Kurdistan Journal of Applied Research (KJAR) 

Print-ISSN: 2411-7684 | Electronic-ISSN: 2411-7706  

 

Website: Kjar.spu.edu.iq | Email: kjar@spu.edu.iq 

 
 

 

 

Kurdistan Journal of Applied Research | Volume 7 – Issue 2 – December 2022 | 26 

 

 

The Impact of Non-constant Inertia 

and Nonlinear Damping on the 

Torsional Vibration Characteristics 

of Internal Combustion Engine 

Including External Forces 

 
Hameed D. Lafta 

Mechanical Engineering Department 

Technical College of Engineering 

Sulaimani Polytechnic University 

Sulaimani, Iraq 

Hameed.lafta@spu.edu.iq 

 

 

Article Info  ABSTRACT 

Volume 7 -  Issue 2- 

December 2022 

DOI: 
10.24017/Science.2022.2.3 

Article history: 

Received: 27/08/2022 

Accepted: 07/10/2022 

 
A nonlinear torsional vibration is an essential problem arises in 

internal combustion engine due to cyclic variation in inertia and 

gas pressure forces. A linear or lumped model of engine effective 

inertia reveals obvious simulation errors. An accurate and 

sophisticated nonlinear dynamic detailed model overcomes the 

obvious simulation errors of the linearized models. The present 

work studies the effect of the non-conservative forces and 

nonlinear damping on the torsional vibration of single-cylinder 

internal combustion engines. Comprehensive dynamic modeling 

based on a developed expression for the instantaneous kinetic 

energy of the reciprocating parts and a general model of the 

overall kinetic energy of the system in terms of the inertia 

parameters were derived.   The nonlinear dynamic equations are 

derived using the Lagrange’s equation considering the nonlinear 

parameters of the variable inertia of reciprocating parts and the 

nonlinear internal damping factors of the crank assembly. The 

damped-forced response of a slider-crank assembly of the engine 

considering the non constant inertia is investigated using the 

numerical integration technique. The results of the nonlinear 

torsional vibration analysis show that the cyclic variation in 

engine effective inertia and natural frequency arise due inertia 

effect activate the phenomenon of secondary rolling excitation 

torque. As well as, the amplitude of the external excitation torque 

is strengthened by the secondary excitation inertia torque and 

introduces multi resonance amplitudes phenomenon and 

widening the critical range of engine speed which results in 

producing of dangerous vibrational stress amplitudes. Also, the 

damped forced results indicate that the presents of damping lead 

Keywords: 

Internal combustion engines, 

Nonlinear torsional vibration, 

Variable inertia, nonlinear 

damping, excitation torques. 



 

 Kurdistan Journal of Applied Research | Volume 7 – Issue 2 – December 2022 | 27 

 

to a vital reduction in the amplitude of torsional displacement 

and excitation torques. The present work attempts to enhance the 

knowledge of the nonlinear linear torsional vibration analysis for 

the reciprocating engine crankshaft assembly. 

 

 

1. INTRODUCTION 

Nonlinear characteristics of torsional vibration in internal combustion engines (ICE)are 

important phenomena that occurs due to excitation forces, cyclic variations resulting from the 

pressure of the ignited  gases, and changes in inertia loads produced  by engine reciprocating 

parts through the conversion of rotary motion into the linear one [1]. By assuming that the 

dynamical parameters are to be constant, represent the main source of the errors in theoretical 

dynamic modeling of the crankshaft assembly and more sophisticated models are often 

required [2].Due to the torsional vibrations in ICE there is a danger for this engine in that the 

cyclic variation in the speed of the reciprocating parts and their associated inertia give a rise 

for appearing of a range of resonance frequencies in the normal range of operating rotational 

speed of the engine which produces amplitudes of torsional vibrations of crankshaft may 

exceed acceptable values [3]. Also, material nonlinearity, nonlinear friction, and nonlinear 

viscous friction result in nonlinear stiffness and damping [4].Normally, linearization of 

nonlinear dynamic parameters in practical modeling can definitely leads to induce an 

inaccurate modeling; thus, finding of non-linear dynamical parameters with accurate 

expressions is vital and important in developing more adequate equations describing the 

nonlinear torsional vibration of the internal combustion engines [5, 6]. 

