Enhancing the handling of standard substitution weights on a hydrostatic weighing apparatus


ACTA IMEKO | www.imeko.org December 2020 | Volume 9 | Number 5 | 27 

ACTA IMEKO 
ISSN: 2221-870X 
December 2020, Volume 9, Number 5, 27 - 32 

 
 

ENHANCING THE HANDLING OF STANDARD SUBSTITUTION 

WEIGHTS ON A HYDROSTATIC WEIGHING APPARATUS 
 

D. Simões 1, A. Furtado 2, J. Pereira2, Rui F. Martins3 

 
1Faculdade de Ciências e Tecnologia, Universidade Nova de Lisboa, Caparica, Portugal, dm.simoes@campus.fct.unl.pt  

2 Instituto Português da Qualidade (IPQ), I.P., Caparica, Portugal, afurtado@ipq.pt    
3 UNIDEMI, Faculdade de Ciências e Tecnologia, Universidade Nova de Lisboa, Caparica, Portugal, rfspm@fct.unl.pt  

 

Abstract: 

The hydrostatic weighing apparatus is a first-

level method used to measure the density of solids 

and liquids. At the Portuguese Institute for Quality 

the hydrostatic tests for the determination of the 

liquids’ density implies the multiple manual 

placement of a set of standard substitution weights 

on apparatus’ balance pan, leading to undesirable 

perturbations in the weighing environment. Also, by 

this the eccentricity effects on the balance, 

contribute for an unwanted increase of the 

measurement uncertainty. To overcome these 

phenomena, an automated mechanism was designed 

and implemented, which allows the placing and 

recollection of the set of standard weights on the 

balance. To validate the new mechanism, tests to 

ultrapure water at 20 °C were executed before and 

after its implementation and the obtained results 

were compared. 

Despite a 3-fold increase in density 

measurement uncertainty, mainly due to the 

exposure of the standard substitution weights to air 

convection currents, the execution of hydrostatic 

tests was improved, not only for the fact that the 

measurements are now carried out quicker, but also 

for the fact that the operator only needs to intervene 

once during the whole test. Moreover, the smaller 

absolute deviation from the reference density value 

for ultrapure water at 20 °C results also in a 

normalised error 𝐸n lower than 1 (0.3), thus 
validating the designed automated mechanism for 

the handling of standard substitution weights. 

Keywords: hydrostatic weighing; density of 

liquids; automated mechanism; substitution weights 

1. INTRODUCTION 

The hydrostatic weighing is the first-level 

measurement method used to determine the density 

of liquids. It is essentially based in the Archimedes’ 

principle, which states that a body, partially or fully 

immersed in a fluid, is subject to an upward force 

named buoyancy, equal to the weight of the fluid 

displaced by the body. Therefore, the loss in its 

apparent mass is the mass of fluid displaced by the 

body. In this method, the immersed body is 

commonly known as “sinker”. The buoyancy force, 

B, can be calculated as follows:  

𝐵 = 𝜌L 𝑉S 𝑔 (1)   

where 𝜌L is the density of the liquid, in kg m
-3, 𝑉S 

the volume of the sinker, in m3, and 𝑔  the 
gravitational acceleration, in m s-2. As the buoyancy 

force acts against the weight of the sinker, the 

resulting force measured by a balance to which it is 

connected results in a smaller value of mass when 

compared to the mass of the sinker when it is 

weighed in the air. This smaller value of mass is 

often named as the “apparent mass”. If the real mass 

𝑚S and volume 𝑉S of the sinker are known, and its 
“apparent mass” 𝑚S

a  determined by hydrostatic 

weighing, the density of the liquid in which the 

sinker is immersed can be calculated according to 

equation (2). 

𝜌L =
𝑚S − 𝑚S

a

𝑉S
 (2)   

A measuring instrument or apparatus which uses 

a comparison method against standards allows a 

decrease in the uncertainty by eliminating 

environmental and instrumental variations. In the 

case of the hydrostatic weighing apparatus, the 

apparent mass of the sinker is compared against the 

mass of standard substitution weights. This 

procedure is named “substitution method”. 

