CN102778317B - System and method for measuring pressure of shock wave in laser shock processing process - Google Patents

Abstract

The invention discloses a shock wave pressure measurement system in the laser shock peening process, comprising: a first resistor, a third resistor and a second resistor connected in series in sequence, and one end of a PVDF sensor is connected between the first resistor and the third resistor , the other end is connected between the first resistor and the second resistor; one end of the oscilloscope is connected between the third resistor and the second resistor, and the other end is connected between the first resistor and the second resistor. The invention also discloses a method for measuring shock wave pressure by using the measurement system. The invention reduces the voltage measured by the acquisition device. At the same time, since the measurement circuit adopts the resistance mode, it ensures that the equivalent resistance at both ends of the oscilloscope is less than 50 ohms, and solves the limitation when PVDF measures higher pressure.

Description

A system and method for measuring shock wave pressure during laser shock peening

technical field

The invention relates to a shock wave pressure measurement system and method in the process of laser shock strengthening.

Background technique

Laser shock peening (LSP) is a mechanical treatment method that effectively uses high-power-density lasers to modify the surface of metal materials. It uses the high-amplitude shock wave formed by the interaction between the short-pulse laser and the surface of the metal material to process and harden the area near the impact surface of the material to form residual compressive stress to improve the mechanical properties of the material such as fatigue life, wear resistance and corrosion resistance.

The laser-induced pressure feature is a key factor in the process of laser shock peening, which directly determines the effect of shock peening. It is generally believed that when the laser-induced peak pressure is 2-2.5 times the Hugoniot elastic limit (HEL) value of the material, a better strengthening effect can be obtained. When it exceeds 2.5 times the HEL, the maximum plastic strain on the surface of the impact area is reversed. decrease.

The pressure characteristics induced by laser shock peening have the characteristics of short transient and high amplitude. As shown in Figure 1, under the action of a short-pulse intense laser with a full width at half maximum (FWHM) on the order of ns and a peak power density on the order of GW/cm ² , the half width at half maximum of the induced plasma pressure characteristic The time is about 2 to 3 times the FWHM of the laser power density, and the peak pressure can reach several GPa.

In order to understand the pressure characteristics in the laser shock strengthening process, it is necessary to carry out experimental measurements on the laser-induced pressure characteristics, including peak pressure and half-peak width time, so as to obtain the influence of process parameters on the pressure characteristics, and to directly measure the effect of laser shock strengthening. Evaluate.

From the above description, it can be seen that the laser-induced pressure feature has the characteristics of short time, high peak pressure, fast pressure change and high frequency. When measuring the impact pressure with such characteristics, conventional measurement methods such as quartz crystal pressure gauges and manganin gauges are used. Due to the constraints of the measurement range, measurement accuracy and frequency response, it is difficult to measure accurately in the experiment.

In recent years, PVDF piezoelectric film sensing has been gradually applied to laser-induced pressure measurement. PVDF piezoelectric film sensor uses the piezoelectric properties of PVDF film to measure pressure. When the PVDF film is affected by the external pressure difference Δp, the positive and negative charges inside the PVDF film will separate and accumulate on the upper and lower surfaces respectively. By measuring the amount of charge, the relationship between the pressure Δp and the surface charge ΔQ can be established ΔQ=K·A·Δp. Among them, A is the effective area of pressure action, and K is the piezoelectric coefficient of PVDF film, indicating the amount of charge generated per unit area and unit pressure, and the unit is C/N. The charge generated by the PVDF sensor due to the pressure difference between the two surfaces needs to be collected by an external circuit, so that the shock wave characteristics can be obtained by using the piezoelectric characteristics.

