Helicopter tail rotor / tail beam coupled substructure direct comprehensive modeling method

Abstract

The invention provides a direct comprehensive modeling method for a helicopter tail rotor / tail beam coupled substructure, which comprises the following steps of: constructing a blade model, and dispersing the blade model to obtain a mass matrix, a rigidity matrix, a damping matrix and a generalized force vector expression of a single blade; establishing a structural kinetic equation of the tail beam; and finally, according to the displacement boundary condition of the joint of each paddle and the tail beam, converting the paddle structure dynamic equation, and assembling the rigidity matrix, the damping matrix, the mass matrix and the generalized force vector of each paddle after conversion with the rigidity matrix and the mass matrix of the tail beam respectively to obtain the structure dynamic equation of the tail rotor / tail beam coupling system. And establishing a structural dynamic model of the tail rotor / tail beam coupling system. The method is simple and reliable in modeling process and high in universality, is not limited by the distribution type of the tail rotor, can be applied to the non-uniformly distributed tail rotor, and provides a new thought for coupling analysis of the tail rotor / tail rotor and rotor wing / fuselage of the helicopter, dynamic characteristic solving of the tail rotor and analysis of stability.

Description

Technical Field

[0001] The invention belongs to the technical field of structural dynamics, and in particular relates to a direct comprehensive modeling method of a helicopter tail rotor / tail beam coupled substructure. Background Art

[0002] The main research directions of helicopter tail rotor dynamics include the dynamic characteristics of the tail rotor and the aeroelasticity of the tail rotor, among which the dynamic characteristics of the tail rotor are the basis of the aeroelasticity of the tail rotor. When studying the dynamics of the tail rotor, the main approach to the effect of the tail boom is to treat it as a rigid body. However, with the development of helicopters, lightweighting has become the mainstream, and the effect of the elasticity of the tail boom on the dynamic characteristics of the tail rotor needs to be considered.

[0003] In response to the tail rotor / tail beam coupling problem, Xu Ning and Ling Aimin from the Helicopter Research Institute studied the vibration stability problem of the helicopter tail rotor / tail beam coupling in the "Study on the Analysis of the Coupling Stability of the Helicopter Tail Rotor / Tail Beam" for the design of the swing soft tail rotor, but the swinging motion of the tail rotor was considered in the analysis process; Chen Xinhua et al. carried out structural parameter optimization considering the dynamic characteristics of the whole helicopter based on the structural dynamic modeling of the whole helicopter. Similarly, in the analysis process, only the swinging mode of the tail rotor was considered, and the influence of the elasticity of the tail beam on the dynamic characteristics of the tail rotor was not considered, resulting in inaccurate analysis results of the tail rotor / tail beam coupling stability. Zhang Lixin et al. from Northwestern Polytechnical University used the Fourier coordinate transformation method to transform the structural dynamic equations of the uniformly arranged tail rotor from the rotating coordinate system to the fixed coordinate system to achieve the analysis of the tail rotor / tail beam coupling stability in response to the problem of the helicopter tail rotor / tail beam coupling dynamic stability in the hovering state. However, the Fourier coordinate transformation method is only applicable to uniformly arranged tail rotors, such as cross propellers, etc. For non-uniformly arranged tail rotors, such as scissor propellers, this method is no longer effective. Summary of the invention

[0004] The purpose of the present invention is to overcome the shortcomings of the existing tail rotor / tail boom coupling stability analysis process, that is, the influence of the elasticity of the tail boom on the dynamic characteristics of the tail rotor is not considered, the analysis results are inaccurate, and the application range of the established dynamic model is small, and a direct comprehensive modeling method for the substructure of the helicopter tail rotor / tail boom coupling is provided.

[0005] To achieve the above purpose, the technical solution provided by the present invention is:

[0006] A substructure direct integrated modeling method for helicopter tail rotor / tail boom coupling includes the following steps:

[0007] Step 1: Considering the influence of the inertial force caused by the rotation speed, establish the structural dynamics equation of each single blade;

[0008] Step 2: Establish the structural dynamic equation of the tail boom;

[0009] Step 3: According to the displacement boundary conditions at the connection between each blade and the tail beam, the independent degrees of freedom in the blade structural dynamic equation are converted into the degrees of freedom corresponding to the tail beam, and the structural dynamic equation of the blade after conversion is obtained in blocks;

[0010] Step 4: Obtain the stiffness matrix, damping matrix, mass matrix and generalized force vector of each blade from the structural dynamic equations expressed in blocks after blade conversion, and assemble the stiffness matrix, damping matrix, mass matrix and generalized force vector of each blade with the stiffness matrix and mass matrix of the tail boom respectively according to the installation position of each blade on the tail boom, obtain the stiffness matrix, damping matrix, mass matrix and generalized force vector of the tail rotor / tail boom coupling system, and establish the structural dynamic model of the tail rotor / tail boom coupling system.

