JP2013061158A - Apparatus and method for measuring gravity of earth - Google Patents

Abstract

PROBLEM TO BE SOLVED: To provide an apparatus for measuring the gravity of the earth that is composed of: a single pendulum; a high-speed camera; and a measuring unit body for calculating the acceleration of the gravity through analysis of a camera image, acquires the acceleration of the gravity by acquiring swing periods and angles and processing them, and prevents a result from being influenced by a swing angle as well as prevents the amplitude thereof from being changed by air resistance when measuring the acceleration of the gravity by moving the single pendulum forward and backward a plurality of time.SOLUTION: The amplitude and period of a single pendulum for each forward/backward motion are captured by the high-speed camera, and the amplitude and the period are properly calculated by image processing to acquire the acceleration of the gravity.

Description

The invention relates to an apparatus and method for measuring the gravity of the earth.

Conventional methods for measuring gravity include the following methods.
1) Method using springs An example of a gravity measurement method using springs is shown in FIG. This is a method of measuring the acceleration g of gravity using the characteristic that the displacement ΔX of the spring is proportional to the magnitude F (= mg) of the load. When the spring constant of the spring is K and the mass of the weight is m, the following equation is established.
F = mg = K · ΔX (1)
If the spring constant K of the spring and the mass m of the weight are known, the acceleration g of gravity can be obtained by the following equation by measuring the displacement ΔX.
g = K · ΔX ₁ / m (2)

As shown in FIG. 3, when the spring displacement ΔX ₁ is known at the point of known gravity acceleration g ₁ , the spring displacement ΔX ₂ at the point of unknown gravity acceleration is measured. it can be obtained acceleration g ₂ unknown gravity by.
mg ₁ = K · ΔX ₁
mg ₂ = K · ΔX ₂

From the above two equations, g ₂ = (ΔX ₂ / ΔX ₁ ) g ₁ (3)
In this case, the spring constant and mass need not be known.

2) KATER reversible pendulum FIG. 4 shows an example of a gravity measurement method known as a KATER reversible pendulum. N _1, N ₂ is the center of rotation of the pendulum, characterized in this pendulum, so that the period tau ₁ and the rotation center when the center of rotation was N ₁ is the period tau ₂ when the N ₂ equals The positions of the weights B ₁ and B ₂ are adjusted. The distance between N ₁ and N ₂ is set equal to the rotation radius k, where k is the rotation radius with N ₁ as the center. N ₁ and N ₂ are called reversible pendulums because they have the same relationship even if they are replaced with each other.

When rotating around a horizontal fixed axis N ₁ under the action of gravity, the distance between N ₁ and the center of gravity G is H ₁ , the angle between the pendulum and the vertical line is θ, and the pendulum (physical pendulum on the axis) ) Where k is the radius of rotation k ² (d ² θ / dt ² ) = − gH ₁ sin θ≈−gH ₁ θ (4)

Pendulum motion equation L (d ² θ / dt ² ) = − g sin θ≈−g θ when mass is concentrated at one point (5)
Comparing equation (4) and equation (5), L = k ² / H ₁ (6)
L is the length of the equivalent simple pendulum.

If the amplitude is very small, the period τ is obtained by solving the differential equation (5).


















Get.

If the radius of rotation around the center of gravity is k _G , k ² = k _G ² + H ₁ ² (8)
Rewriting equation (6): L−k ² / H ₁ = L− (k _G ² + H ₁ ² ) / H _{1
^{= L- (k G 2 / H}} 1 + H 1) = 0
∴L−H ₁ = k _G ² / H ₁ (9)

Therefore, the length L of the equivalent single pendulum is always larger than the distance H ₁ between the fixed axis N ₁ and the center of gravity G.
When a point N ₂ having a distance L between N ₁ and the fixed axis N ₁ and the center of gravity G is taken, L = H ₁ + H ₂
And
L−H ₁ = H ₂ (10)
Therefore, the following equation is obtained from the equations (9) and (10).
H ₁ · H ₂ = k _G ² (11)
Such points N ₂ center of oscillation, is referred to as the center of the hanging N _1.
Rewriting equation (7) yields an equation for determining the acceleration of gravity by the following equation.
g = 4π ² L / τ ² (12)
An acceleration g of gravity is obtained by measuring the periods τ and L according to the equation (12). In this case, there is a problem that τ depends on the maximum deflection angle.