The literatures in the subsequent sections summarize the effect of variable inertia-

reciprocating masses, the nonlinear damping, and the gas force variation on the nonlinear 

torsional vibration analysis of ICE. For instance, Ying et al. [7] developed variable inertia 

model to study the nonlinear dynamic characteristics of a crankshaft assembly in an internal 

combustion engine. Their study is carried on equivalence method of the instantaneous kinetic 

energy. The dynamic equations are derived by using the Lagrange rule taking into account the 

nonlinear parameters. The Eigenvector method is used to determine the system natural 

frequencies and mode shapes, and the forced vibration response is investigated by using the 

numerical integration technique. Based on the system matrix method, the simulation results 

are compared with the detailed and lumped mass models. The results explain that the 

secondary excitation torque generated by the variable inertia excites the 2
nd

order rolling 

vibration. Also the additional damping torque due to the variation in inertia represents the 

main nonlinear factor, and the increase in the torsional displacement of the system caused by 

the high-order secondary excitation torque. N. K. Joshi and V. K. Pravin [8] analyzed the 

effect of constant and non constant inertia on the torsional vibration of the engine of marine. 

Their analysis involves modal, order, harmonic, and stress analyses correspondent to the 

engine critical speeds. They concluded that when the variable inertia factor is considered, the 

range of the critical speed was widened and the system may undergo severe vibration stress 

amplitudes. In analyzing the torsional vibrations of reciprocating machines, identification, 

modeling, and a better understanding of the damping characteristics are of particular 

importance and complexity, and, this complexity has led to the analysis of machines either 

without damping or simple damping with approximations of actual state [4, 9].Y. Wang and 

T. C. Lim [4] developed a comprehensive expression for the damping of reciprocating engines 

experimentally. Their study decomposed the internal damping in reciprocating engines into 

viscous friction, material, and non-friction (including impactive) types. The experimental 

results indicate that the non-friction damping contribution is very significant and most of them 

come from the revolute joints at both ends of the connecting rod. This study also shows that 

the measured damping levels in the reciprocating engine are sensitive to clearance in the joints 

of the connecting rod, and are insensitive to the types of lubricating oil and material used. M. 

S. Pasricha [10] presented the equation of motion based on a lumped model of reciprocating 

https://www.sciencedirect.com/science/article/pii/S2215098621000379#b0065


 

 Kurdistan Journal of Applied Research | Volume 7 – Issue 2 – December 2022 | 28 

 

masses with considering the variation in the inertia of the system with key non dimensional 

parameters to include both damping and external excitations to predict the complete response 

of an equivalent single-cylinder engine system. The results show a secondary resonance effect 

arising from the variable inertia characteristics of the system, and under certain circumstances, 

it can have a serious impact on torsional vibration. Nenad Vulić et al. [11] studied the engine 

excitation and damping on the torsional vibration of ship propulsion system including 

moments of inertia of effective masses, shaft stiffness, the actual damping of the shaft 

components and the propulsion engine excitation moments and forces. They systematically 

compared different models of damping in order to enable designers to correctly selected 

damping model. An important outcome of their analyses showed that the continuous operation 

of the engine drive-propulsion  at the range of speed of rotating shaft of approximately 60 rpm 

should be evaded (barred-shape speed zone) to minimize the conceivable damage of  the 

system caused by the excitation at resonance conditions.Tao Li et al. [12] studied the impact 

of frictional damping between the engine cylinder and the slider piston on the nonlinear 

torsional vibration of a reciprocating compressor considering the variation in the inertia of the 

slider-crank mechanism under undamped and damped conditions. The results indicate that 

when considering the friction between sliding-rotating parts, the shafting system second order 

natural frequency show a "maximum-minimum-maximum" cyclictrend; also, with the increase 

in the friction coefficient, the shafting system amplitude and the maximum  vibration speed 

show an upward trend. While with increases in engine speeds, the shaft system vibration 

altered from a non-periodic to the suggested periodic situation, however, the amplitude of 

vibration tend to be decreased in magnitude dramatically.  

Based on the literature above the nonlinear dynamic parameters have a great effect on the 

nonlinear torsional vibration characteristics of slider-crank assembly and to develop a satisfied 

functional relationship between the variable inertia and the crank shaft angle is an improved 

and effective method. Thus, the present work aims to derive a dynamical relationship 

establishes the interaction between the secondary resonance effects and the harmonic 

excitations considering  the variation in the inertia of the reciprocating parts under the effect 

of nonlinear damping and gas forces. A detailed model of the instantaneous kinetic energy of 

reciprocating parts is derived taking into account the variation in the inertia of the crank shaft 

assembly. A more general form of nonlinear torsional vibration equations set is derived  in 

terms of inertia and damping parameters using the Lagrange’s equation for free-damped and 

forced-damped models. The influence of non- constant inertia and nonlinear damping on the 

torsional vibration characteristics of the crankshaft-slider mechanism are investigated by 

applying the numerical integration technique. The simulation results are analyzed and 

discussed for free-damped and forced-damped models over a range of crank angle and 

frequency ratios to establish the influence of the nonlinear dynamic parameters. 

