Precision balances: general considerations on sensitivity and precision.

The ideal balance

The ideal balance is composed of a rigid body, moving without friction around a segment, defined by the contact of the central knife with the horizontal hard stone plates, and subjected to weight forces exerted on symmetrical, parallel and coplanar segments, defined by the contact of the pans hard stone plates with the inverted knifes at each extremity of the balance arms.
The baricenter of the ideal balance is exactly on the vertical and on the center of the central knife contact segment.
Under these conditions, intrinsic precision of the balance is absolute, and errors are only due to the imperfect detection of the equilibrium point, that is to the limits in sensitivity.
For these reason, only sensitivity is evaluated here.
Sensitivity can be expressed as the weight unbalance which generates the minimum observable deviation from the equilibrium position of the balance, or more generally, as weight per angular deflection of the balance body.
By this unit sensitivity becomes independent from the means used to detect the equilibrium point, naked eye or microscope or other instrumental devices.
It can be demonstrated that sensitivity, in the case of perfect balance, is dependent only on the position and value of the baricenter of the balance body.
If the baricenter of weight W is situated at a distance b from the central knife contact segment, and a weight w is added to one of the pans, at the extremity of the arm of length d, then the arm will lower and rotate by an angle α, according to the expression

d * cos α * w = W * b * sin α


and for α very low

d * w = W * b * α


and finally

α = d * w /( b * W)


or

w/ α = W * b / d


To fix the ideas, if W = 30 grams, b = 0.5 cm, d = 10 cm

w/ α = 30*0.5/10 = 1.5 grams/radians


and if the minimum rotation detectable corresponds to a lowering of the arm of 0.05 cm, α is then 0.05/10 or 0.005 radians, due to an unbalance of 1.5*0.005 = 0.0075 grams, or 7.5 milligrams.
We can also use the formula

w*d = W*b*delta/d


from which, delta being the minimum appreciable deviation from rest

w = W*b*delta/(d*d)


The real balance

Influence of weights position on the pans.

The real balance is a deformable body, moving with some friction around a segment (or may be some points along that segment), defined by the contact of the central knife with an approximately horizontal pair of hard stone plates, and subjected to weight forces exerted on generally non symmetrical, non parallel and non planar segments.
Despite these imperfections, it is however possible to build a reasonably precise real balance, by compensating or minimizing the defects derived from the imperfect construction.
The following analysis will clarify the effect of each defect on the precision and will suggest the method to follow in order to minimize or nullify such effects by proper adjustments.

To start, let us neglect the effects of deformability and friction, assuming that the weights applied to the balance body are light with respect to its rigidity, that the temperature is reasonably constant along the adjustments and the measurements, and that the friction is very limited by using sharp knifes acting on very hard plates.
We are then limited to consider the relative position of the three contact segments, the central on the two lateral.
Let us start from the ideal situation, and introduce gradually more and more deviations from that.

Suppose that the three segments are still parallel, but do not lay in the same plane; for instance the pan segments being in a plane at some distance from the central segment, but still equidistant from it.
In this case the accuracy of the balance is unmodified, only sensitivity is affected, in the sense that it is increased, till unstability, if the the plane is above the central segment; it is decreased if the plane is below.
Since that distance is generally low, the change in sensitivity can be neglected.
In addition to that, suppose now that the distance of the two segments from the center is not equal.
In this case the accuracy of the balance is affected in proportion to that inequality, but a good weight measurement can still be performed by the method of the double weighing, according to which sample and weights are applied to the same pan, then on the same arm of obviously identical length.
Moreover, even the classical way of weighing shows, for small length imperfections, reasonable results, even in the case of differential weighing, since the error affects in the same way the tare and the sample weight.
If for instance an empty vial weighs 5 grams and weighs 5.7 grams when filled, if an error of 1% is present and positive in sign, the empty and filled vials will weigh 5*1.01 = 5,050 grams and 5.2*1.01 = 5.252 grams respectively. The difference, corresponding to the material added, amounts to 5.252-5.050 = 0.202 instead of 0.200, or again a 1% error, quite reasonable.
The most dangerous error is generated by the lack of parallelism between the segments. This defect results in a dependence of the equilibrium condition from the position of the sample or of the weights on the pans.
In fact, if the weight is placed near one side of the pans, the force on the arm segment is not applied centrally, but more on one extremity.
If the arm segment is not parallel to the central one, the momentum will be different for the two extremes, and the resulting weight measurements will differ accordingly.
In the case of the mentioned vial weighing around 5 grams, for a sample of 200 mg a maximum weight difference of 2 milligrams is tolerated to keep the 1% precision, then the two momentum should not differ more than 2/5000, or 0.04%.
It means for a arm length of 10 cm a maximum difference of 100*0.0004 = 0.04 mm or 40 microns!
The strict parallelism is somewhat difficult to reach, since it requires two adjustments. The analysis of the problem seems to offer some possibility to a simpler adjustment. Let us start with the well recognized case of perfect parallelism, then we will introduce some variation.
Suppose a weight W applied on an arm of d length pivoted on a knife F (the black circle is the trace of the contact segment with the hard stone support. The same holds for P.



