BIOMECHANICS BME.200 EXAM 2

Describe and calculate the vector quantities of displacement, velocity, and acceleration (both average and instantaneous) in both 1D and 2D

Vectors have magnitude and direction (ex: displacement)
scalars have only magnitude (ex: distance)

1D
-Velocity
-Acceleration

-Velocity: Δd/Δt; x'(t)
-accel.: Δv/Δt; v'(t)

1D Acceleration Direction

Same sign - increasing speed
-Both positive: Increasing speed in positive x-direction
-Both negative: Increasing speed in negative x-direction•
Opposite sign - decreasing speed
V +, A - Decreasing speed in positive x-direction
V -, A + Decreasing speed in negative x-direction

2D Velocity and Acceleration

velocity: v = vî + vĵ
v = sqrt(vx^2 + vy^2) where Θ = tan^-1(vy/vx)
acceleration: a = ai + aj+ ak

2D Kinematics w/ Constant Acceleration

Ax = 0, Ay = -g
max height vy = 0, vx = constant

Kinetics

why things move (force, energy, work, momentum)

kinematics

how things move (distance, velocity, acceleration)

1D Kinematics w/ Constant Acceleration

v = v0 + a0*t
x = x0 + v0*t + (1/2)(a0)(t^2)
x = x0 + (1/2)*(v + v0)t
v^2 = v0^2 + 2*a0*(x - x0)

Kinetics - Constant Force

ax = Fx/m = constant

Kinetics - Time-Varying Force

ax(t) = Fx(t)/m

Kinetics - Displacement-Varying Force

dvx/dt = Fx(x)/m

Linear Kinetic Problems

1) Start with simple diagram of system to be analyzed
2) Draw FBD of bodies of interest with all external (known and unknown) forces
3) Designate direction of motion (or guess if unknown)
4) Apply coordinate system and split forces/acceleration into components
5) Apply equations of motion and solve for unknown values
6) Can use kinematic equations to then solve for velocity/position if necessary

Work (+ Dot Product)

W = F * d
W = FcosΘ * d
Net W is sum of all work done by different forces

Positive/Negative Work

+ work: F and d same direction
- work: F and d opposite direction

Work w/ Varying Force

W = F(x)*d*cosΘ,
Use Integration to solve over distance

Energy

the ability to do work
Mechanical has potential or kinetic

kinetic energy

energy of motion
Ek = (1/2)mv^2

potential energy

stored energy that results from the position or shape of an object
Ep = mgh

work-energy theorem

The work done on an object equals the change in kinetic energy of the object

Conservation of Energy

Eki + Epi = Ekf + Epf

Power

the rate at which work is done
P = dW/dt

rectangular coordinates → polar coordinates

(x,y) → (r, Θ)
r = sqrt(x^2 + y^2)
Θ = arctan(y/x)

polar to rectangular coordinates

x=rcos(theta)
y=rsin(theta)

angular displacement, s

- the change in the angle as an object rotates
-units: radians
s = rΘ

angular velocity, ω

-the rate at which the angle is changing in circular motion
-units: rad/s
instant: ω = dΘ/dt
average: ω = ΔΘ/Δt

angular acceleration

-The rate of change of angular velocity
-units: rad/s^2
instant: α = dω/dt, d^2Θ/dt2 (α=Θ'')
average: α = Δω/Δt

Angular-Linear Kinematics
(w/ Constant Acceleration)

Split into 2 components:
Normal/radial direction
Tangential Direction

Angular and linear displacement

Normal: 0
Tangential: s = r*Θ

Angular and Linear Velocity

Normal: vn = 0
Tangential: vt = r*ω

angular and linear net acceleration

a = at(t-direction) + an(n-direction)
a = sqrt[(an)^2 + (at)^2]

angular and linear acceleration

Normal: an = v^2/r = r*ω^2
-Normal points in center
Tangential: at = r*α

Linkage systems (don't need to know how to solve)

-usually double pendulum
-first determine s, vt, at, an for each part
-Use geometry of system to convert to global coordinates (i and j)
-use relative motion to find each unknown

linkage problem example

linkage problem solution

harmonic motion (we probably don't need this but just in case)

motion that repeats in cycles
Θ0 = starting angle
T = period (1/f)

Angular vs Linear Kinetics

Equation of Motion

Newton's 2 Law w/ Angular

Taken from quiz

mass moment of inertia (you'll be given specific formulas for inertia)

the resistance to rotation
- single particle: I = mr^2
- rigid body:
I = Σ(mi)(ri)^2

Parallel Axis Theorem

I = Ic + m(rc)^2
Ic = I of centroid
rc = distance between centroid axis and new axis

Angular Kinetic Energy

Ek = (1/2)*I*ω^2

Angular Work (IN RADIANS)

- constant M: W=MΘ
- vary M: W = ∫MdΘ (from Θ1 to Θ2)
- given velocity:
W = (1/2)*I*ωf^2 - (1/2)*I*ωi^2

Angular Power

P = Mω

Dynamic Analysis of Joints

- use simplified models + "fictious" forces (N's 3 Law)/forces that oppose motion and in opposite dir. of at and an

How to for Dynamic Ana. Joints

• Step 1: Draw diagram
• Step 2: Identify and add forces
• Step 3: Identify what quantities can be measured• Kinematic analysis of body segment using various experimental techniques• Measure angular displacement overtime, and then can calculate ω and α.
• Step 4: Make appropriate assumptions/simplifications• Same as with statics but also need to assume moment of inertia and axis of rotation are known
• Step 5: Solve for unknowns using equations of motion/static equilibrium equations. Typically Fm and Fjrf

Dynamic Ana. Joints Question

lecture 18 slide question

Dynamic Ana. Joints Answer

lecture 18 slide answer

Momentum

ρ = m x v
- units: (kg*m)/s
- magnitude and direction

Impulse-Momentum Theorem

- The impulse on an object is equal to the change in its momentum
- constant force: more time = more change in mom.
- constant change in mom.: more time = smaller force

Impulse-Momentum Theorem w/ constant force

Δρ = FΔt

Impulse-Momentum Theorem w/ time-vary force

Integral of force-v-time fxn.

