The study presents a formulation and solution for the problem of damped vibration in T-type frame. The physical model took into consideration the energy dissipation in a vibrating frame as a result of constructional damping in the points of the frame mounting and the supports. As the solution of the problem an influence of constructional damping and system geometry changes on first three eigenvalues of the frame are shown (damped frequencies and vibration amplitude decay rates)

Piotr Rosikoń

Sandra Topczewska

Wojciech Sochacki

English

Journal of Vibroengineering

 1866 © VIBROENGINEERING. JOURNAL OF VIBROENGINEERING. DECEMBER 2013. VOLUME 15, ISSUE 4. ISSN 1392-8716  
1096. Constructional damping mounting influence on 
T type frame vibrations 
Wojciech Sochacki1, Piotr Rosikoń2, Sandra Topczewska3 
Częstochowa University of Technology, Institute of Mechanics and Machine Design Foundations, Poland 
E-mail: 1sochacki@imipkm.pcz.czest.pl, 2p.rosikon@imipkm.pcz.pl, 3s.topczewska@imipkm.pcz.pl  
(Received 19 July 2013; accepted 5 November 2013) 
Abstract. The study presents a formulation and solution for the problem of damped vibration in 
T-type frame. The physical model took into consideration the energy dissipation in a vibrating 
frame as a result of constructional damping in the points of the frame mounting and the supports. 
As the solution of the problem an influence of constructional damping and system geometry 
changes on first three eigenvalues of the frame are shown (damped frequencies and vibration 
amplitude decay rates).  
Keywords: damped vibration, T-type frame, eigenvalue, amplitude decay factor.  
1. Introduction 
Vibration damping processes belong to a group of complex and not fully described issues. The 
purpose of this paper is to analyse the impact of constructional damping of mounting on three rod 
T-type frame vibration. Modelling of damped vibrations of such frames is important in the 
engineering practice. It can imitate a range of support structures in which, due to the various 
factors, motion resistance in joint mounts can occur. Truck crane boom reach changes could be an 
example of such construction. A simplified model of this system consists of T-type frame with 
one free end of the horizontal beam. Vibrations of various flat frames types have been repeatedly 
analysed by different authors. These studies mostly focused on the impact of various types of 
geometry and load on the frequency of vibration of such systems. In addition the effect of 
supplementary clustered elements in the form of masses or rotational and translational springs on 
the dynamics of the frame was studied. 
The problem of stability of flat frames was considered in the works [1-3]. In paper [1] the type 
of instability of a T-type frame with joint mass 𝑀 subjected to a compressive follower force 𝑃 
applied at the joint was investigated. Areas of loss of divergent and flutter frame stability were 
identified. Similar in the work [2] static stability analysis of elastically restrained structures under 
follower forces was investigated. An influence of rigidity of supports modelled by springs on the 
type of the loss of stability of the two types of frames, loaded by the follower force, was 
determined. In the work [3] the effect of deviations from right angle of rectangular frames was 
studied. Also the individual and primary coupling effects of several parameters upon the load-
carrying capacity of this frame were discussed. 
Constructions of Γ and T-type frames have quite an extensive literature on the subject. In the 
research [4] critical load value of closed planar frames was determined. Furthermore course of 
natural frequencies changes in external load function was appointed. Also adopted construction 
solutions were taken into account. 
In the work [5] the results of theoretical, numerical and experimental research on free 
vibrations of T-type frame, loaded by longitudinal force directed to its bolt, were presented. The 
same theoretical, numerical and experimental study associated with a rectangular two-rod frame 
with reference to stability and free vibrations has been carried out in the work [6]. Post of 
researched frame realizes load by the force directed toward the pole. 
Variational method for investigating the stability of a rectangular two-bar frame was applied 
in the work [7]. The frame was acted upon by a subfollower force applied at the joint of both 
members. Due to the nonlinearity behaviour of the system, the problem has been solved using the 
perturbation method. Research on the modal analysis of the simple Bernoulli-Euler beam also was 
carried out. In the paper [8] numerical procedure for the complex frequencies and vibration modes 
1096. CONSTRUCTIONAL DAMPING MOUNTING INFLUENCE ON T TYPE FRAME VIBRATIONS.  
WOJCIECH SOCHACKI, PIOTR ROSIKOŃ, SANDRA TOPCZEWSKA 
 © VIBROENGINEERING. JOURNAL OF VIBROENGINEERING. DECEMBER 2013. VOLUME 15, ISSUE 4. ISSN 1392-8716 1867 
evaluation was presented. In addition, in order to decouple the equation of motion, the appropriate 
orthogonality conditions have been established. In the modal analysis viscous rotary dampers were 
taken into account. 
An example of a rotary dampers application can be found in work [9], in which the influence 
of constructional damping of actuator supports on its eigenvalues was examined. This paper 
presents an analysis of the impact of constructional damping of supports (modeled by using hyper 
viscous rotary dampers) on the dynamics of three-rod flat frame of T-type. The results refer to the 
influence of the geometry and the size of damping of individual supports on the eigenvalues of 
the system under study. 
2. Physical and mathematical model of T-type frame damped vibrations  
The physical model of researched system is shown on Fig. 1. Considered frame consists of 
three parts of the bending stiffness (𝐸𝐽)1, (𝐸𝐽)2, (𝐸𝐽)3, respectively. Values adopted as mass per 
length unit were (𝜌0𝐴)1,  (𝜌0𝐴)2,  (𝜌0𝐴)3.  Furthermore it was assumed that  
(𝐸𝐽)1 = (𝐸𝐽)2 = (𝐸𝐽)3 and (𝜌0𝐴)1 = (𝜌0𝐴)2 = (𝜌0𝐴)3. The damping vibration was modeled by 
placing the individual hyper viscous rotary dampers supports with damping coefficients 𝐶𝑅𝑖, 
where (𝑖 = 1, 2, 3). 
 