3) Throw-up type gravity measurement method As shown in FIG. 5, the object thrown right above is reduced in speed due to gravity and finally begins to fall. The interval between the two measurement surfaces is H, the time interval Δt ₁ from when the object crosses the lower measurement surface to when it crosses again, and the same time interval for the upper measurement surface is Δt ₂ . Also, the speeds of the objects when passing through the lower and upper measurement surfaces are V ₁ and V ₂ , respectively. If air resistance is ignored, the falling speed and rising speed are the same. If the acceleration of gravity is constant in this interval, the following equation is obtained.
² gH = v ₁ ² −V ₂ ² (13)
V ₁ − (− V ₁ ) = gΔt ₁ (14)
V ₂ − (− V ₂ ) = gΔt ₂ (15)
From the above three formulas, V1 and V2 are eliminated to obtain the following formula.
g = 8H / (Δt ₁ ² −Δt ₂ ² ) (16)
Since the distance H between the measurement surfaces is known, the acceleration of gravity is obtained by measuring the time difference Δt ₁ and the time difference Δt ₂ .

There are 1) using a spring, 2) a KATER reversible pendulum, 3) a throw-up type, etc., but here, the acceleration of gravity by a single pendulum as shown in FIG. Think about the measurement method. In this case, if the arm length of the simple pendulum is L, the mass of the weight is m, and the angle between the vertical line and the arm is θ, the equation of motion of the simple pendulum can be expressed by the following equation.
gsin θ-L (d ² θ / dt ² ) = 0 (17)

If the above equation is set as sinθ≈θ, it is known that the solution of the differential equation can be solved easily and is a periodic function having the period shown in the following equation.

Using this formula, if you measure the period τ,
Since the arm length L is known, the acceleration g of gravity can be easily obtained. However, the problem is that the result necessarily includes an error in order to include the approximate expression. The problem remains as long as the approximate expression is used.

Therefore, the equation of motion of the single pendulum not based on the approximate expression will be examined below with reference to FIG.
The mass of the weight of the pendulum is m, and the length of the arm of the pendulum is L. Point A is the lowest point, point C is the position where the angular velocity (dθ / dt) is 0, and is the highest point of the weight, and the angle at that time is α (maximum deflection angle). The point B is an arbitrary position between the points A and C, and the deflection angle at that time is θ.

Considering the balance of forces in the direction of motion of the pendulum weight, the following equation holds: mL (d ² θ / dt ² ) = − mg · sin θ (19)

Considering the balance of force in the direction perpendicular to the direction of motion, N-mg · cos θ-mL (dθ / dt) ² = 0 (20)
Where N: tension applied to the arm
The second term on the left side is the gravity force of the weight
The third term on the left side is centrifugal force
Can be written.

Multiplying both sides of equation (19) by L (dθ / dt) and collecting them on the left side,
mL ² (d ² θ / dt ² ) (dθ / dt) + mgL · (dθ / dt) sin θ = 0
If integrated with respect to t, mL ² (dθ / dt) ² / 2-mgL · cos θ = E (21)
E is an integral constant, and Equation (21) represents a mechanical energy conservation law for the weight of the pendulum. E is the total energy.

Assuming that the weight speed at the lowest position A is V, (Ldθ / dt) _{θ = 0} = (v) _{θ = 0} = V. Therefore, if _{θ = 0 in the} equation (21), E = mV ² / 2-mgL
It becomes. When this value is used, the equation (21) can be expressed as L ² (dθ / dt) ² = V ² -2 gL (1-cos θ) (22)
It can be rewritten as

If (dθ / dt) is eliminated from the equations (20) and (22), N = mV ² / L + mg (3 cos θ−2) (23)
In the acceleration measurement of gravity by the pendulum, the condition is α ≦ π / 2.

Here, the relationship between the maximum deflection angle α and the pendulum speed V at the lowest point is obtained. Position of the energy mgL from the energy relationship of motion ^{(1-cosα) = mV 2} /2
Can be obtained and transformed
cosα = ( ² gL−V ² ) / ² gL (24)
Is obtained.

Rewrite equation (22) using equation (24).
(dθ / dt) ² = 2g (cosθ−cosα) / L (25)
Get.

When the weight is rising from point A to point C, dt / dθ> 0.


If the origin is the moment when the weight departs from point A,

This integral
sin (θ / 2) = sin (α / 2) sin (ψ) (30)
Then, it is changed to the integral about ψ.
cos (θ / 2) ・ (1/2) dθ = sin (α / 2) cosψdψ (31)
From


Where k ² = sin ² (α / 2) (33)

Also,

Substituting Equations (32) and (34) into Equation (29)

The time required to reach point C is 1/4 of the pendulum period τ, and at point C, θ = α.
ψ = π / 2
Therefore, the period τ can be expressed by the following equation.

here,

If

on the other hand,

From the relationship

The following formula for each term in the series of the above formula

When applying

K = π / 2 ・ {1+ (1/2) ²・ k ² + ((1 ・ 3) / (2 ・ 4)) ²・ k ⁴
+ ((1 ・ 3 ・ 5) / (2 ・ 4 ・ 6)) ²・ k ⁶ + ・ ・ ・ ・} (42)
Where k = sin (α / 2) (43)
Get.
Therefore

When α is a small angle, the infinite series part of equation (42) may ignore the second and subsequent terms, and calculating only with the first term, equation (44), which is the calculation formula of the period, τ = 2π · SQRT (L / g), which coincides with the equation (18), which is a formula for calculating the period by the approximate equation.

