Analytical solution of vibration of simply supported beam under the action of centralized moving mass and two numerical methods using life and death element method and displacement contact method are analyzed in this paper. The results show that vertical acceleration resulted from speed and centrifugal acceleration resulted from load moving must be taken into consideration for large quality and high speed. The characteristics and applicable situations of the two numerical methods are also studied to provide a basis for analyzing and considering structural dynamic problems of moving load mass

Huimin Han

Runbo Bai

Xiumei Qiu

Zongmei Xu

English

Vibroengineering PROCEDIA

 446 © JVE INTERNATIONAL LTD. VIBROENGINEERING PROCEDIA. SEP 2015, VOLUME 5. ISSN 2345-0533  
Vibration analysis of the beam structure under the 
moving mass 
Huimin Han1, Xiumei Qiu2, Zongmei Xu3, Runbo Bai4 
College of Water Conservancy and Civil Engineering,  
Shandong Agricultural University, Tai’an 271018, China 
2, 4Corresponding author 
E-mail: 1hanhm1990@163.com, 2qxmxr@126.com, 3xuzongmei_1999@163.com, 4bai1982@126.com 
(Accepted 11 August 2015) 
Abstract. Analytical solution of vibration of simply supported beam under the action of 
centralized moving mass and two numerical methods using life and death element method and 
displacement contact method are analyzed in this paper. The results show that vertical acceleration 
resulted from speed and centrifugal acceleration resulted from load moving must be taken into 
consideration for large quality and high speed. The characteristics and applicable situations of the 
two numerical methods are also studied to provide a basis for analyzing and considering structural 
dynamic problems of moving load mass. 
Keywords: moving mass, vibration, beam, analytical solution, numerical solution. 
1. Introduction 
In modern society, it is important to develop national economy and improve people’s living 
standard by establishing transportation networks. The speed of the train has been constantly 
improved and the heavy-haul transportation has been developed in more and more countries in 
recent years. When taking action on the structure (mass of the heavy object cannot be neglected), 
these heavy moving objects and the structure make up an interactional time-varying vibration 
system. Therefore, it is of important significance for guarantying the train’s safety and stability to 
consider the effect of moving object mass on structural vibration [1-3].  
Displacement and acceleration are important parameters of structural dynamic response. They 
can subtly reflect vibration of the structure and can be used to detect structural damage [4, 5]. In 
this paper, the authors make a comparison between analytical solution of vibration of simply 
supported beam under the action of centralized moving mass and two numerical analysis methods 
using ANSYS [6, 7]. The time history curve of displacement response and speed response are 
given to realize vibration analysis of moving mass. Finally, the authors make a summary of 
characteristics and applicable situations of each method to provide a basis for analyzing and 
considering structural dynamic problems of moving load mass.  
2. Analytical solution of simply supported beam under moving mass 
2.1. Dynamic equilibrium equation and solution 
Moving mass acting on the simply supported beam can be simplified into the model shown in 
Fig. 1. Assuming simply supported beam is a uniform section (Bending moment of inertia is a 
constant) and wheel mass ܯ moves along beam length but not out of beam body.  
 