The external measurement circuit of PVDF is divided into current mode and charge mode (or voltage mode). In the current mode, a 50Ω resistor is connected in parallel with PVDF to discharge, and the oscilloscope collects the voltage across the resistor to indirectly obtain the discharge charge. The advantage of this circuit is that the frequency response is high, but when measuring high voltage, the measured voltage value is easy to exceed the range. The charge mode directly uses the charge integrator to measure the charge, but for high-frequency response pressure measurement, due to the limitation of the bandwidth of the charge integrator, it cannot accurately reflect the time distribution characteristics of the pressure. The pressure characteristic amplitude induced by laser shock peening is high and the frequency response is fast.

Therefore, the pressure signature is measured in current mode. The pressure impacting the position on the back surface of the target is usually measured in amperometric mode. Since the shock wave needs to propagate to the back surface through the target body, its pressure amplitude and half-peak width time will change, so it is necessary to establish a direct measurement method for the shock surface pressure. In addition, due to the high amplitude of the laser-induced pressure, when the current measurement mode is used to measure the laser-induced pressure, there will be a problem of measurement limitation.

Contents of the invention

In view of the problems existing in the prior art, the purpose of the present invention is to provide a shock wave pressure measurement system and method in the process of laser shock peening, which can solve the problem of PVDF limiting the higher pressure measurement while ensuring the pressure measurement accuracy.

A shock wave pressure measurement system in the laser shock peening process of the present invention includes: a first resistor, a third resistor and a second resistor connected in series in sequence, one end of the PVDF sensor is connected between the first resistor and the third resistor, and the other One end is connected between the first resistance and the second resistance; one end of the oscilloscope is connected between the third resistance and the second resistance, and the other end is connected between the first resistance and the second resistance; wherein, The first resistor, the second resistor and the third resistor meet the following conditions:

${\mathrm{<span\; class="notranslate">\; R</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; /</span><span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; R</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; R</span>}}_{\mathrm{<span\; class="notranslate">\; 3</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; <</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; ,</span>\frac{{\mathrm{<span\; class="notranslate">\; R</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; \&times;</span><span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; R</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; R</span>}}_{\mathrm{<span\; class="notranslate">\; 3</span>}}<span\; class="notranslate">\; )</span>}{{\mathrm{<span\; class="notranslate">\; R</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; R</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; R</span>}}_{\mathrm{<span\; class="notranslate">\; 3</span>}}}<span\; class="notranslate">\; <</span>\mathrm{<span\; class="notranslate">\; 50</span>}\mathrm{<span\; class="notranslate">\; \&Omega;</span>}<span\; class="notranslate">\; .</span>$

A method for measuring shock wave pressure in a laser shock peening process of the present invention comprises the following steps:

1) Place the PVDF sensors of the shock wave pressure measurement system in the laser shock peening process on the bottom of the aluminum film with thickness h1 and h2 respectively, and then irradiate the aluminum film with laser to generate plasma pressure;

2) After measuring the aluminum film with a thickness of h ₁ and h ₂ , the peak pressure measured when the thickness is h ₁ is σ ₁ , and the half-peak width time is τ ₁ ; the peak pressure measured when the thickness is h ₂ is σ ₂ , and the half-peak width time is τ ₂ ; then the laser-induced pressure peak σ _m and half-peak width time τ _m are:

${\mathrm{<span\; class="notranslate">\; \&sigma;</span>}}_{\mathrm{<span\; class="notranslate">\; m</span>}}<span\; class="notranslate">\; =</span>\frac{{\mathrm{<span\; class="notranslate">\; h</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; \&sigma;</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; *</span>{\mathrm{<span\; class="notranslate">\; h</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}{\mathrm{<span\; class="notranslate">\; \&sigma;</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}}{{\mathrm{<span\; class="notranslate">\; h</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; h</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}}$

${\mathrm{<span\; class="notranslate">\; \&tau;</span>}}_{\mathrm{<span\; class="notranslate">\; m</span>}}<span\; class="notranslate">\; =</span>\frac{{\mathrm{<span\; class="notranslate">\; h</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; \&tau;</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; h</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}{\mathrm{<span\; class="notranslate">\; \&tau;</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}}{{\mathrm{<span\; class="notranslate">\; h</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; h</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}}$