[0011] Furthermore, the step 1 includes the following sub-steps:

[0012] Step 1.1: Establish the structural dynamics model of a single blade;

[0013] Step 1.2: Discretize the blade structure dynamics model to obtain multiple beam elements, and obtain the stiffness matrix, damping matrix, mass matrix and generalized force vector expression of each beam element respectively;

[0014] Step 1.3: According to the corresponding positions of each beam unit, the stiffness matrix, damping matrix, mass matrix and generalized force vector expression of each beam unit are combined to obtain the overall stiffness matrix, damping matrix, mass matrix and generalized force vector expression of the single blade;

[0015] Step 1.4: According to the overall stiffness matrix, damping matrix, mass matrix and generalized force vector expression of the single blade, the structural dynamic equation of the single blade is established.

[0016] Furthermore, in step 1.1, the variational expressions of the kinetic energy and strain energy of the blade are first obtained, and then the structural dynamics model of the single blade is established based on the Hamiltonian principle.

[0017] Furthermore, the step 2 includes the following sub-steps:

[0018] Step 2.1: Establish the structural dynamics model of the tail boom;

[0019] Step 2.2: Discretize the tail beam structure dynamics model to obtain multiple tail beam units, and obtain the stiffness matrix and mass matrix expressions of each tail beam unit respectively;

[0020] Step 2.3: According to the corresponding position of each tail beam unit, the stiffness matrix and mass matrix expressions of each tail beam unit are combined to obtain the overall stiffness matrix and mass matrix expressions of the tail beam;

[0021] Step 2.4: According to the expressions of the overall stiffness matrix and mass matrix of the tail boom, the structural dynamic equation of the tail boom is established.

[0022] Furthermore, in step 2.1, a structural dynamics model of the tail beam is established based on the Euler-Bernoulli beam theory.

[0023] Furthermore, the specific process of step 3 is as follows:

[0024] Step 3.1: Set the displacement boundary conditions of the blade and tail beam to q O2 =Lq O1 ,q O2 is the displacement of the blade connection, q O1 is the displacement of the tail beam connection, q I2 is the internal displacement of the blade;

[0025] Define the structural dynamic equation of the blade established in step 1 as Where q2, and are the displacement, velocity and acceleration of a single blade respectively;

[0026] The structural dynamics equation of the blade established in step 1 is processed in blocks, and the structural dynamics equation of the blade after block processing is obtained as follows:

[0027]

[0028] Among them, the parameters with subscript II2 represent the stiffness matrix, mass matrix and damping matrix corresponding to the inside of the blade, the parameters with subscripts IO2 and OI2 represent the coupled stiffness matrix, coupled mass matrix and coupled damping matrix inside the blade and at the blade connection, and the parameters with subscript OO2 represent the stiffness matrix, mass matrix and damping matrix corresponding to the blade connection; F I2 is the generalized force along the internal displacement of the blade, F O2 is the generalized force along the displacement of the blade connection;

[0029] Step 3.2: Convert the displacement, velocity and acceleration of the blade to the fixed coordinate system and obtain:

[0030]

[0031] Among them, T is the transformation matrix, obtained from the matrix L,

[0032]

[0033] Where θ is the rotation angle of the blade;

[0034] Step 3.3: According to step 3.1 and step 3.2, the structural dynamic equation of the converted blade is:

[0035] M2′=T T M2T

[0036]

[0037] F2′=T T F2

[0038]

[0039] Where: M2′ is the converted blade stiffness matrix, C2′ is the converted blade damping matrix, K2′ is the converted blade stiffness matrix, and F2′ is the converted blade generalized force vector;

[0040] Step 3.4, the structural dynamics equation of the converted blade obtained in step 3.3 is processed in blocks to obtain:

[0041]

[0042] Step 3.5, follow the process of steps 3.1-3.4 to obtain the structural dynamic equations expressed in blocks after all blades are transformed.