Equation (44) is an infinite polynomial and causes a truncation error. However, it can be considered that the term to be cut off to the required accuracy can be adjusted and the problem of accuracy can be solved. Here, it should be noted that the maximum deflection angle α appears in the calculation of the period. The truncation error is shown in FIG.

Although the calculation formula of the cycle that does not include errors due to approximation was obtained, in order to measure the cycle with a single pendulum, in order to improve accuracy, it is common to swing the pendulum several times and take the average value Is the method. However, there is a problem that the amplitude decreases due to air resistance during measurement. Even if the equation (44), which requires little consideration of the error, is used, the maximum deflection angle α becomes a different value every time it swings, so it is necessary to measure an accurate amplitude every cycle. Become. In the case of the pendulum, there is a problem that the acceleration of gravity cannot be measured accurately unless the amplitude and period are accurately measured every time.
Mechanics 1, Shujiro Kunii et al., Maruzen, P223-232 Internet, http://www.kagaku.info/gravity990510/

FIG. 1 shows a configuration diagram of an apparatus for measuring the acceleration of gravity using a single pendulum. In this method, the motion of the pendulum is photographed with a high-speed camera, the image is analyzed, and the period and amplitude at that time can be accurately measured for each round trip. If it is possible to accurately measure the amplitude and period of n reciprocations, an accurate acceleration of gravity can be obtained as in the following equation.

From data for each cycle

However, K _i = π / 2 · {1+ (1/2) ² k _i ² + (1 · 3/2 · 4) ² k _i ⁴ +.
k _i = sinα _i

Add the left and right sides of the formula (45) together

When the above equation is transformed

However, K _i = π / 2 · {1+ (1/2) ² k _i ² + (1 · 3/2 · 4) ² k _i ⁴ + ······,
k _i = sinα _i / 2
It will be understood that when τ _i and α _i for each reciprocation are accurately determined, an accurate acceleration of gravity can be obtained from the above equation. If you want to increase the accuracy, you can increase the term to calculate the infinite sequence. The standard is shown in FIG. For example, when α is 10 °, the fourth item is an error when calculating up to the third term, which is 4.27 * 10 ⁻⁸ . In the calculation up to the fourth item, the error is 2.48 * 10 ⁻¹⁰ in the fifth item.

FIG. 1 shows an embodiment of the present invention. Reference numeral 1 denotes a measuring instrument main body which takes in an image from the high-speed camera 6 and is structured so that the amplitude and period of the pendulum can be measured by image processing. 2 is a pendulum fulcrum, 3 is a pendulum arm (thread), 4 is a pendulum weight, and 5 is a scale plate for reading the position of the pendulum, which is made of a translucent material and has a scale drawn on the surface. . On the back side, an illumination 7 (see FIG. 2) is installed so that weights and scales can be easily recognized on the camera image. 6 is a high-speed camera. Since it is such a structure, the motion state of a pendulum can be acquired as a digital image with a high-speed camera. The digital image data of the high-speed camera is taken into the measuring instrument main body 1 so that the amplitude of the pendulum and the required time (cycle) for each cycle can be accurately acquired by image analysis.

FIG. 2 is a side view of the apparatus. An illumination 7 is installed behind the translucent scale plate 5 to make it easy to recognize the weight and scale of the pendulum with a high-speed camera. Set the high-speed camera as far as possible and shoot with a telephoto lens. Even if it does in this way, there exists a possibility of producing an error because there is a difference between the distance from the camera to the weight of the pendulum and the distance to the scale plate. For this reason, it is assumed that error correction is performed by the following equation during image processing.
Amplitude after correction (distance from center) = Amplitude before correction (distance from center) * A / (A + B)
Where A is the distance between the camera and the weight
Let B be the distance between the weight and the scale plate surface.
By doing in this way, generation | occurrence | production of the error which arises because the weight of a pendulum and a scale plate are not the same surface was prevented.

FIG. 7 shows details of the measuring instrument main body. 8 is a central processing unit (CPU), 10 is a main memory, 16 is a processor dedicated to image processing, 14 is a display mechanism, and 9 is a system bus. In addition, 13 is a keyboard, 15 is an external memory, and 11 is a USB interface. Digital image data is fetched from the high-speed camera 6 via the USB interface 11 so as to be fetched into the external memory 15.

