Amodel is proposed to predict the response of tracks and belts to vibrations ...
Keywords:
Ai, B i
,
coefficients ofthe equation Eq. (12) that describes theangulardisplacementofthe node connecting the torsion spring and damper ofthe i-th Maxwell-elementofthejointmodelshowninFig. 10b resulting from asinusiodallyvaryingangulardisplacementappliedtotheentireMaxwell-element, [rad]
,
through b9, damping coefficient ofthe torsion damper representing structural damping ofthe track modelinFig. 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]
,
D, DA, DB
,
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 ofa continuous beam member, [Nmm2]
,
errAp, errB p, error functions analysed to estimate the spring-damping parameters ofthemodelshowninFig. 10 for a real track or belt (see Section 3.3)
,
the error exhibited by themodelshowninFig. 10 in predicting the amplitude ofthereactionmomentinducedina real rubber track dueto cyclic bending
,
the error exhibited by themodelshowninFig. 10 in predicting the phase shift between the sinusoidally varying deflection angle and reactionmomentinducedina real rubber track
,
the scaling factor used to calculate thereactionmoment captured by the load cell installed inthe test stand showninFig. 12, [Nm/(mV/V)]
,
the bending frequency appliedtoa track or belt inthe p-th of n cyclic bending tests, [Hz]
,
the scaling factor used to calculate thereactionmoment captured by the acceleration sensor installed inthe test stand showninFig. 12, [kg·m2·s2·(mV)-1]
,
thickness ofasingle aluminium bar discussed in Sections 4.4 and 5.1, [mm]
,
iterator indicating the spring- or damping-element number
,
the area momentof inertia ofthe cross-section ofa beam, [mm4]
,
k n , k 0 through k 9, stiffness ofthe torsion spring representing the elastic properties ofa real track or belt inthemodelshowninFig. 10, [Nm/rad]
,
stiffness ofa torsion spring implemented ina discrete beam model, [Nm/rad]
,
length ofthe aluminium bar-section subjected to bending inthe preliminary tests discussed in Sections 4.4 and 5.1, [mm]
,
length ofasingle section ofa discrete modelofa one-dimensional mechanical structure, [mm]
,
themoment that acts on the node connecting the torsion spring and damper ofthe i-th Maxwell-elementofthejointmodelshowninFig. 10bduetoa non-zero deflection rate ofthe torsion damper, [Nm]
,
the total reactionmomentinducedina real track or belt subjected toa sinusoidally varying deflection angle, the total reactionmomentinducedinthejointmodelshowninFig. 10bduetoa sinusoidally varyingangulardisplacementoftheentirejoint, or thereactionmoment measured using the test stand inFig. 12 or predicted with the track modelinFig. 10, [Nm]
,
the component ofthereactionmoment measured by the acceleration sensor installed inthe test stand showninFig. 12, [Nm]
,
the component ofthereactionmoment measured by the load cell installed inthe test stand showninFig. 12, [Nm]
,
thereactionmomentinducedinasingleMaxwell-elementofthejointmodelshowninFig. 10bduetoasinusiodallyvaryingangulardisplacementappliedtotheentireMaxwell-element, [Nm]
,
themomentofthe spring that acts on the node connecting the torsion spring and damper ofthe i-th Maxwell-elementofthejointmodelshowninFig. 10b, [Nm]
,
thereactionmomentinducedinthe spring-element ofthejointmodelshowninFig. 10bduetoa sinusoidally varying deflection ofthe spring, [Nm]
,
the number of: springs or dampers ofthemodeljointshowninFig. 10b, spring or damping elements ofa rheological model, 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 ofthe acceleration sensor installed inthe test stand showninFig. 12, [mV]
,
the signal ofthe load cell installed inthe test stand showninFig. 12, [mV/V]
,
linear velocity ofa vehicle, [m/s]
,
width ofasingle aluminium bar discussed in Sections 4.4 and 5.1, [mm]
,
the amplitude ofthe sinusoidally varyingangulardisplacementinthejointofthe track or belt modelshowninFig. 10, [rad, °]
,
the amplitude ofthe deflection angle exerted on the aluminium bars during the preliminary tests discussed in Sections 4.2 and 5.1, [rad, °]
,
the amplitude ofthe deflection angle exerted on a real track or belt during a cyclic bending test at theangular frequency ωp, [rad, °]
,
the amplitude ofthereactionmoment resulting from a sinusoidally varyingangulardisplacementinthemodeljointshowninFig. 10b, [Nm]
,
the amplitude ofthereactionmomentinducedinthe aluminium bar under sinusoidally varying deflection discussed in Sections 4.4 and 5.1, [Nm]
,
the amplitude ofthereactionmoment captured only by the load cell installed inthe test bench for cyclic bending tests showninFig. 12, [Nm]
,
the amplitude ofthereactionmoment captured by the load cell and the acceleration sensor installed inthe test bench for cyclic bending tests showninFig. 12, [Nm]
,
the amplitude ofthereactionmomentinducedina real track or belt during the cyclic bending test at theangular frequency ωp and the deflection angle amplitude Xωp, [Nm]
,
the amplitude ofthereactionmoment predicted by themodelshowninFig. 10 for a track or belt under cyclic bending at theangular frequency ωp and the deflection angle amplitude Xωp, [Nm]
,
slope ofa linear function, [-]
,
the viscosity ofthe Newton material model or a Newtonian rheological element, [MPa·s]
,
stress inducedina solid or liquid material, [MPa]
,
the ratio ofthe stiffness coefficient ki tothe damping coefficient bi ofthe elements ofthe i-th Maxwell-elementofthejointmodelshowninFig. 10b, [s−1]
,
theangulardisplacementintheentirejointofthe track or belt modelshowninFig. 10, [rad]
,
theangulardisplacementofthe node that connects the spring and damper oftheMaxwell-elementofthejointmodelshowninFig. 10b, [rad]
,
theangular velocity ofthe node that connects the spring and damper ofthe i-th Maxwell-elementofthejointmodelshowninFig. 10b, [rad/s]
,
the phase shift between the sinusoidally varyingangulardisplacement and thereactionmomentinducedinthemodeljointshowninFig. 10b, [rad]
,
the phase shift between the sinusoidally varying deflection angle and thereactionmomentinducedina real track or belt under cyclic bending at theangular frequency ωp, [rad]
,
the phase shift between the deflection angle and thereactionmoment predicted by the discrete modelshowninFig. 10 for a real track or belt under cyclic bending at theangular frequency ωp, [rad]
,
angular frequency of cyclic bending, [rad/s]
,
theangular frequency of excitation appliedtoa real track or belt inthe p-th (of n) cyclic bending test, [rad/s]
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