WO2021088238A1 - Shpb test system-based dynamic lateral strain measurement device and method for test piece - Google Patents

Abstract

An SHPB test system-based dynamic lateral strain measurement device and method for a test piece (7), which belong to the field of testing dynamic mechanical properties of a brittle material such as coal rock by using an SHPB test system. Said device comprises a fixing system, an oil pressure system and a data acquisition system; the fixing system is used for fixing the whole device on an SHPB test stand and adjusting the height of the device; the oil pressure system is used for applying a desired confining pressure to a test piece (7) and converting a volume change signal of the test piece (7) into an oil pressure change signal; and the data acquisition system is used for capturing the oil pressure change signal and deriving a time-course change curve of lateral strain. Said device and method use the oil pressure system to apply a confining pressure to the test piece (7) so as to simulate comprehensive confining pressure loading of the test piece (7), and work together with an axial acting force caused by a split-Hopkinson pressure bar device, thereby achieving the simulation of a complex stress state of the test piece (7); in addition, said device and method determine a relationship between an oil pressure and a lateral strain by means of an oil pressure change signal caused by the lateral strain, so as to calculate the lateral strain.

Description

A Kind of Device and Method for Measuring Dynamic Lateral Strain of Specimen Based on SHPB Test System Technical field

The invention relates to the technical field of testing the dynamic mechanical properties of brittle materials such as coal and rock by using an SHPB test system, and specifically relates to a test piece dynamic lateral strain measuring device and method based on the SHPB test system.

Background technique

Most materials exhibit a certain degree of sensitivity to loading rate or strain rate in terms of mechanical properties such as strength. The strain rate effect of material mechanical properties caused by high-amplitude short-time pulse dynamic load is very important for the structural design and analysis of dynamic load resistance. The separated Hopkinson pressure bar (SHPB) test device is currently the main test method for testing the dynamic stress-strain response relationship of various engineering materials in the strain rate range of 100/s to 10000/s. It has simple structure, convenient operation and measurement. The method is exquisite, and the loading waveform is easy to control. When the device is used to test the dynamic mechanical characteristics of brittle materials such as coal and rock, the axial strain and axial stress of the material can be obtained by converting the dynamic strain time history curve on the waveguide rod. However, there is still a lack of accurate and effective means for measuring the dynamic lateral strain of the specimen.

technical problem

In a large number of Hopkinson pressure bar experiments, strain gauges are attached to the side of the specimen to obtain the dynamic lateral strain of the specimen. This method has certain limitations. First, for brittle materials such as coal and rock, the strain gauge is not easy to stick due to its relatively rough surface, and it is easy to cause the strain gauge to fall off during dynamic loading, which affects the integrity of the experimental data; second , The lateral strain of the specimen under dynamic load impact is often uneven. The method of pasting strain gauges can only measure the lateral strain of a certain point on the side of the specimen, which represents the inaccuracy of the lateral strain of the entire specimen. of.

Technical solutions

Based on the above technical problems, the present invention proposes a test piece dynamic lateral strain measurement device and method based on the SHPB test system.

The technical solution adopted by the present invention is:

A dynamic lateral strain measurement device for specimens based on the SHPB test system, including a hydraulic system, a data acquisition system and a fixed system;

The hydraulic system includes a cylinder body, a cover plate, a sealing gasket and a rubber bladder. The cylinder body is a hollow cylinder. Both ends of the cylinder body are blocked by a cover plate. The center of the cover plate is provided with a rod for incident or transmission. The round hole through which the rod passes; the rubber bladder is arranged on the inner side of the cylinder, the whole of the rubber bladder is hollow cylinder, and the inner axial direction is used to place the test piece, and the side wall of the rubber bladder is a hollow structure for filling Injection of hydraulic oil; the sealing gasket is arranged between the inner wall of the cover plate and the end surface of the rubber bladder;

An oil pressure sensor connection port and an exhaust port are provided on the upper part of the cylinder, and the exhaust port is in communication with the gap between the cylinder and the rubber bladder; an oil inlet and an oil outlet are provided on the lower part of the cylinder, The oil inlet and outlet are connected with hydraulic pipelines for transmitting hydraulic oil, a hydraulic oil pump is arranged on the hydraulic pipeline, and the oil inlet and outlet are also communicated with the hollow space on the side wall of the rubber bladder;