Fig. 1. Physical model of the system 
The equations of individual beams motion of considered frame can be written as: 
(𝐸𝐽)𝑖
𝜕4𝑊𝑖(𝑥𝑖 , 𝑡)
𝜕𝑥𝑖
4 + (𝜌0𝐴)𝑖
𝜕2𝑊𝑖(𝑥𝑖 , 𝑡)
𝜕𝑡2
= 0. (1) 
Geometric boundary conditions and continuities are as following: 
𝑊1(0, 𝑡) = 𝑊2(0, 𝑡) = 𝑊3(𝑙3, 𝑡) = 0, 
𝜕𝑊1(𝑥1, 𝑡)
𝜕𝑥1
|
𝑥1=𝑙1
=
𝜕𝑊2(𝑥2, 𝑡)
𝜕𝑥2
|
𝑥2=𝑙2
=
𝜕𝑊3(𝑥3, 𝑡)
𝜕𝑥3
|
𝑥3=0
, 
𝑊2(𝑙2, 𝑡) = 𝑊3(0, 𝑡) = 0. 
(2) 
The boundary problem is supplemented by the natural boundary conditions in the form of: 
1096. CONSTRUCTIONAL DAMPING MOUNTING INFLUENCE ON T TYPE FRAME VIBRATIONS.  
WOJCIECH SOCHACKI, PIOTR ROSIKOŃ, SANDRA TOPCZEWSKA 
1868 © VIBROENGINEERING. JOURNAL OF VIBROENGINEERING. DECEMBER 2013. VOLUME 15, ISSUE 4. ISSN 1392-8716  
(𝐸𝐽)1
𝜕𝑊1
2(𝑥1, 𝑡)
𝜕𝑥1
2 |
𝑥1=0
= 𝐶𝑅1
𝜕2𝑊1(𝑥1, 𝑡)
𝜕𝑥1𝜕𝑡
|
𝑥1=0
, 
(𝐸𝐽)2
𝜕𝑊2
2(𝑥2, 𝑡)
𝜕𝑥2
2 |
𝑥2=0
= 𝐶𝑅2
𝜕2𝑊2(𝑥2, 𝑡)
𝜕𝑥2𝜕𝑡
|
𝑥2=0
, 
(𝐸𝐽)3
𝜕𝑊3
2(𝑥3, 𝑡)
𝜕𝑥3
2 |
𝑥3=𝑙3
= −𝐶𝑅3
𝜕2𝑊3(𝑥3, 𝑡)
𝜕𝑥3𝜕𝑡
|
𝑥3=𝑙3
, 
(𝐸𝐽)1
𝜕𝑊1
3(𝑥1, 𝑡)
𝜕𝑥1
3 |
𝑥1=𝑙1
= 0, 
(𝐸𝐽)1
𝜕𝑊1
2(𝑥1, 𝑡)
𝜕𝑥1
2 |
𝑥1=𝑙1
+ (𝐸𝐽)2
𝜕𝑊2
2(𝑥2, 𝑡)
𝜕𝑥2
2 |
𝑥2=𝑙2
− (𝐸𝐽)3
𝜕𝑊3
2(𝑥3, 𝑡)
𝜕𝑥3
2 |
𝑥3=0
= 0. 
(3) 
3. The solution to the problem 
The solutions of equations (1) take the form of: 
𝑊𝑖(𝑥𝑖 , 𝑡) = 𝑤𝑖(𝑥𝑖)𝑒
𝑗𝜔∗𝑡 , (4) 
where 𝜔∗ – frequency of complex number, 𝑗 = √−1. 
By substituting eq. (2) into eq. (1) we obtain: 
𝑤𝑖
𝐼𝑉(𝑥) − 𝛾𝑖𝑤𝑖(𝑥) = 0, 𝑖 = 1, 2, 3, where 𝛾𝑖 =
(𝜌𝐴)𝑖𝜔
∗2
(𝐸𝐽)𝑖
. 
The boundary conditions (after the separation of variables) of considered system are as the 
following: 
𝑤1(0) = 𝑤2(0) = 𝑤3(𝑙3) = 0, 
𝑤2(𝑙2) = 𝑤3(0) = 0, 
𝑤1
𝐼(𝑙1) = 𝑤2
𝐼(𝑙2) = 𝑤3
𝐼(0), 
(𝐸𝐽)1𝑤1
𝐼𝐼(0) = 𝐶𝑅1𝑗𝜔
∗𝑤1
𝐼(0), 
(𝐸𝐽)2𝑤2
𝐼𝐼(0) = 𝐶𝑅2𝑗𝜔
∗𝑤2
𝐼(0), 
(𝐸𝐽)3𝑤3
𝐼𝐼(𝑙3) = −𝐶𝑅3𝑗𝜔
∗𝑤3
𝐼(𝑙3), 
(𝐸𝐽)1𝑤1
𝐼𝐼𝐼(𝑙1) = 0, 