 

2. MATHEMATICAL MODELLING  

2.1. Derivation of Instantaneous Inertia Expression 

The kinematical representation of a single-cylinder internal combustion engine is shown in fig 

(1) below.  

 

 
 

Figure 1: Kinematical representation of slider-crank reciprocating engine. 

L
2
 

L
1
 

r
co

 
r

c
 


co

  

M
r
 

G 

o 



 

 Kurdistan Journal of Applied Research | Volume 7 – Issue 2 – December 2022 | 29 

 

The developed expression of the connecting rod instantaneous kinetic energy is based on the 

definition of the instantaneous fiction radial distance that connect the center of mass of the 

connecting rod (G) to the center of rotation of the crankshaft (O), as shown in figure (1). 

Consequently, from the kinematical relationship of slider-crank mechanism, the expression of 

the radial distance (rco) and its associated angular displacement (co) are given by:  

       √         
     √         

          (1)       

          
       

   
 (2)               

Where 

L1: distance from (G) center of mass to the big end of the connecting rod. 

L2: distance from (G) center of mass to the small end of the connecting rod. 

Mr: reciprocating mass. 

r: crank radius. 

: ratio of crank shaft radius to connecting rod length. 

: crank shaft angular displacement. 

The connecting rod instantaneous angular displacement and radial distance, given in equations 

above are periodic functions in terms of the angular displacement(of the crankshaft. Thus in 

the subsequent analysis, their time derivations producedifficult relationscannotbe manipulated 

mathematically to acquire the system torsional vibration equations. To obtain differentiable 

mathematical forms for these expressions, Fourier analysis can be applied [13]. The following 

system parameters are adopted in order to apply the Fourier analysis, as given in table (1) 

below.  

 
Table 1: Crankshaft-reciprocating parts parameters [4]. 

Parameters Symbol Value 

Radius of crankshaft  r 75 mm 

Mass moment of inertia of crankshaft I 0.03843 kg.m
2
 

Connecting rod  length (from crank end) L1 111 mm 

Connecting rod  length L 327 mm 

Connecting rod mass Mco 6.81 kg 

Connecting rod mass moment of inertia Icog 0.032144 kg.m
2
 

Reciprocating mass M 5.6 kg 

 

 

By using the fast Fourier transform (FFT) to rco, the following equations are obtained: 

       
    
 

 ∑                

 

   

 

     
 

 
∫         

  

 
  

     √  
     

   ,       
  

  
          (3)                                 

   
 

 
∫                 

  

 

 

   
 

 
∫                 

  

 

 

By carrying out the numerical integration of the coefficients r0co, an, and bn, the expression of  

rco  can be derived. In the present work, one of the main objectives is to introduce a more 

rigorous general form for the kinetic energy equation of the system, thus the expression of the 

radial distance can be expressed in a more general form as: 

 



 

 Kurdistan Journal of Applied Research | Volume 7 – Issue 2 – December 2022 | 30 

 

                 (4) 
 

In the same manner, by applying the Fourier transformation   the angular displacement of the 

connecting rod co may be given by: 

 

               
     

 
 (5) 

 

Where A, B, D, and F are constants that can be evaluated depending on the system parameters. 

As a result, in the subsequent analysis, all the necessary equations are derived in terms of 

these constants. Thus, the substitution results in obtaining system equations in a more general 

form which subsequently used to analyze the nonlinear torsional vibration of the reciprocating 

machines by inserting the corresponding system parameters. Consequently, by using the 

system parameters adopted in the present study (see table (1)), the value of constants A, B, D, 

and F are (0.25, 1.4907, 0.07508, and 2) respectively. 

 

2.2. Torsional Vibration Governing Equations 
The system of a single cylinder internal combustion engine with flywheel is shown in figure 

(2). 

 
Figure 2: Slider-crank mechanism with flywheel of single cylinder ICE [5].  

M: reciprocating piston and cross-head shoe; Fw: flywheel; co: connecting rod; c: crank shaft. 

 

By assuming the simple harmonic motion for the reciprocating mass, the system total kinetic 

energy of a single cylinder slider-crank mechanism may be given by: 

 

  
 

 
    ̇  

  
 

 
  ̇      

 

 
         ̇  (6) 

 

The connecting rod instantaneous kinetic energy may be expressed inan explicit form by: 

 

    
 

 
            

   ̇  
  (7) 

 

Where Icogrepresents the connecting rod mass moment of inertia of about the centroid and 

Mcorepresents mass of the connecting rod.  

