Highlights A model is proposed to predict the response of tracks and belts to vibrations. A generalised Zener model captures flexural rigidity and damping of tracks and belts. Models with 3 to 5 Maxwell-elements successfully simulate selected rubber tracks. The spring-damping properties of tracks and belts are identified by cyclic bending. A new test stand is developed to test tracks and belts under cyclic bending.

    Abstract In the article, a model for predicting the energy losses caused by the flexural vibrations of rubber tracks, rubber belts, and rubber-bushed metal link-tracks for off-road vehicles is proposed, and a test stand and an experimental procedure are developed to identify the mechanical parameters of this model. The track or belt is represented by a chain of discrete rigid links connected by revolute joints, and a discrete spring-element is placed in parallel with multiple Maxwell-elements in each joint to capture the flexural rigidity and damping of the real track or belt. The mechanical parameters of the joint are found by testing real tracks or belts under cyclic bending. The models consisting of three, four, or five Maxwell-elements per joint are the most successful in predicting the response of a sample rubber track to cyclic bending. The spring-damping properties of tracks and belts identified with the method discussed herein can be applied in simulation studies on the interaction of tracked vehicles and soil. Furthermore, vehicle elements such as rubber bushings for suspension systems, rubber torsion springs, and oil-filled and rubber torsion dampers can be tested with this method to find their spring-damping properties required by vehicle dynamics simulations.


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    Title :

    Modelling and experimental identification of spring-damping properties of the off-road vehicle rubber tracks, rubber belts, and rubber-bushed tracks subjected to flexural vibrations


    Contributors:

    Published in:

    Publication date :

    2023-08-22


    Size :

    22 pages




    Type of media :

    Article (Journal)


    Type of material :

    Electronic Resource


    Language :

    English


    Keywords :