The data acquisition system includes an oil pressure sensor, a signal transmission line, a dynamic hydraulic acquisition instrument and a computer. The oil pressure sensor is arranged at the connection port of the oil pressure sensor. The lower end of the oil pressure sensor is located in the hollow space of the side wall of the rubber bladder. The sensor is connected with the dynamic hydraulic acquisition instrument through the signal transmission line, and the dynamic hydraulic acquisition instrument is connected with the computer;

The fixing system includes a base and a movable column. The base is fixed on the SHPB test bench. The bottom end of the movable column is welded to the base, and the top end of the movable column is welded to the cylinder. Adjusting bolts for lifting.

Preferably, the cover plate and the cylinder are connected by threads, and the outer surface of the cover plate is set as a frosted surface.

Preferably, the oil inlet and the oil outlet are respectively connected to the hydraulic pipeline through a tapered threaded connection.

A method for measuring the dynamic lateral strain of a specimen based on the SHPB test system, using the above-mentioned measuring device, includes the following steps:

(1) Place the measuring device at the position where the specimen is clamped on the SHPB test bench, unscrew the two cover plates and put them on the incident rod and the transmission rod respectively; adjust the height of the movable column by adjusting the bolts so that the cylinder body and the incident rod are in contact The transmission rod is at the same height;

(2) The selected test piece is cylindrical, the diameter is equal to the diameter of the incident rod and the transmission rod; insert the test piece into the cylinder, the side of the test piece is wrapped by a rubber sleeve, adjust the incident rod and the transmission rod, and clamp Test piece; then move the base of the device so that the test piece is in the middle of the cylinder;

(3) Place the sealing washer between the cover plate and the rubber bladder, then screw the cover plates on both sides to the cylinder block, and then fix the base of the device on the SHPB test bench with bolts;

(4) Open the exhaust port and the oil inlet, keep the oil outlet closed, the hydraulic oil pump starts to pressurize the hydraulic oil into the rubber bladder, the oil volume gradually increases, slowly exhaust the gas in the cylinder until the hydraulic oil is full of rubber Capsule, close the exhaust port;

(5) Continue to enter the oil and pressurize until the confining pressure required for the experiment is reached, close the oil inlet, and make the test piece in the required confining pressure state;

(6) Start the SHPB system, the bullet hits the incident rod, and the incident wave passes through the incident rod to the specimen. The specimen is axially compressed and produces lateral strain. The density of oil in the rubber bladder increases, and the oil pressure increases. The signal is collected by the pressure sensor, and the signal is transmitted to the dynamic hydraulic acquisition instrument to form the oil pressure time history curve, and then processed on the computer to obtain the time history curve of lateral strain, analyze the curve, and process the data;

(7) Open the oil outlet, recover the hydraulic oil to the hydraulic oil pump, unscrew the cover, remove the device, and clean the test bench.

The oil pressure time history curve and the lateral strain time history curve described in the above step (6) are converted using the following steps:

Assuming that the time history curve of the oil pressure change obtained by the dynamic hydraulic pressure acquisition instrument is F ( t ), the volume in the _cylinder is V cylinder, and the relationship between the oil pressure and the oil volume in the cylinder is V _oil = f ( F ), try The initial length of the specimen is h , and the initial radius of the specimen is r , which is the same as the radius of the incident rod and the transmission rod; when the stress wave is loaded to time t , the specimen length is the h _axis , the volume of lateral expansion is the V _side , and the axial strain Is the ε _axis ( t ) and the lateral strain is the ε _side ( t ), then:

h _axis = h [1- ε _axis ( t )]

V _oil = f [ F ( t )]

V _side ={ r [1+ ε _side ( t )]} ² π h [1- ε _side ( t )]- π r ² h [1- ε _side ( t )]

According to the constant volume of the _cylinder V cylinder = V _side + V _oil , we can get:

V _cylinder ={ r [1+ ε _side ( t )]} ² π h [1- ε _side ( t )]- π r ² h [1- ε _side ( t )]+ f [ F ( t )]

Can be obtained by conversion:

(1)

After obtaining the time history curve of the oil pressure change, the time history curve of the lateral strain of the test piece is obtained according to the above formula (1).