Substituting for expressions of rco, and co given in equations (4) and (5), yields the following 

system kinetic energy in terms of the crankshaft angular displacement : 

 

  
 

 
    ̇  

  
 

 
  ̇  

 

 
         ̇  

 

 
(                     

 )(        

      )
 
 ̇   (8)  

 

co 

M 

Fw 

c 



 

 Kurdistan Journal of Applied Research | Volume 7 – Issue 2 – December 2022 | 31 

 

Also, the system potential energy may be given by: 

 

  
 

 
         

  (9)  

 

Where kt represents the crankshaft torsional stiffness constant.  

The system torsional vibration equation of motion can be derived by applying the Lagrange’s 

rule as a function of the corresponding coordinate such that: 

 
 

  
(
  

  ̇
)  

  

  
 

  

  
         (10) 

 

By inserting equations (8) and (9), in equation (10) yields: 

 

[
 
 
 
 
 
 
 
 
 
 
 
 (  

   

 
      

 (
    

 
)      

 ((   
 

 
) (

    

 
)     

  

 
   ))  

   

 
        

     
 (

  

 
      

     

 
            )      

   

(

 
 
 
 
 
 
 
 

(  (
    

 
)  

 

 
)      

(

  

 
(   

 

 
)  

    (
    

 
)
)       

(
 

 
(   

 

 
)     

  

 
)       

(      (   
 

 
))           

(
 

 
(  

 

 
))            

 

 
     

)

 
 
 
 
 
 
 
 

]
 
 
 
 
 
 
 
 
 
 
 
 

 ̈  

[
 
 
 
 
 
 
 
 
 
 
 

   

 
             

 ( 
  

 
                                   )  

    
   

(

 
 
 
 
 
 
 

( 
 

 
(  (

    

 
)  

 

 
)     )  (

  

 
(   

 

 
)      (

    

 
))       

 ((   
 

 
)     

  

 
)       (

 

 
 

 

 
)            

(      (   
 

 
))           

 

 
(      (   

 

 
))           

 (
 

 
 

 

 
)            

 

 
                    

)

 
 
 
 
 
 
 

]
 
 
 
 
 
 
 
 
 
 
 

 ̇  

                              (11) 
 

The equation above represents the general form of the nonlinear torsional vibration of a single 

cylinder engine tokening into account the variable inertiaeffect of the reciprocating masses 

and non-conservative forces. At current work, the non-conservative forces represented by the 

internal damping and gas forces. Thus in the subsequent sections, their effects incorporated 

with the non-constant inertia effect are explored in detail. 

2.3. Nonlinear Torsional Vibration Modelling:Free Vibration  
The effect of nonconstant inertia the reciprocating masses in single cylinder can be detected 

by using equation (11) with assuming the term Qnoncon to be equal zero. A non-dimensional 

conversion can be achieved for equation (11) by introducing the following expressions: 

         
        
      
    

  
  

    

  
        



 

 Kurdistan Journal of Applied Research | Volume 7 – Issue 2 – December 2022 | 32 

 

   
  

      
 

  

  
 
   

    (  
   

 
      

 (   
 )

 
)      

   ((   
 

 
) (

    

 
)  

  

 
   ) (12) 

   
 

 

   

   
  

   
     

 

   
  

   
    

   

   
  

Where: 

fr: frequency ratio. 

Ieq: system equivalent mass moment of inertia. 

: torsional vibration displacement. 

crank shaft angular speed. 

n: natural frequency of the system. 

: dimensionless time. 

1, 2, 3: inertia ratios of reciprocating masses, mass moment of inertia of connecting rod and 

mass of connecting rod respectively. 

By inserting equation (12) in equation (11) and disregarding the product terms, derivative 

terms, and higher order terms, so that equation (11) yields: 

 

(

 
 
              (                    

 

 
     )    

(

 
 

               
             
            
             
           )

 
 

)

 
 

    

 

(

  
 
             (

        
  

 
         

                  
)    

(

  
 

  

 
                      

  

 
                      

                          

 
  

 
                      )

  
 

)

  
 

   

(              (
         

               
      

)    (

  

 
              

                        
                             

)  
 

  
 )   

            (
        

  

 
          

                 
)    

(

  
 

  

 
                      

  

 
                      

                           
  

 
                      )

  
 

 (13) 

 

Where the constants A1 to A8 are given by: 

   
  

 
 ,       

   
  

 
(   

 

 
)      (

    

 
) 

   
 

 
(   

 

 
)     

  

 
 

     
    (   

 

 
) 



 

 Kurdistan Journal of Applied Research | Volume 7 – Issue 2 – December 2022 | 33 

 

   
 

 
(  

 

 
) 

   
 

 
 

Equation (13) given above represents the general form of free nonlinear torsional vibration 

model including all the non-constant inertia terms arises due to the motion of reciprocating 

parts. 