[0043] The advantages of the present invention are:

[0044] The substructure direct comprehensive modeling method of the helicopter tail rotor / tail beam coupling of the present invention takes into account the movements of the tail rotor and the tail beam in different directions when establishing the tail rotor / tail beam coupling structural dynamic equation, thereby ensuring the integrity of the tail rotor / tail beam coupling model, and performs modeling in a physical coordinate system without converting to a modal coordinate system for tail rotor / tail beam coupling.

[0045] The tail rotor / tail boom coupling dynamic model constructed in the present invention is not only applicable to arranged tail rotors and tail booms, but also to non-uniformly arranged tail rotors and tail booms, has a wide range of applications, and a simple modeling process.

[0046] The tail rotor / tail boom coupling model constructed by the method of the present invention can be used for tail rotor / tail boom coupling stability analysis as well as for solving modal characteristics and response characteristics. It has broad application prospects and provides a new idea for future helicopter tail rotor / tail rotor and rotor / fuselage coupling analysis. BRIEF DESCRIPTION OF THE DRAWINGS

[0047] The above and / or additional aspects and advantages of the present invention will become apparent and easily understood from the description of the embodiments in conjunction with the following drawings, in which:

[0048] Figure 1 It is a flowchart of the direct comprehensive modeling method of the substructure of the helicopter tail rotor / tail beam coupling of the present invention;

[0049] Figure 2is a schematic diagram of discretizing a blade model using the finite element method in the present invention;

[0050] Figure 3 It is a schematic diagram of the assembly of the overall mass array of the blades in the present invention;

[0051] Figure 4 It is a schematic diagram of the cross tail rotor / tail boom response verification condition;

[0052] Figure 5 The method of the present invention and the multi-body dynamics software Adams simulation method are respectively used to Figure 4 Comparison of the tip dynamic response results of the first blade and the second blade in the working condition;

[0053] Figure 6 The method of the present invention and the multi-body dynamics software Adams simulation method are respectively used to Figure 4 Comparison of the tip dynamic response results of the third and fourth blades in the working condition.

[0054] Explanation of the reference numerals: 1 - first blade, 2 - second blade, 3 - third blade, 4 - fourth blade, 5 - tail beam. DETAILED DESCRIPTION

[0055] Embodiments of the present invention are described in detail below. The embodiments are exemplary and intended to be used to explain the present invention, but should not be construed as limiting the present invention.

[0056] Reference Figure 1 This embodiment provides a direct comprehensive modeling method for a helicopter tail rotor / tail boom coupled substructure, comprising the following steps:

[0057] Step 1: Establish the structural dynamics equation of each single blade.

[0058] First, the influence of the inertial force caused by the rotation speed is considered to model the single blade. In this embodiment, the variational expressions of the kinetic energy and strain energy of the blade are first obtained, and then the structural dynamics model of the single blade is established based on the Hamiltonian principle.

[0059] Then, the finite element method is used to discretize the blade structure dynamic model into several beam elements, such as Figure 2As shown in the figure, each beam element contains 2 nodes and 12 degrees of freedom. Among them, u1 and u2 are the axial displacements of the nodes on both sides respectively; v1 and v1′ are the displacement and rotation angle of the first node along the swing direction of the beam element, v2 and v′2 are the displacement and rotation angle of the second node along the swing direction of the beam element; w1 and w1′ are the displacement and rotation angle of the first node along the swing direction of the beam element, w2 and w′2 are the displacement and rotation angle of the second node along the swing direction of the beam element; φ1 and φ2 are the torsion angles of the nodes on both sides of the two elements. The stiffness matrix, damping matrix, mass matrix and generalized force vector expressions of each beam element are obtained, and the structural dynamics equation expression of each beam element is:

[0060]

[0061] Where i = 1, 2, 3, ..., n, n is the total number of discrete beam elements; M ei is the unit mass matrix of the ith beam element, K ei is the element stiffness matrix of the i-th beam element, C ei is the element damping matrix of the ith beam element, F ei is the element generalized force vector of the ith beam element, q i , and are the displacement, velocity and acceleration of the i-th beam element respectively.