Therefore, the apparent mass of the sinker can be 

calculated according to equation 5 [1]. 

𝑚S
a = 𝑊W + (𝑊Sread − 𝑊Wread ) (1 −

𝜌A
8000

) (3)   

In the previous equation, 𝑊W  denotes the mass 
value of the used set of standard substitution 

weights, in kg; 𝑊Sread  and 𝑊Wread  denote the 
readings of the balance when loaded with the sinker 

and with the set of standard weights, in kg, 

respectively. The term (1 −
𝜌A

8000
) represents the air 

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ACTA IMEKO | www.imeko.org December 2020 | Volume 9 | Number 5 | 28 

buoyancy correction, where 𝜌A is the density of air, 
in kg m-3, in the weighing region. 

The mass value of the used set of standard 

weights can be calculated based on the parameters 

given by the respective calibration certificate and 

according to equation 4 [1]. 

𝑊W = 𝑚W (1 −
𝜌A
𝜌W

) (4)   

where 𝑚W and 𝜌W correspond to the mass and the 
density values of the standard weights given by their 

calibration certificate, in kg and kg m-3, 

respectively. 

The density of air is calculated with the CIPM-

2007 formula [2]. 

2. MATERIALS AND METHODS  

A hydrostatic weighing apparatus features 

essentially a balance, suitable for underfloor 

weighing, an underfloor mounted suspension, with 

a holder for the sinker, and a vessel which contains 

the test liquid (Figure 1).  

A mechanism for IPQ’s hydrostatic weighing 

apparatus was designed to allow the automated 

loading/unloading of the standard substitution 

weights on the balance (Figure 2). This mechanism 

mainly consists of a fork system, vertically actuated 

by an electric linear axis, which picks and places a 

standard disc-shaped weight set on a holder, 

connected to the hook located underneath the 

balance. The standard weights are loaded on the fork 

by a qualified operator. It is needed only once before 

the density measurement. 

 
Figure 1: Complete hydrostatic weighing apparatus in 

use at IPQ. 

 
Figure 2: Overview of the automated mechanism 

designed for the handling of standard substitution 

weights implemented in IPQ’s hydrostatic weighing 

apparatus. 

The mechanism is located between the balance 

and the thermostatic bath. All the designed 

components of the mechanism were manufactured 

in AISI 316 austenitic stainless steel. 

The design and simulation of the mechanism 

were accomplished through the solid modelling 

computer-aided design (CAD) and computer-aided 

engineering (CAE) program SolidWorks® and 

SolidWorks Simulation®, respectively. 

2.1. Holder for the Standard Substitution 
Weights 

The holder for the standard substitution weights 

(Figure 3) consists in a base, made of a 5 mm thick 

rectangular sheet, where the set of standard weights 

is placed. There are four 8.5 mm wide slots in the 

base which allow the passing of the tines of the fork 

when this element is picking or placing the weights 

on the holder. The holder sustains the weights 

through three 2.5 mm wide tines. In order to 

accommodate the standard weights, there are two 

1.25 mm deep cavities at the base for the larger 

diameter discs (∅48 mm and ∅22 mm), with a 
1.5 mm radial clearance between each cavity and 

disc. 

The holder has a wire structure welded to the 

base with a circular ∅2.5 mm diameter cross-section, 
having a hook at the top to connect this component 

to the balance and a ring at the bottom to link it to 

the suspension. 

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ACTA IMEKO | www.imeko.org December 2020 | Volume 9 | Number 5 | 29 

 

Figure 3: CAD model of the base of the holder for the 

standard substitution weights (at left) and end-product of 

the component (at right). 

