Investigations on nonlinear deformation phenomena in a force calibration system


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ACTA IMEKO 
ISSN: 2221-870X 
December 2020, Volume 9, Number 5, 150 - 155 

 
 

INVESTIGATIONS ON NONLINEAR DEFORMATION PHENOMENA IN A 

FORCE CALIBRATION SYSTEM 
 

Y. Wang1, N. Rogge2, S. Vasilyan2, T. Fröhlich2 

 
1 Tianjin University, Tianjin 300072, China, 3016202030@tju.edu.cn  

2 Technische Universität Ilmenau, 98693 Ilmenau, Germany, norbert.rogge@tu-ilmenau.de  

 

Abstract: 

Previous investigations show, in force 

calibration and measurement systems that are based 

on electromagnetic force compensation (EMFC), 

deformations of the flexure hinges can cause 

significant contributions to the uncertainty of the 

system. A Simulink model of the Planck-Balance 2 

(PB2) is established in MATLAB according to the 

CAD model. The results are compared to simplified 

analytical models and the results obtained from 

measurement. 

Keywords: Planck-Balance; EMFC balance; 

mechanical nonlinearities; mechanical simulation; 

force calibration 

1. INTRODUCTION 

EMFC balances are state of the art for weighing 

systems with low uncertainties. However, they can 

also be modified and used as force measurement 

systems. A recent example is the calibration of laser 

power by the measurement of the photon 

momentum [1]. Another application, where a 

modified EMFC load cell is used, is the Planck-

Balance (PB). It is a table top Kibble balance that 

was developed in cooperation between the 

Physikalisch-Technische Bundesanstalt (PTB) and 

the Technische Universität Ilmenau. In contrast to a 

standard EMFC balance, an interferometer and a 

second voice coil actuator are attached to the load 

carrier. The additional actuator is calibrated in the 

velocity mode and utilized in the force mode to 

compensate the gravitational force of the test weight 

[2].  

The problem that is addressed in this work will 

be exemplary shown for the PB2 system. This 

system is the version of the Planck-Balance that 

aims to achieve uncertainties suitable for calibration 

class E2 mass standards in the range from 1 mg to 

100 g. In the force mode of the current PB2 system, 

the gravitational force is compensated by an 

external voice coil actuator that is attached to the 

load carrier of the system, while the position that 

indicates a force equilibrium is generated by the 

internal position sensor of the balance. Preliminary 

investigations showed that the there is a different 

displacement on the load carrier side of the lever 

depending on the load that is placed onto it. As 

expected, these displacements are smaller than in 

the normal use of the EMFC balance where the 

compensation force is generated at the position 

sensor side of the lever but, as shown in Figure 1, 

they can be nonlinear towards the applied load. 

 
Figure 1: Difference of load carrier displacement for load 

changes of different masses 

The shown difference of the displacement 

between loaded and unloaded state is measured by 

the interferometer of the PB2 system, where the 

error bars represent the standard deviation of the 

repeated determinations of this difference. 

The models that are presented in this work are 

used to identify how deformations of the 

mechanical system contribute to the observed 

nonlinear behaviour. Furthermore, an improved 

understanding of the effects will help to improve the 

design of the system and the correction of errors 

caused by the deformation of the mechanical 

subsystem. 

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2. MODEL OF THE BALANCE 

In order to investigate the nonlinear relation 

between mass and load carrier displacement, a rigid 

body model based on the one described in [3] is set 

up and modified. The model focuses on the 

investigation of geometrical nonlinearities, which 

are able to cause the observed deformation effects. 

The mechanical model is extended by the electronic 

hardware and the PID algorithm, which is used to 

balance the transmission lever after a change of the 

external mass on the weighing pan (see Figure 2). 

The model is mainly made up of five sections: 

mechanical system, position sensor, PID controller, 

U/I converter and voice coil actuator. 

 

 
Figure 2: Main components of the balance model 

2.1. Mechanical Subsystem 
The mechanical system is based on the CAD 

model and mechanical drawings of the device. 

 
Figure 3: Simulink model of the mechanical subsystem 

The design of the mechanical system module is 

shown in Figure 3. The simulation environment is 

ideal. The value of the acceleration of gravity g is 

taken from a previous measurement, and the effects 

of temperature, humidity or air damping in the 

environment are not considered. 

The damping of Planck-Balance during the 

measurement mainly comes from the following 

three sources: the friction of the flexure hinge, air 

friction and electromagnetic damping of the coil. 

Compared to the other two types of damping, the air 

friction damping has a relatively small effect and 

can be ignored. The electromagnetic damping will 

be discussed later. The damping caused by the 

friction force in the flexure hinge is corrected by an 

empirical value obtained from fitting the 

experimental data to the simulation result.  

