Underwater rotating machinery multi-component vibration signal decoupling method based on order analysis

Abstract

The invention discloses an underwater rotating machinery multi-component vibration signal decoupling method based on order analysis. The method comprises the following steps: firstly, acquiring a discrete time domain vibration signal of the underwater rotating machine, wherein the discrete time domain vibration signal is formed by superposing multiple harmonic components; converting the discrete time domain vibration signal into an angle domain signal, executing discrete Fourier transform, and extracting an amplitude spectrum of each order component; on the basis, a multi-peak detection algorithm is adopted to carry out spectrum peak identification on the order amplitude spectrum, and a main order component order set is screened out; and finally, extracting a time domain waveform corresponding to the main order component through an order decoupling method. According to the method, the multi-component vibration signals in the underwater rotating machinery can be effectively decoupled, and a quantitative basis is provided for excitation source positioning, excitation source characteristic analysis or fault diagnosis.

Description

Technical Field

[0001] The present invention relates to a vibration signal decoupling method, and relates to the technical field of underwater rotating machinery vibration signal analysis, and in particular to an underwater rotating machinery multi-component vibration signal decoupling method based on order analysis. Background Art

[0002] In the field of marine equipment, the vibration signals of underwater rotating machinery (such as propellers, drive motors, pumps, etc.) contain key information that characterizes the operating status of the equipment. Due to the common variable speed and variable load operating conditions of underwater machinery, its vibration signals exhibit significant non-stationary and multi-component coupling characteristics. Existing vibration signal analysis methods (such as wavelet transform and empirical mode decomposition) face the following technical bottlenecks in underwater equipment applications: 1) When the machinery is in a time-varying speed condition, the non-stationary characteristics of the signal will cause the spectral energy to diffuse, resulting in a band broadening effect. Vibration signal analysis technology will produce spectral blurring, resulting in a significant reduction in the frequency resolution of the harmonic components, making it difficult to effectively track speed-related features; 2) The underwater environment has stronger environmental noise. Its unique broadband noise couples with the harmonic vibrations of the mechanical vibration signal, making it difficult for conventional signal analysis methods to effectively separate these signals, resulting in a decrease in the signal-to-noise ratio during the vibration signal feature extraction process; 3) Existing adaptive signal decomposition methods (such as empirical mode decomposition and variational mode decomposition) have poor adaptability when processing multi-component signals, making it difficult to accurately extract the order components directly related to the rotational excitation. In addition, the lack of prior knowledge of the characteristics of the rotating machinery vibration signal leads to unsatisfactory component separation. Especially when the signal contains multiple components of similar orders or instantaneous frequency mutations, traditional methods often find it difficult to achieve component separation with clear physical meaning.

[0003] Therefore, there is an urgent need for an analysis method that can adaptively track time-varying speed and accurately decouple multi-order components to address the analysis limitations of traditional methods under underwater variable speed and variable load conditions. Summary of the Invention

[0004] To address the problems in the background art, the present invention provides a method for decoupling multi-component vibration signals of underwater rotating machinery based on order analysis. This method can accurately extract specific order components, providing a quantitative basis for locating the excitation source, analyzing the excitation source characteristics, or performing fault diagnosis.

[0005] The technical solution adopted in the present invention is:

[0006] The method for decoupling multi-component vibration signals of underwater rotating machinery based on order analysis of the present invention comprises:

[0007] Step a) First, a multi-component vibration signal is characterized to obtain a discrete time-domain vibration signal of the underwater rotating machinery that is superimposed by multiple harmonic components.

[0008] Step b) Based on the real-time speed signal of the underwater rotating machinery, the discrete time-domain vibration signal is subjected to angle-domain Fourier transform to obtain the amplitude spectrum of each order harmonic component.

[0009] Step c) using a multi-peak detection algorithm to identify the main-order components on the amplitude spectrum of each order harmonic component, and then screening out the main-order harmonic component order set.

[0010] Step d) Based on the real-time speed signal of the underwater rotating machinery and the order set of the main-order harmonic components, the order component decoupling method is used to adaptively track and decouple each main-order harmonic component, and finally the time domain waveform of each main-order harmonic component is extracted to identify the excitation source of each main-order harmonic component, thereby realizing the decoupling of the multi-component vibration signal of the underwater rotating machinery.

