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I am building a test-rig for a device. I have operational data showing the device experiences impact loads resembling a half-sine pulse with width dt and magnitude F. The test rig's purpose is to replicate this force to gather life cycle data. The device mass and peak force give an equivalent target acceleration a.

Suppose I anchor the device fitted with an accelerometer to a sliding rail which allows it to kickback with close to no initial damping and then strike it with a projectile of mass m and impact velocity V (both projectile and impact surface or made of aluminum, so the collision will be fairly elastic) and found the peak acceleration to be a/2, how can I adjust the projectile impact velocity to reach my desired impulse?

My understanding of this problem is that there are complicated (difficult to predict analytically) deformations and interactions between the projectile and contact surface that occur on impact. However, with experimental data I should be able to isolate these and achieve a relationship between peak acceleration and something. Intuitively I would think the second variable was momentum, hence my use of impact velocity to characterize the impact. I would like to conduct a literature review before embarking on any concrete design work but have had tremendous difficulty finding relevant research.

Any information or research into elastic collisions and resulting acceleration impulses of the initially-stationary object would be highly appreciated, as I have had no luck thus far.

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  • $\begingroup$ Sounds like an Izod tester $\endgroup$
    – Solar Mike
    Commented Feb 17, 2023 at 4:59
  • $\begingroup$ is there any spring on the sliding rail? $\endgroup$
    – NMech
    Commented Feb 17, 2023 at 9:05

2 Answers 2

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DISCLAIMER: Your proposed setup is not entirely clear to me. Also I can't understand the term "initial damping".


Cautionary notes

I'll start with the cautions:

  • It worries me that you assume that there will be no deformation on the projectile or device. I don't know the measure of the impact, so you might be correct, but generally the size would have a lot of issues.

  • What you need to be very clear about this is that momentum is conserved, and mechanical energy is NOT necessarily conserved (i.e. although total energy will be conserved there will be some losses).

  • Finally you can't really expect to have the same duration $dt$ on the test rig with the impact (the duration during impact is probably much smaller, so you won't be able to replicate the load. IMHO, You can at best simulate the maximum force/acceleration with this projectile approach.

Coefficient of restitution

A quantity that is very important for your problem is the coefficient of restitution, which essentially given by $e= -\frac{\text{relative velocity after collision}}{\text{relative velocity before collision}}$.

The big issue is that the coefficient of restitution is depended on material and to a lesser degree on geometry (i.e. if you have a smaller contact area that it is more likely to have higher stresses and a partially plastic collision, which would reduce the coefficient of restitution).

acceleration and impulse.

The impulse $I$ will be calculated by the change of velocity of the device: $$I = m_d\cdot (v_2 - v_1) \tag{eq.1}$$

which is also equal to: $$I = \int_0^{\Delta t} F(t)\; dt\tag{eq.2}$$ where: $\Delta t $ is the duration of the impact (which should be small)

Since, you know $F(t)$ is sinusoidal with time, you'd get something like $F(t)= F_{max}\sin(\frac{\pi}{\Delta t}t)$, so the integration above would yield:

$$I = \int_0^{\Delta t} F(t)\; dt = 2 F_{max} \left[N\cdot s\right]\tag{eq.3}$$

From Eqs.1 and 3, you'd get

$$m_d\cdot (v_2 - v_1) \tag{eq.4} = 2 F_{max} \left[N\cdot s\right]$$

From this equation you can obtain the maximum acceleration that the device will experience, i.e. $a_{d,max} = \frac{F_{max}}{m_d} = \frac{ (v_2 - v_1) }{ 2} $

After you read and understood the 3rd cautionary comment above, then you can proceed with tweaking the experiment. I.e.: the projectile speed.

There are two more equations you can use:

  • conservation of momentum (hopefully the impact is central impact - not oblique)

$$ \text{momentum before} = \text{momentum after} \rightarrow m_d v_1 + m_p v_1 = m_d v_2 + m_p v_2p$$

An from the coefficient of restitution you would get:

$$e = -\frac{v_2- v_{2p}}{v_1- v_{1p}}$$

where:

  • $v_1, v_2$ are the device velocities before and after collision
  • $v_{2p}$are the projectile velocities before and after collision

This will be a very much trial and error, because $e$ is not really known. So, you'd have to start with a projectile velocity $v_{1p}$ and measure/estimate the velocities $v_{2p}, v_2$, in order to calculate the coefficient of restitution. Then you can adjust the velocity, and at the end of the next experiment verify that the coefficient of restitution has not changed significantly .

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Momentum is good, but since you are assuming an elastic collision energy is better. In real life crash events all those weird interactions at the interface become banal details, the physics of the motion of the CGs of the colliding bodies are robust calculations.

So you basically have two masses, a spring, and two initial velocities. It really is that simple.

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