Analysis

FORCE ANALYSIS: GIVENS & CONVERSIONS

For the force analysis of this system, several known values and conversions must be considered first.

Given Values

Several given values for the force analysis were taken from the text Theory of Elasticity [1]. In addition, ATK provided several other values from the on-site GCU. These variables include the radius and mass of both the bolt carrier and SVR as well as the approach velocity of impact. Information from our sponsor at ATK has yielded several given values. These values are tabulated in Table 1 below.

Table I – Given values from ATK

	Variable	Value	Units
Mass of Bolt Carrier	m1	67.816	kg
Mass of SVR*	m2	0.821	kg
Radius of Bolt Carrier	R1	35.88	mm
Radius of SVR	R2	21.96	mm
Approach Velocity of Impact	v	2.85	m/s
Max acceleration of SVR	a1	108.7	g
Cross-Sectional Area of Impact Face	A	36	mm^2

*Mass of SVR includes mass of (3) Duracell CR123A 3V batteries.

Other given values were found in Timoshenko’s Theory of Elasticity [1]. These values are tabulated in Table 2 below.

Table 2 – Given values from Theory of Elasticity [1]

	Variable	Value	Units
Poisson’s ratio for bolt carrier	ν1	0.3	-
Poisson’s ratio for SVR	ν2	0.3	-
Modulus of elasticity for bolt carrier	E1	3x10^7	psi
Modulus of elasticity for SVR	E2	3x10^7	psi

Conversions

In order to ensure accuracy and precision of the force analysis, several conversions must be taken into consideration. The mass of the SVR in Table 1 includes the mass of the batteries. The total weight of the SVR with batteries is 1.81 lbs. The weight of the bolt carrier is 149.51 lbs. Using Equation 1 gives the following results:

142.51 lbs = 67.816 kg = m₁

1.81 lbs = .821 kg = m₂

Where:

W = Weight [lbs]

m = Mass [kg]

In addition, the cross-sectional area of the impact face is 36mm². Dividing this value by (1000)² converts the mm² to m² which will be used later in force analysis calculations. The new value

FORCE ANALYSIS: CALCULATIONS

ATK ran several tests using an ABS plastic model of the SVR. This “dummy round” was cycled through the MK44 Bushmaster Cannon gun system and was ejected similar to an SVR. The plastic model included an embedded accelerometer that was able to collect acceleration data in the x-direction and y-z plane during cycling. The primary forces acting on the SVR were analyzed with this data. The max acceleration was determined to be during the “Ram-Fire-Extract” event and was found to be (-108.7g). Figure 3 below shows an image of the actual ABS plastic model that was used for testing at the ATK facility.

Figure 3 – ABS Plastic SVR

The force analysis was determined using this value of (-108.7g). Equation 2 shows the method of calculating the force component exerted onto the ABS plastic round.

Where:

F = Force [N] a₁ = Max acceleration [m/s²]

The acceleration component was first multiplied by 9.81 m/s² in order to convert from g’s to m/s². The following results were obtained by substituting in values:

Multiplying mass by the max acceleration measured on the ABS plastic model resulted in a force of 875.47N. This represents the max force on the plastic model during cycling through the gun system. In order to determine the force exerted on the actual SVR when cycled through the gun system, further analysis is needed due to the difference in materials. For this analysis, Maxwell’s Viscoelastic equations are applied.

Maxwell’s Viscoelastic Model

Maxwell’s Viscoelastic Model can be used to represent the most realistic impact force analysis. With these upper and lower bounds the values found using Maxwell’s Model can be verified. It is important to discuss the model that Maxwell presents prior to performing any calculations. This model assumes a Maxwell viscoelastic element consisting of a spring damper in series. The system being analyzed follows this model thereby allowing an impact force analysis using this specific model. To begin analyzing the forces, first the effective mass must be calculated. Equation 10 is used to find this value.

Where:

m = Effective mass [kg]

The effective mass was found to be .811 kg. Next, the natural frequency is needed. This can be solved by using Equation 11, below.

Where:

w₀ = Natural frequency [Hz]

K_bc = Axial stiffness

The axial stiffness in Equation 11 has not been calculated. This value can be determined by taking the modulus of elasticity and multiplying it by the cross-sectional area. This quick calculation results in a value of 1.67x10⁵ pounds for the axial stiffness. Substituting axial stiffness into Equation 11 yields a natural frequency of 454.3 Hz. A summary of the initial values needed is helpful in further calculations. These values are tabulated in Table 3, below.

