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Unformatted text preview: ME 577/BME 595 Homework 3 Due 16 February 2011 at Midnight 1. Analysis of the JFK Assassination
Step 1 – Free Body Diagram Begin with a free body diagram of Kennedy’s skull (denoting the skull with a subscript, s) and the bullet (denoted with a subscript, b). Step 2 – Governing Equation As the governing equation, we will employ the conservation of angular momentum with the position vectors referenced to the base of the neck, Hs / O2 + Hb / O2 = Hs / O1 + Hb / O1 , employing the assumptions described above. Under what conditions can we use this equation? Step 3 – Kinematics and Constraints Initially, the skull has negligible angular momentum, and the post
impact velocity is assumed to be the same for both particles, resulting in, ( ) r × ms + mb v 2 = r × mb v b1 . In order to utilize this equation, it is necessary to describe the kinematics. Initially, lets assume that cylindrical coordinates can be used to describe the motion of the skull after the impact v 2 = Lsθ eθ , where Ls is the distance from the base of the neck to the center of mass of the skull. The initial velocity of the bullet is best described in Cartesian coordinates, v b1 = vb11E1 + vb12E2 + vb13E3 , (12) where the extra subscripts denote the 1, 2, and 3 components of the velocity vector (if you come up with better notation, please use it). Step 4
Solution Solving this problem and determining the initial velocity vector of the third bullet requires us to determine a variety of parameters (Table 1). Before his death, the President weighed approximately 160 lbs. which corresponds to a mass of 72.55 kg. If we then assume that the mass of the head/neck complex is approximately 8.26% of the total mass, we can determine ms. With a reported height of 6’1/2”, (1.84 m) it is also possible to estimate the distance from the base of the neck to the center of mass of the head, Ls = 0.198 m. Table 1 – Assumed values for variables involved in the JFK Assassination Parameter Value Source m 72.55 kg Approximate ms 5.99 kg 0.0826*m mb 160 grains = 0.0104 kg
θ 2 4.72 rad/s Ls 0.198 m vb 2,200 ft/s = 609.6 m/s Estimated from Zapruder film 0.1075*H The initial impact of the third bullet occurs at frame 313 of the Zapruder film and the rotation of the President’s head continues through frame 317 yielding an estimate for the angular velocity of the skull after impact, π ⎛ 60o ⎞ 18 frames θ2 =
= 4.72rad / s . s
180o ⎜ 4 frames ⎟
⎝
⎠ (13) The reported muzzle velocity of the Carcano rifle used by Lee Harvey Oswald, is 2,200 ft/s (609.6 m/s), but it is worth noting that, if we are trying to determine the direction of the shot, it may not be appropriate to assume that this value is given a priori. In order to determine the source of the third bullet, it is necessary to solve our equations for the bullet velocity, vb. Derive the following equation, ( ) L2 ms + mb θ 2 e z = r × mb v b1 . s
Using these data, can you figure out the initial velocity and direction of the bullet? Before we claim to have solved the problem we should consider some of the assumptions that were made (Table 2). Which do you think would have the most effect on the solution? Table 2 – Assumptions made in the Analysis of the JFK Assassination Assumption Description 1 The drag on the bullet does not substantially affect its impact speed. 2 The bullet impact occurs at the center of mass of the skull. 3 The value of θ 2 can be estimated based on the final position of the head. 4 Skull fracture does not remove significant energy from the system. 5 The skull moves only in the r
θ plane. 6 The mass of the skull does not change appreciably. 7 The head slumping forward may prevent forward motion of the head. 8 The skull can be modeled as a particle on a massless rod. 2. Assume that a person is lifting a weight in a quasi
static manner using the muscles shown in the diagram below. The hand holds a 15 kg mass a distance of 30 cm (the length of the forearm) from the elbow joint. The forearm weighs 15 N. 2a. Calculate the joint reactions. 2b. For the single force – no moment model, determine the muscle load if we assume only the biceps is acting. Then, assume only the brachialis is acting. Then, assume only the brachioradialus is acting. 2c. Determine the muscle forces for the multi
force
no moment model (assume each muscle develops the same stress). This is a common technique for solving these kinds of problems, but not necessarily the best one. Adapted from Orthopaedic Biomechanics by Mow and Hayes. Muscle Moment Arm (cm) PCSA (cm2) Angle θ (degrees) Biceps brachii (BIC) 80.3 4.6 4.6 Brachialis (BRA) 68.7 3.4 7.0 Brachioradialis (BRR) 23.0 3.0 1.5 PCSA = Physiological Cross
Sectional Area 3. Repeat the problem above assuming that the elbow makes an angle of φ = πt2/2 – π/4. At what time will the forearm be horizontal? At that time, perform the following calculations: 3a. Find the center of mass of the system and the mass moment of inertia relative to the center of the elbow joint. 3b. Calculate the joint reactions. 3c. For the single force – no moment model, determine the muscle load if we assume only the biceps is acting. Then, assume only the brachialis is acting. Then, assume only the brachioradialus is acting. 3d. Determine the muscle forces for the multi
force
no moment model (assume each muscle develops the same stress). This is a common technique for solving these kinds of problems, but not necessarily the best one. ANTHROPOMETRIC DATA – KINEMATICS
(sections 3.03.1 of Winter, Biomechanics and Motor Control of Human Motion) Anthropometry is the study of physical measurements of the human body to determine differences in individuals and groups. A wide variety of physical measurements are required to describe and differentiate the characteristics of race, sex, age and body type. Human motion studies require quantitative information on linear, area and volume measures as well as masses, moments of inertia and their locations. In addition, we need to have information related to joint centers of rotation, the origin and insertion locations of muscles, the pull angles for tendons and the cross
sectional area of muscles.