    Rubber belt , Rubber track , Rubber-bushed track , Flexural rigidity , Structural damping , Experimental test , Track system , Tracked undercarriage , Flexural vibrations , Energy dissipation , <italic>A<inf>i</inf>, B</italic> <inf><italic>i</italic></inf> , coefficients of the equation Eq. (12) that describes the angular displacement of the node connecting the torsion spring and damper of the <italic>i</italic>-th Maxwell-element of the joint model shown in Fig. 10b resulting from a sinusiodally varying angular displacement applied to the entire Maxwell-element, [rad] , <italic>b<inf>i</inf>, b</italic> <inf><italic>n</italic></inf> <italic>, b</italic> <inf><italic>1</italic></inf> , through <italic>b<inf>9</inf></italic>, damping coefficient of the torsion damper representing structural damping of the track model in Fig. 10 [Nm·s/rad] , <italic>C<inf>1</inf>, C<inf>1</inf>′</italic> , a constant and a coefficient involved in the general solution of Eq. (11) (see Appendix A) , <italic>c</italic> , the stress in the Coulomb material model or a Coulomb rheological element, [MPa] , <italic>D, D<inf>A</inf>, D<inf>B</inf></italic> , determinants derived to solve the system of equations Eq. (A.13) , <italic>E, E<inf>0</inf>, E<inf>1</inf>, E<inf>2</inf>, E<inf>3</inf>, E<inf>n</inf></italic> , the modulus of elasticity of the Hooke material model or a Hookean rheological element, [MPa] , <italic>EJ</italic> , flexural rigidity of a continuous beam member, [Nmm<sup>2</sup>] , <italic>err<inf>ovl</inf></italic> , <italic>errA<inf>p</inf>, errB</italic> <inf><italic>p</italic></inf>, error functions analysed to estimate the spring-damping parameters of the model shown in Fig. 10 for a real track or belt (see Section 3.3) , <italic>err</italic> <inf><italic>Y</italic></inf> , the error exhibited by the model shown in Fig. 10 in predicting the amplitude of the reaction moment induced in a real rubber track due to cyclic bending , <italic>err<inf>φ</inf></italic> , the error exhibited by the model shown in Fig. 10 in predicting the phase shift between the sinusoidally varying deflection angle and reaction moment induced in a real rubber track , <italic>F</italic> , the scaling factor used to calculate the reaction moment captured by the load cell installed in the test stand shown in Fig. 12, [Nm/(mV/V)] , <italic>f</italic> , bending frequency, [Hz] , <italic>f<inf>ωp</inf></italic> , the bending frequency applied to a track or belt in the <italic>p</italic>-th of <italic>n</italic> cyclic bending tests, [Hz] , <italic>G</italic> , the scaling factor used to calculate the reaction moment captured by the acceleration sensor installed in the test stand shown in Fig. 12, [kg·m<sup>2</sup>·s<sup>2</sup>·(mV)<sup>-1</sup>] , <italic>h</italic> , thickness of a single aluminium bar discussed in Sections 4.4 and 5.1, [mm] , <italic>i</italic> , iterator indicating the spring- or damping-element number , <italic>J</italic> , the area moment of inertia of the cross-section of a beam, [mm<sup>4</sup>] , <italic>k<inf>i</inf></italic> , <italic>k</italic> <inf><italic>n</italic></inf> <italic>, k</italic> <inf><italic>0</italic></inf> through <italic>k</italic> <inf><italic>9</italic></inf>, stiffness of the torsion spring representing the elastic properties of a real track or belt in the model shown in Fig. 10, [Nm/rad] , <italic>k<inf>s</inf></italic> , stiffness of a torsion spring implemented in a discrete beam model, [Nm/rad] , <italic>l</italic> , length of the aluminium bar-section subjected to bending in the preliminary tests discussed in Sections 4.4 and 5.1, [mm] , <italic>l<inf>s</inf></italic> , length of a single section of a discrete model of a one-dimensional mechanical structure, [mm] , <italic>M<inf>bi</inf></italic> , the moment that acts on the node connecting the torsion spring and damper of the <italic>i</italic>-th Maxwell-element of the joint model shown in Fig. 10b due to a non-zero deflection rate of the torsion damper, [Nm] , <italic>M<inf>C</inf></italic> , the total reaction moment induced in a real track or belt subjected to a sinusoidally varying deflection angle, the total reaction moment induced in the joint model shown in Fig. 10b due to a sinusoidally varying angular displacement of the entire joint, or the reaction moment measured using the test stand in Fig. 12 or predicted with the track model in Fig. 10, [Nm] , <italic>M<inf>C</inf></italic> <inf><italic>,acc</italic></inf> , the component of the reaction moment measured by the acceleration sensor installed in the test stand shown in Fig. 12, [Nm] , <italic>M<inf>C</inf></italic> <inf><italic>,lc</italic></inf> , the component of the reaction moment measured by the load cell installed in the test stand shown in Fig. 12, [Nm] , <italic>M<inf>i</inf>, M</italic> <inf><italic>n</italic></inf> <italic>, M</italic> <inf><italic>1</italic></inf> , the reaction moment induced in a single Maxwell-element of the joint model shown in Fig. 10b due to a sinusiodally varying angular displacement applied to the entire Maxwell-element, [Nm] , <italic>M<inf>si</inf></italic> , the moment of the spring that acts on the node connecting the torsion spring and damper of the <italic>i</italic>-th Maxwell-element of the joint model shown in Fig. 10b, [Nm] , <italic>M<inf>0</inf></italic> , the reaction moment induced in the spring-element of the joint model shown in Fig. 10b due to a sinusoidally varying deflection of the spring, [Nm] , <italic>n</italic> , the number of: springs or dampers of the model joint shown in Fig. 10b, spring or damping elements of a rheological model, or experiments in a test series , <italic>p</italic> , the iterator that indicates an individual experiment within the series of <italic>n</italic> cyclic bending tests on a real track or belt , <italic>t</italic> , time, [s] , <italic>U<inf>acc</inf></italic> , the signal of the acceleration sensor installed in the test stand shown in Fig. 12, [mV] , <italic>U<inf>lc</inf></italic> , the signal of the load cell installed in the test stand shown in Fig. 12, [mV/V] , <italic>v</italic> , linear velocity of a vehicle, [m/s] , <italic>w</italic> , width of a single aluminium bar discussed in Sections 4.4 and 5.1, [mm] , <italic>X</italic> , the amplitude of the sinusoidally varying angular displacement in the joint of the track or belt model shown in Fig. 10, [rad, °] , <italic>X<inf>alu</inf></italic> , the amplitude of the deflection angle exerted on the aluminium bars during the preliminary tests discussed in Sections 4.2 and 5.1, [rad, °] , <italic>X<inf>ωp</inf></italic> , the amplitude of the deflection angle exerted on a real track or belt during a cyclic bending test at the angular frequency <italic>ω<inf>p</inf></italic>, [rad, °] , <italic>Y</italic> , the amplitude of the reaction moment resulting from a sinusoidally varying angular displacement in the model joint shown in Fig. 10b, [Nm] , <italic>Y<inf>alu</inf></italic> , the amplitude of the reaction moment induced in the aluminium bar under sinusoidally varying deflection discussed in Sections 4.4 and 5.1, [Nm] , <italic>Y<inf>lc</inf></italic> , the amplitude of the reaction moment captured only by the load cell installed in the test bench for cyclic bending tests shown in Fig. 12, [Nm] , <italic>Y<inf>lc</inf>+Y<inf>acc</inf></italic> , the amplitude of the reaction moment captured by the load cell and the acceleration sensor installed in the test bench for cyclic bending tests shown in Fig. 12, [Nm] , <italic>Y</italic> <inf><italic>ω</italic></inf> <italic><inf>p</inf></italic> <inf>,</inf> <italic>Y</italic> <inf><italic>ωp,test</italic></inf> , the amplitude of the reaction moment induced in a real track or belt during the cyclic bending test at the angular frequency <italic>ω<inf>p</inf></italic> and the deflection angle amplitude <italic>X<inf>ωp</inf></italic>, [Nm] , <italic>Y<inf>ωp,est</inf></italic> , the amplitude of the reaction moment predicted by the model shown in Fig. 10 for a track or belt under cyclic bending at the angular frequency <italic>ω<inf>p</inf></italic> and the deflection angle amplitude <italic>X<inf>ωp</inf></italic>, [Nm] , tg(<italic>α</italic>) , slope of a linear function, [-] , <italic>ε</italic> , strain, [-] , <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mover><mi>ε</mi> <mo>̇</mo></mover></mrow></math> , strain rate, [s<sup>−1</sup>] , <italic>η, η</italic> <inf><italic>0</italic></inf> <italic>, η</italic> <inf><italic>1</italic></inf> <italic>, η</italic> <inf><italic>2</italic></inf> <italic>, η</italic> <inf><italic>3</italic></inf> <italic>, η</italic> <inf><italic>n</italic></inf> , the viscosity of the Newton material model or a Newtonian rheological element, [MPa·s] , <italic>σ</italic> , stress induced in a solid or liquid material, [MPa] , <italic>τ<inf>i</inf></italic> , the ratio of the stiffness coefficient <italic>k<inf>i</inf></italic> to the damping coefficient <italic>b<inf>i</inf></italic> of the elements of the <italic>i</italic>-th Maxwell-element of the joint model shown in Fig. 10b, [s<sup>−1</sup>] , <italic>θ</italic> , the angular displacement in the entire joint of the track or belt model shown in Fig. 10, [rad] , <italic>θ<inf>1</inf> ,θ</italic> <inf><italic>2</italic></inf> <italic>, θ</italic> <inf><italic>i</italic></inf> <italic>, θ</italic> <inf><italic>n</italic></inf> , the angular displacement of the node that connects the spring and damper of the Maxwell-element of the joint model shown in Fig. 10b, [rad] , <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mover><msub><mi>θ</mi> <mi>i</mi></msub> <mo>̇</mo></mover></mrow></math> , the angular velocity of the node that connects the spring and damper of the <italic>i</italic>-th Maxwell-element of the joint model shown in Fig. 10b, [rad/s] , <italic>φ<inf>0</inf></italic> , the phase shift between the sinusoidally varying angular displacement and the reaction moment induced in the model joint shown in Fig. 10b, [rad] , <italic>φ<inf>0,</inf></italic> <inf><italic>ω</italic></inf> <italic><inf>p</inf>, φ<inf>0,</inf></italic> <inf><italic>ω</italic></inf> <italic><inf>p,test</inf></italic> , the phase shift between the sinusoidally varying deflection angle and the reaction moment induced in a real track or belt under cyclic bending at the angular frequency <italic>ω<inf>p</inf></italic>, [rad] , <italic>φ<inf>0,</inf></italic> <inf><italic>ω</italic></inf> <italic><inf>p,est</inf></italic> , the phase shift between the deflection angle and the reaction moment predicted by the discrete model shown in Fig. 10 for a real track or belt under cyclic bending at the angular frequency <italic>ω<inf>p</inf></italic>, [rad] , <italic>ω</italic> , angular frequency of cyclic bending, [rad/s] , <italic>ω<inf>p</inf></italic> , the angular frequency of excitation applied to a real track or belt in the <italic>p</italic>-th (of <italic>n</italic>) cyclic bending test, [rad/s]



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