Calibration of cyclic force with inertial force correction to a fatigue testing machine


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ACTA IMEKO 
ISSN: 2221-870X 
December 2020, Volume 9, Number 5, 124 - 128 

 
 

CALIBRATION OF CYCLIC FORCE WITH INERTIAL FORCE 

CORRECTION TO A FATIGUE TESTING MACHINE 
 

Tian Feng1, Yin Xiao1, Li Bo1 

 
1 Changcheng Institute of Metrology and Measurement, Beijing, China, tfpu@sina.com  

 

 

Abstract: 

The accuracy between a dynamic force and a 

static force applied on a specimen by a fatigue 

machine is usually not the same. By establishing 

physical vibration models of fatigue machines, it is 

concluded that the error of a cyclic force is mainly 

caused by the inertial force of the vibration mass 

between the machine sensor and the specimen. After 

the inertial force is exactly corrected, the force 

displayed on the machine would be consistent with 

the real force on the specimen. A standard dynamic 

force calibration sensor (DFCS) with an inertial 

force correction method has been used to do 

calibration of fatigue testing machines in this paper. 

Compared with the replica test-piece method, the 

two calibration results are close to each other. 

Keywords: fatigue machine; calibration; 

dynamic force; inertial force; correction 

1. INTRODUCTION 

The purpose of cyclic force calibration is to 

make the indication force of a machine sensor 

consistent with the actual force loaded on the 

specimen. Replica test-piece method in ISO 4965-1 

and ASTM E467-08 has been widely used in fatigue 

machine calibration [1, 2]. In this method, there are 

some strict limitations of a calibration device 

regarding to stiffness and mass. Because standard 

force sensors are widely used in various fields, they 

are expected to be used to calibrate fatigue machines, 

even though they usually don’t have the same 

stiffness and mass with a real specimen. By 

analysing vibration models of some fatigue 

machines, it is concluded that some types of 

machines do not need to have the same stiffness of 

a DFCS as the real specimen. It also suggests that 

although use a calibration device giving different 

mass and stiffness from that of a real specimen, 

doing inertial force correction would lead to the 

same result to the replica test-piece. The difference 

between the results given by the two methods is 

within 1 %. 

2. ANALYSIS OF THE INERTIAL FORCE 
CORRECTION MODEL 

The dynamic error of a cyclic force generally 

comes from two aspects: the performance of the 

machine force measuring system and the influence 

of the inertial force. The cyclic force error caused 

by the performance of a machine force measuring 

system could be avoided or eliminated by selecting 

a force measuring system with a better dynamic 

performance. The error caused by an inertial force 

cannot be avoided and eliminated, it could only be 

induced as little as possible or corrected as 

accurately as possible. The difference between a 

machine sensor force and a specimen real force is 

due to the fact that the machine sensor and the 

specimen are not in the same position. There is a 

component mass that cannot be ignored between 

them, and the connection stiffness between them is 

limited rather than infinite. During a dynamic test, 

the mass of the component between them has an 

acceleration, so that the inertial force is generated. 

For the reason that the inertial force does not act on 

the sensor and the specimen symmetrically, an error 

of the cyclic force is produced. If the exact value of 

the inertial force influence is known, the actual 

force loaded on a specimen could be derived, or 

calibration to a testing machine force could be done. 

An analysis of the vibration system model of a 

fatigue testing machine is needed to determine the 

influence of an inertial force. 

Local modelling concerned with the force 

calibration is carried out. The local vibration 

segment contains all the parts between the testing 

machine sensor and the specimen. Only the mass 

and stiffness within this segment are considered, 

while those beyond this segment are ignored. In this 

case, any fatigue testing machine could be analysed 

simply as a two-degree-of-freedom vibration model, 

as shown in Figure 1 [3]. 

