Background technology
The principle of work of metal resistance strain gauge is resistance strain effect, and namely tinsel is when being subject to effects of strain, and its resistance, along with the size of occurred mechanically deform (stretching or compression), corresponding change occurs.The theoretical formula of resistance strain effect is as follows:
$R=\mathrm{\ρ}\frac{L}{S}---\left(1\right)$
Wherein R is its resistance value, and ρ is metal material resistivity, and L is metal material length, and S is metal material sectional area.Tinsel occurs in the process of mechanically deform bearing strain, and ρ, L, S three will change, thus will inevitably cause the change of metal material resistance value.When metal material is stretched, length increases, and sectional area reduces, and resistance value increases; When by compression, length reduces, and sectional area increases, and resistance value reduces.Therefore, as long as the change of resistance value can be measured, just known strained situation wiry.Metal material resistance change rate formula can be derived by relevant knowledges such as formula (1) and the mechanics of materials
$\frac{\mathrm{\Δ}R}{R}=K\frac{\mathrm{\Δ}L}{L}=K\mathrm{\ϵ}---\left(2\right)$
Wherein Δ R is resistance variation amount, and Δ L is the variable quantity of metal material length on pulling force or pressure action direction, and ε is that the strain on same direction is usually called axial strain, and K is metal material ga(u)ge factor.
In actual applications, metal resistance strain gauge is pasted onto the surface of Sensor Elastic Element or tested mechanical component.When the flexible member in sensor or tested mechanical component produce strain by acting force, also there is identical mechanically deform in the foil gauge pasted thereon thereupon, causes foil gauge resistance that corresponding change occurs.At this moment, mechanical quantity is just converted to the variable quantity output of resistance by resistance strain gage.
But we also need the partial derivative understanding workpiece strain sometimes, such as have three kinds of occasions below, but are not limited thereto three, need to use surface of the work strain partial derivative:
The first, concentrate owing to there will be strain near workpiece shapes sudden change place, often become workpiece and first occur damaging part, the strain partial derivative near sudden change place of monitoring shape, this place strain intensity can be obtained intuitively.
Second, exist in a large number by comer pieces in building, bridge, plant equipment, mechanics of materials relevant knowledge tells us, bent beam surface axial strain is directly proportional to section turn moment, the axial partial derivative of section turn moment is directly proportional to cross section shearing strain, namely can know cross section shearing strain by the axial partial derivative of surperficial axial strain, and this shearing strain directly cannot measure at surface of the work with foil gauge;
3rd, during applied elasticity research workpiece strain, internal strain is decided by partial differential equation, and equation solution needs boundary condition, and surface of the work strain partial derivative is exactly one of boundary condition, and this is that general foil gauge cannot provide.
Summary of the invention
The deficiency of strain local derviation cannot be detected in order to overcome existing metal strain plate, the invention provides a kind of can monitor strain more can effectively detect the axial single order of surface strain and second order local derviation can measured surface strain axis to the two interdigitated metal foil gauge of axial deviation full-bridge of local derviation.
The technical solution adopted for the present invention to solve the technical problems is:
A kind of can measured surface strain axis to the two interdigitated metal foil gauge of axial deviation full-bridge of local derviation, comprise substrate, metal strain plate described in it also comprises four sensitive grids, and the two ends of each sensitive grid connect an extension line respectively, described substrate is fixed described four sensitive grids;
Each sensitive grid comprises sensitive segment and transition section, the two ends of described sensitive segment are transition section, described sensitive segment is elongated strip shaped, described transition section is tubbiness shape, the resistance of described sensitive segment is much larger than the resistance of described transition section, under same strain state, the increased resistance value of described sensitive segment is much larger than the increased resistance value of described transition section, and the increased resistance value of described transition section is close to 0;
All xsect centres of form of each sensitive segment form sensitive segment axis, this sensitive segment axis is straight line section, the axis being parallel of each sensitive segment and be arranged in same plane, and sensitive segment axis is determined in plane, along described sensitive segment axis direction namely axially, be laterally with axially vertical direction; All shape of cross section consistent size of each sensitive segment; Get the axis point midway of each sensitive segment and with this sensitive segment resistance value for nominal mass is formed the nominal particle of place sensitive segment, the centroid position that the nominal particle of each sensitive segment is formed jointly is the center of sensitive grid;
The sensitive segment all-in resistance of four sensitive grids is consistent, described four sensitive grids all-in resistance changing value of sensitive segment under identical strain is consistent, four sensitive grids be centrally located on straight line, this straight line is parallel to four any sensitive segment axis of sensitive grid, four sensitive grids are called left sensitive grid from left to right along this rectilinear direction, left and right sensitive grid, right left sensitive grid and right sensitive grid; Each sensitive segment axis is determined in plane, interdigital layout between left sensitive grid and left and right sensitive grid, also in interdigital layout between right left sensitive grid and right sensitive grid;
There is deviation at four sensitive grid centers in the axial direction, bias free in the horizontal, and the distance at left sensitive grid center and sensitive grid center, left and right is Δ x
_{1}; The distance at sensitive grid center, left and right and right left sensitive grid center is Δ x
_{2}, right left sensitive grid center and right sensitive grid centre distance are Δ x
_{3}, the distance at left sensitive grid center and right left sensitive grid center is Δ x
_{4}=Δ x
_{1}+ Δ x
_{2}, sensitive grid center, left and right and right sensitive grid centre distance are Δ x
_{5}=Δ x
_{2}+ Δ x
_{3}, left sensitive grid center and right sensitive grid centre distance are Δ x
_{6}=Δ x
_{1}+ Δ x
_{2}+ Δ x
_{3}.
