Kinematic Design and Control Analysis of A Subsonic Ejector Nozzle with Omni-Directional Thrust Vectoring for Afterburning Turbofan Engines

Because no practical prototypes exist worldwide that fully address the technical problem under consideration, the variable subsonic nozzle of the AI-222-25F turbofan engine was selected as the baseline design. Fig. 1 shows a component of the AI-222-25F variable nozzle, which adjusts the throat area by repositioning its flaps. A hydraulic cylinder controls each flap through a lever and a connecting triangle, which is pinned to the afterburner (AB) casing. Each flap is actuated mechanically and independently from the others.

Fig. 1.

Subsonic variable nozzle of the AI-222-25F engine.

To achieve mechanical synchronization – beyond synchronization of the hydraulic system itself – a special bracket with levers is installed. This bracket is attached to the AB casing and links two adjacent flaps. If one cylinder fails, the affected flap, although deprived of direct actuation, is forced by the bracket to follow the motion of the neighboring flap, thereby preserving overall nozzle geometry to the extent possible.

The kinematic scheme shown in Fig. 2 preserves the same principle of changing the throat area and allows the thrust vector to be deflected by turning the nozzle flaps. To ensure the thrust vector deflection, a control element is introduced – a ring (12), the position in space of which is uniquely determined by three hydraulic cylinders. The ring is connected to the triangle (9) through the lever (11), and they are connected to each other by means of a spherical hinge (5).

Fig. 2.

Kinematic scheme of subsonic nozzle with TVC.

The spherical hinge compensates for the difference between the rotation radii of the control ring and the connecting triangle. To increase the degrees of freedom in the movable joints, universal hinges (6) are used in place of conventional pin joints at the connections between the triangle, the subsonic flap (1), and the AB casing (13). These universal hinges allow the components to rotate in two planes. The connecting triangle (9) and the subsonic flap (1) are linked by a lever (10), which is attached to the flap via a spherical hinge (5) and to the triangle via a pin joint (8). The connection scheme for flaps 2, 3, 4, and the remaining flaps to the control ring (12) follows the same principle as that for flap 1.

To illustrate the operation of the system, a temporary axis O is introduced into the control ring (12), positioned in the vertical plane, as shown in Fig. 2. The ring rotates about this axis, defining the trajectory of motion for each nozzle flap. It should be noted that the motion of each flap differs from the others, depending on its spatial position relative to the rotation axis. The rotation axis of flap 1 lies in the same plane as the rotation axis of the ring. Flaps 2, 3, and 4 are positioned at angles relative to this axis, with flap 4 offset the most.

As shown in Fig. 3, flap 4 undergoes the largest deflection in the plane parallel to the triangle profile, while flap 1 does not move in this plane. Conversely, flap 1 shows the greatest deflection in the plane perpendicular to the triangle profile, whereas flap 4 shows the smallest deflection, as illustrated in Fig. 4. The complete three-dimensional kinematic scheme is shown in Fig. 5.

Fig. 3.

Deflection of nozzle elements in a plane parallel to the triangle profile.

Fig. 4.

Deflection of nozzle elements in a plane perpendicular to the triangle profile.

Fig. 5.

Subsonic nozzle with TVC, deflected by 18°.

Figure 5 illustrates a subsonic nozzle achieving an 18° thrust-vector deflection, while the control ring is rotated only 2° about the introduced axis. Thus, the mechanism exhibits a gear ratio of 9, which is a major advantage of this kinematic scheme. This ratio can be easily increased or decreased by adjusting the distance between the rotation axis of the ring and that of the flap, or by modifying the relative lengths of the main elements – namely, the connecting triangle and the lever that links it to the ring.

The design of the ejector nozzle with TVC is based on the subsonic TVC nozzle described above. Structurally, the ejector nozzle must include the following components:

• ejector flaps that form the downstream gas-flow channel beyond the throat area;

• cover flaps that form the outer casing that encloses the secondary air used for ejection;

• a system of levers and hinges required for fastening and for coordinated movement of all components.

