Dynamics of Heavy Mechanized Bridge During Span-Lowering Process

: Heavy mechanized bridge is utilized to be rapidly deployed as a temporary bridge to facilitate the movement of vehicles and personnel across obstacles. This paper presents a dynamic model of the TMM-3M heavy mechanized bridge during the span-lowering process, including considerations of cable deformation, rear outriggers, elasticity of tire, and front suspension system. Based on the dynamic model, the authors establish a system of differential equations describing the oscillation of the system using Lagrangian equations of the second kind. The article provides a fundamental basis for studying the entire process of deploying the TMM-3M bridge and aims to improve the rear outriggers and cable winding system to reduce bridge deployment time.


Introduction
Heavy mechanized bridge, in general, is specialized machinery that can be used for both military and civilian purposes to rapidly deploy a temporary bridge, facilitating the quick establishment of a mobile route for vehicles and pedestrians to overcome obstacles such as shallow gaps, rivers, and streams, or to support rescue and relief efforts by creating passageways over challenging terrain.Heavy mechanized bridges are specially manufactured and utilized in countries such as the United States, India, Germany, Russia, the Czech Republic, and China [1,2].The TMM-3M bridge (Fig. 1) is a heavy mechanized bridge produced by Russia."TMMˮ is an abbreviation for the Russian phrase "Тяжелый механизированный мостˮ -heavy mechanized bridge, while "3Mˮ refers to the version of the TMM bridge manufactured in the 1970s-1980s.166 DOI 10.3849/aimt.01878 The chassis of this equipment can be either wheeled or tracked.For wheeled chassisbased heavy mechanized bridges deployed in an upright configuration, the deployment process consists of four main stages in the following order: the frame lifting stage, the span opening stage, the lowering stage, and the intermediate bridge support lowering stage.When deploying a single-span bridge, there are only the first three stages.The deployment process of wheeled chassis-based heavy mechanized bridges depends on the geological conditions at the deployment site, allowing the equipment to be deployed only when ensuring the soil stiffness and the horizontal and vertical slopes of the terrain.
Research on heavy mechanized bridges up to the present moment remains highly limited in publication.This is partly due to the inherently secretive nature of military operations in various countries and the limited dissemination of usage beyond practical applications [3][4][5].Most publications primarily take the form of overviews regarding the development of military bridges by various countries.There is a lack of publications on the dynamics of the operational processes of this equipment.Considering the working characteristics of wheeled chassis-based heavy mechanized bridges during the bridge-laying phase, we can observe similarities with the operation of wheeled cranes in the process of lifting and lowering loads [6][7][8][9][10][11]. Research on the dynamics of wheeled or tracked cranes has garnered significant attention from scientists, with numerous studies dedicated to understanding their mechanical aspects.In [6-8, 11, 12], the authors investigated the dynamics of wheeled or tracked cranes during their operational processes, including the deformation of cables during lifting and lowering tasks.

Fig. 1 Heavy mechanized bridge TMM-3M
However, the dynamic analysis of the operational processes of a specialized system, such as the lifting framework -bridge span -intermediate support bridge on heavy mechanized bridges has not been clearly published or documented.In this article, the authors conducted a dynamic analysis of the heavy mechanized bridge TMM-3M during the span-lowering process.The research results serve as a basis for designing a control strategy for the bridge deployment process and improving the rear outriggers of the equipment.

System Description
The span-lowering process during the deployment of a heavy mechanized bridge occurs with no movement of the base vehicle.During the span-lowering phase, it is assumed that the TMM-3M heavy mechanized bridge is on a flat and perfectly rigid surface.The two rear outriggers are lowered to the ground to eliminate lateral and longitudinal inclinations.At this point, the rear bridge of the base vehicle is considered to have no impact on the oscillation of the vehicle.The lifting frame is linked with the vehicle chassis through hinge joints and is pushed by two cylinders while being pulled by two linkages.In this process, the lifting frame is considered rigidly connected to the vehicle chassis, suitable for practical working conditions.The masses are considered as perfectly rigid and placed at the center of each part, including the nonsuspended front bridge mass m1, the suspended mass on the base vehicle m2, the lifting frame mass m3, and the mass of the bridge span -intermediate bridge support ms.The model describes the dynamics of the TMM-3M heavy mechanized bridge deployment during the span-lowering process as a 2D model (Fig. 2).

Fig. 2 Dynamics of heavy mechanized bridge during span-lowering process 1 -front axle; 2 -chassis; 3 -rear outriggers; 4 -linkages; 5 -lifting frame; 6 -front half-span of the bridge; 7 -rear half-span of the bridge; 8 -intermediate bridge support
The entire mechanical system is positioned within the fixed coordinate system Ox0y0.The vehicle body performs simultaneous translational motion along the vertical axis and rotation about an axis passing through the center of mass, perpendicular to the symmetric vertical plane of the chassis.The bridge span undergoes rotational motion around the hinge joint F, connecting the span and the lifting frame.The vehicle body is considered to oscillate along the vertical axis.The cable drum receives torque from DOI 10.3849/aimt.01878 the engine through the power transmission system to drive the process of lowering the bridge span.
Besides the remaining geometric dimensions described in Fig. 2, several other symbols are conventionally defined as follows: b1 and k1 -represent the damping coefficient and stiffness coefficient of the front tires, respectively; b2 and k2 -the damping coefficient and stiffness coefficient of the front suspension system, respectively; b3 and k3 -represent the damping coefficient and stiffness coefficient of the rear outriggers, respectively; b4 and k4 -represent the damping coefficient and stiffness coefficient of the main cable, respectively; G1, G2, G3 and Gs -correspondingly represent the center of mass positions of masses m1, m2, m3, ms in a system; J2, J3, Js and Jt -respectively represent the moments of inertia of the body vehicle, the lifting frame, the bridge span and the cable winding drum, respectively; H1 and H2 -respectively represent the initial heights of the center of mass of masses m1 and m2, respectively.Some geometric factors are symbolized as follows: in which G2L always has a vertical direction along the chassis.