In the present invention, because measuring bridge has four brachium pontis, just can be arranged in four electric bridges respectively by a graded by four sensitive grids, so claim this foil gauge to be full-bridge.Such as, the xsect of the sensitive segment of four sensitive grids is identical, and material is consistent, and the summation of the length of the sensitive segment of four sensitive grids is equal.
The distance, delta x at the center of left sensitive grid and left and right sensitive grid
_{1}be generally less than even much smaller than the length of each sensitive segment, described interdigital layout refers to: each sensitive segment axis institute of two sensitive grids in the plane, in the sensitive segment distribution straggly with two sensitive grids on sensitive segment axes normal direction, the order occur respectively the sensitive segment of two sensitive grids in the direction in which and number of times do not limit.Equally, between right left sensitive grid and right sensitive grid be also interdigital layout.But left sensitive grid and right left sensitive grid and right sensitive grid are all in interdigital layout; Left and right sensitive grid and right left sensitive grid and right sensitive grid are all in interdigital layout; Right left sensitive grid and left sensitive grid and left and right sensitive grid are all in interdigital layout; Right sensitive grid and left sensitive grid and left and right sensitive grid are all in interdigital layout.Because the relative position of left sensitive grid, left and right sensitive grid, right left sensitive grid and right sensitive grid is ensured quite accurately to be secured by foil gauge production technology, this is also that the present invention can detect the key one of of workpiece strain axis to partial derivative.
Utilize the linear relationship between metal material increased resistance value and strain, the first, as common foil gauge, may be used for monitor strain; In second, four sensitive grids, the ratio of distances constant at the resistance difference of any two and the center of these two sensitive grids reflects the axial local derviation of strain; 3rd, right sensitive grid and left sensitive grid resistance sum deduct left and right sensitive grid and are directly proportional to the axial second order local derviation of strain to the right difference of left sensitive grid resistance sum.
Technique should being noted, keeping each sensitive grid transition section all-in resistance and the transition section resistance variable quantity one under external strain to show heightens measuring accuracy, if the resistance of transition section and the lower resistance change of strain be can not ignore, also can be eliminated when detecting as systematic error.
Further, described metal strain plate also comprises cover plate, and described cover plate is covered in described sensitive grid and substrate.
Further again, described sensitive grid is wire form, foil, diaphragm type or thick-film type sensitive grid.
Further, described substrate is glued membrane substrate, glass fabric substrates, asbestos substrate, metallic substrates or temporary substrate.
Described four sensitive grids are arranged in substrate from left to right.Certainly, also can be other arrangement.
Beneficial effect of the present invention is mainly manifested in: can not only measuring workpieces surface strain, more effectively can detect the axial single order of surface strain and second-order partial differential coefficient.
Embodiment
Below in conjunction with accompanying drawing, the invention will be further described.
