Simulating transmission

A number of blocks of the Mechanics library are designed to simulate various kinds of gearboxes and mechanical gears with a constant gear ratio. In general, a conversion between rotary and translatory motion is possible, however, for clarity, the approach to simulating transmission will be described using the example of a gearbox, both shafts of which rotate. For generalization, the physical dimensions of the quantities will be omitted, since the described approach is applicable to both types of motion.

The gear (speed) ratio U connects the motion profiles of the driving and driven shafts of the gearbox and the torques on them with the following ratios:

The first three ratios will hereafter be referred to as “kinematic ratios”.

In the blocks simulating transmission, the driving and driven shafts correspond to the mechanical ports by which the block is connected to other blocks of the model. The block ports can be input and output ones. Depending on the type of ports, three models are generally used.

The input port is the driving shaft

In this case, the input port receives the motion profile of the driving shaft from the block connected to it. The motion profile of the driven shaft is calculated explicitly by dividing by the gear ratio:

The value of the load resistance torque is read from the output mechanical port corresponding to the driven shaft. The torque reduced to the driving shaft is calculated by dividing by the gear ratio:

The input port is the driven shaft

In this case, the input port receives the motion profile of the driven shaft from the block connected to it. The drive shaft motion profile is calculated explicitly by multiplying by the gear ratio:

The value of the load resistance torque is read from the output mechanical port corresponding to the driven shaft. The torque reduced to the driving shaft is calculated by dividing by the gear ratio:

Both shafts are input ports

In this case, the input ports receive the driving and driven shaft motion profiles from the blocks connected to them. The library design requires torque values to be returned to these ports. Their values should ensure the consistent movement of the shafts (satisfaction of the kinematic relations described above). Torque values are calculated implicitly by solving the nonlinear algebraic equation (NE) of coupling:

where:

• Δφ, Δω, Δε(T₁, T₂) – coordinate, speed and acceleration errors
• τ – time constant for solving the nonlinear algebraic equation (NE)

The error values are generally calculated by formulas based on the kinematic ratios of the gearbox:

If the initial coordinates, speeds or accelerations of the shafts do not satisfy the kinematic relations, then at the beginning of the simulation a hopping transient will be observed, bringing the values of these values to values that satisfy the kinematic relations. The transient time is equal to the time constant for the solution of the nonlinear algebraic equation.