[0062] Finally, if Figure 3 As shown in the figure, the mass matrix, damping matrix, stiffness matrix and generalized force vector of all beam elements are assembled according to the principle of "matching numbers to seats", and the overall mass matrix M2, damping matrix C2, stiffness matrix K2 and generalized force vector F2 of a single blade are obtained, and the structural dynamic equation of a single blade is obtained based on this, as shown below:

[0063]

[0064] In the formula, q2, and are the displacement, velocity and acceleration of a single blade respectively.

[0065] According to the above process, the structural dynamic equation of each blade is obtained.

[0066] Step 2: Establish the structural dynamic equation of the tail boom.

[0067] First, a structural dynamics model of the tail beam is established. In this embodiment, the expressions of stress and strain of the tail beam are first obtained, and then the structural dynamics model of the tail beam is established based on the Euler-Bernoulli beam theory.

[0068] Then, the finite element method is used to discretize the tail beam structure dynamic model into several tail beam units, and the structural dynamic equation of each unit is obtained as follows:

[0069]

[0070] Where j = 1, 2, 3, ..., m, m is the total number of discrete tail beam units; M e1j is the unit mass matrix of the jth tail beam unit, K e1j is the element stiffness matrix of the jth tail beam element, q j is the displacement of the jth tail beam element, is the acceleration of the jth tail beam element.

[0071] Finally, the obtained tail beam unit mass matrix and stiffness matrix are assembled according to the principle of "matching numbers to seats" to obtain the tail beam structure dynamic equation, as shown below:

[0072]

[0073] Where M1 is the overall mass matrix of the tail beam, K1 is the overall stiffness matrix of the tail beam, q1 and are the displacement and acceleration of the tail boom respectively.

[0074] The assembly principle of the tail beam unit is the same as that of the beam unit of the blade.

[0075] Step 3: According to the displacement boundary conditions at the connection between each blade and the tail beam, the independent degrees of freedom in each blade structural dynamic equation are converted into the corresponding degrees of freedom on the tail beam, and the structural dynamic equation of the blade after conversion is obtained. The specific process is as follows:

[0076] Step 3.1: Define the displacement boundary conditions of the blade and tail beam as q O2 =Lq O1 ,q O2 is the displacement of the blade connection, q O1 is the displacement of the tail beam connection, q I2 is the internal displacement of the blade.

[0077] The structural dynamics equation of a single blade shown in formula (2) is divided into blocks and written as follows:

[0078]

[0079] Among them, the parameters with subscript II2 represent the stiffness matrix, mass matrix and damping matrix corresponding to the inside of the blade, the parameters with subscripts IO2 and OI2 represent the coupled stiffness matrix, coupled mass matrix and coupled damping matrix inside the blade and at the blade connection, and the parameters with subscript OO2 represent the stiffness matrix, mass matrix and damping matrix corresponding to the blade connection; F I2 is the generalized force along the internal displacement of the blade, F O2 is the generalized force along the displacement of the blade connection.

[0080] Step 3.2: Convert the displacement, velocity and acceleration of the blade to the fixed coordinate system as follows:

[0081]

[0082] Among them, T is the transformation matrix, which is obtained from the matrix L. The expressions of the transformation matrix T and the matrix L are as follows:

[0083]

[0084] Step 3.3: According to the formulas in steps 3.1 and 3.2, the structural dynamic equation of the blade after conversion is obtained:

[0085] M2′=T T M2T

[0086]

[0087] F2′=T T F2 (9)

[0088]

[0089] Among them: M2′ is the converted blade stiffness matrix, C2′ is the converted blade damping matrix, K2′ is the converted blade stiffness matrix, and F2′ is the converted blade generalized force vector.

[0090] Step 3.4, process formula (10) to obtain the structural dynamic equation expressed in blocks after blade conversion, which is in the form of:

[0091]

[0092] Step 3.5: Follow the process of steps 3.1-3.4 to obtain the structural dynamic equations expressed in blocks after all blades are transformed.