With reference to Fig. 1 ~ Fig. 3, can measured surface strain axis to the two interdigitated metal foil gauge of axial deviation full-bridge of local derviation, comprise substrate, metal strain plate described in it also comprises four sensitive grids, the two ends of each sensitive grid connect an extension line respectively, described substrate are fixed described four sensitive grids;
Each sensitive grid comprises sensitive segment and transition section, the two ends of described sensitive segment are transition section, described sensitive segment is elongated strip shaped, described transition section is tubbiness shape, the resistance of described sensitive segment is much larger than the resistance of described transition section, under same strain state, the increased resistance value of described sensitive segment is much larger than the increased resistance value of described transition section, and the increased resistance value of described transition section is close to 0;
All xsect centres of form of each sensitive segment form sensitive segment axis, this sensitive segment axis is straight line section, the axis being parallel of each sensitive segment and be arranged in same plane, and sensitive segment axis is determined in plane, along described sensitive segment axis direction namely axially, be laterally with axially vertical direction; All shape of cross section consistent size of each sensitive segment; Get the axis point midway of each sensitive segment and with this sensitive segment resistance value for nominal mass is formed the nominal particle of place sensitive segment, the centroid position that the nominal particle of each sensitive segment is formed jointly is the center of sensitive grid;
The sensitive segment all-in resistance of four sensitive grids is consistent, described four sensitive grids all-in resistance changing value of sensitive segment under identical strain is consistent, four sensitive grids be centrally located on straight line, this straight line is parallel to four any sensitive segment axis of sensitive grid, four sensitive grids are called left sensitive grid from left to right along this rectilinear direction, left and right sensitive grid, right left sensitive grid and right sensitive grid; Each sensitive segment axis is determined in plane, interdigital layout between left sensitive grid and left and right sensitive grid, also in interdigital layout between right left sensitive grid and right sensitive grid;
There is deviation at four sensitive grid centers in the axial direction, bias free in the horizontal, and the distance at left sensitive grid center and sensitive grid center, left and right is Δ x
_{1}; The distance at sensitive grid center, left and right and right left sensitive grid center is Δ x
_{2}, right left sensitive grid center and right sensitive grid centre distance are Δ x
_{3}, the distance at left sensitive grid center and right left sensitive grid center is Δ x
_{4}=Δ x
_{1}+ Δ x
_{2}, sensitive grid center, left and right and right sensitive grid centre distance are Δ x
_{5}=Δ x
_{2}+ Δ x
_{3}, left sensitive grid center and right sensitive grid centre distance are Δ x
_{6}=Δ x
_{1}+ Δ x
_{2}+ Δ x
_{3}.
The present embodiment can measured surface strain axis to the two interdigitated metal foil gauge of axial deviation full-bridge of local derviation, comprise a substrate 1, left sensitive grid 2 is had by the left and right order of Fig. 2, left and right sensitive grid 3, right left sensitive grid 4, right sensitive grid 5, eight extension lines 6, can also have cover plate (not giving expression in each accompanying drawing).
Left sensitive grid 2 can be fixed on substrate 1, left and right sensitive grid 3, right left sensitive grid 4 and right sensitive grid 5, for keep each sensitive grid fixing shape, position and size; Substrate 1 is very thin, thus the strain of surface of test piece is delivered to left sensitive grid 2 exactly, left and right sensitive grid 3, right left sensitive grid 4 and right sensitive grid 5.Substrate 1 can be glued membrane substrate, glass fabric substrates, asbestos substrate, metallic substrates and temporary substrate.Usually with cohering, weld, substrate is fixed on the tested position of test block by the mode such as ceramic spraying.Substrate 1 also can be printed on the lines that some are located for foil gauge.
The materials such as cover plate paper using or glue are made, and are covered in left sensitive grid 2, left and right sensitive grid 3, in right left sensitive grid 4, right sensitive grid 5 and substrate 1, play the protective seam of the effects such as protection against the tide, corrosion protection, loss prevention.
Lead-in wire 6 is for connecting sensitive grid and metering circuit, left sensitive grid 2, left and right sensitive grid 3, right left sensitive grid 4 and right sensitive grid 5 respectively have two to go between 6, to with foil and membrane type foil gauge, the left sensitive grid 2 that lead-in wire 6 is connected with it, left and right sensitive grid 3, right left sensitive grid 4 or right sensitive grid 5 combine as a whole.The pin of left sensitive grid 2 is 6-1 and 6-2, and the pin of left and right sensitive grid 3 is 6-3 and 6-4, and the pin of right left sensitive grid 4 is 6-5 and 6-6, and the pin of right sensitive grid 5 is 6-7 and 6-8.
