Executive summary:
================
E max lost to sound: 0.5% per 200 clacks (100 cycles)
E max loss to air drag: 1.7% per 100 cycles
E lost to string: 0.8% per cycle (not measured carefully)
E lost to heat in balls and strings: 3% per 1 clack
Variables: 5 cm drop (1 m/s), 44 g steel balls 1.1 cm radius.
12 cm high strings.
Observation: about 0.5 cm distance sideways after 200 clacks (100 swings)
This is 12*cos(asin(0.5/12)) = 0.01 cm height from initial 5 cm = 0.2% remaining energy
(not measured carefully)
Two balls were dropped simultaneously from both sides, so it's 0.1% of initial energy
E=mgh * 2 = 44 mJ initial energy both balls.
max Energy lost in sound, 0.5% per 200 clacks:
====== proof =======:
Assume worst case, it's like noisy office at 5 meters for 8 ms per clack.
10E6 W/m^2 * 4pi 5^2 * 0.008 secs= 2.5 uJ per clack
For 200 clacks assume max loud for 100 clacks, 0.250 mJ
Max energy in sound, 0.25/44 = 0.5%
max Energy loss to air drag, 3.41% per 200 clacks :
======= proof =========
F=1/2 * C * density * Area * v^2
C=0.5 sphere
air 1.2 kg/m^3
v = 0.75 m/s for 5 cm drop in first few clacks
Area= pi * 0.011^2
F = 62 uN each ball
12 cm string height 0.68 secs per 2 clacks.
5 cm drop => ~12 cm travel per 1 clack
E=Fd = 62 uN * 0.12 * 100 clacks (100 as a high average even for 200 clacks) = 0.74 mJ per 200 clacks max
Max energy in air friction = 0.74/44 =1.7% per 100 cycles absolute max
Can I measure the heat increase with an infrared thermometer?
Steel heat capacity: 2.2 Celsius * gr/J * 0.022 J / 44 g = 0.001 C increase if all energy lost to heat in 1 ball in 1st clack. Can't measure it.
Energy loss in string resistance 0.8% per cycle,
====== proof =======
Measured 80% energy loss after 100 complete swings of ball on the string (no clacks).
(10.80)^(1/200) = 99.2% => 0.8% loss in string per cycle (2 clacks when in cradle).
Energy loss from all sources, measured 3.4% per clack
===================
Observed about 200 max clacks (100 cycles) before stopping =>
0.001^(1/200) = 0.966 energy retained after each clack. ( 0.966^200 = 0.001)
Energy lost to heat in balls (or in strings due to offcenter rotational effects): 3.4%0.8%/2  1.7%/200  0.5/200=3% per clack
=================
96% energy retained is what I measured for steel on glass block, very close to this 96.6%, but that is without a string, possibly offsetting the error of not bouncing steel on steel.
Liquid metal seemed to be 1/2 the height after 44 bounces. 0.5^(1/44) = 98.4% retained energy per bounce.
Friday, February 24, 2017
Sunday, February 19, 2017
The best glues
Glue stuff I wish I had known earlier in life. Actually, only two of these existed when I was young.
Surface preparation for plastics, glass, and metal: roughen surface, soap and water, rubbing alcohol, eyeglass cloth. Nitric acid is used to prepare some metals for epoxy even after sanding (100 grit).
In general: epoxies and acrylic are best for glass and metal. Silicone and polyurethanes can work on them, but these are best on plastics.
glass and metal: 3M VHB doublesided tapes, >40 mil thickenss ones. 5952 seems most popular. To be ideal glass needs silane pretreatment and copper and brass need lacquer or varnish (they oxidize even after it's applied). Needs 15 psi pressure. 50% strength bond in 20 minutes. Has 20 pound/inch peel adhesion.
Drying time: usually 18 hours is needed for epoxies, polyurethanes, and thick layers of super glue is needed to reach 1/2 of their final strength.
Glass: If 3M tape can't be used: Super glues are not so great. Thinner is better, but a thick layer on top instead of in between parts can work. Epoxies for glass are best. The best is Aralite for glass.
Metal: Locktite Metal Concrete and Quick Steel are a lot better than JB weld and Gorilla glue. There are a lot of videos concluding this. Super glues and polyurethanes (goop, aquaseal) can work too.
Plastics and rubber: Goop and Aquaseal. (polyurethanes)
Wood: wood glue and hot glue sticks for temporary holding while the wood clue dries.
Hot glue: The newer, hotter, slowdrying glue sticks are pretty impressive, often doing a goodenough job on all of the above. Arrow SuperPower slow setting is one.
Surface preparation for plastics, glass, and metal: roughen surface, soap and water, rubbing alcohol, eyeglass cloth. Nitric acid is used to prepare some metals for epoxy even after sanding (100 grit).
In general: epoxies and acrylic are best for glass and metal. Silicone and polyurethanes can work on them, but these are best on plastics.
glass and metal: 3M VHB doublesided tapes, >40 mil thickenss ones. 5952 seems most popular. To be ideal glass needs silane pretreatment and copper and brass need lacquer or varnish (they oxidize even after it's applied). Needs 15 psi pressure. 50% strength bond in 20 minutes. Has 20 pound/inch peel adhesion.
Drying time: usually 18 hours is needed for epoxies, polyurethanes, and thick layers of super glue is needed to reach 1/2 of their final strength.