Beneficial effect

(1) The present invention can adapt to the condition that the lateral strain of the test piece changes unevenly under the impact of dynamic load, converts the volume signal of the lateral strain of the test piece into a liquid pressure signal, and the experimental data is more intuitive.

(2) The present invention can measure the complete lateral strain of the entire specimen, rather than the lateral strain of a certain point or a certain area. Compared with the measurement method of pasting strain gauges, the lateral strain measurement data is more accurate.

(3) The present invention does not need to stick strain gauges, and can avoid the limitations caused by sticking strain gauges. For example, for some specimens with rough surfaces, the strain gauges are not easy to stick, fall off, are easily disturbed, and the wire connections are easy to break. And so on, and the present invention better solves the above-mentioned problems.

(4) The principle of the present invention is simple and convenient to operate, and can be used repeatedly.

Description of the drawings

The present invention will be further explained below in conjunction with the drawings and specific embodiments:

FIG. 1 is a schematic diagram of the structure of the measuring device of the present invention in a cutaway;

Figure 2 is a cross-sectional view of the measuring device of the present invention;

Figure 3 is a left side view of the measuring device of the present invention;

Figure 4 is a general assembly diagram of the measuring device of the present invention in the SHPB test system;

Fig. 5 is a schematic diagram of parameter labeling involved in the conversion formula of the present invention;

Figure 6 is a flow chart of the method of the present invention.

1. Incident rod, 2. Cover plate, 3. Oil pressure sensor, 4. Cylinder body, 5. Exhaust port, 6. Gasket, 7. Specimen, 8. Transmission rod, 9. Rubber bladder, 10. Oil inlet, 11. Live column, 12. Base, 13. Adjusting bolt, 14. Oil outlet, 15. Impact loading system, 16. Laser speed measuring system, 17. Hydraulic oil pump, 18. Hydraulic pipeline, 19. Dynamic Hydraulic collector, 20. Computer, 21. Absorption rod, 22. Axial pressure system, 23. Test bench base, 24. Bullet, 25. High-pressure nitrogen cylinder.

Embodiments of the present invention

The invention relates to a test piece dynamic lateral strain measuring device and method based on an SHPB test system. The measuring device includes a fixing system, an oil pressure system and a data acquisition system. The fixing system is used to fix the entire device on the SHPB test bench and adjust the height of the device; the oil pressure system is used to apply the required confining pressure to the test piece and to The volume change signal of the test piece is converted into an oil pressure change signal; the data acquisition system is used to capture the oil pressure change signal and obtain the time history change curve of the lateral strain. The invention uses the oil pressure system to apply confining pressure to the test piece, simulates the comprehensive loading of the confining pressure of the test piece, and acts together with the axial force caused by the separated Hopkinson pressure rod device, thereby realizing the complex stress state of the test piece Simulation. The signal of the oil pressure change caused by the lateral strain is used to obtain the relationship between the oil pressure and the lateral strain to calculate the lateral strain, and even the dynamic Poisson's ratio can be calculated in combination with the axial strain.

The device and method for measuring the dynamic lateral strain of the test piece of the present invention will be described in detail below.

With reference to the attached drawings, a dynamic lateral strain measurement device for specimens based on the SHPB test system, including a fixed system, an oil pressure system and a data acquisition system.

The fixing system is composed of a base 12, a movable post 11 and an adjusting bolt 13. The base 12 is arranged on the SHPB test bench, and two fixing holes are arranged on both sides of the base, which is convenient for fixing it with the SHPB test bench by using nuts. The upper part of the movable column is welded to the cylinder body 4, and the lower part is welded to the base 12. There are adjusting bolts 13 on the movable column. The movable column can be raised and lowered by adjusting the bolts to ensure that the device, the tested part, SHPB waveguide rods (incident rod and transmission rod) meet the same height.