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Figure 1: Local vibration model of fatigue testing 

machines 

According to D’Alembert’s principle, the two 

differential equations of the vibration system are: 

𝑚1�̈�1 + 𝑘A(𝑥1 − 𝑥2) = 𝐹0 sin 𝜔𝑡 

𝑚2�̈�2 + 𝑘B𝑥2 − 𝑘A(𝑥1 − 𝑥2) = 0 
(1) 

The inertial force 𝐹i is as follows: 

𝐹i = 𝑘BΔ𝑥2 − 𝑘AΔ𝑥1 = 𝑚2Δ𝑥2𝜔
2 (2) 

The relative inertial force 𝐹rel,i is as follows: 

𝐹rel,i =
𝐹i
𝐹B

=
𝑚2Δ𝑥2𝜔

2

𝑘BΔ𝑥2
=

𝑚2𝜔
2

𝑘B
 (3) 

where: 

𝑚1, 𝑚2 - the equivalent mass at positions A and 
B, in kg; 

𝑘A , 𝑘B  - the stiffness at positions A and B, in 
N·m-1; 

𝑥1, 𝑥2 - the instantaneous displacements of the 
elastic elements at positions A and B, in mm; 

Δ𝑥1, Δ𝑥2 - the instantaneous deformation of the 
elastic elements  at positions A and B, in mm; 

𝜔 - the angular frequency of a fatigue testing 
machine, in rad·s-1; 

𝐹A, 𝐹B - the real force between positions A and 
B, in kN; 

𝐹i - the inertial force between positions A and B, 
in kN. 

Equation (3) above indicates that a relative 

inertial force is associated with 𝑚2 and 𝑘B. Figure 1 
illustrates two positions, position A and B. Position 

A is far from the motionless end, on the contrary, 

position B is close to the motionless end. 

Correspondingly, there are usually two 

configurations of fatigue machines. One is that the 

machine sensor lies at position A and the test 

specimen lies at position B – Type 1 configuration. 

The other one is exactly the opposite, that is the test 

specimen lies at position A and the machine sensor 

lies at position B – Type 2 configuration.  

For Type 1 configuration that A is the machine 

sensor and B is the test specimen, both the stiffness 

and mass of a specimen are the influence factors of 

an inertial force. The mass and stiffness of every 

different specimen have to be measured, and the 

correction of the inertial force induced by DFCS, 

which has different mass and stiffness from the real 

specimen, has to be done. The function between the 

real force loaded on the specimen and the machine 

sensor force is expressed by equation (4), whereas 

the function between the DFCS force and the 

machine sensor force is expressed by equation (5). 

𝐹A1 = 𝐹B1 (1 −
𝑚2𝜔

2

𝑘B
)

 
(4) 

𝐹A2 = 𝐹B2 (1 −
𝑚2

′ 𝜔′
2

𝑘B
′ ) 

= 𝐹B2 [1 −
(𝑚2 + Δ𝑚)(𝜔 + Δ𝜔)

2

𝑘B + Δ𝑘
] 

= 𝐹B2 [1 −
𝑚2𝜔

2

𝑘B
−

𝑚2(2𝜔Δ𝜔 + Δ𝜔
2)

𝑘B

−
(Δ𝑚𝑘B − 𝑚2Δ𝑘)(𝜔 + Δ𝜔)

2

𝑘B(𝑘B + Δ𝑘)
] 

(5) 

In equation (5), Δ𝑚 is the increment of 𝑚2 after 
the real specimen is replaced by a DFCS in 

calibration. Similarly, Δ𝑘  is the increment of 𝑘B , 
and Δ𝜔  is the increment of 𝜔 . Where for non-
resonant fatigue machine, Δ𝜔 = 0. 

From equation (4), the dynamic error of a 

machine force obtained by a replica test piece is as 

follows: 

𝛿 =
𝐹A1 − 𝐹B1

𝐹B1
= −

𝑚2𝜔
2

𝑘B
 (6) 

From equation (5), the dynamic error of a 

machine force obtained by a DFCS is as follows. 

Note that equation (7) is equivalent to equation (6).  

𝛿 =
1

𝐹B2
[
 
 
 
 𝐹A2 − 𝐹B2 + 𝐹B2

𝑚2(2𝜔Δ𝜔 + Δ𝜔
2)

𝑘B

+𝐹B2
(Δ𝑚𝑘B − 𝑚2Δ𝑘)(𝜔 + Δ𝜔)

2

𝑘B(𝑘B + Δ𝑘) ]
 
 
 
 

 (7) 

For Type 2 configuration that A is a test 

specimen and B is a machine sensor, 𝐹A is the real 
force of the specimen, and 𝐹B  is the machine sensor 
force. Because the specimen stiffness is 𝑘A instead 
of 𝑘B in this case, it is no longer the influence factor 

of the inertial force, given in equation (3). Therefore, 

for this type of machine, the DFCS does not have to 

have the same stiffness as the real specimen. 