(𝐸𝐽)1𝑤1
𝐼𝐼(𝑙1) + (𝐸𝐽)2𝑤2
𝐼𝐼(𝑙2) − (𝐸𝐽)3𝑤3
𝐼𝐼(0) = 0. 
(5) 
The solution of equations (3) are the functions: 
𝑤𝑖(𝑥) = 𝐶1𝑖 sinh(𝜆𝑖𝑥) + 𝐶2𝑖 cosh(𝜆𝑖𝑥) + 𝐶3𝑖 sin(𝜆𝑖𝑥) + 𝐶4𝑖 cos(𝜆𝑖𝑥), (6) 
where 𝜆𝑖 = √√𝛾𝑖. 
After substituting the equations (6) to the boundary conditions (5), system of equations with 
respect to the unknown constants 𝐶𝑘𝑖 was obtained. In the matrix form this system can be written 
as: 
[𝐴](𝜔∗)𝐶 = 0, (7) 
where 𝐴(𝜔∗) = [𝑎𝑝𝑔], (𝑝, 𝑞 = 1, 2, … ,12) and 𝐶 = [𝐶𝑘𝑖]
𝑇 , (𝑘 = 1, 2, … ,4; 𝑖 = 1, 2, 3). 
The system of equations has a nontrivial solution if the determinant of the matrix by constant 
1096. CONSTRUCTIONAL DAMPING MOUNTING INFLUENCE ON T TYPE FRAME VIBRATIONS.  
WOJCIECH SOCHACKI, PIOTR ROSIKOŃ, SANDRA TOPCZEWSKA 
 © VIBROENGINEERING. JOURNAL OF VIBROENGINEERING. DECEMBER 2013. VOLUME 15, ISSUE 4. ISSN 1392-8716 1869 
𝐶𝑘𝑖 equals zero: 
𝑑𝑒𝑡𝐴 = 0. (8) 
As a result of solution of boundary value problem the vibration eigenvalues 𝜔∗ were obtained. 
Searched roots are complex numbers in which damped vibration frequencies are presented by the 
real parts 𝑅𝑒(𝜔∗) while the system damping is characterized by the imaginary parts 𝐼𝑚(𝜔∗). 
4. The results of numerical computations 
The studies of damped frame vibrations were performed for the following geometrical and 
material data: (𝐸𝐽)𝑖 = 6.443  Nm
2 and (𝜌𝐴)𝑖 = 15.443  kg/m. The length of each rod frame 
changed in the range of 𝑙𝑖 = 0.2 m to 𝑙𝑖 = 2 m. For the calculation the dimensionless damping 
parameters were defined as:  
𝜇𝑅𝑖 =
𝑐𝑅𝑖
𝑙𝑖√(𝜌𝐴)𝑖(𝐸𝐽)𝑖  
. (9) 
In Figures 2-7 further eigenvalues (both 𝑅𝑒(𝜔𝑖
∗) and 𝐼𝑚(𝜔𝑖
∗)) were denoted ((𝜔1
∗)  , 
(𝜔2
∗) , (𝜔3
∗ ) ) respectively.  
 