To simulate the stresses and strains on the 

holder, a static load of 2.5 N was applied at the 

surface of the cavity which supports the larger 

diameter discs. This value corresponds to the mass 

of the silicon sphere when weighed in the air 

(approximately 234 g) with a safety factor applied. 

The effect of the gravitational acceleration was also 

taken into account (g = 9.81 m·s-2). 

Figure 4 shows the static stress-strain analysis of 

the holder for the standard substitution weights. 

 

Figure 4: Stress (at left) and strain (at right) analyses of 

the holder for the standard substitution weights. The 

deformed shape is not at the real scale. 

To prevent the material from yield, the condition 

given by equation (5) must be verified: 

𝜎v ≤ 𝜎max =
𝜎y

𝑆𝐹
 (5)   

where 𝜎v is the maximum von Mises stress, in MPa, 
determined by the Finite Element Method (FEM), 

𝜎max is the maximum allowable stress of design, in 
MPa, 𝑆𝐹 is a safety factor of 𝑆𝐹=1.5 and 𝜎y is the 

Yield Stress (290 MPa). Given that the maximum 

value of the von Mises stress, as observed in Figure 

4, is approximately 𝜎v =20 MPa, the von Mises 
stress is substantially lower than the maximum 

allowable design stress (𝜎v = 20 MPa ≪  𝜎max =
193.33 MPa)  , hence concluding that the holder 
operates in the elastic domain. 

Given the required precision to the motion of the 

fork to allow its tines to pass through the slots of the 

holder in order to successfully exchange the 

standard weights between these two components, it 

is crucial that the displacements caused by the 

weight of the discs when supported on these 

elements are not excessive. This factor could 

potentially lead to an undesirable contact between 

the fork and the holder, preventing the correct 

operation of the mechanism. It was verified that the 

maximum displacement on the holder had a value of 

approximately 0.24 mm and is located on the 

bottom ring for suspension connection. This value 

does not compromise the functioning of the 

mechanism. 

2.2. Fork for the Handling of Standard 
Substitution Weights 

The fork for the handling of standard substitution 

weights (Figure 5) consists of an arm, manufactured 

from a 5 mm thick rectangular sheet with 

374.65 mm long. The end of the arm, where the 

standard weights are supported, has four tines with 

a width of 5.5 mm each, ensuring a total horizontal 

clearance of 3 mm between these tines and the 

existing slots in the holder for the standard weights. 

This same clearance is also verified between the 

tines of the holder and the slots in the fork. In order 

to compensate for any possible misalignment 

between the fork and the holder from their working 

positions due to the movement of the mechanism, 

the existence of these considerable clearances is 

mandatory. The fork is also provided with two 

cavities of similar dimensions to the ones designed 

for the holder of the standard weights. 

The fork is connected to the carriage of the 

electric linear axis through a rectangular sheet of 

dimensions 60 mm × 54 mm × 5 mm, welded to the 

arm. This fixing sheet is then fastened to the 

carriage of the linear axis through four M4 bolts. To 

reduce the deformations at the end of the fork due 

to the weight of the discs, a rib was welded at the 

bottom of this component to the arm and the fixing 

sheet. 

 

Figure 5: CAD model of the tip of the fork for the 

handling of standard substitution weights (at left) and the 

respective end-product (at right). 

Figure 6 shows the stress-strain analysis of the 

entire mechanism (set consisted by the fork, a 

simplified geometry model of the electric linear axis 

and its respective fixing bars for fastening the 

mechanism to the supporting structure of the 

balance). 

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ACTA IMEKO | www.imeko.org December 2020 | Volume 9 | Number 5 | 30 

 

Figure 6: Stress (at left) and strain (at right) analyses of 

the entire mechanism, consisted by the fork, a simplified 

geometry model of the electric linear axis and its 

respective fixing bars. The deformed shape is not at the 

real scale. 

