Spring Piston Simulator Project

[MartyMcFly]

It’s been a while since I’ve been active on GTA, and I thought this might be a good time to re-engage with a project I’ve always been passionate about but lacked the time and expertise to fully explore. My interest began years ago when I read a book by the Cardews on the physics of spring-powered airguns. While it was fascinating, it left me wanting more. The spring-piston airgun market has seen little innovation for decades, likely due to the rise in popularity of PCPs. Additionally, there’s a lack of openly accessible, non-proprietary knowledge for inventors and hobbyists to build upon. I suspect that many of the designs still in use today were optimized through trial and error in the and late 80s and early 90s. While the mechanical principles behind springers are straightforward, the internal physics are complex, which leads me to believe there’s still room for meaningful optimization.

The ultimate goal of this thread is to motivate myself to write some code and better understand the trade-offs involved in designing spring-powered airguns. Although I’m not a trained engineer, I hope that with research and input from this forum, I can develop a model that closely approximates real-world performance. Ideally, this model will shed light on how various design choices affect the performance of springers.

For context, I’m not starting entirely from scratch. A pivotal paper published in 2015 by Domingo Tavella provides a solid foundation. While I won’t be replicating his work verbatim, I plan to use it as a guide. Dr. Tavella’s paper is thorough, but the lack of open-source code limits its practical application for DIY inventors. This gap forces hobbyists to rely on costly and time-consuming prototypes, effectively perpetuating the trial-and-error method of innovation.

Approach:

• Develop a High-Level Framework: Create and code a framework that firstly models the major dynamics inside an airgun.
• Validate Predictions: Test the model’s predictions against real-world performance.
• Incremental Complexity: Gradually add more detailed physics to capture the various phenomena within a springer.
• Enable Experimentation: Build functionality for easy experimentation and comparisons between different designs.

Tools:
The simulation will primarily be a one-dimensional (1D) model implemented in Python. There are three key reasons for this choice:

• Accessibility: I lack access to industrial CFD modeling software. While I’ve experimented with ANSYS in the past, the steep learning curve and high subscription costs make it impractical.
• Open-Source Collaboration: I want the code to be freely available for others to use and improve upon.
• Practicality: I believe a well-constructed 1D model can still provide valuable insights, offering hobbyists a clear understanding of 90% of the key dynamics at play.

The simulation will rely on numerical integration of ordinary differential equations. Since the components in the system interact and influence one another, the only feasible approach is to simulate all components simultaneously. Initially, the model will be constructed to simulate the dynamics of an Air Arms TX200 Mk3. That said, it will be trivial to change the dimensional and spring characteristics to match other types of rifles for validating against other designs.

Questions:
There are many questions that interest me and I don't how close we can come in answering them in a simple 1D simulation, nevertheless I'd like to explore the following:

• What are the peak temperature and pressures generated by a springer?
• How much energy or efficiency is lost to various frictions and heat sinks?
• Clarify the exact timing of events, such as where is the pellet in relation to piston position.
• Is there a material difference in performance when comparing a spring and gas ram design?
• How do transfer port diameter and length impact performance and recoil?
• Is there a link between dimensional/power characteristics vs. pellet accuracy?
• What is an ideal barrel length?

I hope that this project will not only educate me, but also engage others to think about novel solutions and experiments. Maybe the commercial age of springers is past us, with no new springer designs coming to market, but that doesn't mean that we can't try to improve on these wonderful machines ourselves. Moreover, I'm tremendously encouraged by the continued work of tuners and especially the lone innovators in England, folks like Tony Leach, for pushing the envelope of what is possible with a little grit and imagination. I hope this project encourages others to push the limits of springer design and performance.

PS. I hope to keep this thread active as much as possible, but I do have a job and family, so pauses in progress should be expected. That said, don't let that stop YOU from contributing ideas/solutions if you are willing and able.

-Marty

[subscriber]

Marty,

This sounds like quite an undertaking.  I think that power hungry Yanks are not interested in 5 FPE springers, nor in 25+ FPE springers because they are too bouncy.  Hence the gravitation towards PCPs.