 The position sensor is placed at the end of the 

transmission lever in the Planck-Balance 

mechanical system, in order to detect the 

displacement of the transmission lever. Its 

characteristics are obtained from experimental data 

and a saturation is introduced to the model to 

represent the mechanical end stops. 

2.2. PID Controller 
The parameter setting of this module mainly 

refers to the Ziegler-Nichols method [4]. At first, set 

both the integral parameter KI and the differential 

parameter KD to zero. Then adjust the proportional 

parameter KP from small to large until the stable 

oscillation occurs. Read the KP value at this time and 

record it as Ku. Record the oscillation period as Tu. 

Then calculate the theoretical parameter values of 

KP, KI and KD according to the Ziegler-Nichols 

method parameter table. Finally, adjust the 

parameters based on the simulation results. 

2.3. Voice Coil Actuator 

 
Figure 4: Simulink model for voice coil actuator 

The Simulink model of the voice coil actuator is 

shown in Figure 4. In terms of the working principle, 

the module is a voice coil actuator, which can 

convert the current signal input into an 

electromagnetic force. The output is returned to the 

load carrier in the mechanical system module to 

achieve a closed loop of the entire system. The 

module consists of two parts: coil and permanent 

magnet. The coil moves in the magnetic field 

generated by the permanent magnet, and the force 

generated by the coil actuator compensates for the 

gravitational force of the load carrier and the weight 

to be measured, which makes the PB2 system 

balance. 

The force on the electromagnetic coil is the 

Lorentz force, which is the force generated by the 

interaction of the current flowing through the 

electromagnetic coil in the magnetic field. At the 

same time, when the coil moves in the magnetic 

field, a dynamic electromotive force is also 

generated inside the coil. 

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ACTA IMEKO | www.imeko.org December 2020 | Volume 9 | Number 5 | 152 

𝐹 = 𝐼 · 𝑙 × 𝐵. (1) 

𝑉 = 𝑢 · 𝑙 × 𝐵. (2) 

F – Lorentz force 

I – Current flowing through the solenoid 

l – Voice coil winding length 

B – Magnetic induction 

V – Dynamic electromotive force 

u – Velocity of coil 

The parameter K needs to be set. K is the cross 

product of the coil winding length l and the 

magnetic field magnetic induction B, and the unit is 

N/A or V/(m/s). In this model, the value of K is 

189 N/A. As for the internal actuator, the value of K 

changes to -50 N/A. Another parameter that plays 

an important role during the motion of the coil is the 

translational damping coefficient. It is determined 

as 9 N/(m/s) by fitting the model behaviour to the 

experimental data. 

The Simulink model of the voltage/current 

converter (U/I Converter) is equivalent to a 500 Ω 

resistor. When the input voltage signal U is within 

the set saturation voltage range, the output current 

signal can be equivalent to the current signal passing 

through the 500 Ω resistor. The purpose of setting 

the saturation value of the voltage signal is to 

prevent the output current signal from being too 

large, which will cause a large deflection of the 

Planck-Balance module. This setting improves the 

stability of the simulation model. 

3. RESULT OF SIMULATION 

Based on the analysis of the working principle of 

Planck-Balance, the Simulink model of Planck-

Balance is established in MATLAB according to the 

CAD model, including the mechanical structure 

subsystem, position sensor, U/I converter, 

electromagnetic coil actuator and PID controller. 

The simulation model can be used to study the 

movement of the model in several non-ideal 

situations. 

3.1. Ideal Condition 
The simulation model can accurately represent 

measurement process of the Planck-Balance. Under 

ideal simulation conditions, a measurement cycle of 

20 s is used. In the first half of the cycle, the 

weighing pan is unloaded. In the second half of the 

cycle, the weighting pan is loaded. The movement 

of the load carrier is observed and the output voltage 

of the PID controller is recorded. 

The displacement of the position sensor 

oscillates when the load alternates to unload and 

then quickly returns to zero. The output voltage of 

the PID controller is stable at unload and load 

respectively. The voltage difference between the 

two is proportional to the mass of the weight. The 

load carrier displacement for loaded and unloaded 

state of the balance is shown in Figure 5. Similar to 

the observed behaviour of the real PB2 system, both 

states show a different displacement, but the 

difference is smaller than the measured one.  

 
Figure 5: Displacement of load carrier 

3.2. Bending of the Transmission Lever 
In the actual measurement process, the 

transmission lever will bend due to the influence of 

gravity. In order to study the influence of this 

deformation on the measurement, in this section, the 

transmission lever will be bent by applying an 

external force. Then, the movement of the load 

carrier is observed and recorded.  