However, the correction of the inertial force could 

still be done using equation (7), where Δ𝑘 = 0. 

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The inertial force 𝐹i could also be expressed as a 
function of the acceleration as follows: 

𝐹i = 𝑘BΔ𝑥2 − 𝑘AΔ𝑥1 = 𝑚2Δ𝑥2𝜔
2 = 𝑚𝑎 (8) 

Equations (4) and (5) can be expressed 

equivalently as equation (10) and (11). Where in 

equation (11), Δ𝑎 is the acceleration increment of 
𝑚2 after the real specimen is replaced by a DFCS. 

𝐹A1 = 𝐹B1 − 𝑚2𝑎 (9) 

𝐹A2 = 𝐹B2 − 𝑚2
′ 𝑎′ 

= 𝐹B2 − (𝑚2 + Δ𝑚)(𝑎 + Δ𝑎) 

= 𝐹B2 − 𝑚2𝑎 − Δ𝑚(𝑎 + Δ𝑎) − 𝑚2Δ𝑎 

(10) 

Equations (6) and (7) can be expressed 

equivalently as equation (12) and (13).
 

𝛿 =
𝐹A1 − 𝐹B1

𝐹B1
= −

𝑚2𝑎

𝐹B1
 (11) 

𝛿 =
1

𝐹B1
[𝐹A2 + Δ𝑚(𝑎 + Δ𝑎) + 𝑚2Δ𝑎 − 𝐹B2] (12) 

For Type 2 configuration of non-resonant 

fatigue machine, when 𝐹B1 = 𝐹B2 in equation (13), 
Δ𝑎 = 0. 

In a cyclic force calibration, the real specimen is 

replaced with a DFCS. Thus, the extra inertial force 

induced by the difference of the mass and stiffness 

of the DFCS from the real specimen should be 

corrected.  

To do the correction to the cyclic force, the first 

step is to measure 𝑚2 and 𝑘B, then the second step 
is to measure Δ𝑚 and Δ𝑘 for Type 1 configuration, 
or only Δ𝑚 for Type 2 configuration. 

Similarly, for acceleration method correction, 

the first step is to measure 𝑚2 and 𝑎, and the second 
step is to measure Δ𝑚  and Δ𝑎  for Type 1 
configuration, or only Δ𝑚 for Type 2 configuration. 

3. EXPERIMENTS AND DISCUSSION 

3.1. Composition of the Calibration System 
A cyclic force calibration system with the 

inertial force correction function consists of a 

dynamic force measurement subsystem, an inertial 

force correction subsystem, an alignment 

measurement subsystem, and a data acquisition and 

processing subsystem. The dynamic force 

measurement subsystem includes a DFCS 

subsystem and a replica test-piece subsystem. A set 

of DFCS consists of some standard dynamic force 

sensors with different force ranges, as shown in 

Figure 2. The inertial force correction subsystem 

includes an acceleration measurement subsystem 

and a stiffness/displacement measurement 

subsystem, as shown in Figure 3. 

 
Figure 2: DFCS 

 
Figure 3:The inertial force correction subsystem  

3.2. Methodology and Procedure of Tests 
Tests were done in an electromagnetic resonance 

fatigue machine with controlled force, as shown in 

Figure 4. There is an inertial mass compensation 

function for this machine, but this function is not 

used in tests so that the inertial force compensation 

function of the DFCS could be verified. 

 

Acceleration 
transducer

Grip A

DFCS

Grip B

Machine 
sensor

 
Figure 4: Calibration to an electromagnetic resonance 

fatigue testing machine with a DFCS 

Firstly, use a replica test piece for calibration, and 

secondly, use a DFCS. When a DFCS is being used, 

a stiffness method or an acceleration method for 

additional inertial force correction to the DFCS are 

simultaneously used. The DFCS is connected to the 

fatigue testing machine with proper adapters, and 

the acceleration transducer is installed on the 

vibration mass between the DFCS and the testing 

machine sensor. Set up the dynamic test force of the 

fatigue testing machine and start the machine. When 

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ACTA IMEKO | www.imeko.org  December 2020 | Volume 9 | Number 5 | 127 

the dynamic test force reaches the target value and 

the sine wave is stable, collect the peak and valley 

values of 50 cycle waveforms, and record the test 

frequency and the acceleration value. The 

displacement data are collected at the same time 

with the former static force calibration, and then the 

stiffness value is calculated. 