2.4. Nonlinear Torsional Vibration Modelling: Damped-Forced Vibration  
The interaction between the variable inertia effect, nonlinear damping, and gas forces on 

nonlinear torsional vibration modeling can be investigated by inserting the mathematical 

models of nonlinear damping and gas forces in equation (11). A comprehensive model for the 

internal damping of a single cylinder ICE including all the types of damping due to viscous 

friction, material hysteresis, and non-friction was presented by [4], and it is adopted in the 

current analysis. The model of nonlinear damping is given by: 

 

                                                              (14) 
 

Where C to C5 are damping constants given by: 

 

          (
 

 
 

  
 

 
) (       )     (

  
 

 
 

  
 

 
)     (

      
 

 
 

  
 

 
) 

   
     

 

 
 

         
 

 
 

         

 
 

              

   
  
 

 
          

   
  
 

 
          

    (                   ) 

And  

  
 

     
    ,       

  

 
 ,      

  

 
 

 

For the external excitation torque, many researchers have shown that the excitation torque due 

to gas forces can be given as a function of crankshaft angular displacement ()[14-15]: 

 

   
   

 

 
    (     

 

 

     

√  
  

 
      

) (15) 

Where: 

Db: cylinder bore dimeter. 

P(): combustion gas pressure as a function of crank angle (). 

For simplicity of analysis, the gas pressure is assumed to be of averaged value (Pavg) to 

compare its effects on excitation torque. Thus, by substituting equations (14) and (15) in 

equation (11) with using expressions given in equation (12), and disregarding the product 

terms, derivative terms, and higher order terms, so that equation (11) yields: 



 

 Kurdistan Journal of Applied Research | Volume 7 – Issue 2 – December 2022 | 34 

 

(

            

  (                    
 

 
     )    (

               
                    

                        
)
)       

 

(

 
 
 
 

             (        
  

 
                           )  

  (

  

 
                      

  

 
                      

                           
  

 
                      

)  

 

  
                                                   )

 
 
 
 

   

(

 
 
 
 

                                               

  (
  

 
                       

                                            
)  

    

  
                      

  

  
 (     

      

√  
  

 
 

     

 

)  
 

  
 

)

 
 
 
 

  

            (
        

  

 
         

                  
)    

(

  
 

  

 
                      

  

 
                      

                           
  

 
                      )

  
 

 

  

  
(

                      
                           

)  
  

  
  (16) 

 

Where the damping ratios  to 5 are given by: 

  
 

      
,     

  

 
 ,      

  

 
    

  

 
,      

  

 
 

Also, the constant Kg and Kr is given by: 

   
   

 

 
     

   
  

  
 

Equation (16) represents the forced–damped torsional vibration equation of motion of single 

cylinder ICE taking into consideration the mutual effect of the excitation gas cylinder torque, 

the effect of internal nonlinear damping and variable inertia of reciprocating parts in a general 

comprehensive form in terms of the system inertia parameters, damping parameters, and gas 

force parameters.  

 

3. SIMULATION RESULTS 

The parametric study can be achieved to examine effects of different system parameters on the 

dynamic characteristics of the nonlinear torsional vibration of single cylinder internal 

combustion engine for the three cases of free vibration, damped-free vibration, and damped-

forced vibration. 

3.1. Free-Damped Vibration Results 
A numerical solution for the dynamic model presented by equation (13) has been developed 

by using Matlab/Simulink. The simulation results show the variations in the effective inertia 

of the engine sliding and connecting rod parts and the variation in the first mode natural 

frequency in figures (3) and (4) respectively. The cyclic fluctuations of the engine’s effective 



 

 Kurdistan Journal of Applied Research | Volume 7 – Issue 2 – December 2022 | 35 

 

mass moment of inertia in each crankshaftangular motion give rise to a periodical variation of 

frequencies and their corresponding amplitudes, which in turn results in widening the critical 

speeds range of the engine and induces the phenomenon of secondary excitation rolling 

torque.  