Left sensitive grid 2, left and right sensitive grid 3, right left sensitive grid 4 and right sensitive grid 5, according to the difference of its metal sensitive material and processing technology, can be wire form, foil, diaphragm type, thick-film type.No matter which kind of left sensitive grid 2, left and right sensitive grid 3, the thickness of right left sensitive grid 4 and right sensitive grid 5 is all very little, makes left sensitive grid 2, left and right sensitive grid 3, the axial length of right left sensitive grid 4 and right sensitive grid 5 with it depend on the deformation of workpiece and change.The basic innovation of the present invention is left sensitive grid 2, left and right sensitive grid 3, and the cooperation between right left sensitive grid 4 and right sensitive grid 5, has following main points:
The first, four sensitive grids are arranged in substrate, is called left sensitive grid 2, left and right sensitive grid 3, right left sensitive grid 4 and right sensitive grid 5.
The second, left sensitive grid 2, left and right sensitive grid 3, right left sensitive grid 4 and right sensitive grid 5 all can be divided into multiple transition section 7 and multiple sensitive segment 8, and each sensitive segment 8 is connected to form sensitive grid by each transition section 7.Comparatively speaking, sensitive segment 8 is in elongated shape, and resistance is comparatively large and its resistance is comparatively responsive to strain; Described transition section 7, substantially in tubbiness shape, makes the resistance of described transition section very little and insensitive to strain, and under duty, resistance variations is close to 0, and therefore the summation of sensitive segment resistance is the all-in resistance of single sensitive grid substantially.Fig. 2 has marked sensitive segment 8 and transition section 7 in more detail from angle more clearly.
3rd, left sensitive grid 2, left and right sensitive grid 3, right left sensitive grid 4 is all identical with the xsect of the sensitive segment 8 of right sensitive grid 5, and left sensitive grid 2, left and right sensitive grid 3, the summation of right left sensitive grid 4 and respective sensitive segment 8 length of right sensitive grid 5 is identical.Ignore the resistance of transition section 7, the all-in resistance of left sensitive grid 2, left and right sensitive grid 3, right left sensitive grid 4 and right sensitive grid 5 is all equal, and four sensitive grids under identical strain, sensitive segment all-in resistance variable quantity should be consistent.
4th, the sensitive segment of each sensitive grid 8 is in elongated strip, and all xsect centres of form of each sensitive segment 8 form sensitive segment axis, and this sensitive segment 8 axis is straight line section, the axis being parallel of each sensitive segment 8 and be arranged in same plane.All shape of cross section consistent size of each sensitive segment 8.The axis point midway of each sensitive segment and sensitive segment resistance value form the nominal particle of place sensitive segment, and the centroid position that the nominal particle of each sensitive segment is formed jointly is the center of sensitive grid.
5th, overlook left sensitive grid 2, left and right sensitive grid 3, right left sensitive grid 4 and right sensitive grid 5, they all have axis of symmetry and axis of symmetry overlaps (x-axis in Fig. 2), left sensitive grid 2, left and right sensitive grid 3, right left sensitive grid 4 sensitive segment 8 respective with right sensitive grid 5 is all parallel with this axis of symmetry, and the sensitive segment 8 of each sensitive grid all distributes about this rotational symmetry.Therefore, left sensitive grid 2, left and right sensitive grid 3, right left sensitive grid 4 and right sensitive grid 5 are coaxial, namely axial deviation is only had without lateral deviation between their center, and left sensitive grid 2, left and right sensitive grid 3, the center of right left sensitive grid 4 and right sensitive grid 5 is all in x-axis.According to the vertical view of foil gauge in Fig. 2, the center of left sensitive grid 2 is at x-axis and y
_{lL}the intersection point of axle, the center of left and right sensitive grid 3 is at x-axis and y
_{lR}the intersection point of axle, the center of right left sensitive grid 4 is at x-axis and y
_{rL}the intersection point of axle, the center of right sensitive grid 5 is at x-axis and y
_{rR}the intersection point of axle.The line mid point at the center of left sensitive grid 2 and the center of left and right sensitive grid 3 is x-axis and y
_{l}the intersection point of axle, the line mid point at the center of right left sensitive grid 4 and the center of right sensitive grid 5 is x-axis and y
_{r}the intersection point of axle.