Both the cover flaps and the ejector flaps must deflect in all available planes at the same angles as the subsonic flaps, following their motion synchronously. The design concept is essentially a “layered” extension of the existing subsonic TVC nozzle, incorporating additional components based on established solutions [23, 24].

To ensure synchronized movement and equal deflection angles across all flaps, the chain of connections shown in Fig. 6 is used. Cover flap 17 is attached to the connecting triangle 9 by a pin joint and to subsonic flap 1 by lever 14 through spherical hinges. To achieve equal rotation angles of subsonic flap 1 and triangle 9 in the plane perpendicular to the triangle profile, the universal hinges are shifted so that they share a common rotation axis in the plane parallel to the triangle profile. Ejector flap 16 is connected to cover flap 17 by a pin joint, and to subsonic flap 1 by lever 15 with a spherical joint. This kinematic scheme allows the position of all elements to be uniquely determined by the rotation of the control ring.

Fig. 6.

Kinematic scheme of ejector nozzle with TVC.

The inclination of the cover and ejector flaps relative to the engine axis is determined based on gas-dynamic calculations.

A key challenge arises when attempting to seal the gaps between the subsonicflap set and the ejector-flap set, in order to close the gas and air paths they form. According to the design requirements, the nozzle must have minimal weight and compact dimensions, while gas losses in the deflected jet must also remain minimal [25]. This naturally suggests reducing the number of components by eliminating blocking flaps and arranging the existing flaps in an overlapping “scale” configuration. The outer row of flaps on the variable nozzle of the F-18A, for example, is constructed this way.

However, what works for a conventional variable nozzle is far more difficult to implement in a nozzle with thrust vector control. Because adjacent flaps experience different deflection angles in two planes, the gaps between “scale-type” flaps either widen or close to the point of jamming, depending on the direction of rotation. To ensure that the sealing spacer can “follow” the motion of neighboring flaps and adopt an intermediate position that maintains proper contact, two brackets are mounted on the spacer. These brackets press against the nozzle flaps, and their movement along the flap surfaces is constrained by panel walls formed on the flaps themselves [26, 27]. The dimensions of these panels and the bracket length are, in fact, the primary limitations on the maximum achievable thrust-vector angle.

For the subsonic TVC nozzle, the optimal number of flaps was determined to be sixteen. A three-dimensional model incorporating sealing spacers is shown in Fig. 7, which also presents the assembly in four operating positions. In the upper-left view, the nozzle is deflected 8° to the left; in the upper-right view, the nozzle is deflected by the same angle, but the throat diameter is increased by a factor of 1.3. In the lower-left view, the thrust vector is deflected 15° to the right.

Fig. 7.

Subsonic nozzle with TVC in different positions.

The preliminary design of the subsonic ejector nozzle with sealing spacers is shown in Fig. 8. In the lower-right view, the nozzle is deflected 15° to the right. The method used to seal the gaps is the same as that applied to the row of subsonic flaps. The blocking flaps incorporate two brackets that extend into corresponding panels on the main flaps. As noted earlier, the dimensions of these panels and the length of the brackets define the primary limits on the achievable thrust-vector angle, as well as on the allowable throat-area range. Figure 9 illustrates the ejector nozzle in its minimum and maximum throat-area positions.

Fig. 8.

Ejector nozzle with TVC.

Fig. 9.

Ejector nozzle with TVC during expansion and contraction.

In the left image, the nozzle is deflected 8° to the left and has a throat-area diameter of 686 mm. In the right image, the nozzle is deflected by the same angle but with a reduced diameter of 516 mm. This corresponds to a total diameter-change factor of 1.33. According to the technical specifications, however, the diameter must change by a factor of 1.45. This required ratio can be achieved by increasing the panel dimensions.

For the purpose of analysis, the entire mechanism must be divided into two parts. The first part is examined in a plane parallel to the profile of triangle 9, which makes it possible to determine how the subsonic-flap deflection angle in this plane depends on the stroke of the piston rod. The second part considers the section of the mechanism that governs the subsonic-flap deflection angle in a plane perpendicular to the profile of triangle 9 [28, 29].