[0011] In the step b), an interpolation algorithm is first used to convert the discrete time-domain vibration signal into an angle-domain multi-component vibration signal, where the angle-domain multi-component vibration signal includes a cumulative angle obtained based on the real-time speed signal of the underwater rotating machinery; then a discrete Fourier transform is performed to extract the complex amplitude of each order harmonic component, thereby obtaining an amplitude spectrum.

[0012] Described step c) is specifically as follows:

[0013] Step c1) For each order harmonic component, first-order difference processing is performed on the amplitude spectrum of the current order harmonic component to obtain the amplitude change rate of adjacent frequency points of the current order harmonic component.

[0014] Step c2) performs local maximum discrimination to construct a candidate peak set C.

[0015] Step c3) Based on a preset vibration amplitude threshold value A, a main-order harmonic component order set P is screened out from the candidate peak set C to eliminate pseudo peaks caused by harmonic side lobes.

[0016] In step c2), the candidate peak set C is specifically as follows:

[0017] C={k|Δ|Y[k]|<-ε∩Δ|Y[k-1]|>ε}

[0018] Δ|Y[k]|=|Y[k]|-|Y[k-1]|,k≥2

[0019] Among them, Δ|Y[k]| and Δ|Y[k-1]| are the amplitude change rates of adjacent frequency points of the harmonic components of the k-order and k-1-order vibration signals, respectively, |Y[k]| and |Y[k-1]| are the amplitudes of the k-order and k-1-order harmonic components, respectively, and Y[k] and Y[k-1] are the complex amplitudes of the k-order and k-1-order harmonic components, respectively; ε is the preset noise suppression threshold.

[0020] In step c3), the main-order harmonic component order set P is specifically as follows:

[0021] P={k∈C||Y[k]|≥A}

[0022] Where |Y[k]| is the amplitude of the k-order harmonic component, and Y[k] is the complex amplitude of the k-order harmonic component.

[0023] Described step d) is specifically as follows:

[0024] Step d1) obtaining the order of each main-order harmonic component according to the main-order harmonic component order set P, and then obtaining the frequency trajectory of each main-order harmonic component.

[0025] Step d2) Based on the frequency trajectory of each main-order harmonic component, an order tracking method is used to track each main-order harmonic component, thereby constructing a minimization target model of each main-order harmonic component, decoupling each main-order harmonic component, and extracting the time domain waveform of each main-order harmonic component to achieve adaptive tracking and decoupling of each main-order harmonic component.

[0026] In the step d1), for each main-order harmonic component, the frequency trajectory F of the main-order harmonic component r [n]Specific details are as follows:

[0027]

[0028] Among them, pr is the order of the current main-order harmonic component; R[n] is the real-time speed of the underwater rotating machinery under the n-th sampling.

[0029] In step d2), the minimization target model of each main-order harmonic component is specifically as follows:

[0030]

[0031] Tracking the main order components using the order tracking method can be expressed as:

[0032] Δ 2 x pr [n]=ε[n]

[0033]

[0034] Where y[n] is the time domain waveform of each main order harmonic component; Δ 2 is the second-order difference operator; x pr [n] is the pr main-order harmonic component at the n-th sampling; y[n] is the observation signal at the n-th sampling; ε[n] is the process noise at the n-th sampling, which controls the smoothness of the vibration signal after decoupling; η[n] is the measurement noise at the n-th sampling, which is used to control the fitting error between the decoupled signal and the original observation signal.

[0035] By optimizing and minimizing the target model, the main order component x p The smoothness and fitting accuracy of [n] are balanced, and adaptive tracking and decoupling of the main order components are achieved.

[0036] The electronic device of the present invention comprises: a memory and a processor coupled to each other, wherein the memory stores program data, and the processor calls the program data to execute the method described above.

[0037] The computer-readable storage medium of the present invention stores program data thereon, and when the program data is executed by a processor, the method described above is implemented.

[0038] The method of the present invention has the following innovations: first, the order-based signal analysis method can enhance the physical interpretability of the signal and convert the non-stationary signal with time-varying frequency into a stationary signal with stable order; second, by focusing on specific order components, background noise unrelated to the rotational speed is filtered out, thereby improving the feature extraction accuracy and robustness; third, under time-varying conditions, it can accurately decouple homologous vibration components and separate the independent vibration signal components of underwater rotating machinery.