 

 
 

Figure 3: Variation in the effective inertia of the system versus crank angle at fr = 0.32. 

 

 
 

Figure 4: Variation in the natural frequency of the system versus crank angle at fr = 0.32. 

 

Also, the results indicated that the consideration of the non-constant inertia in the dynamical 

analysis leads to improving modeling predictions accuracies for torsional vibration of both 

single cylinder and multi cylinder engines and can enhance the theory of nonlinear dynamics 

of reciprocating machines [10]. 

The variation in the response of the nonlinear torsional vibration displacements at different 

frequency ratios are shown in figures (5), (6), and (7) respectively. The results displayed that 

the cyclic variation in the natural frequency of the system produces an unsteady torsional 

amplitude response with obviously dominant beating oscillations, and the response changed to 

Crank angle (deg.) 

N
a

tu
r
a

l 
fr

e
q

u
e
n

c
y
 (

H
z
) 

Crank angle (deg.) 

E
ff

e
c
ti

v
e
 i

n
e
r
ti

a
 (

k
g

.m
2
) 



 

 Kurdistan Journal of Applied Research | Volume 7 – Issue 2 – December 2022 | 36 

 

sustained oscillations of different amplitude and harmonic frequency with increases in the 

engine speeds. Also, a strong coupling between the torsional angular displacement response 

and the engine speed is revealed, which show that the critical range of the engine speed is 

increased and widened in their zones. 

 

 
 

Figure 5: Angular displacement amplitude versus ()at fr = 0.33. 

 

 
 

Figure 6: Angular displacement amplitude versus () at fr = 0.99. 

 

Step iteration () 

A
n

g
u

la
r
 d

is
p

la
c
e
m

e
n

t 
(d

e
g
.)

 
A

n
g

u
la

r
 d

is
p

la
c
e
m

e
n

t 
 (

d
e
g

.)
 

 

Step iteration () 



 

 Kurdistan Journal of Applied Research | Volume 7 – Issue 2 – December 2022 | 37 

 

 
 

Figure 7: Angular displacement amplitude versus () at fr = 1.32. 

 

The variation in the secondary harmonic excitation torque is shown in figure (8), which is 

indicated that the non-constant inertia consideration leads to arise of the self-excitation 

secondary excitation torque which in turn results in enhancement of the effective excitation 

external torque load and produces a range of critical speeds. 
 

 
 

Figure 8: Secondary excitation torque versus crank angle at fr = 0.33. 

 

In the same sense the excitation inertia torque amplitude increases dramatically with 

increasing in the frequency ratio (speed of engine), as depicted in figure (9) and this attributed 

to the fact that the inertia torque proportional directly with squareof the speed of the engine. 

Consequently, the additional increase in the inertia torque caused by the non-constant inertia 

terms enforces the influence of the inertial torque, which produces irregular variation of the 

angular torsional vibration displacement. 

 

Crank angle (deg.) 

S
e
c
o

n
d

a
r
y
 e

x
c
it

a
ti

o
n

 

to
r
q

u
e
 (

N
.m

) 

Step iteration () 

A
n

g
u

la
r
 d

is
p

la
c
e
m

e
n

t 
(d

e
g
.)

 



 

 Kurdistan Journal of Applied Research | Volume 7 – Issue 2 – December 2022 | 38 

 

 
 

Figure 9: Inertia excitation torque amplitude versusfrequency ratio (fr). 

 

The effect of frequency ratio (engine speed) on the free undamped torsional vibration 

amplitude over a range of harmonic motion (crank angles) is shown in figure (10). The figure 

depicted that increasesof the speed of the engine produces an increase in the amplitude of 

vibration and phenomena accompanying them especially with the absence of the damping 

effect; produces relatively large amplitude induces dangerous vibration stresses leads to a vital 

failure of the crank assembly. 

 

 
 

Figure 10: Maximum angular displacement amplitude versus crank angle at different frequency ratios. 

 

The results of free-damped vibration can be presented by inserting the damping parameters of 

the present model in equation (15) and equating the external excitation torque (gas torque) to 

zero. Table (2) shows the damping parameters adopted in the present study to evaluate their 

effects on the nonlinear vibration analysis of the internal combustion engine considering the 

variable inertia of the reciprocating parts. 

Crank angle (deg.) 
Frequency ratio (fr) 

A
n

g
u

la
r
 d

is
p

la
c
e
m

e
n

t 

a
m

p
li

tu
d

e
 (

d
e
g

.)
 