Table 3 – Initial values for Maxwell Model analysis

	Variable	Value	Units
Mass of Bolt Carrier	m1	67.816	kg
Mass of SVR	m2	0.821	kg
Effective mass	m	.811	kg
Axial Stiffness	Kbc	1.67e5	pounds
Approach Velocity of Impact	v	2.85	m/s
Natural Frequency	w0	454.3	Hz

Next, the damping ratio will be used to determine the damped natural frequency. However, this damping ratio is unknown and must be calculated using an iterative process. This iterative process will be used to determine the appropriate damping ratio based on the coefficient of restitution. ATK provided a coefficient of restitution range of (0.6 – 0.9) for this particular impact. Using this range as well as Equation 12, below, the following values were determined.

Where:

C_r = Coefficient of restitution

= Damping ratio

Table 4 – Iterative process, values for damping ratio

ζ	Cr
0.03	0.910
0.04	0.882
0.05	0.855
0.06	0.828
0.07	0.802
0.08	0.777
0.09	0.753
0.1	0.729
0.11	0.706
0.12	0.684
0.13	0.663
0.14	0.641
0.15	0.621
0.16	0.601
0.17	0.582

Narrowing the range down into upper and lower bounds gives a more specific set of values. These values can be seen in Table 5, below.

Table 5 – Damping ratio values to be used in force calculations

ζ	Cr
0.0335	0.9001
….	….
0.1605	0.6001

These values will be used to find the damped natural frequency of the system. Equation 13 shows the correlation between the damped natural frequency and the damping ratio.

Where:

w_d = Damped natural frequency [Hz]

Using the damping ratios from Table 5 gives a damped natural frequency range of (454.02 – 448.40) Hz. Once the damped natural frequency has been calculated, the time of peak force between colliding bodies can be found. This time is calculated using the following equation.

Where:

t_c = Time of peak force between colliding bodies [sec]

The time of peak force between colliding forces for the range specified above was found to be between (0.00339 – 0.00314) seconds. Finally, the peak force between bodies is calculated using Equation 15, below.

Where:

F = Peak force between bodies [N]

The range of forces was found to be between (997.5 – 835.1) N. A summary of these results can be seen in Table 6, below.

Table 6 – Range of Peak Force

Damping Ratio	Coefficient of Restitution	Force [N]
0.0335	0.900	997.48
….	….	….
0.1605	0.600	835.09

Comparing this range of forces to the force on the ABS plastic round calculated on page 7 ( confirms that these values are accurate.

Drop Test Experiment

In order to further verify the values calculated using an analytical approach, a drop test experiment was utilized. Material samples of equivalent weight were manufactured: a 2.537 inch cube of 7075-T7 aluminum, a 1.800 inch cube of 4340 steel corresponding to an ABS plastic cube weighing 1.65 pounds. Sections of cylindrical metal tubing were affixed to the side of each cube so that they could be dropped along a guiding rod (thus preventing the samples from tipping before impact). A hole was screwed into the top of each sample so that the accelerometer could be secured to each sample during drop testing.

Notes:

· Drilling holes and affixing sleeves to each sample created weight differences that would need to be accounted for before drop testing.

Initial drop impacts were conducted with the ABS plastic sample to generate sample data with the accelerometer. The plastic cube was drop three times from heights of 1, 3, and 5 feet as shown in Figure 4.

Figure 4-Drop Test Setup

to the right. After this initial drop testing, the accelerometer fatally malfunctioned. Due to time constraints, drop testing was abandoned, however, if more time were allowed drop testing would serve to verify theoretical results.

Spring Constant Analysis

A separate analysis of the spring constant is needed to determine pre-loaded forces on the batteries while in the battery compartment of the SVR. Conducting a weight/deflection test on the SVR spring yielded a linear trendline. This was done by adding various weights to the spring and measuring the deflection for each. Averaging the endpoints and converting to metric units gave the following results:

When the nose cone is screwed onto the top of the SVR, the batteries inside have a force exerted on them due to the compression of the spring. This spring constant was checked with ATK to ensure that findings were accurate during the weight/deflection test.

Figure 4, below, shows a close up view of the SVR nose cone.

Figure 5 – Close up view of SVR nose cone

With a spring constant of 3850 N/m, the force applied to the batteries can be found. The length of the top battery is equal to the length of the threads in the casing. Measuring the length of the threading on the nose piece will give the battery displacement on the spring. This length was found to be 1.1 cm = 0.011 m. The following calculation gives the force applied to each battery when the batteries are at rest.

(1850 N/m)* (0.011m) = 42.35N

The resultant displacement from screwing the nose cap over the batteries corresponds to a constant static force of 42.35N. This force can be added to the force being applied the batteries through the SVR due to the bolt carrier during cycling.