The three node numerical integration scheme with the displacement and velocity as the nodal variables for integration of the second order differential equation using finite elements in time on the basis of the method of least squares is proposed. The precision of integration is investigated and compared with the corresponding procedure based on the method of Galerkin. The error measure is introduced which shows the higher precision of the least squares techniqu

Aleksa, A.

Bubulis, A.

Ragulskis, K.

Ragulskis, M.

JVE International

© VIBROMECHANIKA.  JOURNAL OF  VIBROENGINEERING.  2007 OCTOBER/DECEMBER,   VOLUME  9,  ISSUE  4,  ISSN 1392-8716   73 324. Calculation of Vibrations of a Single Degree of Freedom System  M. Ragulskis1, A. Aleksa2, K. Ragulskis3, A. Bubulis4 1 Kaunas University of Technology, Department of Mathematical Research in Systems, Studentų 50-222, Kaunas, Lithuania E-mail: minvydas.ragulskis@ktu.lt Phone: +37069822456 Fax: +37037330446 2 Kaunas University of Technology, Department of Mathematical Research in Systems, Studentų 50-222, Kaunas, Lithuania E-mail: algiment.aleksa@ktu.lt 3 Kaunas University of Technology, Kaunas, Lithuania E-mail: kazimieras3@hotmail.com, kazimieras3@yahoo.com 4 Kaunas University of Technology, Center of Mechatronics, Mickevičiaus 37, Kaunas, Lithuania E-mail: algimantas.bubulis@ktu.lt  (Received 11 October 2007; accepted 2 December 2007)    Abstract. The three node numerical integration scheme with the displacement and velocity as the nodal variables for integration of the second order differential equation using finite elements in time on the basis of the method of least squares is proposed. The precision of integration is investigated and compared with the corresponding procedure based on the method of Galerkin. The error measure is introduced which shows the higher precision of the least squares technique.  Keywords: one degree of freedom, vibrations, finite elements, least squares.  Introduction  The use of finite elements in time for performing numerical integration is described in [1, 2, 3, 4]. The comparison of various numerical integration schemes for the second order differential equations is performed in [5]. The three node numerical integration scheme with the displacement and velocity as the nodal variables for integration of the second order differential equation using finite elements in time on the basis of the method of least squares is proposed. The precision of integration is investigated and compared with the corresponding procedure based on the method of Galerkin.   Numerical integration scheme  The dynamics of a single degree of freedom vibrating system is described by the equation:  ,22fkxdtdxcdtxdm =++        (1)  where m, c, k and f are the mass, damping, stiffness and force; x is the displacement; t is the time variable. The nodes of the finite element correspond to the values -1, 0 and 1 of the local coordinate ξ. So the numerical integration is performed on the basis of the following matrix equation of second order:  ,2122211211⎭⎬⎫⎩⎨⎧=⎭⎬⎫⎩⎨⎧⎥⎦⎤⎢⎣⎡ffxxkkkkTT           (2)  where T is the time step, the upper dot denotes differentiation with respect to time and:  ∫ ∫∫− −−+++=1111555511252511,ξξξξξdNWkTdddNWcddNdWTmk                  (3)  ∫ ∫∫− −−+++=1111565611252621,ξξξξξdNWkTdddNWcddNdWTmk                  (4)  ∫ ∫∫− −−+++=11116526511262512,ξξξξξdNWkTdddNWcTddNdWmk            (5) 324. CALCULATION OF VIBRATIONS OF A SINGLE DEGREE OF FREEDOM SYSTEM. M. RAGULSKIS, A. ALEKSA, K. RAGULSKIS, A. BUBULIS  © VIBROMECHANIKA.  JOURNAL OF  VIBROENGINEERING.  