(a) 
 
(b) 
Fig. 2. The dependence between the first three eigenvalues (real (a) and imaginary (b) parts)  
and the damping parameter 𝜇1 
 
(a) 
 
(b) 
Fig. 3. The dependence between the first three eigenvalues (real (a) and imaginary (b) parts)  
and the length of the vertical frame rod 𝑙1 
1096. CONSTRUCTIONAL DAMPING MOUNTING INFLUENCE ON T TYPE FRAME VIBRATIONS.  
WOJCIECH SOCHACKI, PIOTR ROSIKOŃ, SANDRA TOPCZEWSKA 
1870 © VIBROENGINEERING. JOURNAL OF VIBROENGINEERING. DECEMBER 2013. VOLUME 15, ISSUE 4. ISSN 1392-8716  
The results of eigenvalue changes research, including damping changes at the vertical rod 
frame mounting, are shown in Fig. 2. All research results performed for the variable parameters 
of damping were also performed for the length of each rod frame equal 𝑙1 = 𝑙2 = 𝑙3 = 2 m.  
On the other hand, Fig. 3 shows the research results of frame eigenvalue changes related with 
changes in 𝑙1  rod length. In this case the support damping was constant, 𝜇1 = 0.3. The same 
damping values were always assumed during the research of the influence of each rod length on 
the eigenvalues of the frame. 
The research results on eigenvalue and damping changes in the points of vertical frame rod 
mounting are shown on Fig. 4. The dependence of first three eigenvalues (𝑅𝑒(𝜔𝑖
∗) and 𝐼𝑚(𝜔𝑖
∗)) 
on horizontal frame rods 𝑙2 = 𝑙3 = 0.2 − 2 m is shown on Fig. 5. 
 
(a) 
 
(b) 
Fig. 4. The dependence between the first three eigenvalues (real (a) and imaginary (b) parts)  
and the damping parameters 𝜇2 and 𝜇3 
 
(a) 
 
(b) 
Fig. 5. The dependence between the first three eigenvalues (real (a) and imaginary (b) parts)  
and the length of the horizontal frame rods 𝑙2 = 𝑙3 
Fig. 6 presents frame eigenvalues changes with simultaneous change of all damping 
coefficients 𝜇𝑖. Fig. 7 also presents frame eigenvalues changes but with simultaneous change of 
all rods lengths, with 𝜇𝑖 = 0.3. 
1096. CONSTRUCTIONAL DAMPING MOUNTING INFLUENCE ON T TYPE FRAME VIBRATIONS.  
WOJCIECH SOCHACKI, PIOTR ROSIKOŃ, SANDRA TOPCZEWSKA 
 © VIBROENGINEERING. JOURNAL OF VIBROENGINEERING. DECEMBER 2013. VOLUME 15, ISSUE 4. ISSN 1392-8716 1871 
 