[0039] The beneficial effects of the present invention are:

[0040] The method of the present invention models the time-domain vibration signal as a superposition of multiple harmonic components, and combines the direct relationship between time-varying frequency and the rotational speed of the rotating machinery to accurately characterize the multi-component vibration characteristics under non-steady-state speed. The method overcomes the limitations of traditional Fourier transform under variable speed conditions and improves the accuracy of frequency analysis. At the same time, it maintains stable tracking of the target frequency under complex working conditions, solving the decoupling failure problem caused by time-varying frequency in traditional vibration signal analysis methods. The present invention can effectively decouple multi-component vibration signals in underwater rotating machinery, providing a quantitative basis for locating the excitation source, analyzing the excitation source characteristics, or performing fault diagnosis. BRIEF DESCRIPTION OF THE DRAWINGS

[0041] Figure 1 is a flow chart of the method of the present invention;

[0042] Figure 2 This is a waveform diagram of the amplitude of the angle domain Fourier transform according to an embodiment of the present invention;

[0043] Figure 3 This is a frequency trajectory diagram corresponding to the main-order harmonic component of an embodiment of the present invention;

[0044] Figure 4 4 is a time domain waveform diagram corresponding to the main-order harmonic component of an embodiment of the present invention. DETAILED DESCRIPTION

[0045] In order to enable those skilled in the art to better understand the solutions of the present invention, the technical solutions in the embodiments of the present invention will be clearly and completely described below in conjunction with the accompanying drawings. However, this embodiment and the accompanying drawings do not limit the scope of protection of the present invention. Obviously, the described embodiment is only one embodiment of the implementation of the present invention. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without making creative efforts should fall within the scope of protection of the present invention.

[0046] like Figure 1 As shown, the multi-component vibration signal decoupling method of underwater rotating machinery based on order analysis of the present invention is as follows:

[0047] Step a) First, characterize the multi-component vibration signal to obtain the discrete time domain vibration signal of the underwater rotating machinery composed of multiple harmonic components. In the specific implementation, the sensor arrangement is as follows: a piezoelectric acceleration sensor (sensitivity: 10mV / m·s) is installed at the radial position of the deep-sea motor housing of the underwater electro-hydraulic actuator. 2 The signal acquisition system (with a range of ±50g, a frequency response of 0.5–7000Hz, and a resolution of 0.0005g) is used to collect vibration signals from the motor surface. The signal acquisition system uses a 24-bit high-precision data acquisition card, continuously sampling the motor vibration signals at a sampling frequency of 10kHz. During the experiment, the motor speed was linearly increased from 1000RPM to 2000RPM using a variable frequency control system, with an acceleration rate of 50RPM / s. Deep-sea motor noise is primarily caused by the electromagnetic-mechanical coupling between the stator and rotor. When the stator winding magnetic field interacts with the magnetic field of the rotor permanent magnet, a specific spatial and temporal distribution of electromagnetic fields is generated in the air gap. This causes the stator teeth to deform under the action of alternating magnetic pull, resulting in periodic vibration. When the motor structure is fixed, the vibration characteristics are dominated by speed-dependent harmonic components. However, when the motor is accelerating, the rotor angle θ varies with time, and the vibration signal exhibits nonstationary characteristics. Its discrete form has the same structure as the multi-component vibration signal. Therefore, the multi-component signal analysis method can effectively extract the various frequency components in the motor vibration signal and further analyze the changing trend of each harmonic component.

[0048] Each harmonic component consists of a time-varying amplitude, a time-varying frequency related to the real-time speed of the underwater rotating machinery, and an initial phase. The time-varying frequency is associated with the rotational frequency through the order fundamental wave. The vibration waveform of the underwater rotating machinery is closely related to the harmonic components generated by its periodic rotation. Its discrete time-domain vibration signal x[n] can be expressed as the superposition of multiple harmonic components, namely:

[0049]

[0050] x k [n]=A k [n]cos(2πf k [n]·n+φ k )

[0051] Among them, x k [n] is the k-order harmonic component under the n-th sampling; A k [n] is the time-varying amplitude of the k-th order harmonic component under the n-th sampling; f k [n] is the time-varying frequency of the k-th order harmonic component under the n-th sampling related to the speed, φ k is the initial phase of the k-order harmonic component.