The above diagram shows the invariance of the momentum for a balance arm at two different application P and P' of the pan weight W, based on a perfect parallelism of the pan knife P and that of the balance body F.
Let us now consider the situation when the knife P' is not perfectly parallel to F, but inclined vertically. By applying the above formulas to the upper and lower extremes of the knife we find that the momentum on F is unchanged. The only condition to respect is that the pan knife lay in a plane parallel to the knife F.
This result holds also if the knife F is not horizontal. In fact by tilting the above image the relations remain invariable, only the force W is reduced to its projection on the plane perpendicular to F.
All this analysis seems to prove that the effect of the position of the bodies on the pans can be completely nullified, by positioning the knife in a plane perpendicular to the knife F, and not only on a line parallel to F.
Practically it means that by adjusting the position of one extreme of the knife P along only one direction it is possible to reach such a condition.
A line parallelism would require adjustments in two directions, much more inconvenient and mechanically more complex.
Having examined in detail and found a possible remedy for the most annoying cause of imprecision, I felt confident to design and build a balance of adequate precision for my chemical experiments, which, I recall, require the evaluation of the weight of a sample in the range of 200 mg held in a vial or crucible weighing up to 10 grams, with a precision of 1 percent.
In the design I tried to employ materials and supplies anybody can find easily.
More ambitious design can be derived from the exposed one, and I will at each crucial step suggest any possible improvement, which I warmly solicit from anybody expert in precision instrument, and I see many among AmSci members, to reach the precision offered by professional balances.
The cost of the balance I designed is practically neglegible, I believe about 50 $, but the construction will require somewhat like five or six weekends of careful and patient work. The result however compensates the efforts!

Appendix 1

Dependence of sensitivity from weights.
An ideal balance, having parallel and coplanar body and pan knifes, has a sensitivity which depends only from the distance between body knife and baricenter.
A real balance with a single direction pan knife adjustment does not have perfectly coplanar knifes, and this fact reflects itself in a dependence of sensitivity on the weights placed on the pans.
In short, if the pan knifes are on a plane above the body knife, the sensitivity increases with the weights, until an instability point is reached.
On the contrary, the sensitivity decreases with the applied weights, a convenient behaviour, since it assures a relatively constant relative sensitivity, which is what is generally needed. But let us see it in more precise terms.

Ideal case (coplanar knives)




d*(P+w) * cos α = b * W * sin α + d * P * cos α
d*(P+w) = b*W*tg α + d*P
d*w = b*W*tg α
w = (b/d) * W * tg α

Sensitivity = lim w/α = lim w/tg α = W*b/d
                     α->0         α->0

Real case 1 : pan knives higher then body knife (instable case)




d*(P+w)*cos(α - β) = b*W*sin α + d*P* cos(α + β)
d*(P+w)*(cos α*cos β+sin α*sin β) = b*W*sin α + d*P* (cos αa *cos β- sin α*sin β)
d*P*cos α*cos β+ d*P*sin α*sin β + d*w*cos(α- β) =
b*W*sin α + d*P* cos α *cos β- d*P* sin α*sin β
d*w*cos(α-β) = b*W*sin α - 2*d*P* sin α*sin β
w = (b*W*sin α - 2*d*P* sin α*sin β)/ (d *cos(α- β))
w/sin α = (b*W - 2*d*P*sin β)/(d*cos(α-β))
A positive sensitivity (metastable equilibrium) is present if (cos β being always positive)

b*W > 2*d*P*sin α


Sensitivity = lim w/α = lim w/sin α = (b*W - 2*d*P*sin β)/(d*cosβ)
                     α->0         α->0

The balance can be used for

P < b*W/2*d*sin α


A practical case


Let b = 0.5 cm, d = 10 cm, W = 30 grams, α = 0,1 (1 centimeter above the body knife)

P < 0.5*30/(*10*sin(0.1))= 15/(20*0. 1) = 15/2 = 7.5 grams


If a 1 gram sample is being weighed the sensitivity is

Sensitivity = (0,5*30 * - 2*10*1*0,1)/10*1 = 1.3 grams/rad


If a 7 gram sample is being weighed the sensitivity is

Sensitivity = (0,5*30 * - 2*10*7*0,1)/10*1 = 0.1 grams/rad

This means ten times higher.