3.3. Data of Tests 
The calibration data is shown in Table 1 and 

Table 2. The calculations for the relative error of the 

cyclic force are given in equations (6), (7), (12), and 

(13) and the calculation parameters are shown in 

Table 3. The sample rate is 5 kHz, Bessel filter 

frequency is 500 Hz. 

 

Table 1: Calibration data of the replica test piece 

Data of 

the 

replica 

test 

piece 

Test 

frequency 

 

 

/ Hz 

Peak force 

of the 

machine 

/ kN 

Valley 

force of 

the 

machine 

/ kN 

Amplitude 

of the 

machine 

force 

/ kN 

Peak force 

of the test 

piece 

/ kN 

Valley 

force of 

the test 

piece 

/ kN 

Amplitude 

of the test 

piece force 

/ kN 

Error of 

the 

machine 

amplitude  

/ % 

Difference 

from 

DFCS 

 

/ % 

Test 1 110.3 300.01 99.99 100.01 301.80 99.56 101.12 -1.10 -0.41 

Test 2 110.2 99.95 -100.05 100.00 102.02 -101.14 101.58 -1.55 -0.26 

Test 3 110.3 300.01 99.99 100.01 301.80 99.56 101.12 -1.10 -0.63 

Test 4 110.3 99.95 -100.05 100.00 102.02 -101.14 101.58 -1.55 -0.46 

Table 2: Calibration data of the DFCS 

Data of 

the 

DFCS 

 

 

 

Test 

frequency 

𝒇 
 

 

/ Hz 

Peak 

force of 

the 

machine 

  

/ kN 

Valley 

force of 

the 

machine 

  

/ kN 

Amplitude 

of the 

machine 

force 

 

/ kN 

Peak 

force of 

the 

DFCS 

   

/ kN 

Valley 

force of 

the 

DFCS 

  

/ kN 

Amplitude 

of the DFCS 

force 

 

/ kN 

Error of the 

machine 

amplitude 

(uncorrected) 

 

/ % 

Error of the 

machine 

amplitude 

(corrected) 

/ % 

Test 1 97.2 300.00  99.98 100.01 302.73 97.66 102.54 -2.46 -1.51 

Test 2 97.0  99.97  -100.04 100.01 102.94 -102.71 102.82 -2.74 -1.81 

Test 3 97.2 300.00  99.98 100.01 302.73 97.66 102.54 -2.46 -1.73 

Test 4 97.0  99.97  -100.04 100.01 102.94 -102.71 102.82 -2.74 -2.01 

Table 3: Calculation parameters in additional inertial force correction of the DFCS 

Calculation parameters of the stiffness 

method correction 

𝒎𝟐 

/ kg 

𝚫𝒎 

/ kg 

𝒌
B

 

/ (N·m-1) 

𝚫𝒌 

/ (N·m-1) 

𝝎 

/ (rad·s-1) 

𝚫𝝎 

/ (rad·s-1) 

Test 1 35 6 3.1E+08 -7.0E+07 693.0 -82.3 

Test 2 35 6 3.1E+08 -7.0E+07 692.4 -82.9 

Calculation parameters of the 

acceleration method correction 

𝒎𝟐 

/ kg 

𝚫𝒎 

/ kg 

𝒂 

/ (m·s-2) 

𝜟𝒂 

/ (m·s-2) 
  

Test 3 35 6 43.8 12.6   

Test 4 35 6 43.8 12.6   

 

3.4. Discussion 

The Analysis of Data 

According to the test results, the differences 

between the DFCS results with inertial force 

corrections and the replica test-piece results are 

mostly within 0.5 %. The differences of individual 

comparison data exceed 0.5 %, which might be 

caused by the inconsistency between the real 

measurement position and the ideal measurement 

position - centre of equivalent inertial mass - in the 

acceleration measurement. Further studies about the 

acceleration distribution along the axis of the 

applied force and the acceleration measurement 

method are needed. 