In this section, we will consider published data that is particularly related to kinematic analysis. In a later section, we will also consider published anthropometric data related to muscles. Segment Lengths
For the study of human motion kinematics, the most fundamental body dimensions are
“segment lengths” (the length of the body segments between joints). Admittedly, these
dimensions vary from individual to individual, and particularly with variations among
body builds, gender and racial origin. However, for general studies of human motion
average values for segment lengths are useful in the understanding of trends in motion.
Shown in Figures 1 and 2 are summaries for average segment lengths for the body and
for the hand, respectively. In Figure 1, the segment lengths are given as a fraction of the
body height H. In Figure 2, the segment lengths, SLj, (along with other bone/segment
dimensions) are identified for the joints of the hand where numerical values for the joint
center lengths (distance from proximal end to the distal joint center), JCj, can be
determined from information given in Table 1. Note that the segment lengths are
important measurements for kinematic analysis.
Sample calculations Figure 1: Body segment lengths as a fraction of the body height H (from Winter,
Biomechanics and Motor Control of Human Motion). Figure 2: Segment lengths, SLj, bone lengths, BLj, and joint center lengths (distances
from proximal end of bone to joint center), JCj, for the hand (from Katsiorsky,
Kinematics of Human Motion). Legend: CMC – carpometacarpal joints, MCP
– metacarpophalangeal joints, PIP – proximal interphalangeal joints, DIP –
distal interphalangeal joints. Table 1: Joint center length coefficients, Aj, (with ± standard deviations) for bones of the
hand, where JCj = Aj x BLj (from Katsiorsky, Kinematics of Human Motion). For
Digit I, joint 1 is the CMC, joint 2 is the MCP and joint 3 is the IP. For digits IIV, joint 1 is the MCP, joint 2 is the PIP and joint 3 is the DIP. Segment Properties
Several other body properties will be needed in our studies of human motion kinematics and kinetics, including segment mass, segment center of mass and segment radius of gyration. Some average measures of these properties taken from cadaver measurements are summarized in Table 2. Radius of gyration
Recall that the mass moment of inertia IA of a body segment (of mass m) about an axis passing through point A can be found from its corresponding radius of gyration kA as: 2
IA = m kA In addition, we can use the parallel axis theorem to relate the mass moment of inertia about point A to the mass moment of inertia about the centroid G: IA = IG + m d 2 where d is the distance from A to G. From this, we can see that the radius of gyration about A is related to the radius of gyration about G through: 2
kA = kG + d 2 Sample Calculations
A
G
d Table 2: Segment
Hand Average anthropometric data for segment mass, location of segment center
of mass and segment radius of gyration (from Winter, Biomechanics and
Motor Control of Human Motion). Upper arm . Definition
Wrist axis/knuckle II middle finger Elbow axis/ulnar styloid Glenohumeral axis/elbow axis Forearm and hand Elbow axis/ulnar styloid Forearm Total arm Glenohumeral joint/ulnar styloid Foot Lateral malleolus/head metatarsal Il Leg Femoral condyles/medial malleolus Thigh Greater trochanter/femoral condyles Foot and leg Femoral condyles/medial malleolus Total leg Greater trochanter/medial malleolus Head and neck C7 TI and 1st rib/ear canal Shoulder mass Sternoclavicular joint/glenohumeral axis Thorax C7 Tl/T12
Ll and diaphragm Abdomen T12
Ll/L4
L5 Segment
Center of Mass/
Mass/
Segment Length
Total Body Proximal
Distal
Mass
0.006 0.506 0.494 Radius of Gyration/
Segment Length
C of M Proximal Distal
0.297 0.587 0.577 0.016 0.430 0.570 0.303 0.526 0.647 0.028 0.436 0.564 0.322 0.542 0.645 0.022 0.682 0.318 0.468 0.827 0.565 0.050 0.530 0.470 0.368 0.645 0.596 0.0145 0.50 0.50 0.475 0.690 0.690 0.0465 0.433 0.567 0.302 0.528 0.64 0.100 0.433 0.567 0.323 0.540 0.653 0.061 0.606 0.394 0.416 0.735 0.572 0.161 0.447 0.553 0.326 0.560 0.650 0.081 1.000
0.495 1.116
0.712 0.288
0.216 0.82 0.18
0.139 0.44 0.56
L4
L5/greater trochanter C7 Tl/L4
L5 0.142 0.105 0.895
0.355 0.63 0.37
Abdomen and pelvis T12
Ll/greater trochanter 0.281 0.27 0.73
Trunk Greater trochanter/glenohu
meral joint Greater trochanter/glenohu
meral joint Greater trochanter/glenohu
meral joint Greater 0.497 0.50 0.50
0.578 0.66 0.34 0.503 0.830 0.607 0.678 0.626 0.374 0.496 0.798 0.621 0.678 1.142
0.903 1.456
Pelvis Thorax and abdomen Trunk head neck HAT HAT trochanter/mid rib ...
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 Fall '09

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