FIG. 8 shows the process flow of the main program of the main measurement process. Box A10 is a process for capturing initial data such as the length L of the pendulum arm and the frame rate FPS of the high-speed camera. Waiting for the start of the process, if there is an instruction from the keyboard 13, the process proceeds to the next process (box A20). Box A30 is processing for capturing digital image data from the high-speed camera. Box A40 is a process for acquiring and analyzing the acquired digital image screen by screen and calculating the amplitude and period.
Box A50 is an amplitude / period calculation process. Box A60 is a process of calculating the acceleration of gravity using the above processing result.

FIG. 9 shows the detailed processing of the box A30 in a flow. The camera image is captured (box B10), and the image data is written in the storage memory (box B20).

FIG. 10 shows the detailed processing of the box A40. It is a processing flow of a camera image analysis processing program. Camera images are captured screen by screen (box C30), and when the pendulum weight passes the leftmost end, the coordinates of the weight are calculated and stored (box C30). Next, when the weight passes through the center, the frame number of the image is recorded (box C70), and when the pendulum weight passes through the rightmost end, the coordinates of the weight are recorded (box C90). Next, in the case of passing through the center line, the file number at the time of passing through the center is recorded (box C110).
The processing result is written in a table 20 as shown in FIG. Along with the coordinate value, the frame number (FNO) of the image and the data code (codes such as L, R, and C indicating the division of the left end, right end, and center) are written in the table 20. Further, the coordinate value is corrected by the difference between the scale surface and the moving surface of the weight. That is,
Correction coordinate value = Coordinate value before correction * A / (A + B) (41)
Here, A is the distance between the camera and the weight, and B is the distance between the weight and the scale plate surface.

FIG. 12 shows the detailed processing of box A50. It is a flow of an amplitude and period calculation processing program. Data for one round-trip is fetched from the image analysis result storage table 20 (box D20), and amplitude and period are calculated (box D30). The result is written in the amplitude / period storage table 22 (box D40).

FIG. 13 shows the contents of the amplitude / period storage table 22. Here, the period is calculated from the frame number (FNO) and the frame rate. The frame rate of the high-speed camera can be selected from 300, 1200, etc. When the frame rate 300 is selected and the frame difference is 242, the time difference in that section is 242/300 = 0.800 seconds.

FIG. 14 shows the detailed processing of box A60. It is the flow of the calculation process of the acceleration of gravity. In box F10, calculation of the total time (Στi) in the denominator of equation (47) is performed. In box F20, the summation of the series part in the numerator of equation (47) is calculated.
In box F30, the acceleration of gravity is calculated using the above calculation result, and in box F40, the result is output.

The gravitational acceleration measuring instrument of the present invention is intended to measure gravitational acceleration with high accuracy and automatically.

1)
The period error due to the amplitude of the pendulum is corrected, the exact period can be used in the calculation formula, and the measurement accuracy of the acceleration of gravity is good.
2)
Since a digital image of a simple pendulum is taken with a high-speed camera and processed, the operation is easy.
3)
High-precision measurement is possible with a simple configuration of a simple pendulum, high-speed camera, and personal computer.

Overall configuration diagram of the apparatus of the present invention View from the side of the device of the present invention An example of a spring-type gravimeter Kater's reversible pendulum Throw-up type gravity measurement method Simple pendulum Details of the measuring device Main program processing flow Flow of image data import processing program Camera image analysis program flow Structure of image analysis result storage table Amplitude / period calculation program flow Amplitude / period storage table structure Gravity acceleration calculation processing program flow Gravity acceleration calculation error evaluation data (error when the upper term is truncated in the calculation of K)

1 …… Measuring instrument body 3 …… Pendulum arm (thread)
4 ... Pendulum weight 5 ... Scale plate 6 ... High-speed camera 7 ... Lighting

Claims (2)

It consists of a simple pendulum, a high-speed camera, and the main body of the instrument. The motion state of the simple pendulum is captured by the high-speed camera, and the image data is imported from the memory of the high-speed camera into the memory of the instrument body, and the captured image is analyzed. Then, a gravitational acceleration measuring device that obtains an amplitude and a period for each period of the pendulum and calculates the acceleration of gravity based on the amplitude data of the period. It consists of a simple pendulum, a high-speed camera, and the main body of the instrument. The motion state of the simple pendulum is captured by the high-speed camera, and the image data is imported from the memory of the high-speed camera into the memory of the instrument body, and the captured image is analyzed. Then, a method for measuring the acceleration of gravity, such as obtaining the amplitude and period for each period of the pendulum and calculating the acceleration of gravity based on the amplitude data of the period.