[0093] Step 4: According to the installation position of each blade on the tail beam, assemble the converted mass matrix, damping matrix, stiffness matrix and generalized force vector of all blades with the mass matrix and stiffness matrix of the tail beam according to the rule of "matching the number to the seat", and obtain the mass matrix, damping matrix, stiffness matrix and generalized force vector of the tail rotor / tail beam coupling system, and then obtain the structural dynamic equation of the tail rotor / tail beam coupling system. Taking the coupling of a single blade and tail beam as an example, the structural dynamic equation of the single blade / tail beam coupling system obtained after assembly is:

[0094]

[0095] From the above modeling process, it can be concluded that the method of the present invention has no requirements on the layout type of the tail rotor, and is applicable to both uniformly distributed tail rotors, such as a cross-shaped tail rotor, and non-uniformly distributed tail rotors, such as a scissor-shaped tail rotor.

[0096] To verify the reliability of the modeling method of the present invention, this embodiment calculates the tail rotor / tail beam coupling frequency characteristics in the tail rotor / tail beam coupling model constructed by the present invention under the condition of a speed of 0 according to the eigenvalue analysis method. The calculation results are compared with the calculation results of the Nastran finite element software as shown in Table 1.

[0097] Table 1 Comparison of calculation results of tail rotor / tail beam coupling natural frequency

[0098] Degree Nastran / Hz The present invention / Hz 1 1.910 1.910 2 3.82 3.82 3 11.97 11.97 4 23.94 23.94 5 33.52 33.52 6 65.69 65.70

[0099] It can be seen from the table that the tail rotor / tail beam coupling natural frequency obtained by the coupling model constructed based on the method of the present invention is consistent with the tail rotor / tail beam coupling natural frequency calculated by using the Nastran finite element software.

[0100] Figure 4 An example of a cross tail rotor / tail beam response verification working condition is given, in which the speed of the cross tail rotor is πrad / s, the cross tail rotor includes a first blade 1, a second blade 2, a third blade 3 and a fourth blade 4, each blade is 0.2 meters long, and the cross section is a rectangular cross section with a length of 0.005 meters and a width of 0.0025 meters, and the length of the tail beam 5 is 0.8 meters; F in the figure is the load applied to the fourth blade 4. Firstly, the coupling model of the cross tail rotor / tail beam under this working condition is established by the method of the present invention, and the Newmark method is used to calculate the coupled dynamic response of the tail rotor / tail beam, and then the multi-body dynamics software Adams is used alone to calculate the coupling dynamic response of the cross tail rotor / tail beam. Figure 4 The simulation analysis is performed on the working conditions shown in the figure. The calculation results of the two methods are as follows Figure 5 and Figure 6 As shown, Figure 5 This is a comparison chart of the dynamic response results of the blade tip of the first blade 1 and the second blade 2. Figure 6It is a comparison diagram of the tip dynamic response results of the third blade 3 and the fourth blade 4. As can be seen from the figure, the dynamic response of the tail rotor / tail beam coupling structure calculated by the method of the present invention is almost completely consistent with the calculation result of the multi-body dynamics software Adams, indicating that the tail rotor / tail beam coupling model constructed by the method of the present invention is highly accurate and suitable for engineering applications.

[0101] In addition, the tail rotor / tail boom coupling model constructed using the method of the present invention can also be used for tail rotor / tail boom stability analysis, tail rotor / tail boom coupling aeroelastic response analysis, tail rotor / tail boom coupling internal force and support reaction force calculation, and has a wide range of applications.

[0102] The above description is only a specific implementation mode of the present invention, but the protection scope of the present invention is not limited thereto. Any technician familiar with the technical field can easily think of various equivalent modifications or substitutions within the technical scope disclosed in the present invention, and these modifications or substitutions should be included in the protection scope of the present invention.

Claims

1. A direct comprehensive modeling method for substructure of helicopter tail rotor / tail boom coupling, characterized in that: The following steps are involved: Step 1: Considering the influence of the inertial force caused by the rotation speed, establish the structural dynamics equation of each single blade; Step 2: Establish the structural dynamic equation of the tail boom; Step 3: According to the displacement boundary conditions at the connection between each blade and the tail beam, the independent degrees of freedom in the blade structural dynamic equation are converted into the degrees of freedom corresponding to the tail beam, and the structural dynamic equation of the blade after conversion is obtained in blocks; Step 4: Obtain the stiffness matrix, damping matrix, mass matrix and generalized force vector of each blade from the structural dynamic equations expressed in blocks after blade conversion, and assemble the stiffness matrix, damping matrix, mass matrix and generalized force vector of each blade with the stiffness matrix and mass matrix of the tail boom respectively according to the installation position of each blade on the tail boom, obtain the stiffness matrix, damping matrix, mass matrix and generalized force vector of the tail rotor / tail boom coupling system, and establish the structural dynamic model of the tail rotor / tail boom coupling system.