[0052] The time-varying frequency f at the nth sampling k [n] is further defined based on the relationship between harmonics and speed:

[0053]

[0054] Among them, R[n] is the real-time speed of the underwater rotating machinery under the nth sampling, O R It is the order fundamental wave with the rotation frequency as the fundamental frequency.

[0055] Step b) Based on the real-time speed signal of the underwater rotating machinery, the discrete time domain vibration signal is subjected to an angle domain Fourier transform to obtain the amplitude spectrum of each order harmonic component. Specifically, the discrete time domain vibration signal is first converted into an angle domain multi-component vibration signal using an interpolation algorithm, and the angle domain multi-component vibration signal contains the cumulative angle obtained based on the real-time speed signal of the underwater rotating machinery; then, a discrete Fourier transform is performed to extract the complex amplitude of each order harmonic component, thereby obtaining the amplitude spectrum.

[0056] The angle domain Fourier transform is as follows:

[0057] First, the angle domain signal is acquired: the time domain vibration signal is converted into the angle domain multi-component vibration signal y[n] under the n-th sampling through the interpolation algorithm, as follows:

[0058]

[0059] Wherein, N is the total number of sampling points of the vibration signal; h(·) is the interpolation kernel function, such as linear interpolation, spline interpolation, etc. In this embodiment, the linear interpolation method is used for interpolation processing; θ[n] is the cumulative angle of the underwater rotating machinery under the n-th sampling; Δθ is the angular resolution. In this embodiment, the angular resolution Δθ = 0.002496.

[0060] Then, perform order domain spectrum analysis: perform discrete Fourier transform on the angle domain multi-component vibration signal y[n] to extract the order components, as follows:

[0061]

[0062] Where Y[k] is the complex amplitude of the k-th order harmonic component, and its absolute value is the amplitude |Y[k]|, which represents the intensity of the order component; j is an imaginary unit.

[0063] like Figure 2 The following figure shows the angular domain Fourier transform amplitude waveform of an embodiment of the present invention, clearly demonstrating the distribution characteristics of each order component. In particular, the amplitudes at orders 23, 25, 31, and 38 show significant peaks, indicating that the primary components of the vibration signal are concentrated at these specific orders. By analyzing the order amplitudes of multi-component vibration signals, the primary frequency components of the signal can be effectively extracted.

[0064] Step c) uses a multi-peak detection algorithm to identify the main order components of the amplitude spectrum of each order harmonic component, and then screens out the main order harmonic component order set, as follows:

[0065] Step c1) For each order harmonic component, first-order difference processing is performed on the amplitude spectrum of the current order harmonic component to obtain the amplitude change rate of adjacent frequency points of the current order harmonic component.

[0066] Step c2) performs local maximum discrimination to construct a candidate peak set C, as follows:

[0067] C={k|Δ|Y[k]|<-ε∩Δ|Y[k-1]|>ε}

[0068] Δ|Y[k]|=|Y[k]|-|Y[k-1]|,k≥2

[0069] Among them, Δ|Y[k]| and Δ|Y[k-1]| are the amplitude change rates of adjacent frequency points of the harmonic components of the k-order and k-1-order vibration signals, respectively, |Y[k]| and |Y[k-1]| are the amplitudes of the k-order and k-1-order harmonic components, respectively, and Y[k] and Y[k-1] are the complex amplitudes of the k-order and k-1-order harmonic components, respectively; ε is the preset noise suppression threshold. In this embodiment, the noise suppression threshold ε=0.1.

[0070] Step c3) Based on the preset vibration amplitude threshold A, the order set P of each main-order harmonic component is screened from the candidate peak set C to eliminate the pseudo peaks caused by the harmonic sidelobes, as follows:

[0071] P={k∈C||Y[k]|≥A}

[0072] Wherein, |Y[k]| is the amplitude of the k-order harmonic component, and Y[k] is the complex amplitude of the k-order harmonic component. In this embodiment, the preset threshold value A=3m / s 2 .

[0073] Finally, after multi-peak detection, 23.2, 24.8, 31, and 38.2 were selected as the main order components.