The oil pressure system is composed of a cover plate 2, a cylinder block 4, a sealing gasket 6, a rubber bladder 9, a hydraulic pipeline 18 and a hydraulic oil pump 17. Cylinder body 4 is a hollow cylinder. The function of the cylinder body is to form a closed oil storage space. Both ends of the cylinder body are blocked by cover plate 2. There are two cover plates 2 which are located on both sides of cylinder body 4. 2 and the cylinder 4 are connected by a screw thread, and sealed by a sealing gasket 6. Two sealing gaskets 6 are provided, located on both sides of the cylinder body, close to the rubber bladder and cover plate, the inner ring of the sealing gasket is in close contact with the incident rod or the transmission rod passing through, and the sealing gasket is used for sealing the cylinder body. The center of the cover plate 2 is provided with a circular hole through which the incident rod 1 or the transmission rod 8 passes. The diameter of the circular hole is slightly larger than the diameter of the incident rod or the transmission rod, so as not to hinder the horizontal movement of the incident rod or the transmission rod. The outer surface of the cover plate 2 is processed into a frosted surface to facilitate loading and unloading. The rubber bladder 9 is arranged on the inner side of the cylinder. The whole of the rubber bladder has a hollow cylindrical shape, and the inner axial direction is used to place the test piece 7, and the side wall of the rubber bladder has a hollow structure for filling hydraulic oil.

An oil pressure sensor connection port and an exhaust port 5 are provided on the upper part of the cylinder, which are used to connect the oil pressure sensor and exhaust respectively. The exhaust port is sealed with ordinary high-strength bolts and copper gaskets, that is, ordinary seals can be used. , The exhaust port 5 communicates with the gap between the cylinder 4 and the rubber bladder 9. The lower part of the cylinder 4 is provided with an oil inlet 10 and an oil outlet 14. One end of the oil inlet and the oil outlet is connected with the hollow space of the side wall of the rubber bladder, and the other end of the oil inlet and the oil outlet is connected with the transmission The hydraulic pipeline 18 of the hydraulic oil is connected, and is used for respectively injecting and discharging oil into the rubber bladder. After the rubber bladder enters the oil, the rubber bladder expands and closely adheres to the surface of the test piece, and then the required confining pressure is applied to the test piece. A hydraulic oil pump 17 is provided on the hydraulic pipeline 18, the hydraulic pipeline is used for the transmission of hydraulic oil, and the hydraulic oil pump is used for supply and recovery of hydraulic oil. The above-mentioned oil inlet and outlet and the hydraulic pipeline are preferably connected by tapered threads. The tapered thread is used because it has a certain taper, which can be tightened as it is tightened to eliminate gaps, achieve interference fit, and ensure the seal of the cylinder. Better performance and higher pressure resistance.

The data acquisition system is mainly composed of an oil pressure sensor 3, a signal transmission line, a dynamic hydraulic acquisition instrument 19 and a computer 20. The oil pressure sensor 3 is installed at the connection port of the oil pressure sensor, the lower end of the oil pressure sensor is located in the oil in the rubber bladder, and the dynamic hydraulic pressure collector is connected with the oil pressure sensor directly above the cylinder. The oil pressure sensor is used to sense oil pressure signals in real time; the signal transmission line is used to connect the oil pressure sensor and the dynamic hydraulic acquisition instrument and transmit data; the dynamic hydraulic acquisition instrument is used to capture oil pressure change signals for a long time , And generate the time history curve of the oil pressure change in the cylinder, and then display the time history curve of the lateral strain in the computer processing.

The test piece dynamic lateral strain measuring device of the present invention is used in combination with the existing SHPB test system, as shown in Figure 4, the SHPB test system includes an impact loading system 15, a laser speed measurement system 16, an axial compression system 22, and a bullet 24 And high-pressure nitrogen bottle 25 and so on.

The present invention also provides a method for measuring dynamic lateral strain of a specimen based on the SHPB test system. The method adopts the above-mentioned measuring device and includes the following steps:

(1) Place the measuring device at the position where the specimen is clamped on the SHPB test bench, unscrew the two cover plates 2 and put them on the incident rod 1 and the transmission rod 8 respectively; adjust the height of the movable post 11 by adjusting the bolt 13 to make the cylinder The body 4 is at the same height as the incident rod 1 and the transmission rod 8.

(2) The selected specimen 7 is cylindrical, with a diameter equal to the diameter of the incident rod and the transmission rod. Insert the test piece into the cylinder, the side of the test piece is wrapped by the rubber bladder 9, adjust the incident rod 1 and the transmission rod 8, and clamp the test piece 7; then move the device base 12 so that the test piece 7 is in the front of the cylinder 4 centre position.

(3) Place the sealing washer 6 between the cover plate 2 and the rubber bladder 9, then screw the cover plates 2 on both sides to the cylinder 4, and then fix the base of the device on the SHPB test bench with bolts.

