Algorithm for determination the type of bilateral motions of a carrier with a mobile load along a horizontal plane

The motion of a mechanical system consisting of a carrier and a load is considered. The carrier, located all the time in a horizontal plane, can move translationally along a rectilinear trajectory. The carrier has a rectilinear channel through which the load can move. The load is considered further on as a material point. The load can move in the channel according to a predetermined motion law. The channel axis is located in a vertical plane passing through the trajectory of the carrier. The Coulomb dry friction model is applied for simulation the forces of resistance to the motion of the carrier from the side of the underlying plane. In the conditions of the carrier motion along horizontal plane without detachment, the carrier motion differential equations are a system of three linear second-order differential equations. The influence of the system parameters on the motion of the carrier from the rest state is studied. In the considered mechanical system (carrier + load) only two types of motion are possible: regular motions in which the relative moments of time being the switching times from one of the three differential equations to the other are unchanged, and the irregular motions when such switching times are different in each of the half-periods of the load motion. Definitive expressions allowing determine the type of carrier motion for the studied systems are derived. An algorithm for classifying two-sided carrier motions for a predetermined load motion law in a channel is presented based on the values of the defining expressions. Examples of specific systems are considered and the results of computational experiments are presented.


Carrier motion differential equations
The motion of a mechanical system consisting of a carrier and a load is considered [1][2][3][4][5]. The carrier, located all the time on a horizontal plane, moves translationally along a rectilinear trajectory. The carrier has a rectilinear channel through which the load can move. The channel axis is located in a vertical plane passing through the trajectory of the carrier.
Let the law of motion of the load in the channel be given in the form x 2 (t) = · sin(ωt), where = const, ω = const, and the forces of resistance of the medium to the motion of the carrier are modeled by forces of the Coulomb friction type, then the carrier motion differential equations (CMDE), according to [1,2], arë x = β · (cos ϕ + f · sin ϕ) · sin (ωt) − γ forẋ > 0 ; (1) x = β · (cos ϕ − f · sin ϕ) · sin (ωt) + γ forẋ < 0 ; where x is the carrier coordinate; ϕ is channel setting angle; β = · ω 2 · m m + M ; γ = g · f ; M is mass of the carrier; m is weight of load; f is coefficient of sliding friction in motion (equal to the coefficient of sliding friction at rest) for a pair of materials "carrier-underlying horizontal plane". It is assumed that the inequality [3] β · |sin ϕ| ≤ g is satisfied. Inequality (4) guarantees the uninterrupted motion of the carrier from the horizontal plane. If β ≥ g, then from (4) there follows a restriction on the channel setting angle 2. Conditions for the motion of a carrier from a state of rest A necessary and sufficient condition for the CMDE (1) to take place in the real dynamics of the carrier is the inequality γ < β · (cos ϕ + f · sin ϕ) , and for the realization of the CMDE (2) in real dynamics there is an inequality The replacement z = tg ϕ 2 leads these inequalities to the following form, respectively Inequality (6) (or (7)) is satisfied, if the channel setting angle is such that tg ϕ 2 ∈ (z 1 ; z 2 ), where z 1 and z 2 are the real roots of the square trinomial corresponding to (8) (or (9)). These roots are real when β ≥ γ Therefore, if (10) holds, and then the carrier can move from a state of rest in the positive direction of the axis Ox. If (10) is satisfied, and ϕ ∈ , then the carrier can move from the state of rest in the negative direction of the axis Ox. Simultaneous fulfillment of both conditions (6) and (7) (i.e. (8) and (9)) means that the intersection of intervals − → Φ 0 and ← − Φ 0 is not empty, i.e. − → ϕ 1 < 0 , which leads to the requirement Therefore, if (11) takes place, and the channel setting angle is such that If the parameter β is such that Thus, it is determined that if the parameter of the investigated system is such that then in the real dynamics of the carrier it is possible either only one-way motion in the positive direction of the axis Ox when the channel setting angle where − → ϕ 1 > 0 , − → ϕ 2 > 0 , or only one-way motion in the negative direction of the axis Ox when the installation angle where If the parameter β > γ , i.e. (11) is satisfied, then one-way carrier motion from the state of rest only in the positive direction is possible when ϕ ∈ , and the oneway carrier motion in the negative direction of the axis Ox when the channel setting angle In a further analysis of the dynamics of the carrier, lets study its possible bilateral motions from a state of rest with a positive value of the channel setting angle, i.e. select from the set Here β is the root of equation Note. In the considered case one-way carrier motions are possible only in the positive direction of the axis Ox when β < β , and the case of unilateral motions of the carrier for β ≥ β is impossible in principle.

MMBVPA
, and take as the first and the second defining expressions (20) From the system of equations which reduces to the form where θ > τ − , we find the third defining expression The algorithm for determination the type of CM for the considered cases has the form (here DAC means that "dynamics analysis is completed"): Step 1. Compute the value I 1 .
Step 1.1. If I 1 < 0, then the CM has a type R2 and DAC.
Step 1.2. If I 1 = 0, then the CM has a type R3 and DAC.
Step 2. Compute the value I 2 .
Step 2.1. If I 2 < 0, then the CM has a type R3 and DAC.
Step 2.2. If I 2 = 0, then the CM has a type R3 and DAC.
Step 3. Compute the value I 3 .
Step 3.1. If I 3 < 0, then the CM has a type R3 and DAC.
Step 3.2. If I 3 = 0, then the CM has a type R5 and DAC.
Step 3.3. If I 3 > 0, then the CM has a type NR and DAC. Here, the CM of type R2, R3 and R5 [1,2] are realized in carrier dynamics with the following regular sequences of alternation of CMDE (1) The algorithm for determination the type of CM for ϕ = 0 is: Step 1. Compute the value I.
Step 1.1. If I < 0, then the CM has a type R2 and DAC.
Step 1.2. If I = 0, then the CM has a type R5 and DAC.
Step 1.3. If I > 0, then the CM has a type NR and DAC.