In this section, the task is to establish the dependence between the position of each mechanism element and the position of the initial link. The connecting triangle is chosen as this initial link, and its deflection angle from the starting position, ϕ′₁, is taken as the reference parameter (Fig. 10).

Fig. 10.

Positions of triangle 9 in the initial and current state.

To establish the dependence of the angle φ′₁ on the ring rotation angle γ₁ of ring 12 (Fig. 11), the ring itself is represented as a projection onto a plane perpendicular to the axis of its rotation (Fig. 12).

Fig. 11.

Connection scheme of lever 11 with triangle 9.

Fig. 12.

Projection of the ring onto a plane perpendicular to the axis of its rotation.

Where the parameters are defined as follows:

g₂, g₃, g₄, g₅ – distances from a point on the ring’s rotation axis to the location where lever 11 is attached; e₂, e₃, e₄, e₅ – x-coordinates of the attachment points of lever 11 to ring 12 in its deflected position.

As a result, we obtain an approximate dependence describing the relationship between the distances e_i, g_i, and the rotation angle γ₁: 5.1ei=gi·tanγ1{e_i} = {g_i}\;\cdot\;\tan {\gamma _1}

Figure 12 shows the scheme used to determine the coordinates of point B, which represents the connection between lever 11 and triangle 9 in the plane parallel to the triangle profile.

The distance e_i is known from the dependence (5.1), and the value si can be found using the formula: 5.2si=ei·tanγ12·cos(90−θi){s_i} = {e_i}\;\cdot\;\tan {{{\gamma _1}} \over 2}\;\cdot\;\cos \left( {90 - {\theta _i}} \right) where γ₁ – is ring rotation angle, θ_i – the angle between the rotation axis of the i-th flap and the vertical axis z in the plane perpendicular to the triangle profile (known from the nozzle design).

As a result, the connection point of lever 11 to ring 12 has coordinates (e_i, s_i + y₁).

The numerical coordinates of point B can then be obtained by solving the system of equations (5.3), which consists of the equations of circles 1 and 2. Circle 2 is the trajectory described by lever 11, and circle 1 is the trajectory described by triangle 9. The initial coordinates of circle 1 are (x₂, y₂), of circle 2 – (e_i, s_i + y₁).5.3{ (x+ei)2+(y+si+x1)2=L2(x+x2)2+(y+y2)2=R2\left\{ {\matrix{ {{{\left( {x + {e_i}} \right)}^2} + {{\left( {y + {s_i} + {x_1}} \right)}^2} = {L^2}} \hfill \cr {{{\left( {x + {x_2}} \right)}^2} + {{\left( {y + {y_2}} \right)}^2} = {R^2}} \hfill \cr } } \right.

In this system, L – is the length of the lever 11 (Fig. 6), R – is the radius described by triangle 9.

To obtain the triangle-deflection angle φ₁, we must compute the arc length between the connection point of lever 11 and triangle 9 in the deflected position and the point of the same connection B₀ in the mechanism’s initial position. To do this, we express the coordinates of these points relative to the center of the circle circumscribed by triangle 9: 5.4{ h0=y0−y2;k0=x0−x2;h1=y−y2;k1=x−x2.\left\{ {\matrix{ {{h_0} = {y_0} - {y_2};} \hfill \cr {{k_0} = {x_0} - {x_2};} \hfill \cr {{h_1} = y - {y_2};} \hfill \cr {{k_1} = x - {x_2}.} \hfill \cr } } \right.

Using a definite integral, we calculate the length of the arc from point B to B₀ as: 5.5L=∫k0k11+((R2−x2)′)2dxL = \int_{{k_0}}^{{k_1}} {\sqrt {1 + {{\left( {{{(R2 - x2)}^\prime }} \right)}^2}} } dx

Angle φ′₁ is then equal to the ratio of arc length L to the radius R.5.6φ1′=LR.\varphi _1^\prime = {L \over R}.