6th, left sensitive grid 2 and the interdigital layout of left and right sensitive grid 3, right left sensitive grid 4 and the interdigital layout of right sensitive grid 5, the center of these sensitive grids is all in same axis of symmetry x-axis.Can notice, the direct result that interdigital layout causes is that left sensitive grid 2 center and left and right sensitive grid 3 center are comparatively close, and distance is Δ x
_{1}; The distance at left and right sensitive grid 3 center and right left sensitive grid 4 center is Δ x
_{2}, right left sensitive grid 4 center and right sensitive grid 5 centre distance are Δ x
_{3}, the distance at left sensitive grid 2 center and right left sensitive grid 4 center is Δ x
_{4}=Δ x
_{1}+ Δ x
_{2}, left and right sensitive grid 3 center and right sensitive grid 5 centre distance are Δ x
_{5}=Δ x
_{2}+ Δ x
_{3}, left sensitive grid 2 center and right sensitive grid 5 centre distance are Δ x
_{6}=Δ x
_{1}+ Δ x
_{2}+ Δ x
_{3}, as shown in Figure 2.Due to left sensitive grid 2, left and right sensitive grid 3, the relative position of right left sensitive grid 4 and right sensitive grid 5 is ensured quite accurately to be secured by foil gauge production technology, and this is also that the present invention can detect the key one of of workpiece strain axis to partial derivative.
Described left sensitive grid 2, left and right sensitive grid 3, right left sensitive grid 4 and right sensitive grid 5 equal and opposite in direction, direction is coaxial without lateral deviation, is formed centrally six kinds of different axial distances in each sensitive grid.
The resistance of left sensitive grid is designated as R
_{lL}, the resistance of left and right sensitive grid is designated as R
_{lR}, the resistance of right left sensitive grid is designated as R
_{rL}, the resistance of right sensitive grid is designated as R
_{rR}.The resistance of four sensitive grids is equal is in a free state R
_{0}.Foil gauge of the present invention is placed in certain when having a strain surface, gets two sensitive grids wherein, must have one left both this, must have one right.It is R that the sensitive grid on the left side is designated as resistance
_{0}+ Δ R
_{l}, the sensitive grid resistance on the right is designated as R
_{0}+ Δ R
_{r}, two sensitive grid centre distances are Δ x
_{i}, i is one of 1 to 6.The difference of two sensitive grid center strains causes the difference of the two resistance change.The relation of sensitive grid resistance and surface strain is utilized to have:
$\frac{\∂\mathrm{\ϵ}}{\∂x}{|}_{x=\stackrel{\‾}{x}}=\underset{{\mathrm{\Δx}}_{i}\→0}{\mathrm{lim}}\frac{{\mathrm{\ϵ}}_{l}-{\mathrm{\ϵ}}_{r}}{{\mathrm{\Δx}}_{i}}\≈\frac{K({\mathrm{\ΔR}}_{l}-{\mathrm{\ΔR}}_{r})}{{\mathrm{\Δx}}_{i}}.---\left(3\right)$
Wherein i=1,2 ..., 6, ε
_{l}for the strain of the sensitive grid center on the left side, ε
_{r}for the strain at the sensitive grid place on the right,
be two sensitive grid line of centres point midways.Namely this is the principle of measured surface strain axis of the present invention to local derviation.Above formula is actual is numerical evaluation to local derviation, and according to the theory of numerical differentiation, this is with Δ x
_{i}/ 2 is step size computation differential, and the error that this local derviation calculates is no more than
rank, is
higher order indefinite small, precision comparison is high.Utilize four foil gauges of foil gauge of the present invention, limit the Δ x of foil gauge
_{1}=Δ x
_{3}, utilize the numerical computation method of the relation of sensitive grid resistance and surface strain and second order local derviation to have:
$\begin{array}{c}\frac{{\∂}^{2}\mathrm{\ϵ}}{\∂{x}^{2}}{|}_{x={x}_{0}}=\underset{{\mathrm{\Δx}}_{2}\→0}{\underset{{\mathrm{\Δx}}_{1}\→0}{\mathrm{lim}}}\left(\frac{{\mathrm{\ϵ}}_{LL}-{\mathrm{\ϵ}}_{LR}}{{\mathrm{\Δx}}_{1}}-\frac{{\mathrm{\ϵ}}_{RL}-{\mathrm{\ϵ}}_{RR}}{{\mathrm{\Δx}}_{1}}\right)/{\mathrm{\Δx}}_{2}\\ =\underset{{\mathrm{\Δx}}_{2}\→0}{\underset{{\mathrm{\Δx}}_{1}\→0}{\mathrm{lim}}}\frac{{\mathrm{\ϵ}}_{LL}-{\mathrm{\ϵ}}_{LR}-{\mathrm{\ϵ}}_{RL}-{\mathrm{\ϵ}}_{RR}}{{\mathrm{\Δx}}_{1}{\mathrm{\Δx}}_{2}}\\ \≈\frac{K\left({\mathrm{\ΔR}}_{LL}-{\mathrm{\ΔR}}_{LR}-{\mathrm{\ΔR}}_{RL}+{\mathrm{\ΔR}}_{RR}\right)}{{\mathrm{\Δx}}_{1}{\mathrm{\Δx}}_{2}}\end{array}---\left(4\right)$
Wherein x
_{0}for the intersection point of x-axis and y-axis in the point midway at foil gauge four sensitive grid centers and Fig. 2, ε
_{lL}for the strain of left sensitive grid center, ε
_{lR}for the strain of sensitive grid center, left and right, ε
_{rL}for the strain of left sensitive grid center, ε
_{rR}for the strain of sensitive grid center, left and right.