Fig. 1. Moving mass at a constant speed acting on simply supported beam  
VIBRATION ANALYSIS OF THE BEAM STRUCTURE UNDER THE MOVING MASS.  
HUIMIN HAN, XIUMEI QIU, ZONGMEI XU, RUNBO BAI 
 © JVE INTERNATIONAL LTD. VIBROENGINEERING PROCEDIA. SEP 2015, VOLUME 5. ISSN 2345-0533 447 
Assuming vertical displacement of the mass ݕ(ݔ, ݐ) , acting force on the beam can be 
represented as: 
ܲ(ݔ, ݐ) = ܯ݃ − ܯ ݀
ଶݕ(ݔ, ݐ)
݀ݐଶ . (1)
Based on the assumption of no separation, the following can be deducted: 
ܲ(ݔ, ݐ) = ܯ݃ − ܯ ቆ∂
ଶݕ(ݔ, ݐ)
∂ݐଶ + 2ݒ
∂ଶݕ(ݔ, ݐ)
∂ݐଶ + ݒ
ଶ ∂ଶݕ(ݔ, ݐ)
∂ݔଶ ቇ. (2)
In the Eq. (2), physical significance of items in the round bracket is as follows: ∂ଶݕ(ݔ, ݐ) ∂ݐଶ⁄  
represents vertical acceleration of beam vibration where the wheel locates; 2ݒ ∂ଶݕ(ݔ, ݐ) ∂ݐଶ⁄  
represents vertical acceleration resulted from change of vertical speed of the beam due to load 
moving; ݒଶ ∂ଶݕ(ݔ, ݐ) ∂ݔଶ⁄  represents centrifugal acceleration generated from load moving along 
the vertical curve due to beam vibration. Therefore, dynamic equilibrium equation of moving mass 
load is considered as: 
ܧܫ ∂
ସݕ(ݔ, ݐ)
∂ݔସ + ρܣ
∂ଶݕ(ݔ, ݐ)
∂ݐଶ + ܿ
∂ݕ(ݔ, ݐ)
∂ݐ = δ(ݔ − ݒݐ)ܲ(ݔ, ݐ), (3)
where ܧ is Young’s modulus of the beam material, ܫ is the moment of inertia of the cross-section, 
ߩ is the density of the beam material, ܣ is the cross-sectional area of the beam, ܿ is the beam 
damping and ߜ is the Dirac function. Using the modal analysis method, the displacement ݕ(ݔ, ݐ) 
can be written in the form of generalized coordinates as: 
ݕ(ݔ, ݐ) = ෍ ݍ௜
ஶ
௜ୀଵ
(ݐ)߮௜(ݔ), (4)
where ߶௜(ݔ) is the ݅th order mode of vibration and ݍ௜(ݐ) is the generalized coordinates of the 
mode of vibration. Substitute Eq. (4) into Eq. (3) and calculate the integral to get the following: 
ቈߩܣ න ߶௡ଶ(ݔ)݀ݔ
௅
଴
቉ ݀
ଶݍ௡(ݐ)
݀ݐଶ + ቈܿ න ߶௡
ଶ(ݔ)݀ݔ
௅
଴
቉ ݀ݍ௡(ݐ)݀ݐ + ቈܧܫ න ߶௡(ݔ)
݀ସ߶௡(ݔ)
݀ݔସ ݀ݔ
௅
଴
቉ ݍ௡(ݐ) 
     = ܯ݃߶௡(ݔ) − ܯ ෍
݀ଶݍ௜(ݐ)
݀ݐଶ ߶௜(ݔ)߶௡
ஶ
௜ୀଵ
(ݔ).
(5)
Rearrange Eq. (5) and we can get: 
൥ቆߩܣ න ߶௡ଶ(ݔ)݀ݔ
௅
଴
ቇ ݍሷ௡(ݐ) + ܯ ෍ ݍሷ௜(ݐ)߶௜(ݔ)߶௡(ݔ)
ஶ
௜ୀଵ
൩ + ቈ2ߦ௡ߩܣ߱௡ න ߶௡ଶ(ݔ)݀ݔ
௅
଴
቉ ݍሶ௡(ݐ) 
     + ቈܧܫ න ߶௡(ݔ)
݀ସ߶௡(ݔ)
݀ݔସ ݀ݔ
௅
଴
቉ ݍ௡(ݐ) = ܯ݃߶௡(ݔ).
(6)
Given that ߶௡(ݔ) has been calculated, Eq. (6) can be deemed as a differential equation for 
ݍ௡(ݐ) (݊ = 1, 2, 3...). Use Newmark-ߚ method to calculate ݍ௡(ݐ) and the beam response can be 
obtained.  
VIBRATION ANALYSIS OF THE BEAM STRUCTURE UNDER THE MOVING MASS.  
HUIMIN HAN, XIUMEI QIU, ZONGMEI XU, RUNBO BAI 
448 © JVE INTERNATIONAL LTD. VIBROENGINEERING PROCEDIA. SEP 2015, VOLUME 5. ISSN 2345-0533  
2.2. Case study 
Following the procedure described above, in what follows various cases are created on the 
basis of a simply supported base beam with dimensions of length 20 m and 45a I-beam as 
cross-section. The beam mass per unit length and the flexural rigidity EI of the base beam are 
80.4 kg/m and 1.43×10-3 m³, respectively. As examples, masses are selected as 300 kg and 
1000 kg, and the moving speeds are 72 km/h and 144 km/h. Time history curve of mid-span 
displacement and acceleration under the condition of counting and not counting the last two items 
(2ݒ ∂ଶݕ(ݔ, ݐ) ∂ݐଶ⁄ , ݒଶ ∂ଶݕ(ݔ, ݐ) ∂ݔଶ⁄ ) are shown in Figs. 2 and 3, respectively. 
 
