearings (not shown in FIG. 23) are sitting in the holes at the top and bottom of the outer ring 504 along the yaw axis. Two yaw axles are rigidly and perpendicularly connected to the inner ring 505 at the top and bottom, respectively as shown in FIG. 23. The yaw axles go through the bearings sitting in the outer ring 504 to provide a smooth rotation of the block together with the inner ring 505 about the yaw axis. Yet another two bearings are sitting in the holes on stanchion 503a and 503b in a similar way to provide smooth rotation of the outer ring 504 about the pitch axis. The crank 502 is rigidly connected to the outer ring 504 by a shaft (not shown) through a bearing in stanchion 503a and able to apply external pitching moment on the ring 504.
(49) There are two roll restoring and damping assemblies 507a and 507b, and two yaw restoring and damping assemblies 508a and 508b. Each of these assemblies includes a machined torsional spring 531, a torsional damper 530, and a bearing 533 as shown in FIG. 24. The machined torsional spring 531 is capable to be mounted and tied on rotational axle (roll or yaw) on one end by a screw 534 and on the other end the torsional spring has four cylindrical legs 532a, 532b, 532c as shown in FIG. 24, and 532d (not shown) which are capable to slide smoothly into cylindrical holes (not shown) on the outer ring 504 or the inner ring 505 such that this end of the torsional spring has no freedom of motion transversely on the ring (either 504 or 505) and the other end of the spring is capable to rotate with either the roll axle or the yaw axle. For example, FIG. 25 shows the side view of the assembly 507b. The spring 531 is tied on the roll axle 509b by screw 534 and the other end of 531 is connected by the four legs (not shown in FIG. 25) on the outer ring 505. The roll axle 509b is rigidly fixed on one end along the roll direction on the block 506 as shown in FIG. 23 and the other end of the roll axle 509b is through a bearing (not shown) which is embedded in the inner ring 505. The rotational damper 530 is mounted outside of the inner ring 505 and on the end of the roll axle 509b as shown in FIG. 25. The restoring and damping assembly 507a is a mirror of the assembly 507b about the YOZ plane as shown in FIG. 23. As another example, FIG. 26 shows the side view of the assembly 508a. The yaw axle 510a is through a bearing (not shown) which is embedded in the outer ring 504. One end of 510a is fixed with the inner ring 505 as shown in FIG. 26. One end of the torsional spring 531 is tied on the yaw axle 510a outside of the outer ring 504. The other end of 531 is connected to the outer ring 504 by the four legs of the spring (not shown) so that the torsional spring is capable to rotate only on one end with the yaw axle 510a. The four legs of the spring are free sliding in the four holes in the ring 504. The torsional damper 530 is mounted at the far ends of the yaw axle 510a in order to be easily taken off for the minimum damping case as discussed before. In this case, the damping of the system is only due to the structural frictions of the apparatus. The restoring and damping assembly 508b is a mirror of the assembly 508a about the XOY plane as shown in FIG. 23. A similar demonstration, as shown in the case of the aircraft model in FIG. 2, about the nonlinear pitch instability of the rectangular block can be performed by this general-case apparatus.
(50) It should be understood that the detailed descriptions and specific examples, while indicating the preferred embodiments, are intended for purposes of illustration only and it should be understood that it may be embodied in a large variety of forms different from the one specifically shown and described without departing from the scope and spirit of the invention. For example, one modification may be as that the top part of the aircraft model in FIG. 3b may be asymmetric with the bottom part of the model, but the principal inertia axe of roll still aligns with the roll axe of the model and all the demonstrations described above could be achieved by such modified model. It should be also understood that the invention is not limited to the specific features shown, but that the means and construction herein disclosed comprise a preferred form of putting the invention into effect, and the invention therefore claimed in any of its forms of modifications within the legitimate and valid scope of the appended claims.