Coordinated by the present embodiment electric bridge to can be used for monitor strain, strain axis to local derviation, suppose that bridge input voltage is u
_{i}, output voltage is u
_{o}, Fig. 3 is shown in by the schematic diagram of measuring bridge.When without workpiece effects of strain, each arm resistance of electric bridge is labeled as R respectively according to clockwise direction
_{1}, R
_{2}, R
_{3}, R
_{4}, when obscuring also with these sign flag resistance place electric bridges.Each electric bridge can be laid sensitive grid or the resistance of foil gauge.Arrange identical with general foil gauge, if settle sensitive grid on multiple brachium pontis, to the order of each installation position, the requirement of strain difinite quality.During without workpiece effects of strain, the Output Voltage Formula of electric bridge is
${u}_{o}=\frac{{R}_{1}{R}_{3}-{R}_{2}{R}_{4}}{({R}_{1}+{R}_{2})({R}_{3}+{R}_{4})}{u}_{i};---\left(5\right)$
Now, bridge balance i.e. u is required
_{o}=0, so so-called bridge balance condition R must be met
_{1}r
_{3}-R
_{2}r
_{4}=0, the electric bridge of employing meets further
R
_{1}＝R
_{2}＝R
_{3}＝R
_{4}，(6)
Because, the first, satisfy condition (6) time, the highest according to relevant theoretical foil gauge sensitivity; The second, monitor strain or strain axis are set up to the equal requirement condition of method (6) of local derviation.When foil gauge, with external world's strain, strain also occurs, above-mentioned bridge balance condition is generally no longer set up, now
$\begin{array}{c}{u}_{o}=\frac{\left({R}_{1}+{\mathrm{\ΔR}}_{1}\right)\left({R}_{3}+{\mathrm{\ΔR}}_{3}\right)-\left({R}_{2}+{\mathrm{\ΔR}}_{2}\right)\left({R}_{4}+{\mathrm{\ΔR}}_{4}\right)}{\left({R}_{1}+{\mathrm{\ΔR}}_{1}+{R}_{2}+{\mathrm{\ΔR}}_{2}\right)\left({R}_{3}+{\mathrm{\ΔR}}_{3}+{R}_{4}+{\mathrm{\ΔR}}_{4}\right)}{u}_{i}\\ \≈\frac{{R}_{3}{\mathrm{\ΔR}}_{1}+{R}_{1}{\mathrm{\ΔR}}_{3}-{R}_{4}{\mathrm{\ΔR}}_{2}-{R}_{2}{\mathrm{\ΔR}}_{4}+{\mathrm{\ΔR}}_{1}{\mathrm{\ΔR}}_{3}-{\mathrm{\ΔR}}_{2}{\mathrm{\ΔR}}_{4}}{\left({R}_{1}+{R}_{2}\right)\left({R}_{3}+{R}_{4}\right)}{u}_{i}\\ \≈\frac{\left({\mathrm{\ΔR}}_{1}-{\mathrm{\ΔR}}_{2}\right)+\left({\mathrm{\ΔR}}_{3}-{\mathrm{\ΔR}}_{4}\right)}{4{R}_{1}}{u}_{i}\end{array}---\left(7\right)$
Due to Δ R
_{i}< < R
_{i}(i=1,2,3,4 events) first ≈, part Δ R that second ≈ ignores
_{1}Δ R
_{3}-Δ R
_{2}Δ R
_{4}also very little, and it can be made in engineering much smaller than comparatively reserve part.The voltage measurement strain that general available formula (7) obtains; Can convolution (3), formula (4) and formula (7) to the axial local derviation of strain, reasonable design arranges each brachium pontis sensitive grid and resistance can obtain with strain axis to single order local derviation or the linear magnitude of voltage u of second order local derviation
_{o}, this voltage is that feeble signal need be amplified.