2007 OCTOBER/DECEMBER,   VOLUME  9,  ISSUE  4,  ISSN 1392-8716   74 ∫ ∫∫− −−+++=11116626611262622,ξξξξξdNWkTdddNWcTddNdWmk        (6) ,0111145245112425011113535112325111125225112225111115151121251151xdNWkTdddNWcTddNdWmxdNWkTdddNWcddNdWTmxdNWkTdddNWcTddNdWmxdNWkTdddNWcddNdWTmfdWTfTT⎟⎟⎟⎟⎟⎠⎞⎜⎜⎜⎜⎜⎝⎛+++−−⎟⎟⎟⎟⎟⎠⎞⎜⎜⎜⎜⎜⎝⎛+++−−⎟⎟⎟⎟⎟⎠⎞⎜⎜⎜⎜⎜⎝⎛+++−−⎟⎟⎟⎟⎟⎠⎞⎜⎜⎜⎜⎜⎝⎛+++−−=∫ ∫∫∫ ∫∫∫ ∫∫∫ ∫∫∫− −−− −−−− −−−− −−−ξξξξξξξξξξξξξξξξξξξξξ (7) −⎟⎟⎟⎟⎟⎠⎞⎜⎜⎜⎜⎜⎝⎛+++−−⎟⎟⎟⎟⎟⎠⎞⎜⎜⎜⎜⎜⎝⎛+++−−=−− −−−− −−−∫ ∫∫∫ ∫∫∫TTxdNWkTdddNWcTddNdWmxdNWkTdddNWcddNdWTmfdWTf111126226112226111116161121261162ξξξξξξξξξξξ       ,0111146246112426011113636112326xdNWkTdddNWcTddNdWmxdNWkTdddNWcddNdWTm⎟⎟⎟⎟⎟⎠⎞⎜⎜⎜⎜⎜⎝⎛+++−−⎟⎟⎟⎟⎟⎠⎞⎜⎜⎜⎜⎜⎝⎛+++−∫ ∫∫∫ ∫∫− −−− −−ξξξξξξξξξξ                (8)  where Ni are the shape functions and W5, W6 are the weighing functions. For the method of Galerkin:  ,ii NW =             (9)  and for the method of least squares:  .222 iiii kNddNTcdNdTmW ++=ξξ     (10)  Investigation of precision of integration  The following equation is analysed:  ,022=+ xdtxd             (11)  with the initial conditions:  ,1)0( =x          (12) .0)0( =dtdx        (13)  The analytical and numerical time histories of motion when 5 steps in the period of oscillations are used for the method of Galerkin are presented in Fig. 1. The analytical time history is grey, while the numerical one is black. The corresponding results for the method of least squares are presented in Fig. 2. The values of the displacements after one period of oscillations as a function of the number of time steps are presented in Fig. 3. For the method of Galerkin they are grey, while for the method of least squares they are black. The corresponding values of the velocities after one period of oscillations as a function of the number of time steps are presented in Fig. 4. For the method of Galerkin 324. CALCULATION OF VIBRATIONS OF A SINGLE DEGREE OF FREEDOM SYSTEM. M. RAGULSKIS, A. ALEKSA, K. RAGULSKIS, A. BUBULIS  © VIBROMECHANIKA.  JOURNAL OF  VIBROENGINEERING.  2007 OCTOBER/DECEMBER,   VOLUME  9,  ISSUE  4,  ISSN 1392-8716   75 they are grey, while for the method of least squares they are black.   Fig. 1. Analytical and numerical time histories of motion for the method of Galerkin (the analytical time history is grey, while the numerical one is black).   Fig. 2. Analytical and numerical time histories of motion for the method of least squares (the analytical time history is grey, while the numerical one is black).   Fig. 3. Displacement after one period of oscillations as a function of the number of time steps (for the method of Galerkin in grey, while for the method of least squares in black).   Fig. 4. Velocity after one period of oscillations as a function of the number of time steps (for the method of Galerkin in grey, while for the method of least squares in black).  The error is defined as:  ( ) ( ),~1~ TdtdxTxE +−=       (14)  where T~ is the period of oscillations. The values of the error as a function of the number of time steps are presented in Fig. 5. For the method of Galerkin they are grey, while for the method of least squares they are black.   Fig. 5. Error as a function of the number of time steps (for the method of Galerkin in grey, while for the method of least squares in black).  Thus the introduced error measure evidently shows the higher precision of the least squares technique.  Conclusions  The three node numerical integration scheme with the displacement and velocity as the nodal variables for integration of the second order differential equation using finite elements in time on the basis of the method of least squares is developed in detail. 324. CALCULATION OF VIBRATIONS OF A SINGLE DEGREE OF FREEDOM SYSTEM. M. RAGULSKIS, A. ALEKSA, K. RAGULSKIS, A. BUBULIS  © VIBROMECHANIKA.  JOURNAL OF  VIBROENGINEERING.  2007 OCTOBER/DECEMBER,   VOLUME  9,  ISSUE  4,  ISSN 1392-8716   76 The precision of integration is compared with the corresponding procedure based on the method of Galerkin. The error measure is introduced which evidently shows the higher precision of the least squares technique. The presented results are used in the process of analysis of vibrating systems with one degree of freedom.   References  [1] Zienkiewicz O. C., Morgan K. Finite Elements and Approximation. Moscow: Mir, 1986. [2] Segerlind L. J. Applied Finite Element Analysis. Moscow: Mir, 1979. [3] Zienkiewicz O. C. The Finite Element Method in Engineering Science. Moscow: Mir, 1975. [4] Connor J. J., Brebbia C. A. Finite Element Techniques for Fluid Flow. Leningrad: Sudostroienie, 1979. [5] Bathe K. J. Finite Element Procedures in Engineering Analysis. New Jersey: Prentice-Hall, 1982.    