Applying the previous methodology, the von 

Mises stress is maximum on the rib of the fork with 

a value of approximately 9.4 MPa. According to the 

design criteria defined by equation (5), it is 

concluded that the fork is distant from yielding 

(𝜎v = 9.4 MPa ≪ 𝜎max = 193.33 MPa). 
The maximum of displacements is on the edge of 

the fork’s tines with a value of 0.36 mm. This is 

small enough to meet the required precision of the 

mechanism. 

2.3. Electric Linear Axis 
Based on the reaction forces at the support of the 

fork, i.e., the fixing sheet, an electric linear axis 

from Igus® (Figure 7) was chosen for the 

mechanism for the handling of the standard weights. 

The linear axis’ carriage is driven by a lead-screw, 

coupled to a NEMA17 Litze stepper motor, being 

the preferred option for the required precision. 

 

Figure 7: CAD model of the lead-screw driven electric 

linear axis from Igus® (size 0630) (at left); Real model 

of the electric linear axis (at right). 

The NEMA17 Litze stepper motor of the linear 

axis is controlled through the TB6600 driver, 

connected to an Arduino UNO R3 board (Figure 8). 

The Arduino platform was chosen to program the 

stepper motor because it is an open-source software, 

having a wide integration with different 

programming languages, particularly with 

LabVIEW®. The stepper motor is powered by a 

24 V DC power supply, shown as well in Figure 8. 

 

Figure 8: TB6600 stepper motor driver (at left), Arduino 

UNO R3 board (at the centre) and 24 V DC power supply 

(at right). 

To hold the electronic components in place, and 

protect the respective connections, these were 

installed in the interior of a proper aluminium 

housing (Figure 9). 

 

Figure 9: Electric control circuit of the stepper motor 

mounted in the interior of the housing (at left), existing 

connections at the back of the housing (at the centre) and 

ventilation holes on the top of the housing (at right). 

The NEMA17 stepper motor of the electric 

linear axis was programmed using the LINX 

interface for LabVIEW®. This interface is 

responsible to establish the connection between the 

Arduino UNO R3 board and the LabVIEW® 

software. The developed LabVIEW® program to 

control the stepper motor was integrated within an 

already existing program which controls the 

hydrostatic test, acquiring all the data from the 

measuring instruments. 

It is important to point out that the carriage of the 

electric linear axis always moves at a constant speed 

of 0.223 mm·s-1. It is essential that the actions of 

placement and removal the standard weight set by 

the fork are carried out at low speeds to minimise 

the oscillations in the holder and to prevent the 

weights from being dropped. 

3. RESULTS AND ITS DISCUSSION 

The validation of the mechanism for the 

handling of standard weights was performed 

through the carrying out of hydrostatic tests to 

ultrapure water (Type 1), at 20 °C. The choice of 

this liquid was because ultrapure water is a density 

standard whose properties are well known at a 

certain temperature range [3]. The sample used in 

the tests was degassed using a vacuum oven. 

The results of density and its respective 

expanded uncertainty, obtained from the 

measurements to ultrapure water at 20 °C, both with 

the hydrostatic weighing apparatus pre and post-

optimised, are given in Table 1. The term “pre-

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ACTA IMEKO | www.imeko.org December 2020 | Volume 9 | Number 5 | 31 

optimisation” refers to the original apparatus, as it 

was, while the “post-optimisation” term refers to the 

improved apparatus with the enhancement of the 

handling of standard substitution weights. The 

reference density value for ultrapure water at this 

temperature, as stated by Tanaka [3], is also listed 

in the same table.  

Table 1: Results of density values 𝜌𝐿  and respective 
expanded uncertainty 𝑈𝜌𝐿  (absolute and relative values) 

obtained from measurements with the hydrostatic 

weighing apparatus, pre and post-optimised, of an 

ultrapure water sample at 20 °C. The reference value 

observed by Tanaka [3] is also shown. 

 

Figure 10 illustrates the previous results in a 

scatter plot to provide for a better comparison 

between them. 