Frequency ratio 

 

E
x

c
it

a
ti

o
n

 t
o

r
q

u
e
 

a
m

p
li

tu
d

e
 (

N
.m

) 

 



 

 Kurdistan Journal of Applied Research | Volume 7 – Issue 2 – December 2022 | 39 

 

Table 2: Crank, connecting rod, and slider damping parameters [4].  

Parameters Symbol Value(
   

       
) 

Crankshaft  total  damping     0.5828 

Slider head damping     0.09 

Cross-head damping     0.0549 

Connecting rod  upper joint damping     0.2864 

Connecting rod lower joint damping     0.3024 

 

One of the critical concerns in the torsional vibration of ICE is the inherent torsional 

resonances and corresponding amplitudes ranges of the crankshaft assembly, and the ability to 

quantify these natural modes more precisely requires a better understanding of the damping 

characteristics [4, 11]. The variation of the effective damping of the crankshaft assembly over 

a crankshaft angular displacement is presented in figure (11). It can be seen that the cyclic 

variation of the damping shows clearly the nonlinear inherent characteristics of the engine 

damping, which in turn leads to modified system frequencies and amplitudes variation over 

crank cycles, as shown in figure (12). 

 

 
 

Figure 11: Variation of the damping ratio versus crank angle. 

 

Crank angle (deg.) 

 

D
a

m
p

in
g

 r
a

ti
o

 
 



 

 Kurdistan Journal of Applied Research | Volume 7 – Issue 2 – December 2022 | 40 

 

 
 

Figure 12: Damped frequency of the system versus crank angle. 

 

Alternatively, the damping inherently leads to a change in both the amplitude of frequencies 

and the harmonic response of the torsional vibration angular displacement and in general 

causes a decrease in the angular displacement amplitudes of the system, as shown in figures 

(13), (14), and (15), at different frequency ratios of 0.33, 0.99, and 1.32 respectively. Also, the 

effect of introducing the nonlinear damping is clearly to be seen for the irregular response of 

the torsional vibration displacement occurred due to the widening and changing the range of 

the critical speed of the engine especially at frequencies near the resonance one (fr =0.33 and 

1.32) which appear as a results after taking into account the effect of variation in the inertia of 

the reciprocating parts in the dynamical analysis.    

 
 

Figure 13:Damped angular displacement response versus () at fr = 0.33. 

 

A
n

g
u

la
r
 d

is
p

la
c
e
m

e
n

t 
(d

e
g
.)

 
 

Iteration step 

() 

N
a

tu
r
a

l 
fr

e
q

u
e
n

c
y
 (

H
z
) 

 

Crank angle (deg.) 

 



 

 Kurdistan Journal of Applied Research | Volume 7 – Issue 2 – December 2022 | 41 

 

 
 

Figure 14:Damped angular displacement response versus () at fr = 0.99. 

 

 
Figure 15:Damped angular displacement response versus () at fr = 1.32. 

 

From point of view of the effect of the nonlinear damping on the excitation torque, the 

damping torque components with the damping torque arise due to variation in the inertia of 

the system contributes to an effective reduction in the amplitude of the secondary excitation 

inertia torque, as shown in figure (16). Consequently, this reduction in the torque amplitude 

leads to an acceptable reduction in the amplitude of the torsional vibrationdisplacement of the 

system, as shown in figure (17). 

 

Iteration step 

() 

A
n

g
u

la
r
 d

is
p

la
c
e
m

e
n

t 
(d

e
g
.)

 
 

Iteration step 

() 

A
n

g
u

la
r
 d

is
p

la
c
e
m

e
n

t 
(d

e
g
.)

 
 



 

 Kurdistan Journal of Applied Research | Volume 7 – Issue 2 – December 2022 | 42 

 

 
 

Figure 16:Secondary harmonic excitation torque versus crankangle at fr = 0.49. 

 

 
 

Figure 17:Free angular displacement amplitude versus frequency ratio (fr)                                            

and secondary excitation torque. 

 

Also with taking into account the variation in the effective inertia of the engine reciprocating 

masses, shows an increase in the range of the speed over which the system experiences 

resonance effects which in turn produces resonance amplitudes of relatively large magnitude. 

Thus, at frequency ratios around the resonance frequency, the amplitude depicted peak values 

over arrange of frequency ratios. For example, at frequency ratios of 0.67 and 1.59, the 

amplitude increased around 155.75% and 167.4% compared with that resonance one for the 

undamped case. While for the damped one, these amplitudes show 141.8 % and 157.7 % 

increases in their magnitudes at the aforementioned frequencies. Consequently, this shows a 

reduction of 13.95 % at low frequency ratio and 9.7 % at moderate frequency ratio in the 

torsional vibration amplitude. These results emphasis on the fact that the variation in the 

inertia of the crank assembly leads to critical change in the characteristics vibration of the 

internal combustion engine especially for resonance frequencies and amplitudes. Thus, the 

validation of theoretical analysis supposed that the major parameter to be verified is the 

damping [4, 15]. 