Once φ′₁ has been so determined, we can proceed to analyze the kinematic scheme in the plane parallel to the triangle profile. For this analysis, a planar representation of the problem is sufficient, so all spherical and universal hinges are replaced by pin joints. First, the scheme is represented as a system of interconnecting levers, with lengths shown in Fig. 13 and denoted as L1, L2, L3, L4, L4’, L5, L6, L7, where:

• L1 and L6 – lengths corresponding to the two sides of the connecting triangle 9 (Fig. 6);

• L2 – the length of the portion of the cover flap 17 (Fig. 6) between its attachment to the triangle and lever 14 (Fig. 6);

• L3 – the length of lever 14 (Fig. 6);

• L4 – the length of the portion of subsonic flap 1 (Fig. 6) between its attachment to the AB casing and to lever 14 (Fig. 6);

• L4′ – the length of the subsonic flap 1 (Fig. 6) between its attachment to the AB casing and to lever 10 (Fig. 6);

• L5 – the length of the lever 10 (Fig. 6);

• L7 – the distance between the fastenings of the triangle and the subsonic flap to AB.

Fig. 13.

Mechanism of hinge multi-link.

The solution of the mechanism-position problem is reduced to analyzing three triangles and one quadrilateral obtained after the introduction of the vectors S and S′. From the scheme shown in Fig. 14, we determine the dependence of angle φ₄ on the rotation of the initial link (angle φ₁). The angle φ₁ is given by φ₁ = φ₀ – φ′₁, where φ₀ is the angle of the initial position of triangle 9.

Fig. 14.

Four-hinge link mechanism.

First, we consider the triangle L6, L7, S to determine the value S and angle φ_S using the cosine theorem: 5.7S2=L62+L72−2·L6·L7·cosφ1;{S^2} = L{6^2} + L{7^2} - 2\;\cdot\;L6\;\cdot\;L7\;\cdot\;\cos {\varphi _1}; 5.8S=L62+L72−2·L6·L7·cosφ1;S = \sqrt {L{6^2} + L{7^2} - 2\;\cdot\;L6\;\cdot\;L7\;\cdot\;\cos {\varphi _1}} ; 5.9cosφS=L72+S2−L622·L7·S.\cos {\varphi _S} = {{L{7^2} + {S^2} - L{6^2}} \over {2\;\cdot\;L7\;\cdot\;S}}.

From the second triangle S, L5, L4, we find the angles φ₄ and φ₅: 5.10L52=S2+L4′2−2·S·L4·cosφ4S;L{5^2} = {S^2} + L{4^{\prime 2}} - 2\;\cdot\;S\;\cdot\;L4\;\cdot\;\cos {\varphi _{4S}}; 5.11cosφ4S=S2+L4′2−L522·L4′·S;\cos {\varphi _{4S}} = {{{S^2} + L{4^{\prime 2}} - L{5^2}} \over {2\;\cdot\;L{4^{\prime \;}}\cdot\;S}}; 5.12L4′2=S2+L52−2·S·L5·cosφ5S;L{4^{\prime 2}} = {S^2} + L{5^2} - 2\;\cdot\;S\;\cdot\;L5\;\cdot\;\cos {\varphi _{5S}}; 5.13cosφ5S=S2+L52−L4′22·L5·S;\cos {\varphi _{5S}} = {{{S^2} + L{5^2} - L{4^{\prime 2}}} \over {2\;\cdot\;L5\;\cdot\;S}}; 5.14φ5S=φS+φ5;{\varphi _{5S}} = {\varphi _S} + {\varphi _5}; 5.15φ4S=φ4−φS;{\varphi _{4S}} = {\varphi _4} - {\varphi _S};

Next, we consider a quadrilateral formed by links L1, L4, L7 and the introduced vector S′ (Figure 15).

Fig. 15.

Four-hinge link mechanism.