(4) Open the exhaust port 5 and the oil inlet 10, the oil outlet 14 remains closed, the hydraulic oil pump 17 starts to pressurize the hydraulic oil to enter the rubber bladder 9, the oil volume gradually increases, and the gas in the cylinder is slowly discharged. Until the hydraulic oil is filled with the rubber bladder 9, close the exhaust port 5.

(5) Continue to pressurize the oil until the confining pressure required for the experiment is reached. Close the oil inlet 10 to make the test piece in the required confining pressure state.

(6) Start the SHPB system, the bullet 24 hits the incident rod 1, and the incident wave passes through the incident rod 1 into the specimen 7. The specimen 7 is axially compressed, resulting in lateral strain, and the density of the oil in the rubber bladder 9 increases. When the oil pressure increases, the oil pressure sensor 3 collects the signal, and the signal is transmitted to the dynamic hydraulic acquisition instrument 19 to form an oil pressure time history curve, which is then processed on the computer 20 to obtain a time history curve of lateral strain, analyze the curve, and process the data .

(7) Open the oil outlet 14, recover the hydraulic oil to the hydraulic oil pump 17, unscrew the cover plate 2, remove the device, and clean the test bench.

The oil pressure time history curve and the lateral strain time history curve described in the above step (6) are converted using the following principle steps:

Assuming that the time history curve of the oil pressure change obtained by the dynamic hydraulic pressure acquisition instrument is F ( t ), the volume in the _cylinder is V cylinder, and the relationship between the oil pressure and the oil volume in the cylinder is V _oil = f ( F ), try The initial length of the specimen is h , and the initial radius of the specimen is r , which is the same as the radius of the incident rod and the transmission rod; when the stress wave is loaded to time t , the specimen length is the h _axis , the volume of lateral expansion is the V _side , and the axial strain Is the ε _axis ( t ) and the lateral strain is the ε _side ( t ), then:

h _axis = h [1- ε _axis ( t )]

V _oil = f [ F ( t )]

V _side ={ r [1+ ε _side ( t )]} ² π h [1- ε _side ( t )]- π r ² h [1- ε _side ( t )]

According to the constant volume of the _cylinder V cylinder = V _side + V _oil , we can get:

V _cylinder ={ r [1+ ε _side ( t )]} ² π h [1- ε _side ( t )]- π r ² h [1- ε _side ( t )]+ f [ F ( t )]

Can be obtained by conversion:

(1)

... After obtaining the time history curve of the oil pressure change, the time history curve of the lateral strain of the test piece is obtained according to the above formula (1).

The parts not mentioned in the above methods can be realized by adopting or learning from existing technologies.

It should be noted that, under the teaching of this specification, any equivalent alternatives or obvious modifications made by those skilled in the art should fall within the protection scope of the present invention.

Claims (5)

1. A test piece dynamic lateral strain measuring device based on the SHPB test system, which is characterized in that it includes a hydraulic system, a data acquisition system and a fixed system;

   The hydraulic system includes a cylinder body, a cover plate, a sealing gasket and a rubber bladder. The cylinder body is a hollow cylinder. Both ends of the cylinder body are blocked by a cover plate. The center of the cover plate is provided with a rod for incident or transmission. The round hole through which the rod passes; the rubber bladder is arranged on the inner side of the cylinder, the whole of the rubber bladder is hollow cylinder, and the inner axial direction is used to place the test piece, and the side wall of the rubber bladder is a hollow structure for filling Injection of hydraulic oil; the sealing gasket is arranged between the inner wall of the cover plate and the end surface of the rubber bladder;

   An oil pressure sensor connection port and an exhaust port are provided on the upper part of the cylinder, and the exhaust port is in communication with the gap between the cylinder and the rubber bladder; an oil inlet and an oil outlet are provided on the lower part of the cylinder, The oil inlet and outlet are connected with hydraulic pipelines for transmitting hydraulic oil, a hydraulic oil pump is arranged on the hydraulic pipeline, and the oil inlet and outlet are also communicated with the hollow space on the side wall of the rubber bladder;

   The data acquisition system includes an oil pressure sensor, a signal transmission line, a dynamic hydraulic acquisition instrument and a computer. The oil pressure sensor is arranged at the connection port of the oil pressure sensor. The lower end of the oil pressure sensor is located in the hollow space of the side wall of the rubber bladder. The sensor is connected with the dynamic hydraulic acquisition instrument through the signal transmission line, and the dynamic hydraulic acquisition instrument is connected with the computer;