Thus, the laser-induced pressure signature is obtained;

Wherein, the shock wave pressure measurement system in the laser shock peening process includes: a first resistor, a third resistor and a second resistor connected in series in sequence, one end of the PVDF sensor is connected between the first resistor and the third resistor, and the other end connected between the first resistor and the second resistor; one end of the oscilloscope is connected between the third resistor and the second resistor, and the other end is connected between the first resistor and the second resistor; wherein, the The first resistor, the second resistor and the third resistor meet the following conditions:

${\mathrm{<span\; class="notranslate">\; R</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; /</span><span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; R</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; R</span>}}_{\mathrm{<span\; class="notranslate">\; 3</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; <</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; ,</span>\frac{{\mathrm{<span\; class="notranslate">\; R</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; \&times;</span><span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; R</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; R</span>}}_{\mathrm{<span\; class="notranslate">\; 3</span>}}<span\; class="notranslate">\; )</span>}{{\mathrm{<span\; class="notranslate">\; R</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; R</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; R</span>}}_{\mathrm{<span\; class="notranslate">\; 3</span>}}}<span\; class="notranslate">\; <</span>\mathrm{<span\; class="notranslate">\; 50</span>}\mathrm{<span\; class="notranslate">\; \&Omega;</span>}<span\; class="notranslate">\; .</span>$

The invention builds an equivalent measurement circuit between the PVDF sensor and the acquisition device (ie, oscilloscope). For the PVDF circuit, the second resistor and the third resistor are connected in parallel with the first resistor after being connected in series; for the acquisition device, the first resistor and the third resistor are connected in parallel with the second resistor after being connected in series. In this way, the current formed by the PVDF two-pole discharge is shunted through the second resistor and the third resistor, reducing the voltage measured by the acquisition device. At the same time, since the measurement circuit adopts the resistance mode, it ensures that the equivalent resistance at both ends of the oscilloscope is less than 50 ohms, and solves the limitation when PVDF measures higher pressure. In the experiment, the first, second and third resistors are 50 ohms, 50 ohms and 500 ohms respectively. The peak voltage measured by this method was 3.5 volts. However, if the traditional measurement method is adopted, the measured voltage peak value is 38.5 volts, far exceeding the measurement range of the oscilloscope.

Description of drawings

Figure 1 is a characteristic curve of pressure induced by a short-pulse intense laser;

Fig. 2 is PVDF measuring system of the present invention;

Fig. 3 is a curve diagram of time distribution of laser power density according to an embodiment of the present invention;

Fig. 4 is when the aluminum film thickness of the embodiment of the present invention is respectively 60 μ m and 100 μ m, the circuit voltage signal graph obtained by PVDF measurement;

Fig. 5 is a pressure-time waveform graph obtained by PVDF measurement when the thickness of the aluminum film in the embodiment of the present invention is 60 μm and 100 μm respectively.

Detailed ways

As shown in Figure 2, the measuring system of the present invention comprises: the first resistor R1, the third resistor R3 and the second resistor R2 connected in series in sequence, one end of the PVDF sensor is connected between the first resistor R1 and the third resistor R3, and the other One end is connected between the first resistor R1 and the second resistor R2; one end of the acquisition device (that is, the oscilloscope) is connected between the third resistor R3 and the second resistor R2, and the other end is connected between the first resistor R1 and the second resistor R2 between. Wherein, the first resistor R1, the second resistor R2 and the third resistor R3 satisfy the following conditions:

