A model is proposed to predict the response of tracks and belts to vibrations ...
Keywords:
coefficients ofthe equation Eq. (12) that describes the angular displacement ofthe node connecting the torsion spring and damper ofthe i-th Maxwell-element ofthe joint model shown in Fig. 10b resulting from a sinusiodally varying angular displacement applied tothe entire Maxwell-element, [rad]
,
through b9, damping coefficient ofthe torsion damper representing structural damping ofthe track model in Fig. 10 [Nm·s/rad]
,
a constant and a coefficient involved inthe general solution of Eq. (11) (see Appendix A)
,
the stress inthe Coulomb material model or a Coulomb rheological element, [MPa]
,
determinants derived to solve the system of equations Eq. (A.13)
,
the modulus of elasticity ofthe Hooke material model or a Hookean rheological element, [MPa]
,
flexural rigidity of a continuous beam member, [Nmm2]
,
errAp, errB p, error functions analysed to estimate the spring-damping parameters ofthe model shown in Fig. 10 for a real track or belt (see Section 3.3)
,
the error exhibited by the model shown in Fig. 10 in predicting the amplitude ofthe reaction moment induced in a real rubber track due to cyclic bending
,
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
,
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)]
,
bending frequency, [Hz]
,
thebending frequency applied to a track or belt inthe p-th of n cyclic bendingtests, [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 discussedinSections4.4and5.1, [mm]
,
iterator indicating the spring- or damping-element number
,
the area moment of inertia ofthe cross-section of a beam, [mm4]
,
k n , k 0 through k 9, stiffness ofthe torsion spring representing the elastic properties of a real track or belt inthe model shown in Fig. 10, [Nm/rad]
,
stiffness of a torsion spring implemented in a discrete beam model, [Nm/rad]
,
lengthofthealuminiumbar-sectionsubjectedtobendinginthepreliminarytestsdiscussedinSections4.4and5.1, [mm]
,
lengthof 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 ofthe i-th Maxwell-element ofthe joint model shown in Fig. 10b due to a non-zero deflection rate ofthe torsion damper, [Nm]
,
the total reaction moment induced in a real track or belt subjectedto a sinusoidally varying deflection angle, the total reaction moment induced inthe joint model shown in Fig. 10b due to a sinusoidally varying angular displacement ofthe entire joint, or the reaction moment measured using the test stand in Fig. 12 or predicted with the track model in Fig. 10, [Nm]
,
the component ofthe reaction moment measured by the acceleration sensor installed inthe test stand shown in Fig. 12, [Nm]
,
the component ofthe reaction moment measured by the load cell installed inthe test stand shown in Fig. 12, [Nm]
,
the reaction moment induced in a single Maxwell-element ofthe joint model shown in Fig. 10b due to a sinusiodally varying angular displacement applied tothe entire Maxwell-element, [Nm]
,
the moment ofthe spring that acts on the node connecting the torsion spring and damper ofthe i-th Maxwell-element ofthe joint model shown in Fig. 10b, [Nm]
,
the reaction moment induced inthe spring-element ofthe joint model shown in Fig. 10b due to a sinusoidally varying deflection ofthe spring, [Nm]
,
the number of: springs or dampers ofthe model joint shown in Fig. 10b, spring or damping elements of a rheological model, or experiments in a test series
,
the iterator that indicates an individual experiment within the series of n cyclic bendingtests on a real track or belt
,
the signal ofthe acceleration sensor installed inthe test stand shown in Fig. 12, [mV]
,
the signal ofthe 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 discussedinSections4.4and5.1, [mm]
,
the amplitude ofthe sinusoidally varying angular displacement inthe joint ofthe track or belt model shown in Fig. 10, [rad, °]
,
the amplitude ofthe deflection angle exerted on thealuminium bars during thepreliminarytestsdiscussedinSections 4.2 and5.1, [rad, °]
,
the amplitude ofthe deflection angle exerted on a real track or belt during a cyclic bending test at the angular frequency ωp, [rad, °]
,
the amplitude ofthe reaction moment resulting from a sinusoidally varying angular displacement inthe model joint shown in Fig. 10b, [Nm]
,
the amplitude ofthe reaction moment induced inthealuminium bar under sinusoidally varying deflection discussedinSections4.4and5.1, [Nm]
,
the amplitude ofthe reaction moment captured only by the load cell installed inthe test bench for cyclic bendingtests shown in Fig. 12, [Nm]
,
the amplitude ofthe reaction moment captured by the load cell andthe acceleration sensor installed inthe test bench for cyclic bendingtests shown in Fig. 12, [Nm]
,
the amplitude ofthe reaction moment induced in a real track or belt during the cyclic bending test at the angular frequency ωp andthe deflection angle amplitude Xωp, [Nm]
,
the amplitude ofthe reaction moment predicted by the model shown in Fig. 10 for a track or belt under cyclic bending at the angular frequency ωp andthe deflection angle amplitude Xωp, [Nm]
,
slope of a linear function, [-]
,
the viscosity ofthe Newton material model or a Newtonian rheological element, [MPa·s]
,
stress induced in a solid or liquid material, [MPa]
,
the ratio ofthe stiffness coefficient ki tothe damping coefficient bi ofthe elements ofthe i-th Maxwell-element ofthe joint model shown in Fig. 10b, [s−1]
,
the angular displacement inthe entire joint ofthe track or belt model shown in Fig. 10, [rad]
,
the angular displacement ofthe node that connects the spring and damper ofthe Maxwell-element ofthe joint model shown in Fig. 10b, [rad]
,
the angular velocity ofthe node that connects the spring and damper ofthe i-th Maxwell-element ofthe joint model shown in Fig. 10b, [rad/s]
,
the phase shift between the sinusoidally varying angular displacement andthe reaction moment induced inthe model joint shown in Fig. 10b, [rad]
,
the phase shift between the sinusoidally varying deflection angle andthe reaction moment induced in a real track or belt under cyclic bending at the angular frequency ωp, [rad]
,
the phase shift between the deflection angle andthe 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]
,
angular frequency of cyclic bending, [rad/s]
,
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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