Frequency ratio (fr) 

 

S
e
c
o

n
d

a
r
y

 e
x
c
it

a
ti

o
n

 

to
r
q

u
e
 (

N
.m

) 

 

A
n

g
u

la
r
 d

is
p

la
c
e
m

e
n

t 

a
m

p
li

tu
d

e
 (

d
e
g

.)
 

 

Crank angle (deg.) 

 

E
x
c
it

a
ti

o
n

 t
o
r
q

u
e
 (

N
.m

) 

 



 

 Kurdistan Journal of Applied Research | Volume 7 – Issue 2 – December 2022 | 43 

 

 

3.2. Forced-Damped Vibration Results 
The forced vibration analysis can be carried out by imposing an expression for the external 

excitation torque gas, as given in equation (15). In the present study, for simulation results 

purposes, the gas torque is expressed in terms of the Kr factor to explore its mutual effect with 

secondary inertia excitation torque on the analysis of the torsional vibration of non-constant 

inertia of a single-cylinder ICE. In general, when the gas torque acts simultaneously with 

secondary excitation torque, their amplitudes are strengthened over a crank cycles, as shown 

in figure (18). 

 

 

Figure 18:Forced excitation torque versus crank angle at fr = 0.49 and Kr = 0.1. 

 

The results also show that the damping effectively reduces the amplitude of the excitation 

torque compared with the undamped one. 

In the same sense, the torsional displacement response is greatly affected by the increase in the 

amplitude of the excitation torques (gas torque and inertia torque), and due to this, the 

torsional displacement amplitudes and harmonics are depicted large values compared with 

their values for the free vibration (no damping) case under the same secondary excitation 

torque, as shown in figure (19). 

 

E
x

c
it

a
ti

o
n

 t
o

r
q

u
e
 (

N
.m

) 

 

Crank angle (deg.) 

 



 

 Kurdistan Journal of Applied Research | Volume 7 – Issue 2 – December 2022 | 44 

 

 
 

Figure19:Forced-damped angular displacement response versus () at fr = 0.49 and Kr = 0.1. 

 

Consequently, these variations in the response of the torsional displacement of the crank 

assembly lead to producing hazardous vibration stress, which in turn may cause the vital 

failure of the engine assembly. As a result, it is essential to provide the system with effective 

damping properties to enhance the operational reliability and durability of the internal 

combustion engine through the full range of engine speeds [1]. 

 

4. CONCLUSION 

1) The present modeling based on the developed expression of the instantaneous kinetic 

energy of the reciprocating parts provided more enhancements for the knowledge of the effect 

of the secondary inertia phenomenon when the non-constant inertia terms included in 

comparison with equivalent or lumped mass models.  

2)-Including of non-constant inertia in dynamic modeling produces the inertia excitation 

torque, damping torque, and additional stiffness. 

3)-The inertia excitation torque activates the secondary rolling vibration phenomenon, which 

results in widening the critical range of the engine speed and altering the resonance amplitudes 

of the crank assembly of the internal combustion engine. 

4) Introducing of nonlinear damping model provides more details for the response of the 

torsional vibration displacement amplitude and harmonic frequencies, and variation in natural 

frequency of the crank – reciprocating assembly. 

5) The nonlinear damping introduces a reduction in the amplitude of the torsional vibration 

displacement of 3.95 % at low-frequency ratio(engine speed) and 9.7 % at moderate frequency 

ratio (engine speed). 

6) The damped free vibration results show that the amplitude of the damped inertia excitation 

torque is less than that of the undamped inertia excitation torque. 

7) The forced vibration results show that the external excitation torque amplitudes are 

strengthened by the inertia excitation torque and can produce hazardous vibration amplitudes 

and stress, and in the spite of this, the damping work well to decrease the amplitude of the 

torsional vibration displacement. 

 

 

 

 

A
n

g
u

la
r
 d

is
p

la
c
e
m

e
n

t 
 (

d
e
g

.)
 

 

Iteration step () 

 



 

 Kurdistan Journal of Applied Research | Volume 7 – Issue 2 – December 2022 | 45 

 

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https://www.researchgate.net/publication/326554249