${\mathrm{<span\; class="notranslate">\; R</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; /</span><span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; R</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; R</span>}}_{\mathrm{<span\; class="notranslate">\; 3</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; <</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; ,</span>\frac{{\mathrm{<span\; class="notranslate">\; R</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; \&times;</span><span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; R</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; R</span>}}_{\mathrm{<span\; class="notranslate">\; 3</span>}}<span\; class="notranslate">\; )</span>}{{\mathrm{<span\; class="notranslate">\; R</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; R</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; R</span>}}_{\mathrm{<span\; class="notranslate">\; 3</span>}}}<span\; class="notranslate">\; <</span>\mathrm{<span\; class="notranslate">\; 50</span>}\mathrm{<span\; class="notranslate">\; \&Omega;</span>}<span\; class="notranslate">\; .</span>$

The invention builds an equivalent measurement circuit between the PVDF sensor and the acquisition device (ie, oscilloscope). For the PVDF circuit, the second resistor R2 and the third resistor R3 are connected in parallel with the first resistor R1 after being connected in series; for the collection device, the first resistor R1 and the third resistor R3 are connected in series and then connected in parallel with the second resistor R2.

In this way, the current formed by the discharge of the two poles of the PVDF is shunted through the second resistor R2 and the third resistor R3, reducing the voltage measured by the acquisition device. Suppose the voltage measured by the oscilloscope is U _m , and the voltage across the PVDF is U. but

U _m /U＝R ₂ /(R ₂ +R ₃ ),

Um can be reduced by setting a larger third resistor R3 to ensure that the voltage value measured by the oscilloscope is not limited, thereby obtaining a complete voltage waveform. In addition, the equivalent measurement circuit is a bridge mode with high frequency response, which ensures the accuracy of measurement, and the accuracy of measurement can reach the traditional current circuit measurement mode.

In the specific measurement process, the above-mentioned PVDF measurement system is used for direct measurement. The PVDF sensors are respectively placed on the bottom of the aluminum film with thickness h1 and h2, and then the aluminum film is irradiated by laser to generate plasma pressure. Laser-induced pressure signatures were obtained by measuring two aluminum films of different thicknesses. The thickness of the aluminum film should be small enough, and the thicknesses of the aluminum films in the two experiments are set to be h1 and h2 respectively. When the thickness is h1, the measured peak pressure is σ ₁ , and the half-width time is τ ₁ ; when the thickness is h2, the measured peak pressure is σ ₂ , and the half-width time is τ ₂ . By linear interpolation, the laser-induced pressure peak σ _m and half-maximum width time τ _m are

${\mathrm{<span\; class="notranslate">\; \&sigma;</span>}}_{\mathrm{<span\; class="notranslate">\; m</span>}}<span\; class="notranslate">\; =</span>\frac{{\mathrm{<span\; class="notranslate">\; h</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; \&sigma;</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; *</span>{\mathrm{<span\; class="notranslate">\; h</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}{\mathrm{<span\; class="notranslate">\; \&sigma;</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}}{{\mathrm{<span\; class="notranslate">\; h</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; h</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}}$

${\mathrm{<span\; class="notranslate">\; \&tau;</span>}}_{\mathrm{<span\; class="notranslate">\; m</span>}}<span\; class="notranslate">\; =</span>\frac{{\mathrm{<span\; class="notranslate">\; h</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; \&tau;</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; h</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}{\mathrm{<span\; class="notranslate">\; \&tau;</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}}{{\mathrm{<span\; class="notranslate">\; h</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; h</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}}$

By this method, the laser-induced pressure characteristics can be obtained directly from the two measurements, which provides an experimental method for pressure measurement in the LSP process.

The plasma pressure under laser action was measured when the thickness was 60μm and 100μm respectively. The laser time waveform in the experiment is shown in Figure 3. The time and space distribution of laser power density is approximately Gaussian distribution, and its half-peak width is about 12.1 ns. The voltage signal output by PVDF was collected 4.0 ns after the laser arrived.

