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Power transmission system with chain and two sprockets
This block represents a power transmission system with a chain and two sprockets. The chain meshes with the sprockets, transmitting rotary motion between the two. Power transmission can occur in reverse, i.e., from driven to driver sprocket, due to external loads. This condition is known as back-driving.
The drive chain is compliant. It can stretch under tension and slacken otherwise. The compliance model consists of a linear spring-damper set in a parallel arrangement. The spring resists tensile strain in the chain. The damper resists tensile motion between chain elements.
The spring and damper forces act directly on the sprockets that the chain connects. The spring force is present when one chain branch is taut. The damper force is present continuously. If you specify a maximum tension value, when the net tensile force in the chain exceeds this value, simulation stops with an error.
The block accounts for viscous friction at the sprocket joint bearings. During motion, viscous friction causes power transmission losses, reducing chain-drive efficiency. These losses compound those due to chain damping. Setting viscous friction and chain damping to zero eliminates power transmission losses in the chain drive.
The tensile strain rate in the chain is the difference between the sprocket tangential velocities, each the product of the angular velocity and pitch radii. Mathematically,
$$\dot{x}={\omega}_{A}{R}_{A}-{\omega}_{B}{R}_{B},$$
where:
x is the tensile strain.
ω_{A}, ω_{B} are the sprocket angular velocities.
R_{A}, R_{B} are the sprocket pitch radii.
The figure shows the relevant equation variables.
The chain tensile force is the net sum of spring and damper forces. The spring force is the product of the tensile strain and the spring stiffness constant. This force is zero whenever the tensile strain is smaller than the chain slack. The damper force is the product of the tensile strain rate and the damping coefficient. Mathematically,
$$F=\{\begin{array}{cc}-\left(x-\frac{S}{2}\right)k-\dot{x}b,& x>\frac{S}{2}\\ -\dot{x}b,& \frac{S}{2}\ge x\ge -\frac{S}{2}\\ -\left(x+\frac{S}{2}\right)k-\dot{x}b,& x<-\frac{S}{2}\end{array},$$
where:
S is the chain slack.
k is the spring stiffness constant.
b is the damper coefficient.
The chain exerts a torque on each sprocket equal to the product of the tensile force and the sprocket pitch radius. The two torques act in opposite directions according to the equations
$${T}_{A}=-F\xb7{R}_{A},$$
and
$${T}_{B}=F\xb7{R}_{B},$$
where:
T_{A} is the torque that the chain applies on sprocket A.
T_{B} is the torque that the chain applies on sprocket B.
The sprocket tooth ratio equals the sprocket pitch radius ratio.
Chain inertia is negligible.
Specify the sprocket and chain dimensions.
Radius of the sprocket A pitch circle. The pitch circle is an imaginary circle passing through the contact point between a chain roller and a sprocket cog at full engagement. The default value is 80 mm.
Radius of the sprocket B pitch circle. The pitch circle is an imaginary circle passing through the contact point between a chain roller and a sprocket cog at full engagement. The default value is 40 mm.
Maximum distance the loose branch of the drive chain can move before it is taut. This distance equals the length difference between actual and fully taut drive chains.
If one sprocket is held in place while the top chain branch is taut, then the slack length is the tangential distance that the second sprocket must rotate before the lower chain branch becomes taut. The default value is 50 mm.
Chain displacement from the neutral position at time zero. In the neutral position (the default value) the chain can move in either direction by half the chain length before it is taut. The default value is 0 mm.
Specify the internal forces acting in the chain and on the sprockets.
Linear spring constant in the chain compliance model. This constant describes the chain resistance to strain. The spring element accounts for elastic energy storage in the chain due to deformation. The default value is 1e+5 N/m.
Linear damping coefficient in the chain compliance model. This coefficient describes the chain resistance to tensile motion between adjacent chain elements. The damper element accounts for power losses in the chain due to deformation. The default value is 5 N/(m/s).
Friction coefficient due to the rolling action of the sprocket A joint bearing in the presence of a viscous lubricant. The default value is 0.001 N*m/(rad/s).
Friction coefficient due to the rolling action of the sprocket B joint bearing in the presence of a viscous lubricant. The default value is 0.001 N*m/(rad/s).
Specify the upper tension limit in the drive chain.
Select whether to constrain the maximum tensile force in the drive chain.
No maximum tension — Chain tension can be arbitrarily large during simulation.
Maximum tension — Chain tension must remain lower than a maximum value. Simulation stops with an error when tension exceeds this value.
Maximum allowed value of the tensile force acting in the chain. The default value is 1e+6 N.
Port | Description |
---|---|
A | Conserving rotational port associated with sprocket A |
B | Conserving rotational port associated with sprocket B |
Open sdl_sheet_metal_feedersdl_sheet_metal_feeder for an example using the Chain Drive block.