Calculation of vibrations of a single degree of freedom system

M. Ragulskis

A. Aleksa

K. Ragulskis

A. Bubulis

Journal of Vibroengineering

© VIBROMECHANIKA.  JOURNAL OF  VIBROENGINEERING.  2007 OCTOBER/DECEMBER,   VOLUME  9,  ISSUE  4,  ISSN 1392-8716 
 
 
73 
324. Calculation of Vibrations of a Single Degree of Freedom System 
 
M. Ragulskis1, A. Aleksa2, K. Ragulskis3, A. Bubulis4 
1
 Kaunas University of Technology, Department of Mathematical Research in Systems, Studentų 50-222, Kaunas, Lithuania 
E-mail: minvydas.ragulskis@ktu.lt 
Phone: +37069822456 
Fax: +37037330446 
2
 Kaunas University of Technology, Department of Mathematical Research in Systems, Studentų 50-222, Kaunas, Lithuania 
E-mail: algiment.aleksa@ktu.lt 
3
 Kaunas University of Technology, Kaunas, Lithuania 
E-mail: kazimieras3@hotmail.com, kazimieras3@yahoo.com 
4
 Kaunas University of Technology, Center of Mechatronics, Mickevičiaus 37, Kaunas, Lithuania 
E-mail: algimantas.bubulis@ktu.lt 
 
(Received 11 October 2007; accepted 2 December 2007) 
 
 
 
Abstract. The three node numerical integration scheme with the displacement and velocity as the nodal variables for 
integration of the second order differential equation using finite elements in time on the basis of the method of least squares is 
proposed. The precision of integration is investigated and compared with the corresponding procedure based on the method of 
Galerkin. The error measure is introduced which shows the higher precision of the least squares technique. 
 
Keywords: one degree of freedom, vibrations, finite elements, least squares. 
 
Introduction 
 
The use of finite elements in time for performing 
numerical integration is described in [1, 2, 3, 4]. The 
comparison of various numerical integration schemes for 
the second order differential equations is performed in 
[5]. 
The three node numerical integration scheme with 
the displacement and velocity as the nodal variables for 
integration of the second order differential equation using 
finite elements in time on the basis of the method of least 
squares is proposed. The precision of integration is 
investigated and compared with the corresponding 
procedure based on the method of Galerkin.  
 
Numerical integration scheme 
 
The dynamics of a single degree of freedom 
vibrating system is described by the equation: 
 
,2
2
fkx
dt
dx
c
dt
xd
m =++        (1) 
 
where m, c, k and f are the mass, damping, stiffness and 
force; x is the displacement; t is the time variable. 
The nodes of the finite element correspond to the 
values -1, 0 and 1 of the local coordinate ξ. 
So the numerical integration is performed on the 
basis of the following matrix equation of second order: 
 
,
2
1
2221
1211
⎭⎬
⎫
⎩⎨
⎧=
⎭⎬
⎫
⎩⎨
⎧⎥⎦
⎤⎢⎣
⎡
f
f
x
x
kk
kk
T
T
?            (2) 
 
where T is the time step, the upper dot denotes 
differentiation with respect to time and: 
 
∫ ∫
∫
− −
−
++
+=
1
1
1
1
55
5
5
1
1
2
5
2
511
,ξξξ
ξξ
dNWkTd
d
dNWc
d
d
NdW
T
mk
                  (3) 
 