We now write a vector equation for the quadrilateral and project it onto the x- and y-axes: 5.16L1¯+S′¯+L4¯−L7¯=0\overline {L1} + \overline {{S^\prime }} + \overline {L4} - \overline {L7} = 0 5.17x:L1·cosφ1S′+S′·cosφS′+L4·cos(360°−φ4)−L7=0x:L1\;\cdot\;\cos {\varphi _{1{S^\prime }}} + {S^\prime }\cdot\cos {\varphi _{{S^\prime }}} + L4\;\cdot\;\cos \left( {{{360}^^\circ } - {\varphi _4}} \right) - L7 = 0 5.18y:L1·sinφ1S′+S′·sinφS′+L4·sin(360°−φ4)=0y:L1\;\cdot\;\sin {\varphi _{1{S^\prime }}} + {S^\prime }\cdot\sin {\varphi _{{S^\prime }}} + L4\;\cdot\;\sin \left( {{{360}^^\circ } - {\varphi _4}} \right) = 0

We next determine the magnitude of vector S′ and angle φ_S′ : 5.19φ1S′=φ1+90{\varphi _{1{S^\prime }}} = {\varphi _1} + 90 5.20tanφS′=−L1·sinφ1S′−L4·sin(360°−φ4)L7−L1·cosφ1S′−L4·cos(360°−φ4)\tan {\varphi _{{S^\prime }}} = {{ - L1\;\cdot\;\sin {\varphi _{1{S^\prime }}} - L4\;\cdot\;\sin \left( {{{360}^^\circ } - {\varphi _4}} \right)} \over {L7 - L1\;\cdot\;\cos {\varphi _{1{S^\prime }}} - L4\;\cdot\;\cos \left( {{{360}^^\circ } - {\varphi _4}} \right)}} 5.21S′=−L1·sinφ1S′−L4·sin(360°−φ4)sinφS′{S^\prime } = {{ - L1\;\cdot\;\sin {\varphi _{1{S^\prime }}} - L4\;\cdot\;\sin \left( {{{360}^^\circ } - {\varphi _4}} \right)} \over {\sin {\varphi _{{S^\prime }}}}}

Finally, we consider the triangle L2, L3, S′ (Fig. 16). Using the cosine theorem, we find the values of the angles φ₂, φ3.5.22φ2S′=φ2−φS′{\varphi _{2{S^\prime }}} = {\varphi _2} - {\varphi _{{S^\prime }}} 5.23φ3S′=φ3−φS′{\varphi _{3{S^\prime }}} = {\varphi _3} - {\varphi _{{S^\prime }}} 5.24L32=L22+S′2−2·S′·L2·cosφ2S′L{3^2} = L{2^2} + {S^{\prime 2}} - 2\;\cdot\;{S^\prime }\cdotL2\;\cdot\;\cos {\varphi _{2{S^\prime }}} 5.25L22=L32+S′2−2·S′·L3·cosφ3S′L{2^2} = L{3^2} + {S^{\prime 2}} - 2\;\cdot\;{S^\prime }\cdotL3\;\cdot\;\cos {\varphi _{3{S^\prime }}} 5.26cosφ2S′=S′2+L22−L322·L2·S′\cos {\varphi _{2{S^\prime }}} = {{{S^{\prime 2}} + L{2^2} - L{3^2}} \over {2\;\cdot\;L2\;\cdot\;{S^\prime }}} 5.27cosφ3S′=S′2+L32−L222·L3·S′\cos {\varphi _{3{S^\prime }}} = {{{S^{\prime 2}} + L{3^2} - L{2^2}} \over {2\;\cdot\;L3\;\cdot\;{S^\prime }}}

Fig. 16.

Three-link hinge mechanism.