I appreciate the simplicity of springers and the challenge of shooting them well, but if you could invent a non-bouncy 10 to 15 FPE springer you will have a winner.  But only if you can do it cheaply.  This may require copying and simplifying the dual opposing piston systems.  Or piston with counterweight systems.  The latter may sound like wasted parts, but plumbing the airflow is much easier with only one piston.

I have not come here to distract from your mission.  If you want an academic or lab understanding, rather than jumping to a marketable product, more power to you.  Perhaps decent lab work would be a better foundation for a better product - as it should be. 

Perhaps springers with a sled approach are better than dual pistol or dual mass ones.  All I know is that over sophistication often increase the weight of the air rifle; as does making it produce more than 20 FPE.   I would be most interested in a "recoilless" springer than weights 7 lb and is 40" long.  It can have any power level from 7 to 15 FPE - if it is light and compact enough.  Peak cocking force should not exceed 30 lb - half of that would be better, but can proportionate to power.  And spring twang should be subdued...  If you can improve efficiency, the above should be easy

[MartyMcFly]

subscriber, I'm glad you found this thread. I hope a few more regulars like you come along for the journey too. That said, I can say confidently we share very similar ideas of what makes an ideal springer. I outgrew my power-hungry fantasies a number of years ago and I would like nothing better than a mid-level springer that is smooth to shoot, doesn't weight a ton, and is efficient at translating elbow power into FPE.

I think that opposing pistons are great from a novelty perspective. They are admirable works of art which, when calibrated and well maintained, give great performance. Unfortunately, they add a level of complexity and mass that makes them impractical for all around use and affordability. I believe there is probably a good halfway point that lies between a simple design and some sort of dummy piston or sled design that gets us 80% of the way with 20% of the price and complexity.

Anyways, my hope is that we can take various designs and show how close they are to their peak efficiency, and if they are far away figure out how to get them there. Thanks for your input and I look forward to your future feedback.

-Marty

[MartyMcFly]

Starting with some basics...

The heart of a springer is of course the spring! Therefore, before we can get anywhere we need to determine the characteristics of the spring in our simulation. The most important information we need is the length of the spring and the spring's constant 'k' (stiffness coefficient). We can plug this information into Hooke's Law, which provides the amount of force a spring produces given a certain level of deflection. Spring k gives the amount of force (Newtons) the spring pushes or pulls per meter. 

I did a google search and according to some UK forums the TX200 has a 200mm spring with a k = 8.923x10^3 Newtons/meter.  I suspect that this is for the 12 FPE version and not the FAC. I don't have a way to test the force generated by my FAC spring (yet), but there is a formula which we can use to derive spring k if we know the material and the spring's dimensions. So, I took the following measurements:

Rest Length = 225mm (0.225 meters)
Outer diameter (OD) = 20.78mm (0.02078 meters)
Inner diameter (ID) = 14.24mm (0.01424 meters)
Wire diameter (WD) = 3.2mm (0.0032 meters)
Mean coil diameter (MCD) - there are two ways of computing this, so I will take the average of the two methods for the time being:

• MCD = OD - WD --> 20.78 - 3.2 = 17.58
• MCD = ID + WD --> 14.24 - 3.2 = 17.44
• AVG MCD = (17.58 + 17.44) = 17.51

Coil count = 29
Active Coil count = 29 -2 = 27 (subtract 2 for the collapsed ends)
Spring mass = 100.5g
Spring Modus of Rigidity (material dependent) - I don't know the exact material of the spring so I'll experiment with the following common ones. Note: 1 GPa = 10^9 Pascal. This is about 145,039 psi.

• Music Wire = 81.4 GPa
• Hard Drawn Steel = 79.3 GPa
• Chrome Steel = 77.2 GPa
• Spring Steel = 69.0 GPa

The formula to determine k:
Spring k = (modulus_rigidity * WD^4) / (8 * MCD^3 * AC). If we plug in the modus of rigidity for Music Wire we get about 7,361 N/m. This seems less than the Spring k noted in the UK forum (8,923 N/m), but it is in the ballpark. I suspect that either the longer length of the spring or its pre-load allows for a less rigid spring to be used in the FAC TX200. Of course, the internet figure could also be wrong. We won't know for sure until we do further testing.

But let's move on to Hooke's Law, which states: F = -kx This simply states that the (spring k) multiplied by the displacement of the spring (i.e. compression or extension) will generate some amount of force at said displacement distance. It's easy to see that the more we compress or stretch a spring from its rest length the more force it will generate.