(a) 
 
(b) 
Fig. 6. The dependence between the first three eigenvalues (real (a) and imaginary (b) parts)  
and the damping parameters 𝜇1, 𝜇2 and 𝜇3 
 
(a) 
 
(b) 
Fig. 7. The dependence between the first three eigenvalues (real (a) and imaginary (b) parts)  
and the length of the frame rods 𝑙1 = 𝑙2 = 𝑙3 
5. Conclusions 
In this paper a T-type frame vibration damping model was presented. The vibration damping 
was implemented by the use of viscous rotary dampers applied in mounting places and supports 
of the frame. Based on the obtained results it can be concluded that including constructional 
damping mounting in frame vibration researches causes significant changes in the frame 
eigenvalues. The change of value of the damping coefficient 𝜇𝑖 significantly effects on the first 
eigenvalue (both for the damped frequency 𝑅𝑒(𝜔1
∗) and the frequency decay factor 𝐼𝑚(𝜔1
∗)). In 
this case the damped frequency 𝑅𝑒(𝜔1
∗)  increases (Fig. 2a) to a value that corresponds with 
vertical rod clamp mounting, while the frequency decay rate increases to a maximum value and 
then tends to 0 when 𝜇1 → ∞ . Similar changes both in the damped frequency (𝑅𝑒(𝜔2
∗ ) and 
𝑅𝑒(𝜔3
∗ )) as well as in the frequency decay factor (𝐼𝑚(𝜔2
∗) and 𝐼𝑚(𝜔3
∗ )) can be seen in the case 
when rotary damper coefficients in horizontal rod mounting places are being changed (Fig. 4a and 
Fig. 4b). The changes concern the second and third eigenvalues of the tested frame. The 
simultaneous change of all damping coefficients in the place of frame mounting and in supports 
also causes similar waveforms of the real and imaginary parts of all researched frame eigenvalues 
(Fig. 6a and Fig. 6b).  
1096. CONSTRUCTIONAL DAMPING MOUNTING INFLUENCE ON T TYPE FRAME VIBRATIONS.  
WOJCIECH SOCHACKI, PIOTR ROSIKOŃ, SANDRA TOPCZEWSKA 
1872 © VIBROENGINEERING. JOURNAL OF VIBROENGINEERING. DECEMBER 2013. VOLUME 15, ISSUE 4. ISSN 1392-8716  
Analysing the effect of the lengths of each of the rods on eigenvalues of researched frame 
(Fig. 3a, Fig. 3b, Fig. 5a and Fig. 5b), it can be concluded that damped vibration frequencies 
𝑅𝑒(𝜔𝑖
∗) decrease by rod elongation, while vibration frequency decay factors rise to maximum and 
then decrease. The simultaneous elongation of all frame rods (Fig. 7a and Fig. 7b) results in a 
permanent reduction in both the damped frequencies 𝑅𝑒(𝜔𝑖
∗) and the frequency decay factors 
𝐼𝑚(𝜔𝑖
∗). The possibility of calculating the length of the rod frame for which frequency decay 
factors are the biggest allows designing rod systems with lengths ensuring the minimal vibration 
amplitudes of such systems. The proposed mathematical model of T-type frame with 
constructional damping mounting, after taking into account appropriate boundary, continuity and 
bonds conditions, can be extended to any more complex system, consisting of a number of simple 
systems (T-type frames).  
Acknowledgements 
This research was supported by the Ministry of Science and Higher Education in 2013, Warsaw, 
Poland. 
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[5] Szmidla J. Vibrations and stability of T-type frame loaded by longitudinal force in relation to its bolt. 
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Constructional damping mounting influence on T type frame vibrations

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Constructional damping mounting influence on T type frame vibrations

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