 

Figure 10: Scatter plot of the density of ultrapure water 

(Type 1), at 20 °C, measured by the hydrostatic weighing 

(HW) apparatus pre and post-optimised and the 

respective value observed by Tanaka [3]. The vertical 

bars represent the values of expanded uncertainty relative 

to each measured density value. 

Another indicator which allows for a clearer 

comparison of results is the relative deviation in 

density 𝛿𝜌 , in ppm. Considering the density value of 

ultrapure water, at 20 °C, observed by Tanaka [3] as 

reference, the relative deviation of the obtained 

results by hydrostatic weighing from the reference 

value is given as follows:  

𝛿𝜌 = |
𝜌HW − 𝜌ref.

𝜌ref.
| × 106 (6)   

where 𝜌HW  and 𝜌ref.  are the density values of the 
liquid experimentally measured by hydrostatic 

weighing and determined by Tanaka, respectively. 

The term |𝜌HW − 𝜌ref.|  can be described as the 

absolute deviation in density, ∆𝜌.  

The validation procedure of a certain measuring 

method implies the calculation of the normalised 

error 𝐸n of a measurement result against a reference 
value. The results are satisfactory if the normalised 

error is less than or equal to 1, i.e., 𝐸n ≤ 1. 
Assuming once again the density value for ultrapure 

water at 20 °C determined by Tanaka [3] as 

reference, the normalised error yielded by the 

results obtained by hydrostatic weighing can be 

calculated according to equation (7). 

𝐸n =
|𝜌HW − 𝜌ref.|

√𝑈𝜌HW
2 + 𝑈𝜌ref.

2

 
(7)   

In the equation above, 𝑈𝜌HW  and 𝑈𝜌ref.  are the 

expanded uncertainty values in the measurements 

by hydrostatic weighing and in the reference value, 

observed by Tanaka [3], respectively. 

The absolute and relative deviations in density, 

of the obtained results by hydrostatic weighing to 

ultrapure water at 20 °C from the reference value 

observed by Tanaka [3], both in the stages of pre 

and post-optimisation of the apparatus, are listed in 

Table 2. The normalised error of the method is also 

shown. 

Table 2: Absolute, ∆𝜌 , and relative, 𝛿𝜌 , deviations in 

density, of the obtained results by hydrostatic weighing 

to ultrapure water (type 1) at 20 °C from the reference 

value observed by Tanaka [3], both in the stages of pre 

and post-optimisation of the apparatus. The normalised 

error, 𝐸𝑛 , is also listed. 

 

Comparing the results obtained in the two 

different stages, with the reference value suggested 

by Tanaka [3], a smaller absolute deviation in the 

density of the liquid ∆𝜌, was verified in the stage of 
post-optimisation (0.004 kg m-3 (3.9 ppm)) 

relatively to the pre-optimisation (0.0126 kg m -3 

(12.6 ppm)). These results yield in a normalised 

error 𝐸n of 2.9 for the stage of pre-optimisation of 
the apparatus and 0.3 for the stage of post-

optimisation. The main factor which could have 

potentially led to this difference is the fact that the 

ultrapure water sample used, despite being the same 

at pre and post- optimisation tests, may varied in its 

gas saturation [3], due to different ambient 

conditions, and therefore resulting in different 

density deviations from the Tanaka’s reference 

value. There are other possible causes which could 

lead to a more significant deviation in density from 

the reference value, such as the lack of knowledge 

of the chemical composition of the sample (in 

Ultrapure 

Water 

𝝆𝐋 𝑼𝝆𝐋  

/ kg·m-3) /kg·m-3 / ppm 

Pre-

optimisation 
998.219 3 

0.004 2 

(k =1.97) 
4.2 

Post-

optimisation 
998.211 

0.013 

(k =2.07) 
13.2 

Tanaka 998.206 7 
0.000 83 

(k =2.00) 
0.8 

Stage 
∆𝝆 𝜹𝝆 𝑬𝐧 

/ kg·m-3 / ppm 

Pre-

optimisation 
0.0126 12.6 2.9 

Post-

optimisation 
0.004 3.9 0.3 

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ACTA IMEKO | www.imeko.org December 2020 | Volume 9 | Number 5 | 32 

particular, the isotopic composition) [3]; or other 

unknown factors, such as mathematical 

distributions, heat or contamination of the sample. 