Impulse: Direction of force in direction of motion

addition of the two

Impulse: Direction of force against direction of motion

difference of the two

Impulse vs Work

Impulse vs Work

Impulse/mom. Kinetic Problem

question of impulse

impulse/mom. kinetic answer

answer to impulse

Difference between elastic, inelastic, and completely inelastic collisions

- MOMENTUM IS CONSERVED IN ALL COLLISIONS
- Elastic → kinetic energy + mom. conserved
- Inelastic → mom. conserved
- completely inelastic → objects stick together w/ same v

conservation of momentum

m1v1i + m2v2i = m1v1f + m2v2f

Perfectly inelastic problem

The diagram below contains a depiction of a ballistic pendulum, which was the primary method for determining the velocity of projectiles which were too difficult to measure on their own. A bullet is fired into a wooden block suspended by ropes, and the height of the block at the peak of its swing is recorded. Define an equation for v i as a function of the mass of the projectile, mass of the block, and height the block moved.

perfect inelastic answer

perfect inelastic answer

Harmonic Displacement

Harmonic Velocity and acceleration

coefficient of restitution

the absolute value of the ratio of the velocity of separation to the velocity of approach

coefficient of restitution (2)

(2) If individual velocities have opposite direction, relative velocity is summation• If individual velocities have same direction, relative velocity is difference

coefficient of restitution for perfectly inelastic

COR: 0

coefficient of restitution for perfectly elastic

COR: 1

Coefficient of restitution problem

problem statement

coefficient of restitution answer

answer to COR

Momentum vs. Impulse

difference between mom. and impulse for linear and angular values

COR for inelastic

iclicker for COR inelastic

Types of Deformation

- Tensile/compressive
- shear
- twist moments/torsion
- bend moments

tensile/compressive forces

- lengthen or shorten object in direction of force
- perpendicular to cross-sectional area

shear forces

-aka tangential forces
-Change angle of line in object with external reference (sliding)
- tangential to object surface

twisting moments (torsion)

- Change in the path of a straight line within an object
- perpendicular to cross-sectional area (internal)

bending moments

- Combination of shear, tension, and compression within different parts of the object
- tangential to object (internal)

normal stress σ

- perpendicular to the surface
- units: Pa (N/m^2)
σ = F/A

shear stress τ

- component of stress that acts tangential to area
- units: Pa (N/m^2)
τ = F/A

Normal strain ε

- Change in length of dimension parallel to force
- units: none
- pos value (+) = tension
- neg value (-) = compression
ε = ΔL/L

shear strain γ

- Change in angle between line of object before and after
- uses radians
γ = tan(γ) = d/L

Shear stress problem

shear stress statement

shear stress answer

shear stress math

Evaluate stress-strain curves and use information from those graphs to compare and characterize different materials. You should know and be able to evaluate all mechanical characteristics of materials covered in the lecture slides.

- each curve can compare different materials/situations
- situations can change materials and values (ex: temperature)

Properties of Materials from Curves

- Stiffness → higher E = stiffer material
- Brittleness → more brittle if ruptures at smaller strain
- Toughness → total area under curve = toughness
- resilience → area under elastic region (σPεP/2 for linearly elastic)
- strength → the max stress withstood before rupture

anatomy of strain/stress curve

• O - starting point with no applied load
• P - Proportionality limit - stress and strain are linearly related to this region
• E - Elastic Limit - Greatest stress that can be applied without permanent deformation
• Y - Yield Point - Point at which large strain can occur with minimal additional stress
• σy - Yield strength - stress at which yield point occurs
• U - Highest point of yield curve
• σU - Ultimate strength - highest stress a material can experience
• R - Rupture - point at which material breaks/fails Note: Yield Point and Elastic Limit are typically the same

Elastic Deformations

- a deformation in which the object returns to its original dimensions due to application of stress
- linearly elastic: P and Y are same point
- nonlinear elastic: P and Y are NOT the same point

Young's Modulus

Relationship between stress and strain (slope of line)
stress = E*strain

Elastic shear stress

- Materials can also undergo elastic deformation due to shear stress
- For linearly elastic shear materials
G = Shear modulus/Modulus of Rigidity

plastic deformation

- permanent deformation caused by strain when stress exceeds E/Y

Necking

- Process known as necking - decrease in cross-sectional area during deformation
• Stress calculated based on original area, but that changes during necking.
• If accounting for area change, graph would not decrease

linear elastic deformation problem

linear elastic deformation statement

linear elastic deformation answer

linear elastic deformation answer and work

Poisson's Ratio (v)

v = - (lateral strain/axial strain)
- how much strain in one direction affects strain in the other direction

How poisson's ratio relates normal stress-strain to shear stress-strain

poisson's ratio creates relationship to get G (shear) from E (normal) and vice versa

Multiaxial Stress/Strain

stress/strain happening in multiple axes at once

Poisson's Ratio problem

Poisson's Ratio problem statement

Poisson's Ratio problem answer

Poisson's Ratio answer and work