The Necessity of the DFCS Method 

The calibration method of a replica test-piece has 

been used according to ISO 4965-1. Replica test-

pieces are made of one or more test-pieces to be 

tested, which are bonded with strain gauges. Replica 

test-pieces are used in dynamic force calibration of 

fatigue machines after they are calibrated on force 

standard machines. Because the calibration state is 

highly consistent with the testing state and there is 

no additional inertial force caused by external 

factors, a high accurate calibration result could be 

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ACTA IMEKO | www.imeko.org  December 2020 | Volume 9 | Number 5 | 128 

achieved using the replica test-piece calibration 

method. But the disadvantages of this method are as 

follows: (1) different replica test-pieces are needed 

to be made for different fatigue machines. Different 

replica test-pieces are also needed to be made for 

some same specification fatigue machines to be 

calibrated, only because the materials and 

specifications of the test-pieces of each fatigue 

machine are different. Therefore, the production 

cost of the replica test-pieces is high; (2) because the 

maximum bearing force of the replica test-piece 

made of the actual test pieces to be tested on a 

fatigue machine usually could not cover the full 

range of the fatigue machine force, the maximum 

force of the dynamic force range of the fatigue 

machine could not be calibrated by its own replica 

test-piece of the fatigue machine; (3) because of the 

factors of the replica test-piece caused by the strain 

gauge bonding process, it is difficult to evaluate the 

uncertainty of the replica test-piece itself. 

The advantages of the DFCS calibration method 

are as follows: (1) the whole dynamic force range of 

a fatigue machine could be calibrated with a proper 

DFCS; (2) a DFCS could be reused, and could be 

used for many different fatigue machines. So it costs 

low in use and is easy to be promoted; (3) the 

technology of force sensor is mature, and its own 

uncertainty could be accurately evaluated for a 

qualified force sensor; (4) the DFCS method is not 

only suitable for a typical metallic material fatigue 

testing machine, but also for some other types of 

fatigue testing machines and for actuators with 

single or combined use. The disadvantage of DFCS 

method is that some additional inertial force 

influences are induced. In order to overcome this 

shortcoming, it is necessary to give the inertial force 

compensation method. 

There are two typical types of fatigue machines, 

which are the resonant fatigue machine and the non-

resonant fatigue machine. For each type machine, 

there are two kinds of configurations, which are 

Type 1 and Type 2 configuration. So there are 

mainly 4 kinds of machines widely used so far. The 

above are the types and structures of the fatigue 

testing machines. Aiming at the influence of inertial 

force always existing in fatigue machines, there are 

two methods of correcting the inertial force ,which 

is the stiffness method and the acceleration method. 

They are two correction methods which could be 

selected to use in a calibration. 

4. SUMMARY 

(1) Using a replica test piece that has the same 

stiffness and mass with the real specimen gives an 

accurate result of a cyclic force calibration because 

no other inertial force is induced except the one by 

the real specimen. 

(2) It would be best if the DFCS has exactly the 

same mass and stiffness as the real specimen, so that 

no additional inertial force is generated, and no 

correction is needed. 

(3) It is not necessary to strictly limit the mass 

and stiffness of the DFCS to be consistent with the 

real specimen. By accurately measuring mass and 

stiffness or acceleration to correct additional inertial 

force, the same accurate calibration results could be 

obtained by using a DFCS different from the real 

specimen. The difference between the DFCS and 

the replica test-piece methods is within ±1 %. 
(4) With an appropriate adaptor, a DFCS could 

be reused in the calibration for different fatigue 

machines. Further studies about the acceleration 

distribution along the axis of the applied force and 

the acceleration measurement method are needed. 

5. REFERENCES 

[1] ISO 4965-1, “Metallic materials - Dynamic force 
calibration for uniaxial fatigue testing - Part 1 

Testing systems”, ISO, Switzerland, 2012. 

[2] ASTM E467-08, “Standard Practice for 
Verification of Constant Amplitude Dynamic 

Forces in an Axial Fatigue Testing System”, ASTM 

International, USA, 2014. 

[3] D. Georgakopoulos, J. M. Williams, A. Knott, T. 
Esward, P.S. Wright, “Dynamic calibration of 

fatigue testing machines”, in Proc. of British 

Electromagnetic Measurements Conference, 

Teddington, UK, 14-17 November 2005. Online 

[accessed 20201105]:  

http://eprintspublications.npl.co.uk/3317/1/BEMC

2005-11.pdf

 

 

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