Fig. 2. Time history curve of vertical displacement of mid-span under moving mass 
 
Fig. 3. Time history curve of acceleration under moving mass 
3. Numerical solution of simply supported beam under moving mass 
3.1. Life and death element method  
Vibration analysis of the beam model under the action of moving mass with life and death 
element method is performed in ANSYS using BEAM 189. First, create mass elements on the 
beam node where moving quality will reach, and kill them all; then as the mass moves to a certain 
node, activate the mass element at this point and kill the previously activated mass element. 
Therefore, structural dynamic analysis under the action of moving mass can be realized through 
VIBRATION ANALYSIS OF THE BEAM STRUCTURE UNDER THE MOVING MASS.  
HUIMIN HAN, XIUMEI QIU, ZONGMEI XU, RUNBO BAI 
 © JVE INTERNATIONAL LTD. VIBROENGINEERING PROCEDIA. SEP 2015, VOLUME 5. ISSN 2345-0533 449 
such a circulation. Time history curve of displacement and acceleration by applying the moving 
load ݉ = 300 kg, ݒ = 144 km/h using the life and death element method and the theory analysis 
resolution are shown in Figs. 4 and 5, respectively. 
 
Fig. 4. Time history curve of displacement by life and death element method 
 
Fig. 5. Time history curve of acceleration by life and death element method 
3.2. Displacement contact method 
Vibration analysis of the beam model under the action of moving mass with displacement 
contact method is performed in ANSYS using BEAM 189 and MASS 21. First create contact pair 
between BEAM 189 element and MASS 21 element, and then apply UX displacement on the node 
of mass element to realize mass moving. Time history curve of displacement and acceleration by 
applying the moving load ݉ = 300 kg, ݒ = 360 km/h using the displacement contact method and 
the theory analysis resolution are shown in Figs. 6 and 7, respectively. 
4. Conclusions 
From dynamic equilibrium equation, it can be seen that the last two items 
(2ݒ ∂ଶݕ(ݔ, ݐ) ∂ݐଶ⁄ , ݒଶ ∂ଶݕ(ݔ, ݐ) ∂ݔଶ⁄ ) in Eq. (3) is related to bridge mass and moving speed of 
load. From Figs. 2 and 3, it can be concluded that when the mass is the same, at low speed, it has 
little impact on the result (displacement) whether the last two items are counting or not, therefore 
these two items can be ignored for general railway bridge (deformation curvature under the action 
VIBRATION ANALYSIS OF THE BEAM STRUCTURE UNDER THE MOVING MASS.  
HUIMIN HAN, XIUMEI QIU, ZONGMEI XU, RUNBO BAI 
450 © JVE INTERNATIONAL LTD. VIBROENGINEERING PROCEDIA. SEP 2015, VOLUME 5. ISSN 2345-0533  
of live load is small) and existing slow train; when moving mass is large and moves at a high 
speed, there are great differences with the same tendency whether the last two items are counting 
or not, therefore the impact of the last two items cannot be ignored for the train at high speed.  
Time history curve of beam response by life and death element method and theoretical result 
not counting the last two items nearly overlap with each other. As node of quality element and 
bridge node is the same node, moving mass response is basically the same with theoretical  
solution. However, the method is complicated for multiple moving masses. 
 
Fig. 6. Displacement response by displacement contact method 
 
Fig. 7. Acceleration response by displacement contact method 
Results of displacement contact method and theoretical results counting the last two items are 
the same. Compared with life and death element method, it is not necessary to consider whether 
moving mass moves to the position of beam node for displacement contact method, therefore 
element length is not limited and only a certain number of elements are required. When the model 
is complex, for example in the situation with multiple factors such as moving mass, dead-weight 
of beam and irregularity influence, displacement contact method is superior.  
Acknowledgements 
The authors gratefully acknowledge the Natural Science Foundations of China 
(No. 51508156), the China Postdoctoral Science Foundation (Grant No. 2014M560386), and the 
Natural Science Foundation of Shandong Province of China (Grant No. ZR2014EL034). 
VIBRATION ANALYSIS OF THE BEAM STRUCTURE UNDER THE MOVING MASS.  
HUIMIN HAN, XIUMEI QIU, ZONGMEI XU, RUNBO BAI 
 © JVE INTERNATIONAL LTD. VIBROENGINEERING PROCEDIA. SEP 2015, VOLUME 5. ISSN 2345-0533 451 
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Vibration analysis of the beam structure under the moving mass

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