Quick example. If we compress our spring by 100mm it will be pushing back by:
F = -kx --> -(7,361) x (0.1) = -736.1 Newtons (the negative sign tells us that its pushing in the opposite direction of the displacement or compression).
It should be noted that spring pre-load will naturally increase the amount of Force pushing back in addition to any compression we add via cocking.

Ok, so now we have some spring basics down. I'll see you on the next post.

-Marty

 

[MartyMcFly]

On to more measurements!

In addition to spring data, we need to gather more information about the internal dimensions of the TX200 Mk3 FAC. Knowing the dimensions will allow us to compute how much air is being compressed and what are the distances over which the parts travel. A quick google search returned the following:

• Piston Diameter = 25mm
• Stroke Length = 96mm
• Piston Mass = 220g
• Piston Washers/Weights = 28g

I don't know if these measures are accurate, so I will have to take my TX200 apart to confirm. Additionally, we need to know how much spring pre-load is present, how big is the diameter of the transfer port, how long is the transfer port, and how long is the barrel (the TX has a shroud so its not obvious).

If anyone has measured these things on their own rifle, please feel free to share your measurements.

-Marty

[WhatUPSbox?]

Marty, Nice to see someone revisiting airgun basic physics. Not sure if that is your plan, but I always thought it would be valuable if someone could code the analysis in the Tavella paper.

Several years ago I did some testing on the shot cycle of a Crosman Titan (gas ram). The cycle timing can be measured, including pellet motion, But the thermodynamics (Pressure, Temperature) are much more elusive.

If you are planning to do a lot of testing, I would suggest using a basic low cost springer so that any planned or unplanned irreversible modifications don't hurt as much.

Looking forward to this thread

[MartyMcFly]

Marty, Nice to see someone revisiting airgun basic physics. Not sure if that is your plan, but I always thought it would valuable if someone could code the analysis in the Tavella paper.

Several years ago I did some testing on the shot cycle of a Crosman Titan (gas ram). The cycle timing can be measured, including pellet motion, But the thermodynamics (Pressure, Temperature) are much more elusive.

If you are planning to do a lot of testing, I would suggest using a basic low cost springer so that any planned or unplanned irreversible modifications don't hurt as much.

Looking forward to this thread

Stan/WhatUPSBox - Indeed, I think I remember you doing some accelerometer readings on the Titan. I don't have one (either same rifle or accelerometer), but I'm sure some of the data you captured will be useful in correlating with the output of the simulation.

Regarding Tavella's paper, I plan to implement as much of it as I can manage. I'm not aiming for 100% replication, but if I can get 80% there I'm sure others will be able to improve and add to it.

-Marty

[MartyMcFly]

On to more measurements!

In addition to spring data, we need to gather more information about the internal dimensions of the TX200 Mk3 FAC. Knowing the dimensions will allow us to compute how much air is being compressed and what are the distances over which the parts travel. A quick google search returned the following:

• Piston Diameter = 25mm
• Stroke Length = 96mm
• Piston Mass = 220g
• Piston Washers/Weights = 28g

I don't know if these measures are accurate, so I will have to take my TX200 apart to confirm. Additionally, we need to know how much spring pre-load is present, how big is the diameter of the transfer port, how long is the transfer port, and how long is the barrel (the TX has a shroud so its not obvious).

If anyone has measured these things on their own rifle, please feel free to share your measurements.

-Marty

OK, so I did some of my own measurements on the piston. They came out slightly different from the internet search, this could be the result of differences in production runs or measurement errors. I still need to measure a few more things but here is what I got so far:

• Piston Mass with seal = 221.5g
• Piston Washers/Weights (Top Hat) = 23.1g
• Transfer Port Length = 9.87mm (internet says it should be 9.8mm)
  
• Transfer Port Diameter = 3.6mm (internet says it should be 3.75mm, possibly my caliper jaws are too thick to get a good reading)
• Spring Pre-load washer = 2.5mm

-Marty

[subscriber]

Apart from spring rate, there is dynamic hysteresis to consider: If you slowly compress a spring with a load cell and displacement measure, then slowly allow the spring to expand, the force in the spring at each incremental length will be less while expanding than at the same spring length, while compressing it.  This is known as hysteresis.  To complicate the matter, we care about how efficiently the spring can generate force while expanding rapidly. 