∫ ∫
∫
− −
−
++
+=
1
1
1
1
56
5
6
1
1
2
5
2
621
,ξξξ
ξξ
dNWkTd
d
dNWc
d
d
NdW
T
mk
                  (4) 
 
∫ ∫
∫
− −
−
++
+=
1
1
1
1
65
26
5
1
1
2
6
2
512
,ξξξ
ξξ
dNWkTd
d
dNWcT
d
d
NdWmk
            (5) 
324. CALCULATION OF VIBRATIONS OF A SINGLE DEGREE OF FREEDOM SYSTEM. M. RAGULSKIS, A. ALEKSA, K. RAGULSKIS, A. BUBULIS 
 
© VIBROMECHANIKA.  JOURNAL OF  VIBROENGINEERING.  2007 OCTOBER/DECEMBER,   VOLUME  9,  ISSUE  4,  ISSN 1392-8716 
 
 
74 
∫ ∫
∫
− −
−
++
+=
1
1
1
1
66
26
6
1
1
2
6
2
622
,ξξξ
ξξ
dNWkTd
d
dNWcT
d
d
NdWmk
        (6) 
,01
1
1
1
45
24
5
1
1
2
4
2
5
01
1
1
1
35
3
5
1
1
2
3
2
5
1
1
1
1
25
22
5
1
1
2
2
2
5
1
1
1
1
15
1
5
1
1
2
1
2
5
1
1
51
x
dNWkTd
d
dNWcT
d
d
NdWm
x
dNWkTd
d
dNWc
d
d
NdW
T
m
x
dNWkTd
d
dNWcT
d
d
NdWm
x
dNWkTd
d
dNWc
d
d
NdW
T
m
fdWTf
T
T
?
?
⎟⎟
⎟⎟
⎟
⎠
⎞
⎜⎜
⎜⎜
⎜
⎝
⎛
++
+
−
−
⎟⎟
⎟⎟
⎟
⎠
⎞
⎜⎜
⎜⎜
⎜
⎝
⎛
++
+
−
−
⎟⎟
⎟⎟
⎟
⎠
⎞
⎜⎜
⎜⎜
⎜
⎝
⎛
++
+
−
−
⎟⎟
⎟⎟
⎟
⎠
⎞
⎜⎜
⎜⎜
⎜
⎝
⎛
++
+
−
−=
∫ ∫
∫
∫ ∫
∫
∫ ∫
∫
∫ ∫
∫
∫
− −
−
− −
−
−
− −
−
−
− −
−
−
ξξξ
ξξ
ξξξ
ξξ
ξξξ
ξξ
ξξξ
ξξ
ξ
 (7) 
−
⎟⎟
⎟⎟
⎟
⎠
⎞
⎜⎜
⎜⎜
⎜
⎝
⎛
++
+
−
−
⎟⎟
⎟⎟
⎟
⎠
⎞
⎜⎜
⎜⎜
⎜
⎝
⎛
++
+
−
−=
−
− −
−
−
− −
−
−
∫ ∫
∫
∫ ∫
∫
∫
T
T
x
dNWkTd
d
dNWcT
d
d
Nd
Wm
x
dNWkTd
d
dNWc
d
d
NdW
T
m
fdWTf
?
1
1
1
1
26
22
6
1
1
2
2
2
6
1
1
1
1
16
1
6
1
1
2
1
2
6
1
1
62
ξξξ
ξξ
ξξξ
ξξ
ξ
       
,01
1
1
1
46
24
6
1
1
2
4
2
6
01
1
1
1
36
3
6
1
1
2
3
2
6
x
dNWkTd
d
dN
WcT
d
d
NdWm
x
dNWkTd
d
dN
Wc
d
d
Nd
W
T
m
?
⎟⎟
⎟⎟
⎟
⎠
⎞
⎜⎜
⎜⎜
⎜
⎝
⎛
++
+
−
−
⎟⎟
⎟⎟
⎟
⎠
⎞
⎜⎜
⎜⎜
⎜
⎝
⎛
++
+
−
∫ ∫
∫
∫ ∫
∫
− −
−
− −
−
ξξξ
ξξ
ξξξ
ξξ
                (8) 
 
where Ni are the shape functions and W5, W6 are the 
weighing functions. 
For the method of Galerkin: 
 
