# Tandem Primary Cylinder

This example shows how to model, parameterize and test a tandem primary cylinder starting from manufacturer datasheet information. First, there is a brief discussion on the mathematical modeling of the system. Given the numerical data extracted from the datasheet, optimization is then used to determine remaining unknown parameters. The model is then simulated and the resulting push rod force - brake pressure relationship curve is compared with the curve provided on the manufacturer datasheet. Understanding the behavior of the tandem primary cylinder is an important prerequisite to selection of other braking system components.

### Model

The following figure shows the tandem primary cylinder test harness. Spring-loaded accumulators are used for loading in the and the circuits in place of disk or drum brakes. The input to the model is the push rod force coming from the lever mechanism or the brake booster. Output of the model is the pressures achieved in the brake circuits 1 and 2 respectively.

### Manufacturer Datasheet Data

The supplier has provided the following data in the datasheet.

*dp*: first and second circuit piston diameter.*stroke1*: first circuit piston stroke.*stroke2*: second circuit piston stroke.*totalStroke*: overall stroke.*disp1*: first circuit displacement.*disp2*: second circuit displacement.*totalDisp*: net displacement.*maxPress*: max pressure.

### Pressure Development Characteristics

The supplier datasheet gives the following function diagram or the push rod force-brake pressure curve for the tandem primary cylinder.

### Equations of Motion (EoM) of System

The following figure shows the tandem primary cylinder. Mass 1 and mass 2 use the coordinate frames shown below. This description only includes the mechanical EoM of the system. The fluid dynamics equations are not shown.

, when

, when

, when

, when

where,

and are positions of piston mass 1 and mass 2 from the left side assumed hard stop.

and are masses of piston 1 and 2 respectively.

and are damping coefficients of piston 1 and 2 respectively.

and are stiffness coefficients of left (Spring 1) and right (spring 2) respectively.

is force applied on push rod of primary cylinder.

and are pressures in brake circuit 1 and 2 respectively.

and are reaction forces on mass 1 and 2 respectively when both masses are at the leftmost position.

and are preloads on left and right side springs respectively.

### Parameter Estimation Method

Some parameters, which are important for functionality assessment, are not provided in the manufacturer's datasheet, therefore to simulate the model it is pertinent to estimate these parameters. The EoM are converted to steady state equations to estimate unknown parameters.

, when

, when

, when

, when

Data for the function diagram is explained with following figure.

where,

and are the push rod forces for points 1,2, and 3 respectively.

,Here is for the brake circuit pressures at points 1,2, and 3 respectively.

and represent the leftmost positions for piston 1 and 2 respectively.

and as per the manufacturer datasheet data.

and are piston positions at data point 2. This point needs to be iterated on during optimization to estimate the unknown parameters.

The estimation scheme is as follows:

The estimation should yield results with the following conditions met:

and should be greater than or equal to 0.

and should be greater than 0.

### Assumed Parameters

Some parameters are assumed as they are neither given in the datasheet nor can they be estimated. The following is the list:

*c1*is damping coefficient for piston 1.*c2*is damping coefficient for piston 2.*mass1*is piston 1 mass.*mass2*is piston 2 mass.*v2D*is brake circuit 1 dead volume.*v2M*is brake circuit 1 maximum volume.*v4D*is brake circuit 2 dead volume.*v4M*is brake circuit 2 maximum volume.*a1Ori*is compensating orifice area.*a2Ori*is brake circuit orifice area

### Load Circuit

It is important to test the system with the correct load. In the test model, loading is done on the tandem primary cylinder using spring-based accumulators. Setting up correct parameters in the accumulators is important. From the datasheet it is known what should be the stroke values for brake circuits 1 and 2 respectively. Iterated parameters and are important for deciding the fluid chamber capacity of the accumulators. Assumed or estimated parameters for the load circuit accumulators are:

vol1 is fluid chamber capacity for brake circuit 1.

vol2 is fluid chamber capacity for brake circuit 2.

loadP1 is pressure at full capacity for brake circuit 1.

loadP2 is pressure at full capacity for brake circuit 2.

### Simulation Results from Simscape Logging

The model generates push rod force-brake pressure plots for selected manufacturer designs. The applied push rod force to the tandem primary cylinder is ramped at a 25 N/sec rate in the simulation.