Measurment system friction adds load to the loadcell while compressing the spring, and subtracts from the loadcell reading while the spring is expanding; because the friction is reversed.  Yes, we also care about system friction is a given airgun design; but it is what is happening inside the spring's material that I am emphasizing. 

The spring has some internal damping inherent in the material, so dynamic hysteresis more pertinent than measure force while the spring is allowed to expand slowly.  Think of it as crystalline matrix inertia.  Or friction between crystals is probably more accurate, except that it is worse at high speed.  The opposite of static and dynamic surface friction.  Rubber has very poor dynamic hysteresis; or a lot of damping, if you like.  In the case of a spring piston airgun, the damping we want on the spring is to reduce the twang.    Different spring materials have different crystal matrix friction or inertia.

I have seen a number of posts about people reducing their spring length a coil or two to reduce power and harshness on their airgun, only to find the power remaining the same, while cocking effort and harshness were reduced.  I suspect that system friction and or crystal matrix restitution suffer when a spring is over compressed or over stressed.  Cutting a coil of a spring reduced the preload, while increasing the spring rate.  Perhaps the strange plateau in pellet FPE is because of an increase in spring restitution efficiency when compressed to a given overall length. 

When a longer spring is almost coil bound the spring and its material may be "unhappy".   Giving the coils more room by cutting it shorter seems to fix that, despite the shorter spring having a higher rate.  A higher rate spring does not only load up faster in compression; it unloads faster in extension.  This begs the question; would a similar effect exist for adding or removing spacer length, without altering the spring?

[MartyMcFly]

subscriber, thanks for the info on hysteresis. I’m not planning to model that behavior yet,  but I do think it will make initial calibration trickier. The expected impact is a 1-5% of loss of power for metal springs. I may include a fudge factor for this since I don’t  how know to model it, yet.

As to why a cut spring can create the same power, I think this can also occur due to choke conditions in the TP. If the spring is over powered choke may limit the transfer of energy to the pellet. But I don’t know for sure and it’s something that would be worth investigating. It may also do with the non-linearity of the piston drag and speed of gas compression under different loads…

-Marty

[subscriber]

Marty,

Too much spring may increase piston bounce, possibly related to TP volume.  TP flow area comes into the equation after the pellet starts moving rapidly.

[rsterne]

Many years ago, I downloaded this file below, which deals with a lot of what your are interested in.... I hope it may be of some assistance....

I have another one in Excel,, but it is too large to attach.... I will try zipping it....

Bob

[MartyMcFly]

Some more thoughts on simulating the spring:

Tavella's paper mentions that he takes a finite element approach to simulating the spring. Basically, he breaks-up the spring into multiple segments each with its own number of coils and mass. He does this to account more accurately for deflection at the end of the stroke. He states that unless this is done the kinetic energy is overestimated, resulting in higher output in the simulations than experienced in real life. Fortunately, as mentioned previously there is also the phenomena of spring hysteresis, which lowers the kinetic energy that the spring gives back on decompression. Therefore, for the initial version of my simulation I will take a simpler approach than Tavella. I will assume that the spring is without mass and that the mass is concentrated at the end of the spring, namely in the piston assembly. This means that I will assume that most of the power overestimation is cancelled out by spring hysteresis. This may or may not be true in real life, but for the moment it should be a good enough simplification to get us to a semi-working model, which can be improved once we start computing some forces.

All that being said, the spring's mass is still important and most of the spring simulation approaches I've read take a third of the spring's mass and add it to the total mass of the piston assembly, which is what I will do (at least initially).

-Marty

[MartyMcFly]

Many years ago, I downloaded this file below, which deals with a lot of what your are interested in.... I hope it may be of some assistance....

I have another one in Excel,, but it is too large to attach.... I will try zipping it....

Bob

Thank you, Bob! I have not read his paper on PCPs, I will definitely take peak.

-Marty

[rsterne]

Here is my attempt to attach the other one....

Bob

[MartyMcFly]

Here is my attempt to attach the other one....