,ii NW =             (9) 
 
and for the method of least squares: 
 
.2
2
2 i
ii
i kNd
dN
T
c
d
Nd
T
mW ++= ξξ      (10) 
 
Investigation of precision of integration 
 
The following equation is analysed: 
 
,02
2
=+ x
dt
xd
             (11) 
 
with the initial conditions: 
 
,1)0( =x          (12) 
.0)0( =
dt
dx
        (13) 
 
The analytical and numerical time histories of motion 
when 5 steps in the period of oscillations are used for the 
method of Galerkin are presented in Fig. 1. The analytical 
time history is grey, while the numerical one is black. 
The corresponding results for the method of least 
squares are presented in Fig. 2. 
The values of the displacements after one period of 
oscillations as a function of the number of time steps are 
presented in Fig. 3. For the method of Galerkin they are 
grey, while for the method of least squares they are black. 
The corresponding values of the velocities after one 
period of oscillations as a function of the number of time 
steps are presented in Fig. 4. For the method of Galerkin 
324. CALCULATION OF VIBRATIONS OF A SINGLE DEGREE OF FREEDOM SYSTEM. M. RAGULSKIS, A. ALEKSA, K. RAGULSKIS, A. BUBULIS 
 
© VIBROMECHANIKA.  JOURNAL OF  VIBROENGINEERING.  2007 OCTOBER/DECEMBER,   VOLUME  9,  ISSUE  4,  ISSN 1392-8716 
 
 
75 
they are grey, while for the method of least squares they 
are black. 
 
 
Fig. 1. Analytical and numerical time histories of motion for 
the method of Galerkin (the analytical time history is grey, 
while the numerical one is black). 
 
 
Fig. 2. Analytical and numerical time histories of motion for 
the method of least squares (the analytical time history is grey, 
while the numerical one is black). 
 
 
Fig. 3. Displacement after one period of oscillations as a 
function of the number of time steps (for the method of 
Galerkin in grey, while for the method of least squares in 
black). 
 
 
Fig. 4. Velocity after one period of oscillations as a function of the 
number of time steps (for the method of Galerkin in grey, while for 
the method of least squares in black). 
 
The error is defined as: 
 
( ) ( ),~1~ T
dt
dxTxE +−=       (14) 
 
where T~  is the period of oscillations. The values of the 
error as a function of the number of time steps are presented 
in Fig. 5. For the method of Galerkin they are grey, while 
for the method of least squares they are black. 
 
 
Fig. 5. Error as a function of the number of time steps (for the 
method of Galerkin in grey, while for the method of least squares 
in black). 
 
Thus the introduced error measure evidently shows the 
higher precision of the least squares technique. 
 
Conclusions 
 
The three node numerical integration scheme with the 
displacement and velocity as the nodal variables for 
integration of the second order differential equation using 
finite elements in time on the basis of the method of least 
squares is developed in detail. 
324. CALCULATION OF VIBRATIONS OF A SINGLE DEGREE OF FREEDOM SYSTEM. M. RAGULSKIS, A. ALEKSA, K. RAGULSKIS, A. BUBULIS 
 
© VIBROMECHANIKA.  JOURNAL OF  VIBROENGINEERING.  2007 OCTOBER/DECEMBER,   VOLUME  9,  ISSUE  4,  ISSN 1392-8716 
 
 
76 
The precision of integration is compared with the 
corresponding procedure based on the method of 
Galerkin. The error measure is introduced which 
evidently shows the higher precision of the least squares 
technique. 
The presented results are used in the process of 
analysis of vibrating systems with one degree of 
freedom. 
  
References 
 
[1] Zienkiewicz O. C., Morgan K. Finite Elements and 
Approximation. Moscow: Mir, 1986. 
[2] Segerlind L. J. Applied Finite Element Analysis. Moscow: 
Mir, 1979. 
[3] Zienkiewicz O. C. The Finite Element Method in 
Engineering Science. Moscow: Mir, 1975. 
[4] Connor J. J., Brebbia C. A. Finite Element Techniques for 
Fluid Flow. Leningrad: Sudostroienie, 1979. 
[5] Bathe K. J. Finite Element Procedures in Engineering 
Analysis. New Jersey: Prentice-Hall, 1982.  
 
 


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Calculation of vibrations of a single degree of freedom system

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