Cylinder Components

Cylinder Component General Description

The pressure created by a fed working fluid is the basis for the operation of a hydraulic cylinder. Flow Simulator provides two types of piston cylinders. In a single-acting hydraulic cylinder, the working fluid comes into the cylinder from one hydraulic tank and presses down on the piston from one side, creating a return stroke of the rod. The return spring actuates a forward stroke.

In a double-acting cylinder, the return spring is absent, its role is performed by the second hydraulic tank, creating pressure on the piston from the other side and, as a result, providing reciprocating motions of the mechanism.

Quick Guide for Cylinder Component Creation in the GUI

There are two Cylinder subtypes under “Compressible Gas Elements” and “Incompressible Liquid Elements” sections in Flow Simulator.

The Single Acting cylinder is attached to one chamber (side 1). The Double Acting cylinder is attached to two chambers.

Cylinder Component Inputs

Table of the inputs for the Cylinder Component.

Cylinder Component Theory Manual

Single Acting Cylinder Schematic

Cylinder Initial Position

The spring forces and pressure forces on the piston must be balanced at the start of the transient. The initial position, x_o, input determines the spring forces and must be set correctly. The solver does not calculate or set this input so you must set it. Ignoring the effects of the wall friction or piston mass, the equation is:

Cylinder Governing Equations

Piston-Rod Force Variable Definitions

Force Balance Equation (Newton’s Second Law)

Where:

The Force Balance Equation can be re-written as a second order linear ODE.

There are two complications:

1. The Coulomb (Dynamic) Friction term is either positive or negative depending on direction of piston motion. This means the governing ODE of piston motion needs to be split into two parts, one for left-to-right motion and the other for right-to-left motion. In both cases, the initial position and velocity are the initial conditions.
   $$F_{linear}=\left(1-C_{cf}\right) \left(F_{p1}+F_{p2}\right)+F_{e1}-F_{e2} if \frac{dx}{dt}\ge 0$$
   $$F_{linear}=\left(1+C_{cf}\right) \left(F_{p1}+F_{p2}\right)+F_{e1}-F_{e2} if \frac{dx}{dt}<0$$

2. The total pressure force and external force () is changing within each timestep. Flow Simulator assumes linear variation within each timestep, which is second order accurate, and helps when you want to take relatively large fluid timesteps. However, it should be noted that you can always achieve greater accuracy by taking smaller fluid timesteps.

   

The governing ODE can be split into four distinct ODE regimes by introducing and parameters. (LR signifies left-to-right motion and RL is right-to-left motion.)

Depending on the ODE coefficients of the two non-trivial ODEs, the ODE solution can follow one of eight different equation forms.

Under-damped when

Solution Parameters based on ODE Coefficients and Initial Conditions:

Solution Form:

Over-damped when

Solution Parameters based on ODE Coefficients and Initial Conditions:

Critically Damped when

Solution Parameters based on ODE Coefficients and Initial Conditions:

Spring-less when K=0

Solution Parameters based on ODE Coefficients and Initial Conditions:

Spring-less & Friction-less when K=0 & C=0

Solution Parameters based on ODE Coefficients and Initial Conditions:

Mass-less when M=0

Solution Parameters based on ODE Coefficients and Initial Conditions:

Mass-less & Friction-less when M=0 & C=0

Solution Parameters based on ODE Coefficients and Initial Conditions:

Mass-less & Spring-less when M=0 and K=0

Solution Parameters based on ODE Coefficients and Initial Conditions:

Adaptive Time-stepping

Detecting changes in four ODE regimes adaptively and adjusting the piston motion timestep and/or fluid timestep is critical to accurate results.

If the “Mechanical Only” adaptive option is chosen, then is equal to . The piston-cylinder solver will shorten the piston motion ODE interval. A second (or sometimes more) piston motion solution intervals will be required to reach . Therefore, there will be piecewise piston motion solution in on the interval from . However, the linearly varying force assumption, which is usually due to time-varying pressures, still applies on the interval from . This means you a not resolving the fluid pressure changes at the shorter timestep.

If the “Coarse” adaptive option is chosen, then is set equal to , so the fluid timestep is shortened as well. This achieves greater accuracy.

The “Fine” adaptive option is like “Coarse”, expect with an addition constraint. If there is a spring, then there is a natural oscillation period based on the spring stiffness and piston-rod mass. The timestep will not be allowed to go higher than 1/N of the natural oscillation period. The default N is 12. You can change this input in the Advanced Options menu. This option should be used for Cylinder components with high coupling sensitivity to the fluid model – and where oscillations of the piston must be captured to correctly model the physical behavior of the system.

Finally, the “Off” option does not shorten either mechanical or fluid timestep. However, it does use a numerical trick to prevent spurious oscillations that occur when you require the piston to leave the end at the “wrong time”. It uses a smoothing option, described in detail in the following section, but only for the time-steps where the piston becomes unstuck.

Smooth Solution Option

The Cylinder Component provides an option to smooth the solution of oscillating pistons. Usually the fluid network will dampen the piston motion. Alternatively, you can specify viscous friction. However, in some cases, these two options may not be sufficient. If you do not know how much friction or fluid mass that is required for damping, but you know the piston motion should be smooth (non-oscillatory), then you can use the Smooth option. If the solution is in the underdamped mode, the smooth option will solve for piston motion including oscillations, and only at the last stage it zeros out the amplitude of the sine and cosine terms. For piston position, this is not usually important. However, for piston velocity, which directly affects cylinder port flow rates that are returned to the fluid solver, this can provide critical stabilization. This will aid both convergence robustness and solution accuracy. For all the other ODE regimes besides underdamped, the Smooth option does not take effect.

Mechanical & Fluid Coupling Method

To couple the piston motion with the fluid solver, the mass flow rate at the piston ports must be calculated. Here we define as the rate of mass entering the cylinder through the port.

The velocity term needs special treatment that is analogous to unwinding in CFD. This helps to achieve a stable and accurate solution.

In other words, Flow Simulator uses the exact solution for velocity to calculate mass flow rate when the piston is not near the end positions. However, when the piston becomes stuck or unstuck from the end position, Flow Simulator uses a conservative differencing formula. This makes sure the rest of fluid circuit experiences the correct amount of mass release or mass removal by the cylinder.

Cylinder Component Outputs

The following listing provides details about Cylinder Component output variables.

Note: Currently, Cylinder Piston Component results are only in ENG unit.