3.2.1. Applying external force on one side of the 
balance 

The simple model shown in Figure 6 describes 

the working state of the balance. G represents the 

gravity of the weight to be tested placed on the load 

tray, and F represents the force exerted on the load 

carrier. The default direction is the same as the 

direction of gravity. The right side of the balance 

model represents the position sensor. When the left 

side of the balance is forced to move, the right side 

of the balance will produce a corresponding reverse 

movement. At the same time, the position sensor 

converts the detected position offset signal into a 

voltage signal and inputs it to the PID controller. At 

the same time, the output voltage of the PID 

controller generates via the voltage-current 

converter (not shown in the simple model) an output 

current signal that enters the external voice coil 

actuator. So, the coil generates an upward 

electromagnetic force to balance the weight and the 

additional applied force. Finally, the position sensor 

is kept in the zero plane and the balance is balanced. 

 
Figure 6: A simple model of the PB2 (Model A) 

A different model is shown in Figure 7, an 

internal coil actuator is used. The torque generated 

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ACTA IMEKO | www.imeko.org December 2020 | Volume 9 | Number 5 | 153 

by the actuator acts on the end of the transmission 

lever. 

 
Figure 7: Model with internal actuator (Model B) 

Figure 8 shows a simpler Planck-Balance model. 

The end of the transmission lever is fixed by a 

mechanical method to keep it in zero plane all the 

time. 

 
Figure 8: Fixed transmission lever (Model C) 

In the simulation, the external force F applied by 

the facility to the load carrier is 5 N, and the 

direction is opposite to the direction of the weight G. 

The three models were used for six sets of 

experiments. The weights used in the six sets of 

simulation were 10 g, 20 g, 50 g, 60 g, 70 g and 

100 g. Each set of simulation measures ten cycles of 

data, each cycle lasting 20 s. In the first half of the 

cycle, the weighing pan is unloaded. In the second 

half of the cycle, the weighing pan is loaded and is 

subjected to an external vertical upward force. 

 
Figure 9: Difference of displacement in Model A/B/C 

 As shown in Figure 9, the red line represents the 

simulation result of Model A. It is not difficult to 

find that the extension line of the red line segment 

crosses the (0,0) point, which means that under this 

condition, the displacement of the load carrier is 

proportional to the mass of the weight. Model B 

shows only the difference that all data has a constant 

offset from the data of the classic model A. The blue 

line represents the simulation result of model C, 

which is almost the same as the result of model B. 

Within the acceptable range of measurement error, 

it can be considered that fixing the end of the 

transmission lever by mechanical method and using 

the internal actuator has the same effect on the 

mechanical deformation of the measuring system. 

This is one of the reasons why the external 

electromagnetic coil driver is selected in the actual 

measurement process. The use of the external 

actuator can make less deformation of the 

mechanical structure during the measurement 

process and can be closer to the ideal state. It can 

reduce measurement errors, improve measurement 

capabilities and resolution. 

3.2.2. Apply external force on both sides of the 
balance 

 

Figure 10: Schematic of the external force applied at both 

ends of the balance 

In order to make the transmission lever have a 

bigger deformation, two forces in the same 

direction to both ends of the transmission lever are 

applied. One force is applied to the centre of gravity 

of the load carrier, and the other force is applied to 

the end of the transmission lever. The force applied 

by the facility to the load carrier is FLC, and the 

force applied to the end of the transmission lever is 

FPS. The direction of these two forces is the same as 

the direction of gravity, and  

FLC = 3.75 ·FPS. (3) 

 
Figure 11: Difference of displacement of load carrier 

when external force is applied at both ends 

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The simulation results are shown in Figure 11. 

The abscissa indicates the mass of the measurement 

weight in grams. The ordinate is the displacement 

of the load carrier in nanometres. The red line 

represents the result of the Simulink model when 

FLC = 3.75 N and FPS = 1 N; the green one 

represents FLC = 37.5 N and FPS = 10 N; the blue 

one represents FLC = 75 N and FPS = 20 N. The 

measurement results show that when the 

transmission lever bends due to the force on both 

sides, the movement of the load carrier will show 

nonlinearity, and the greater the degree of bending, 

the more prominent the nonlinear behaviour. 

However, even when the force is increased to values 

of FLC = 75 N and FPS = 20 N, the maximum 

external displacement of the load carrier still retains 

at about 3 nm, smaller than observed in 

measurement. Therefore, it is considered that the 

bending of the transmission lever is not the main 

reason for the non-linear deformation in the force 

calibration system.  

 
Figure 12: Coordinate system at the centre of the 

weighing pan 

3.3. Off-Centre Load 
In the actual measurement process, the weight is 

picked up by the experimenter with tweezers or a 

mechanical arm and placed on weighing pan. In this 

process, there is no guarantee that the weights are 

placed exactly in the centre of the pan, which may 

cause the nonlinear displacement of the load carrier 

during the measurement. In order to study the 

influence of this phenomenon, as shown in Figure 12, 

a plane coordinate system is established on the 

surface of the weighing pan, and the x-axis and y-

axis are set according to the illustrated direction. In 

the simulation results that are shown in this section, 

the independent variable is the position of the 

weight. The object of observation is the 

displacement difference between load and unload. 

