before starting: Please if you are want to understand this simulation example you have firstly to read and understand all the below otherwise it will be a waste of time. When you feel it is useful for you please, pray for me, my parents and the Iraqi people.


Function description:

The drive piston (1) is actuated in a periodic translator motion by the crank mechanism (2). The force transmission from the drive piston to the percussion piston (3) results from the air cushion (4). Due to this excitation the percussion piston performs an oscillating translator motion and axial percussions on an interposed piston, which itself impacts on the tool (in the following the interposed piston and the tool are neglected).

System equations:

The rotation of the drive crank is assumed to be constant. The position of the drive piston is explicitly described by the length s, which is dependent on the geometric parameters r, q as well as the angular velocity w (omega) of the drive crank {equation no. 1}
The percussion piston can move independently in x-direction.
A friction force (Coulomb’s friction) acts between the percussion piston and the wall of the cylinder, arranged by {equation no. 2}
A is the cross sectional area of the cylinder, M (mue) the dynamic friction coefficient between piston and the wall of the cylinder and Po is the ambient pressure.
A force acts between the drive and the percussion piston due to the air cushion, which depends on the distance h-Xs-s between the piston surfaces and in a nonlinear {equation no. 3}
Applying the principle of linear momentum we obtain the equation of motion {equation no. 4}



Modeling of the impact process:

When the percussion piston impinges on the tool respectively the header, a momentum appears which is approximately described by a spring-damper combination with spring constant c and damping coefficient d {equation no. 5}
However, this force only acts when the percussion piston tries to penetrate into tool (Xs<0). Consider this circumstance with an appropriate logical block in the block diagram (simulation model).

Realization of the simulation:

Develop a block diagram, which can establish a basis for the input in SIMULINK. Initially generate separate block diagrams for the motion of the drive piston, the friction force, the air force as well as the momentum. Afterwards develop a block diagram of the equations of motion, which imply the forces to be combined in subsystems being dependent on Xs(t) and s(t).
Enter the block diagram in SIMULINK and perform simulations of different initial conditions and parameters. Pore over the influence of the rotational velocity of the crank on the function of the hammer in particular.
An important characteristic of the function of the hammer is the impulsion velocity of the percussion.

We will apply the following numerical values for the simulation:

ms = 0.6; % percussion mass [kg]
h = 0.2; % cylinder length [m]
A = 0.00151976; % cross-sectional area of the air cushion [m^2]
l0 = 0.0365; % air cushion height at ambient pressure [m]
p0 = 1e5; % ambient pressure [N/m^2]
r = 0.022; % crank radius [m]
q = 0.076; % length of conrod [m]
xs0 = 1e-4; % initial position of the percussion [m]
mue = 0.2; % friction coefficient [-]
c = 1e8; % spring constant for impact modelling [N/m]
d = 7610.8; % damping constant for impact modelling [Ns/m]
ns = 1300; % input speed [1/min] Hint: we need angular velocity

So, first we will generate Matlab (M-file) with the above numerical values and save this file to define all the parameters which we will use them later in our model (you should see all these parameters in the workspace after running your M-file) this is very important step to be ready to build our model in the next step

Now, we will build our simulation model with Matlab. Go firstly to Matlab and then Simulink and open new project
We start with percussion piston by using {equation no. 1}. Our input will be the time and angular velocity and the output is S(t) according to our equation



Then the impact force and our input are Xs and Xs_dot {equation no. 5} we get


The next step is the friction force and here our input is Xs_dot according to {equation no. 2} thus


Now we can get Xs_2dot (double dot) by adding the forces and dividing by mass {equation no. 4} see the image below



Till now all our forces are completed but we missed some of the inputs, namely Xs_dot and Xs and we can get these two inputs implicitly by using two integration steps


Congradulation !! our model completed and what we have to do in the next few steps just extract our data

Ok, we are going now to extract some data
first we will start with S(t) where our input was the time (clock) and angular velocity



The next will be Xs and it should look like below



We can see also the friction force



And air cushion force



And finally is the most important force, impact force



We can see clearly it is some of impulses acting sequentially with impact time so small, anyway by zooming in the draw you can see the width of the impact time

source:
Duisburg-Essen Universität
Ingenieurwissenschaft Fakulität
Lehrstuhl für Mechatronik
http://www.uni-due.de/mechatronik