Figure 4 shows the voltage signals measured by an oscilloscope when the thickness of the aluminum film is 60 μm and 100 μm respectively. When the pressure is initially applied, the voltage rises rapidly, and the corresponding pressure rises rapidly at this time. After 80.0ns, the voltage becomes negative, corresponding to the unloading stage, and the pressure begins to drop. Figure 5 shows the corresponding pressure characteristics when the thickness of the aluminum film is 60 μm and 100 μm, respectively. When the thickness of the aluminum film is 60μm, the pressure starts to rise at 5.6ns, and the initial time difference Δt ₁ =9.6ns from the laser. The peak pressure σm ₁ =1.32GPa was reached at 52.9ns, and then dropped to zero around 120ns. The half-width of the pressure τ ₁ is 64.8ns, which is 5.35 times of the half-width of the laser. When the thickness of the aluminum film is 100 μm, the pressure starts to rise at 11.9 ns, and the initial time difference from the laser is Δt ₂ =15.9 ns. The pressure peak σm ₂ =1.02GPa was reached at 74.6ns, and then dropped to zero around 180ns. The half-width τ ₂ of the pressure is 92.9ns, which is 7.68 times of the laser half-width.

It is generally believed that the half-width time of the plasma pressure is approximately 2-3 times the half-width time of the laser power density. During the experimental measurement, the laser-induced plasma needs to propagate through the aluminum film, and the half-peak width of the pressure gradually increases due to the difference in elastic-plastic wave velocity. The thicker the aluminum film, the longer the duration of the pressure and the larger the half-peak width.

In addition, the measured pressure peak decays from 1.32GPa to 1.02GPa when the thickness of the aluminum film is 60μm and 100μm, respectively. The pressure measured by the above-mentioned PVDF piezoelectric film sensor is the average pressure within the action range of the light spot. For lasers whose spatial distribution of laser power density is Gaussian, the peak pressure can be approximately considered to be twice the average pressure. Therefore, for aluminum films with a thickness of 60 μm and 100 μm, the actual measured peak pressure can be approximately considered to be 2.64GPa and 2.04GPa. Assuming that the peak pressure and the pressure half-width time decay linearly within 100 μm, it can be determined that the peak plasma pressure on the impact surface is about 3.54 GPa, and the pressure half-peak time is 22.7 ns. In this way, the plasma pressure characteristics of the laser-acting surface are obtained.

Claims (2)

1. shock wave pressure measuring system in a laser impact intensified process, it is characterized in that, comprise: the first resistance, the 3rd resistance and second resistance of series connection successively, one end of PVDF sensor is connected between described the first resistance and the 3rd resistance, and the other end is connected between described the first resistance and the second resistance; Oscillographic one end is connected between described the 3rd resistance and the second resistance, and the other end is connected between described the first resistance and the second resistance; Wherein, described the first resistance, the second resistance and the 3rd resistor satisfied following condition:

2. the method that in laser impact intensified process, shock wave pressure is measured, is characterized in that, comprises the steps:

1) the PVDF sensor of shock wave pressure measuring system in laser impact intensified process being placed in respectively to thickness is h ₁and h ₂the bottom of aluminium film, then by laser, aluminium film is carried out to irradiation, produce plasma pressure;

2) process is h to thickness ₁and h ₂the measurement of aluminium film, obtaining thickness is h ₁time the surge pressure that measures be σ ₁, the half-peak breadth time is τ ₁; Thickness is h ₂time the surge pressure that measures be σ ₂, the half-peak breadth time is τ ₂; The pressure peak σ of induced with laser _mwith half-peak breadth time τ _mfor:

Thereby obtain the pressure characteristic of induced with laser;

Wherein, in described laser impact intensified process, shock wave pressure measuring system comprises: the first resistance, the 3rd resistance and second resistance of series connection successively, one end of PVDF sensor is connected between described the first resistance and the 3rd resistance, and the other end is connected between described the first resistance and the second resistance; Oscillographic one end is connected between described the 3rd resistance and the second resistance, and the other end is connected between described the first resistance and the second resistance; Wherein, described the first resistance, the second resistance and the 3rd resistor satisfied following condition:

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