[0074] Step d) Based on the real-time speed signal of the underwater rotating machinery and the set of main-order harmonic component orders, an order component decoupling method is used to adaptively track and decouple each main-order harmonic component. The time domain waveform of each main-order harmonic component is ultimately extracted to identify the excitation source of each main-order harmonic component, thereby achieving multi-component vibration signal decoupling of the underwater rotating machinery. The details are as follows:

[0075] Step d1) Obtain the order of each main-order harmonic component according to the main-order harmonic component order set P, and then obtain the frequency trajectory of each main-order harmonic component. For each main-order harmonic component, the frequency trajectory F of the main-order harmonic component is r [n]Specific details are as follows:

[0076]

[0077] Wherein, pr is the order of the current main-order harmonic component; R[n] is the real-time rotation speed of the underwater rotating machinery under the n-th sampling; in this embodiment, P={23.2, 24.8, 31, 38.2}.

[0078] Step d2) Based on the frequency trajectory of each main-order harmonic component, an order tracking method is used to track each main-order harmonic component, thereby constructing a minimization target model for each main-order harmonic component, decoupling each main-order harmonic component and extracting the time domain waveform of each main-order harmonic component, achieving adaptive tracking and decoupling of each main-order harmonic component, and using a Vold-Kalman filter to track and decompose the main-order components during order decoupling. The minimization target model for each main-order harmonic component is as follows:

[0079]

[0080] Tracking the main order components using the order tracking method can be expressed as:

[0081] Δ 2 x pr[n]=ε[n]

[0082]

[0083] Where J[n] is the time domain waveform of each main order harmonic component; Δ 2 is the second-order difference operator; x pr [n] is the pr main-order harmonic component at the n-th sampling; y[n] is the observation signal at the n-th sampling; ε[n] is the process noise at the n-th sampling, which controls the smoothness of the vibration signal after decoupling; η[n] is the measurement noise at the n-th sampling, which is used to control the fitting error between the decoupled signal and the original observation signal.

[0084] By optimizing and minimizing the target model, the main order component x p The smoothness and fitting accuracy of [n] are balanced, and adaptive tracking and decoupling of the main order components are achieved.

[0085] like Figure 3 As shown, it is the frequency trajectory corresponding to the main order component of the embodiment of the present invention, showing the evolution law of the four characteristic orders during the speed change process.

[0086] like Figure 4 As shown, the time domain waveforms corresponding to the main order components of the embodiment of the present invention are the time domain waveforms of 23.2, 31, 24.8 and 38.2 order components respectively. By adaptively decoupling signals of different orders, the time domain waveform of each order can be clearly seen, which is consistent with the actual physical characteristics.

[0087] This example verifies the effectiveness of the present invention in decoupling multi-component vibration signals under variable underwater working conditions, as shown in the following:

[0088] 1) The order amplitude spectrum extracted by angle domain Fourier transform can effectively identify the order components related to the rotational speed.

[0089] 2) The adaptive tracking capability of the method of the present invention for target order components under non-stationary signal conditions was verified, solving the decoupling failure problem of traditional methods caused by frequency mutation.

[0090] 3) By decoupling the main-order components, the separated time domain waveforms show clear independence, achieving accurate separation of multi-component vibration signals.

[0091] In summary, the method of the present invention significantly improves the accuracy and reliability of decoupling vibration signals of underwater rotating machinery, and provides strong technical support for equipment vibration analysis under variable speed and variable load conditions.

[0092] The specific embodiments described above further illustrate the objectives, technical solutions and beneficial effects of the present invention in detail. It should be understood that the above are only specific embodiments of the present invention and are not intended to limit the present invention. Any modifications, equivalent substitutions, improvements, etc. made within the spirit and principles of the present invention should be included in the scope of protection of the present invention.

Claims

1. A method for decoupling multi-component vibration signals of underwater rotating machinery based on order analysis, characterized in that: include: Step a) obtaining a discrete time-domain vibration signal of an underwater rotating machinery composed of a superposition of multiple harmonic components; Step b) performing an angle-domain Fourier transform on the discrete time-domain vibration signal based on the real-time speed signal of the underwater rotating machinery to obtain the amplitude spectrum of each order harmonic component; Step c) using a multi-peak detection algorithm to identify the main order components of the amplitude spectrum of each order harmonic component, and then screening out the main order harmonic component order set; Step d) Based on the real-time speed signal of the underwater rotating machinery and the order set of the main-order harmonic components, the order component decoupling method is used to adaptively track and decouple each main-order harmonic component, and finally the time domain waveform of each main-order harmonic component is extracted to achieve multi-component vibration signal decoupling of the underwater rotating machinery.