Real case 2 : pan knives lower then body knife (stable case)




d*(P+w)*cos(α + β) = b*W*sin α + d*P* cos(α - β)
d*(P+w)*(cos α*cos β - sin α*sin β) = b*W*sin α + d*P* (cos α* cos β + sin α*sin β)
d*P*cos α*cos β - d*P*sin α*sinβ + d*w*cos(α- β) =
b*W*sin α + d*P* cos α*cos β + d*P* sin α*sin β
d*w*cos(α-β) = b*W*sin α + 2*d*P* sinα*sinβ
w = (b*W*sin α + 2*d*P* sin α*sin β)/ (d *cos(α- β))
w/sin α = (b*W + 2*d*P*sin β)/(d*cos(α-β))
A practical case
If a 1 gram sample is being weighed the sensitivity is

Sensitivity = (0,5*30 * + 2*10*1*0,1)/10*1 = 1.7 grams/rad


If a 7 gram sample is being weighed the sensitivity is

Sensitivity = (0,5*30 * + 2*10*7*0,1)/10*1 = 2.9 grams/rad


Conclusions
From the above formulas we can summarize as follows:
The sensitivity of a balance is in general

Sensitivity = lim w/ α = lim w/sin α = (b*W - 2*d*P*sin α)/(d*cos α)
                      α→0         α→0

where α is positive if the pan knives are higher than the central, negative if are lower, zero in the perfect balance.
This result can be interpreted also as follows.
The term
b*W - 2*d*P*sin α
can be written also as
((b*W - 2*d*P*sin α)/(b+2*P))*(b+2*P)
but
(b*W - 2*d*P*sin α)/(b+2*P) = b'
is the distance from the body knife of the combined baricenter (body+pans)
and
b+2*P = W'
is the combined weight.
On the other side
d*cos α = d'
is the projection of the arm d on the horizontal,
then sensitivity can be in any case written as
Sensitivity = W' * b' / d'

Notes on low cost balance project


Position of the balance body suspension : 0.4*cos(30°) = 0.3464 cm from the top sheet
Position of the pan suspension: halfway the vertical sheet (2.5 mm from the bottom)=
0.25+cos(30°)= 0.25+0.8660 = 1.116
Then distance between the two = 1.116-0.3464 = 0.7696 cm
Calculation of the baricenter
ElementDistance from the top sheet(cm)Surface (cm2)
Top sheet020+2*0.25 = 20.5
Side sheets0.5 * cos(30°)2*20 = 40
Bottom sheet1*cos(30°)+0.252*10 = 20
Vertical strip5+cos(30°)0.4*10 = 4
Needle10+cos(30°)0.4*1+0.4*0.35=0.54
85.14
Calculation of the distance of the baricenter from top sheet=
h = (0*20.5 + 0.5 * cos(30°)*40 + (cos(30°) + 0.25) * 20)+ (5+cos(30°))* 4 + (10+cos(30°))*0.54/85.14 =
(44.54*cos(30°) + 30.4)/85.14 =(44.54*0.8660 +30.4)/85.14 = 0,81 cm
Distance between baricenter and suspension point (arm of the baricenter)=
b = 0.81 - 0.3464 = 0.4636
Total weight of the body
W = 80.5 * G grams (G for my paper is 0.008 g/cm2 then W = 0.644 g)
Sensitivity ( of the body alone) = W*b/d = 0.644 * 0.4636 * / 10 = 0,02985 g/rad
Calculations of the weight of the pans




ElementSurface (cm2)
Base16 =16
Side2*((4+1)*6.5)=65
top2*12
83
Weight = 83*0.008 = 0,664 g
Sensitivity of the complete balance

SENSITIVITY = (b*W + 2*d*P*sin β)/(d*cos β)


since β in rad is 0.7696/10 = 0.07696 then sin β is 0,0769 cos β is 0.997
sensitivity (at zero weights)
= (0,4636*0,644 + 2*10*0,664*0.0769)/(10*0.997) = 0.1324 g/rad
sensitivity (with 1 gram weight)
= (0,4636*0,644 + 2*10*1,664*0.0769)/(10*0.997) = 0.2866 g/rad
or, for a 0.5 mm displacement, equivalent to 0.005 rad, a weight unbalance of 0.2866*0.005 = 0.0014 g is required.