System Description
Generalized coordinates have been assumed whose vector (Fig. 2) has the form: q y ϕ φ ψ where q [m] -vertical displacement of the unsprung mass of the front axle; y [m] -vertical displacement of the center of mass of the chassis; φ [rad] -pitch angle of the chassis; ϕ [rad] -angle of rotation of the cable winding drum; ψ [rad] -angular displacement of the bridge span.

The kinetic energy of the system:
The kinetic energy of the system includes the kinetic energy of the unsprung mass of the front axle, the kinetic energy of the suspended mass on the chassis, the kinetic energy of the lifting frame, the kinetic energy of the bridge span, and the kinetic energy of the cable winding drum, and is determined by the expression: The potential energy of the system: The potential energy of the system includes gravitational potential energy and elastic potential energy.Firstly, we have the cable elongation during the lowering process, which includes static deformation and dynamic deformation determined by the expressions [6]: The static cable tension force Fcs at any given time is determined through the equation of moment equilibrium with respect to the axis passing through point F, expressed as: in which rc is the distance from F to the line of action of the force vector cs F , and is determined by the expression: The static deformation is determined by the maximum static cable tension force Fcs max occurring in the cable at the end of the span-lowering process, and is given by: csmax s 4 The dynamic deformation of the cable during the span-lowering process is determined as follows: ( ) In the expression (6), S is the length of the cable segment from H to N at any given time, S0 is the initial length of the cable segment from H to N; Rt is the radius of the cable winding drum; ϕ0 and ψ0 are the respective angles of rotation of the cable winding drum and the angle determining the position of the bridge span at the initial time.These quantities are determined by the following expressions: We have the expression defining the total potential energy of the system as: The total dissipative energy of the system The total dissipative energy of the system is determined by the following expression: DOI 10.3849/aimt.01878 In the expression (10), l ∆ ɺ is determined as follows: )

Generalized Forces
The static moment Mt at any given time due to the static cable tension force acting on the cable winding drum is: From the expression ( 12), we can determine the maximum static moment Mt max due to the load acting on the cable winding drum at the end of the span-lowering process when φmin, ψmin.
The brake moment Mp is taken with a safety factor of 1.25.The brake moment ensures that the bridge span can be stopped at any time without falling, ensuring safety.this case, the necessary value of the brake moment to be applied to the cable winding drum is [11]: To drive the cable winding drum, a torque Me is required from the engine to the cable winding drum.Me must overcome the resistance of the brake moment after subtracting the torque due to the load pulling the cable winding drum during lowering.Therefore, the total external torque acting on the cable winding drum to rotate it is determined by the expression: The virtual work done by external forces acting on the system is: Apply the Lagrangeʼs second kind equation to formulate the system of differential equations describing the oscillations of the system in the form: Substituting the expressions for kinetic energy, potential energy, and dissipative functions into Eq.( 16), we obtain the system of differential equations describing the oscillations of the system as follows: ( ) ( ) In expressions ( 20) and ( 21), we have: ) The system of differential Eqs (17-21) is solved using numerical methods based on the application of Matlab simulation software using the Runge-Kutta algorithm.The results are presented in the following section.

Results and Discussion
The input parameter set for solving the system of differential equations describing the oscillations of the system includes:   In Fig. 3, the oscillation of the unsprung mass of the front axle gradually damps, with a stable oscillation amplitude in the seconds from 05 to 20, corresponding to very small oscillation velocities around 0.05 m/s.The final position of the center of mass of m1 at the end of this phase is about 4.5 cm lower than the initial position.
In Fig. 4, we observe that the vertical displacement of the chassis reaches its maximum value of about 16 cm, with a stable oscillation amplitude in the seconds from 4 to 20, corresponding to very small oscillation velocities around 0.06 m/s.The pitch angle of the chassis reaches its maximum value of about 2.6° at the beginning of the lowering phase, and then stabilizes with a small amplitude (Fig. 5), with an angular velocity during the stable phase around 0.08 rad/s.At the end of the lowering phase, the position of the center of mass of m2 is about 10.5 cm lower than the initial position, and the chassis is tilted at an angle of about 1.66° relative to the horizontal.Throughout the entire span-lowering process, the rotation angle of the drum gradually increases from 0 to 98 rad, equivalent to 15.6 revolutions of the drum DOI 10.3849/aimt.01878 (Fig. 6).The length of the cable wound onto the drum can be calculated to be approximately 9.8 m, which is a result consistent with practical operations.
During the span-lowering process, due to the elasticity of the cable, the rotation angle ψ determining the position of the bridge span undergoes oscillations, decreasing from the initial value of 92° to 0°.However, the oscillation amplitude is small (Fig. 7).At the beginning of the lowering phase, the amplitude of the rotation angle ψ undergoes changes corresponding to the oscillation of the chassis.

Conclusion
The paper has presented a dynamic model and established a system of differential equations describing the oscillations of a heavy mechanized bridge during the spanlowering process.With this model, it is possible to study the oscillations of the unsprung mass of the front axle, the chassis, and the bridge span.The obtained results of the paper can serve as a basis for improvement of the rear outriggers of the equipment and solution of both computational design problems, and the control problem of the cable winding drum to increase the lowering speed while ensuring the stability.These ideas will be further explored by the authors in future research.

Fig. 3 Fig. 4
Fig. 3 Vertical displacement and velocity of the center of mass of m1

Fig. 5
Fig. 5 Pitch angle and angular velocity of the chassis