2. The method for decoupling multi-component vibration signals of underwater rotating machinery based on order analysis according to claim 1, characterized in that: In the step b), an interpolation algorithm is first used to convert the discrete time-domain vibration signal into an angle-domain multi-component vibration signal, where the angle-domain multi-component vibration signal includes a cumulative angle obtained based on the real-time speed signal of the underwater rotating machinery; then a discrete Fourier transform is performed to extract the complex amplitude of each order harmonic component, thereby obtaining an amplitude spectrum.

3. The method for decoupling multi-component vibration signals of underwater rotating machinery based on order analysis according to claim 1, characterized in that: Described step c) is specifically as follows: Step c1) for each order harmonic component, performing first-order difference processing on the amplitude spectrum of the current order harmonic component to obtain the amplitude change rate of adjacent frequency points of the current order harmonic component; Step c2) performing local maximum discrimination to construct a candidate peak set C; Step c3) Based on a preset vibration amplitude threshold A, a set P of main-order harmonic component orders is screened out from the candidate peak set C.

4. The method for decoupling multi-component vibration signals of underwater rotating machinery based on order analysis according to claim 3 is characterized in that: In step c2), the candidate peak set C is specifically as follows: C={k|Δ|Y[k]|<-ε∩ΔY[k-1]|>ε} Δ|Y[k]|=|Y[k]|-|Y[k-1]|,k≥2 Among them, Δ|Y[k]| and Δ|Y[k-1]| are the amplitude change rates of adjacent frequency points of the harmonic components of the k-order and k-1-order vibration signals, respectively, |Y[k]| and |Y[k-1]| are the amplitudes of the k-order and k-1-order harmonic components, respectively, and Y[k] and Y[k-1] are the complex amplitudes of the k-order and k-1-order harmonic components, respectively; ε is the preset noise suppression threshold.

5. The method for decoupling multi-component vibration signals of underwater rotating machinery based on order analysis according to claim 3, characterized in that: In step c3), the main-order harmonic component order set P is specifically as follows: P={k∈C||Y[k]|≥A} Where |Y[k]| is the amplitude of the k-order harmonic component, and Y[k] is the complex amplitude of the k-order harmonic component.

6. The method for decoupling multi-component vibration signals of underwater rotating machinery based on order analysis according to claim 1, characterized in that: Described step d) is specifically as follows: Step d1) obtaining the order of each main-order harmonic component according to the main-order harmonic component order set P, and then obtaining the frequency trajectory of each main-order harmonic component; Step d2) Based on the frequency trajectory of each main-order harmonic component, an order tracking method is used to track each main-order harmonic component, thereby constructing a minimization target model of each main-order harmonic component, decoupling each main-order harmonic component, and extracting the time domain waveform of each main-order harmonic component to achieve adaptive tracking and decoupling of each main-order harmonic component.

7. The method for decoupling multi-component vibration signals of underwater rotating machinery based on order analysis according to claim 6, characterized in that: In the step d1), for each main-order harmonic component, the frequency trajectory F of the main-order harmonic component r [n]Specific details are as follows: Among them, pr is the order of the current main-order harmonic component; R[n] is the real-time speed of the underwater rotating machinery under the n-th sampling.

8. The method for decoupling multi-component vibration signals of underwater rotating machinery based on order analysis according to claim 7, characterized in that: In step d2), the minimization target model of each main-order harmonic component is specifically as follows: D 2 x pr [n]=ε[n] Where J[n] is the time domain waveform of each main order harmonic component; Δ 2 is the second-order difference operator; x pr [n] is the main-order harmonic component of pr at the nth sampling; y[n] is the observation signal at the nth sampling; ε[n] is the process noise at the nth sampling; η[n] is the measurement noise at the nth sampling.

9. An electronic device, characterized in that: include: A memory and a processor coupled to each other, wherein the memory stores program data, and the processor calls the program data to execute the method according to any one of claims 1 to 8.

10. A computer-readable storage medium having program data stored thereon, characterized in that: When the program data is executed by a processor, the method according to any one of claims 1 to 8 is implemented.

Images