As in the experiment, this displacement is obtained 

at a position that is offset from centre of gravity of 

the load carrier since the beam of the interferometer 

cannot be pointed there due to design restrictions. 

Therefore, the tilt of the load carrier due to a 

deformation of the mechanical system has an 

important role in the described phenomena. 

3.3.1. Y-axis Offset 

 

Figure 13: The weight is offset in y-axis direction 

As shown in Figure 13, when the weight is placed 

in the centre of the weighing pan, the gravity G is 

equal to the force F generated by the external 

actuator. 

𝐹 = 𝐺. (4) 

When the position of the weight is shifted in the 

y-axis direction, the gravity of weight is still G. The 

difference is that the equivalent force arm is L', 

where L'= L − Δy, and the support force provided by 

the actuator at this time is F', according to the 

principle of lever balance. 

𝐺 · 𝐿′ = 𝐹′ · 𝐿. (5) 

𝐺 · (𝐿 − 𝛥𝑦) = 𝐹′ · 𝐿. (6) 

 
Figure 14: Simple spring model of load carrier 

In Section 3.2, we discussed that when the right 

side of the balance is fixed, the left side can still 

balance the weight of the weight through its own 

deformation without external force. Therefore, the 

right side of the balance can be considered as a 

simple spring model. As shown in Figure 14, set the 

spring coefficient of the model to C. Then the 

support force Fc = C·ΔZ is generated when the load 

carrier is displaced in the z-axis direction. For the 

force balance of the load carrier on the z-axis. 

𝐺 + 𝐹𝐶 = 𝐹′. (7) 

So 

∆𝑍 = −(𝐺 ·  ∆𝑦)/(𝐿 ·  𝐶). (8) 

It can be obtained from equation (8) that in the 

above model, the additional displacement of the 

load carrier will be in accordance with the weight G 

of the weight and the deviation Δy, where L and C 

are constants. The gravitational force G and the 

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ACTA IMEKO | www.imeko.org December 2020 | Volume 9 | Number 5 | 155 

deviation Δy are used as variables to observe the 

movement of the load carrier during the 

measurement process. 

Figure 15 shows the displacement of load carrier 

in different deviation Δy. It was generated with the 

Simulink model which does not show measured 

data. The results verify the rationality of the simple 

spring model as well as the correctness of equation 

(8) in an intuitive and simple way. The slope 

G/(L×C) can be identified from Figure 15, then the 

value of C can be determined. 

 
Figure 15: Displacement of the load carrier when the 

weight is offset in y-axis direction 

3.3.2. X-axis Offset 

 

Figure 16: Displacement of the load carrier when the 

weight is offset in x-axis direction 

As shown in Figure 16, the offset of the weight 

position on the x-axis will also cause additional 

displacement of the load carrier, but it is much 

smaller than the situation when y-axis is offset. 

Therefore, it is considered that when the weights are 

offset on the x-axis and the y-axis at the same time, 

the y-axis offset is the main reason for the additional 

displacement of the load carrier. 

4. SUMMARY 

The investigations show that nonlinear 

deformations can occur in EMFC balances. A rigid 

body model based on previous work about 

modelling of the dynamic characteristics of the 

same balance is proposed in order to investigate the 

contribution of geometrical nonlinearities to the 

observed effects.  

The simulation results show that the weighing 

pan will have additional displacement under several 

conditions such as bending of the transmission lever, 

eccentric position of the weight, which influence the 

relation between mass and load carrier displacement 

that shows nonlinear pattern. However, these 

contributions above are much smaller than the 

deviation in the experiment. These factors are 

responsible for the nonlinear deformation in a force 

calibration system, but not the main reason. 

Therefore, the basic factor which causes a dominant 

performance in the deviation phenomenon still 

needs further investigation. 

5. REFERENCES 

[1] E. Manske, T. Fröhlich, S. Vasilyan, “Photon 
momentum induced precision small forces: a static 

and dynamic check”, Measurement Science and 

Technology, vol. 30, 105004, 2019. 

[2] C. Rothleitner et al., “The Planck-Balance - using a 
fixed value of the Planck constant to calibrate 

E1/E2-weights”, Measurement Science and 

Technology, vol. 29, no. 7, 2018.  

[3] H. Bai, N. Rogge, C. Rothleitner, T. Fröhlich, 
“Model based correction of motion deviations in 

the Planck-Balance”, GMA/ITG-Fachtagung 

Sensoren und Messsysteme, vol. 20, pp. 554-560, 

2019. 

[4] J. G. Ziegler, N. B. Nichols “Optimum setting for 
automatic controllers”, Trans. ASME, vol. 64, 

pp. 759-768, 1942. 
 

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