2. A helicopter tail rotor / tail boom coupled substructure direct comprehensive modeling method according to claim 1, characterized in that: The step 1 comprises the following sub-steps: Step 1.1: Establish the structural dynamics model of a single blade; Step 1.2: Discretize the blade structure dynamics model to obtain multiple beam elements, and obtain the stiffness matrix, damping matrix, mass matrix and generalized force vector expression of each beam element respectively; Step 1.3: According to the corresponding positions of each beam unit, the stiffness matrix, damping matrix, mass matrix and generalized force vector expression of each beam unit are combined to obtain the overall stiffness matrix, damping matrix, mass matrix and generalized force vector expression of the single blade; Step 1.4: According to the overall stiffness matrix, damping matrix, mass matrix and generalized force vector expression of the single blade, the structural dynamic equation of the single blade is established.

3. A helicopter tail rotor / tail boom coupled substructure direct comprehensive modeling method according to claim 2, characterized in that: In step 1.1, the variational expressions of the kinetic energy and strain energy of the blade are first obtained, and then the structural dynamics model of the single blade is established based on the Hamiltonian principle.

4. The method for direct comprehensive modeling of substructures of a helicopter tail rotor / tail boom coupling according to claim 1, characterized in that: The step 2 includes the following sub-steps: Step 2.1: Establish the structural dynamics model of the tail boom; Step 2.2: Discretize the tail beam structure dynamics model to obtain multiple tail beam units, and obtain the stiffness matrix and mass matrix expressions of each tail beam unit respectively; Step 2.3: According to the corresponding position of each tail beam unit, the stiffness matrix and mass matrix expressions of each tail beam unit are combined to obtain the overall stiffness matrix and mass matrix expressions of the tail beam; Step 2.4: According to the expressions of the overall stiffness matrix and mass matrix of the tail boom, the structural dynamic equation of the tail boom is established.

5. A helicopter tail rotor / tail boom coupled substructure direct comprehensive modeling method according to claim 4, characterized in that: In the step 2.1, a structural dynamics model of the tail beam is established based on the Euler-Bernoulli beam theory.

6. The method for direct comprehensive modeling of substructures of helicopter tail rotor / tail boom coupling according to claim 1, characterized in that: The specific process of step 3 is: Step 3.1: Set the displacement boundary conditions of the blade and tail beam to q O2 =Lq O1 ,q O2 is the displacement of the blade connection, q O1 is the displacement of the tail beam connection, q I2 is the internal displacement of the blade; Define the structural dynamic equation of the blade established in step 1 as Where q2, and are the displacement, velocity and acceleration of a single blade respectively; The structural dynamics equation of the blade established in step 1 is processed in blocks, and the structural dynamics equation of the blade after block processing is obtained as follows: Among them, the parameters with subscript II2 represent the stiffness matrix, mass matrix and damping matrix corresponding to the inside of the blade, the parameters with subscripts IO2 and OI2 represent the coupled stiffness matrix, coupled mass matrix and coupled damping matrix inside the blade and at the blade connection, and the parameters with subscript OO2 represent the stiffness matrix, mass matrix and damping matrix corresponding to the blade connection; F I2 is the generalized force along the internal displacement of the blade, F O2 is the generalized force along the displacement of the blade connection; Step 3.2: Convert the displacement, velocity and acceleration of the blade to the fixed coordinate system and obtain: Among them, T is the transformation matrix, obtained from the matrix L, Where θ is the rotation angle of the blade; Step 3.3: According to step 3.1 and step 3.2, the structural dynamic equation of the converted blade is: M2′=T T M2T F2′=T T F2 Where: M2′ is the converted blade stiffness matrix, C2′ is the converted blade damping matrix, K2′ is the converted blade stiffness matrix, and F2′ is the converted blade generalized force vector; Step 3.4, the structural dynamics equation of the converted blade obtained in step 3.3 is processed in blocks to obtain: Step 3.5, follow the process of steps 3.1-3.4 to obtain the structural dynamic equations expressed in blocks after all blades are transformed.

Images