Concerning the density expanded uncertainty of 

the liquid 𝑈𝜌L , a value approximately 3 times 

greater was verified in the hydrostatic tests after the 

optimisation of the apparatus (0.013 kg m-3) when 

compared against the value obtained from the pre-

optimisation stage (0.0042 kg m-3). This increase 

was mainly caused by a greater standard uncertainty 

in the repeatability of the readings of the balance 

𝑢rep., in the stage of post-optimisation (2.6 × 10
-4 g) 

when compared against the pre-optimisation stage 

(5.0 × 10-5 g). This phenomenon was primarily due 

to the exposure of the holder and respective standard 

weights to convection currents and vibrations from 

the bath, disturbing the stability of the set and 

leading to unstable readings of the balance. 

Another modification made to IPQ’s hydrostatic 

weighing apparatus after its optimisation was the 

installation of the pressure, temperature and relative 

humidity sensors for the atmospheric air in the 

operating area of the mechanism for the handling of 

standard weights. These sensors were previously 

installed in the interior of the campanula which 

covers the balance, so they had to be moved to a 

location closer to where the standard weights are 

now placed in order to yield measurements with a 

higher reliability degree, resulting in a more precise 

calculation of the buoyancy force originated by the 

air. Given that the operation region of the 

mechanism is exposed to the surrounding 

environment, a significant variation in the air 

conditions within this region may occur, leading to 

a greater dispersion in the measurements of this 

component, thus, resulting in a larger uncertainty in 

repeatability. Nonetheless, a slightly smaller 

uncertainty in the measurements of the air was still 

verified in the stage of post-optimisation of the 

apparatus. 

4. CONCLUSION 

Despite a 3-fold increase in density 

measurement uncertainty 𝑈𝜌𝐿 , from 0.0042 kg m
-3 

to 0.013 kg m-3, the smaller absolute deviation from 

the reference density value for ultrapure water at 

20 °C results also in a satisfactory normalised error 

𝐸n  of 0.3, thus validating the designed automated 
mechanism for the handling of standard substitution 

weights. Furthermore, the execution of hydrostatic 

tests was improved with the introduction of a 

mechanism for the handling of standard weights, not 

only by the fact that the measurements are now 

carried out quicker, but also that the operator only 

needs to intervene once during the whole test. A 

better stabilisation of the weighing environment 

(i.e., in air conditions) was also observed, leading to 

a slight decrease in the uncertainty in the density of 

the liquid. 

5. ACKNOWLEDGEMENT 

This work was developed under the framework 

of the EMPIR project “17RPT02-rhoLiq” which is 

carried out with funding of European Union under 

the EMPIR. The EMPIR is jointly funded by the 

EMPIR participating countries within EURAMET 

and the European Union. 

6. REFERENCES 

[1] H. Fehlauer, H. Wolf, “Density reference liquids 
certified by the Physikalisch-Technische 

Bundesanstalt”, Meas. Sci. Technol., vol. 17, 

pp. 2588–2592, 2006. 

[2] A. Picard, R. S. Davis, M. Gläser, K. Fujii, 
“Revised formula for the density of moist air 

(CIPM-2007)”, Metrologia, vol. 45, pp. 149–155, 

2008. 

[3] M. Tanaka, G. Girard, R. Davis, A. Peuto, 
N. Bignell, “Recommended table for the density of 

water between 0 °C and 40 °C based on recent 

experimental reports”, Metrologia, vol. 38, 

pp. 301-309, 2001. 
 

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