   The fixing system includes a base and a movable column. The base is fixed on the SHPB test bench. The bottom end of the movable column is welded to the base, and the top end of the movable column is welded to the cylinder. Adjusting bolts for lifting.
2. The test piece dynamic lateral strain measuring device based on the SHPB test system according to claim 1, characterized in that: the cover plate and the cylinder are connected by a screw thread, and the outer surface of the cover plate is set to be frosted surface.
3. The test piece dynamic lateral strain measuring device based on the SHPB test system according to claim 1, characterized in that: the oil inlet and the oil outlet are respectively connected with the hydraulic pipeline through a tapered threaded connection. .
4. A method for measuring dynamic lateral strain of a specimen based on an SHPB test system, using the measuring device according to any one of claims 1 to 3, characterized by comprising the following steps:

   (1) Place the measuring device at the position where the specimen is clamped on the SHPB test bench, unscrew the two cover plates and put them on the incident rod and the transmission rod respectively; adjust the height of the movable column by adjusting the bolts so that the cylinder body and the incident rod are in contact The transmission rod is at the same height;

   (2) The selected test piece is cylindrical, the diameter is equal to the diameter of the incident rod and the transmission rod; insert the test piece into the cylinder, the side of the test piece is wrapped by a rubber sleeve, adjust the incident rod and the transmission rod, and clamp Test piece; then move the base of the device so that the test piece is in the middle of the cylinder;

   (3) Place the sealing washer between the cover plate and the rubber bladder, then screw the cover plates on both sides to the cylinder block, and then fix the base of the device on the SHPB test bench with bolts;

   (4) Open the exhaust port and the oil inlet, keep the oil outlet closed, the hydraulic oil pump starts to pressurize the hydraulic oil into the rubber bladder, the oil volume gradually increases, slowly exhaust the gas in the cylinder until the hydraulic oil is full of rubber Capsule, close the exhaust port;

   (5) Continue to enter the oil and pressurize until the confining pressure required for the experiment is reached, close the oil inlet, and make the test piece in the required confining pressure state;

   (6) Start the SHPB system, the bullet hits the incident rod, and the incident wave passes through the incident rod to the specimen. The specimen is axially compressed and produces lateral strain. The density of oil in the rubber bladder increases, and the oil pressure increases. The signal is collected by the pressure sensor, and the signal is transmitted to the dynamic hydraulic acquisition instrument to form the oil pressure time history curve, and then processed on the computer to obtain the time history curve of lateral strain, analyze the curve, and process the data;

   (7) Open the oil outlet, recover the hydraulic oil to the hydraulic oil pump, unscrew the cover, remove the device, and clean the test bench.
5. The method for measuring dynamic lateral strain of a specimen based on the SHPB test system according to claim 4, wherein the oil pressure time history curve and the lateral strain time history curve described in step (6) adopt the following steps Conversion:

   Assuming that the time history curve of the oil pressure change obtained by the dynamic hydraulic acquisition instrument is F ( t ), the volume in the _cylinder is V cylinder, and the relationship between the oil pressure and the oil volume in the cylinder is V _oil = f ( F ), try The initial length of the specimen is h , the initial radius of the specimen is r , which is the same as the radius of the incident rod and the transmission rod; when the stress wave is loaded to time t , the specimen length is the h _axis , the volume of lateral expansion is the V _side , and the axial strain Is the ε _axis ( t ) and the lateral strain is the ε _side ( t ), then:

   h _axis = h [1- ε _axis ( t )]

   V _oil = f [ F ( t )]

   V _side ={ r [1+ ε _side ( t )]} ² πh [1- ε _side ( t )]- πr ² h [1- ε _side ( t )]

   According to the constant volume of the _cylinder, V cylinder = V _side + V _oil , we can get:

   V _cylinder ={ r [1+ ε _side ( t )]} ² πh [1- ε _side ( t )]- πr ² h [1- ε _side ( t )]+ f [ F ( t )]

   Can be obtained by conversion:

   

   (1)

   After obtaining the time history curve of the oil pressure change, the time history curve of the lateral strain of the test piece is obtained according to the above formula (1).