Amodel is proposed to predict the response of tracks and belts to vibrations ...
Keywords:
Ai, B i
,
coefficients of the equation Eq. (12) that describes the angular displacement of the node connecting the torsion spring and damper of the i-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]
,
through b9, damping coefficient of the torsion damper representing structural damping of the track modelin Fig. 10 [Nm·s/rad]
,
a constant and a coefficient involved inthe general solution of Eq. (11) (see Appendix A)
,
thestressintheCoulombmaterialmodeloraCoulombrheologicalelement, [MPa]
,
D, DA, DB
,
determinants derived to solve the system of equations Eq. (A.13)
,
the modulus of elasticity of the Hooke materialmodelora Hookean rheologicalelement, [MPa]
,
flexural rigidity of a continuous beam member, [Nmm2]
,
errAp, errB p, error functions analysed to estimate the spring-damping parameters of themodel shown in Fig. 10 for a real track or belt (see Section 3.3)
,
the error exhibited by themodel shown in Fig. 10 in predicting the amplitude of the reaction moment induced ina real rubber track due to cyclic bending
,
the error exhibited by themodel shown in Fig. 10 in predicting the phase shift between the sinusoidally varying deflection angle and reaction moment induced ina real rubber track
,
the scaling factor used to calculate the reaction moment captured by the load cell installed inthe test stand shown in Fig. 12, [Nm/(mV/V)]
,
the bending frequency applied to a track or belt inthe p-th of n cyclic bending tests, [Hz]
,
the scaling factor used to calculate the reaction moment captured by the acceleration sensor installed inthe test stand shown in Fig. 12, [kg·m2·s2·(mV)-1]
,
thickness of a single aluminium bar discussed in Sections 4.4 and 5.1, [mm]
,
iterator indicating the spring- or damping-element number
,
the area moment of inertia of the cross-section of a beam, [mm4]
,
k n , k 0 through k 9, stiffness of the torsion spring representing the elastic properties of a real track or belt inthemodel shown in Fig. 10, [Nm/rad]
,
stiffness of a torsion spring implemented ina discrete beam model, [Nm/rad]
,
length of the aluminium bar-section subjected to bending inthe preliminary tests discussed in Sections 4.4 and 5.1, [mm]
,
length of a single section of a discrete model of a one-dimensional mechanical structure, [mm]
,
the moment that acts on the node connecting the torsion spring and damper of the i-th Maxwell-element of the joint model shown in Fig. 10b due to a non-zero deflection rate of the torsion damper, [Nm]
,
the total reaction moment induced ina real track or belt subjected to a sinusoidally varying deflection angle, the total reaction moment induced inthe joint model shown in Fig. 10b due to a sinusoidally varying angular displacement of the entire joint, orthe reaction moment measured using the test stand in Fig. 12 or predicted with the track modelin Fig. 10, [Nm]
,
the component of the reaction moment measured by the acceleration sensor installed inthe test stand shown in Fig. 12, [Nm]
,
the component of the reaction moment measured by the load cell installed inthe test stand shown in Fig. 12, [Nm]
,
the reaction moment induced ina 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]
,
the moment of the spring that acts on the node connecting the torsion spring and damper of the i-th Maxwell-element of the joint model shown in Fig. 10b, [Nm]
,
the reaction moment induced inthe spring-element of the joint model shown in Fig. 10b due to a sinusoidally varying deflection of the spring, [Nm]
,
the number of: springs or dampers of themodel joint shown in Fig. 10b, spring or damping elements of arheologicalmodel, or experiments ina test series
,
the iterator that indicates an individual experiment within the series of n cyclic bending tests on a real track or belt
,
the signal of the acceleration sensor installed inthe test stand shown in Fig. 12, [mV]
,
the signal of the load cell installed inthe test stand shown in Fig. 12, [mV/V]
,
linear velocity of a vehicle, [m/s]
,
width of a single aluminium bar discussed in Sections 4.4 and 5.1, [mm]
,
the amplitude of the sinusoidally varying angular displacement inthe joint of the track or belt model shown in Fig. 10, [rad, °]
,
the amplitude of the deflection angle exerted on the aluminium bars during the preliminary tests discussed in Sections 4.2 and 5.1, [rad, °]
,
the amplitude of the deflection angle exerted on a real track or belt during a cyclic bending test at the angular frequency ωp, [rad, °]
,
the amplitude of the reaction moment resulting from a sinusoidally varying angular displacement inthemodel joint shown in Fig. 10b, [Nm]
,
the amplitude of the reaction moment induced inthe aluminium bar under sinusoidally varying deflection discussed in Sections 4.4 and 5.1, [Nm]
,
the amplitude of the reaction moment captured only by the load cell installed inthe test bench for cyclic bending tests shown in Fig. 12, [Nm]
,
the amplitude of the reaction moment captured by the load cell and the acceleration sensor installed inthe test bench for cyclic bending tests shown in Fig. 12, [Nm]
,
the amplitude of the reaction moment induced ina real track or belt during the cyclic bending test at the angular frequency ωp and the deflection angle amplitude Xωp, [Nm]
,
the amplitude of the reaction moment predicted by themodel shown in Fig. 10 for a track or belt under cyclic bending at the angular frequency ωp and the deflection angle amplitude Xωp, [Nm]
,
slope of a linear function, [-]
,
the viscosity of the Newton materialmodelora Newtonian rheologicalelement, [MPa·s]
,
stress induced ina solid or liquid material, [MPa]
,
the ratio of the stiffness coefficient ki to the damping coefficient bi of the elements of the i-th Maxwell-element of the joint model shown in Fig. 10b, [s−1]
,
the angular displacement inthe entire joint of the track or belt model shown in Fig. 10, [rad]
,
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]
,
the angular velocity of the node that connects the spring and damper of the i-th Maxwell-element of the joint model shown in Fig. 10b, [rad/s]
,
the phase shift between the sinusoidally varying angular displacement and the reaction moment induced inthemodel joint shown in Fig. 10b, [rad]
,
the phase shift between the sinusoidally varying deflection angle and the reaction moment induced ina real track or belt under cyclic bending at the angular frequency ωp, [rad]
,
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 ωp, [rad]
,
the angular frequency of excitation applied to a real track or belt inthe p-th (of n) cyclic bending test, [rad/s]
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