To obtain the equation of the connection between the control (initial link) and other elements of the mechanism, we use formula 5.24 and substitute the previously found values of S′ and φ_S′ (5.20, 5.21): 5.28L22+S′2−L32−2·S′·L2·cosφ2S′=0L{2^2} + {S^{\prime 2}} - L{3^2} - 2\;\cdot\;{S^\prime }\cdotL2\;\cdot\;\cos {\varphi _{2{S^\prime }}} = 0 5.29A=−L1·sin(φ1+90)−L4·sin(360−φ4)sin(arctan(−L1·sin(φ1+90)−L4·sin(360−φ4)L7−L1·cos(φ1+90)−L4·cos(360−φ4)))A = {{ - L1\;\cdot\;\sin \left( {{\varphi _1} + 90} \right) - L4\;\cdot\;\sin \left( {360 - {\varphi _4}} \right)} \over {\sin \left( {\arctan \left( {{{ - L1\;\cdot\;\sin \left( {{\varphi _1} + 90} \right) - L4\;\cdot\;\sin \left( {360 - {\varphi _4}} \right)} \over {L7 - L1\;\cdot\;\cos \left( {{\varphi _1} + 90} \right) - L4\;\cdot\;\cos \left( {360 - {\varphi _4}} \right)}}} \right)} \right)}} 5.30φ4=arccos(S2+L4′2−L522·L4′2·S)+arccos(L72+S2−L622·L7·S){\varphi _4} = \arccos \left( {{{{S^2} + L{4^{\prime 2}} - L{5^2}} \over {2\;\cdot\;L{4^{\prime 2}}\;\cdot\;S}}} \right) + \arccos \left( {{{L{7^2} + {S^2} - L{6^2}} \over {2\cdotL7\;\cdot\;S}}} \right) 5.31L22+A2−L32−2·A·L2·(L22+A2−L322·A·L2)=0L{2^2} + {A^2} - L{3^2} - 2\;\cdotA\;\cdot\;L2\;\cdot\;\left( {{{L{2^2} + {A^2} - L{3^2}} \over {2\;\cdot\;A\;\cdotL2}}} \right) = 0

This nozzle is omnidirectional, which allows it to rotate relative to a single point in different directions. The control ring defines this motion, and the point about which the ring rotates is chosen as the reference point. The angle of the normal vector drawn to the plane of the ring and the thrust vector angle are not the same. This difference is achieved by the transmission mechanism shown in Fig. 17. All elements in this figure are shown in a plane perpendicular to the ring’s axis of rotation:

• 1 – the triangle;

• 2 – axis of rotation of the triangle;

• 3 – the lever connecting the control ring with the triangle;

• 4 – the control ring;

• γ₁ – the angle between lever 3 and the engine axis;

• γ₂ – the angle between the triangle and the engine axis;

• h1 – the distance from the ring rotation axis (reference point) to the triangle rotation axis;

• h2 – the distance from the triangle rotation axis to the connection point between the triangle and lever.

Fig. 17.

Transmission mechanism.

Note that the thrust vector angle is taken as the deflection angle of the subsonic flap whose axis of rotation lies in the same plane as the ring’s rotation axis. Thus, taking into account that, in the plane considered, the rotation angles of this subsonic flap and triangle 1 are equal, the thrust vector angle coincides with the triangle’s rotation angle and is equal to γ₂, while the rotation angle of ring 4 is γ₁. To determine the relationship between angles γ₁ and γ₂, we solve a system of equations composed of the equation of the circle traced by triangle 1 and the equation of the line crossing the circle. The straight line is lever 3, the circle is the trajectory of the connection point of triangle 1 with lever 3. As a result, we have the following system of equations: 5.32{ y=tan(γ1)x(x+h1)2+y2=R2.\left\{ {\matrix{ {y = \tan \left( {{\gamma _1}} \right)x} \hfill \cr {{{(x + h1)}^2} + {y^2} = {R^2}.} \hfill \cr } } \right.

Solving the system (5.32) for x, we obtain the following roots: 5.33{ x=−h1±h12−(1+tan2(γ1)·(h12−R2))y=tan(γ1)x.\left\{ {\matrix{ {x = - h1 \pm \sqrt {h{1^2} - \left( {1 + {{\tan }^2}\left( {{\gamma _1}} \right)\cdot\left( {h{1^2} - {R^2}} \right)} \right)} } \hfill \cr {y = \tan \left( {{\gamma _1}} \right)x.} \hfill \cr } } \right.

Knowing the coordinates of the intersection point of the line and the circle, we can find the triangle rotation angle γ₂, using a right triangle with legs h₂ and y: 5.34h2=x−h1;{h_2}\;{\rm{ = }}\;x - h{\rm{1;}} 5.35γ2=arctan(yh2).{\gamma _2} = \arctan \left( {{y \over {{h_2}}}} \right).