Bob

Thanks for sharing the spreadsheet. Given the limitations of excel, the person that put this together did a great job doing the integration steps manually. This will be useful in cross checking the initial results from the python simulation. What I'd like to do different then this spreadsheet is make the code object oriented so it's easy to add more advanced friction models or change things like spring models or even add piston bounce. Additionally, I want to be able to easily compare changing a single or multiple variables in order to isolate the exact impact of certain tweaks.

-Marty

[MartyMcFly]

Setting up a program skeleton...

The following is just some boiler plate code. It doesn't compute anything but instead serves to provide a layout of where the different pieces will exist. Names of variables and functions will probably change as the sim evolves.

import numpy as np
import matplotlib.pyplot as plt
from scipy.integrate import solve_ivp

class SpringAirgunSimulation:
    def __init__(self):

        # --- Mass measurements ---
        self.spring_mass         = xxx           # kg
        self.piston_mass         = xxx           # kg
        self.piston_weights_mass = xxx      # kg

        # Effective piston assembly mass: 1/3 of spring + piston + weights
        self.pistonAssembly_mass = (1/3)*self.spring_mass + self.piston_mass + self.piston_weights_mass

        self.pellet_mass = xxx     # kg
        self.rifle_mass = xxx       # kg

        # --- Geometry & Dimensions ---
        self.spring_length = xxx   # m
        self.spring_preload = xxx  # m
        self.spring_cocking = xxx  # m

        # Total spring deflection
        self.spring_deflection = self.spring_cocking + self.spring_preload

        self.pellet_caliber = xxx  # m
        self.pellet_head_area = np.pi*(self.pellet_caliber/2)**2

        self.piston_diameter = xxx       # m
        self.piston_radius   = self.piston_diameter / 2 # m
        self.piston_head_area = np.pi * self.piston_radius**2 # m^2

        # Initial "chamber" length set to cocking distance
        self.chamber_length = self.spring_cocking
        self.chamber_volume = np.pi * self.piston_radius**2 * self.chamber_length

        self.transferPort_diameter = xxx # m
        self.transferPort_length = xxx  # m
        self.barrel_length = xxx  # m

        # --- Physical constants & properties ---
        self.spring_k = xxx                    # N/m
        self.ambient_pressure = 101325 # Pa
        self.ambient_temp = 293            # K
        self.gamma  = 1.4                      # Heat index
        self.R = 287.05                          # J/(kg·K)
        self.gravity = 9.801                    # m/s^2

    # --- Spring Model ---
    def SpringForce(self, spring_x):

    # --- Friction Models ---
    def BoreFriction(self, pellet_x):
    def PistonFriction(self, piston_x):
    def PelletDeformation(self, pellet_x):

   
    # ----------------------------------------------------------------------
    #  ODE System - 2nd order equations
    # ----------------------------------------------------------------------
    def system_equations(self, t, y):

    # ----------------------------------------------------------------------
    #  Simulation block
    # ----------------------------------------------------------------------
    def simulate(self, time_span, time_steps):

    # ----------------------------------------------------------------------
    #  Graphing simulation data
    # ----------------------------------------------------------------------
    def plot_results(self, solution):

-Marty

[rsterne]

I thought that spreadsheet included piston bounce, but it's been years since I looked at it....

Bob

[MartyMcFly]

I thought that spreadsheet included piston bounce, but it's been years since I looked at it....

Bob

I didn't look in detail at the spreadsheet logic, I assumed it didn't because there is no variable for restitution force for the piston seal, which I think it necessary to model a bounce properly. That said, the spring velocity does go negative on the graph, which at first looked like a bounce but that could be just the spring oscillation and slowdown from the friction dampening. I'll have to look at it in more detail to say for sure.

-Marty

[WobblyHand]

Don't know if this helps or not, but there's CFD available in FreeCAD, as well as FEA (called the FEM workbench) and there's a CFDoF workbench.  I've used the FEM workbench, and have been able to get CFDoF to run some examples, including supersonic airflow.  The latter (CFDoF) is a lot harder to me to understand and to get running.  I've yet to get one of my CFDoF models to simulate.

I used to program in Python (and NumPy & SciPy) extensively, to design and run radar simulations, from physics models through signal processing.  Can't say I was a Python expert, but I used the tools to get the job done, for designing automotive radars.  It's great to see some active Python programming!