Glass: If 3M tape can't be used: Super glues are not so great. Thinner is better, but a thick layer on top instead of in between parts can work. Epoxies for glass are best. The best is Aralite for glass.
Metal: Locktite Metal Concrete and Quick Steel are a lot better than JB weld and Gorilla glue. There are a lot of videos concluding this. Super glues and polyurethanes (goop, aquaseal) can work too.
Plastics and rubber: Goop and Aquaseal. (polyurethanes)
Wood: wood glue and hot glue sticks for temporary holding while the wood clue dries.
Hot glue: The newer, hotter, slowdrying glue sticks are pretty impressive, often doing a goodenough job on all of the above. Arrow SuperPower slow setting is one.
Saturday, February 11, 2017
Bouncing balls a little beyond the elastic range
This is an edit to wikipedia
==Predicting the coefficient from material properties==
When colliding objects do not have a center of gravity that is inline with their direction of motion and point of impact, energy that would have been available for the postcollision velocity difference will be lost to rotation and friction. This section will consider only spherical objects colliding directly either with other spherical objects or with a flat surface to avoid rotation and friction losses.
When soft objects strike hard objects, most of the energy available for for the postcollision velocity different will be stored in the soft object and the value of the COR will depend on how efficient the soft object is at storing the compression energy without losing it to heat and plastic deformation. A rubber ball will bounce a lot better off concrete than glass, but the COR of glassonglass is a lot better than rubberonrubber. So when wanting to know the COR of an object, it's good to impact it with an object that is much harder. For this reason the [[Leeb rebound hardness test]] impacts test samples with a tip of tungsten carbide, one of the hardest substances available. There is no perfectly hard material, and the COR depends on both objects, so for ideal testing and theory, determining the COR of a material depends on both objects being that same material.
Many materials are assumed to be perfectly elastic when their yield strength is not approached during impact. The impact energy is theoretically stored only in the springeffect of elastic compression and results in e=1. In practice a various stainless steels have a large variation well below e=1. Amorphous metals can achieve e = 0.95 or higher. The elastic range can be exceeded at low velocities because all the kinetic energy is concentrated at the point of impact. If the velocity is above 1 m/s, the yield strength of metals is usually exceeded in part of the contact area, losing energy to "plastic deformation" by not remaining in the elastic region. To account for this, the following method estimates the percent of the initial impact energy that did not get lost. Approximately, it divides how easy a volume of the material can store energy in compression ($1/\{\backslash text\{elastic\; modulus\}\}$) by how well it can stay in the elastic range ($1/\{\backslash text\{yield\; strength\}\}$):
$\%\; \backslash text\{impact\; energy\; available\; for\; restitution\}\; \backslash propto\; \backslash frac\{\backslash text\{yield\; strength\}\}\{\backslash text\{elastic\; modulus\}\}$
For a given material density and velocity this results in:
$\backslash text\{coefficient\; of\; restitution\}\; \backslash propto\; \backslash sqrt\{\backslash frac\{\backslash text\{yield\; strength\}\}\{\backslash text\{elastic\; modulus\}\}\; \}$
To be more precise, these and two more quantities can be shown to be important when predicting the COR at moderate velocities. A high ''yield strength'' allows the material to stay in the elastic region at higher energies. A lower ''elastic modulus'' allows a larger surface area of contact during impact so the energy is distributed to a larger volume at the contact point which helps prevent the yield strength from being exceeded. A ''lower velocity'' increases the coefficient by needing less energy to be absorbed. A ''lower density'' also means less initial energy needs to be absorbed. The density instead of mass is used because the volume of the sphere cancels out with the volume of the affected volume at the contact area.
Combining these four variables, a theoretical estimation of the coefficient of restitution can be made when a ball is dropped onto a surface of the same material.http://itzhak.green.gatech.edu/rotordynamics/Predicting%20the%20coefficient%20of%20restitution%20of%20impacting%20spheres.pdf
* e = coefficient of restitution
* S_{y} = dynamic yield strength (dynamic "elastic limit")
* E' = effective elastic modulus
* ρ = density
* v = velocity at impact
* μ = Poisson's ratio
$e\; =\; 3.1\; \backslash left(\backslash frac\{S\_\{y\}\}\{1\}\backslash right)^\backslash frac\{5\}\{8\}\; \backslash left(\backslash frac\{1\}\{E\text{'}\}\backslash right)^\backslash frac\{1\}\{2\}\; \backslash left(\backslash frac\{1\}\{v\}\backslash right)^\backslash frac\{1\}\{4\}\; \backslash left(\backslash frac\{1\}\{\backslash rho\}\backslash right)^\backslash frac\{1\}\{8\}$
$E\text{'}\; =\; \backslash frac\{E\}\{1\backslash mu^2\}$
This applies for a direct impact and when:
$0.001\; <\; \backslash frac\{\backslash rho\; v^2\}\{S\_y\}\; <\; 0.1$
Although the accuracy of this equation is not good, it is easy to calculate and accurately predicts the relative coefficient for many materials, even at velocities above and below its intended range.
Theoretical coefficient of restitution solid spheres dropped 1 meter (v= 4.5 m/s). Values > 1.0 indicates the equation indicates the equation has errors.http://wwwmdp.eng.cam.ac.uk/web/library/enginfo/cueddatabooks/materials.pdf Yield strength instead of dynamic yield strength was used.
{
'''Metals and Ceramics:'''
'''Predicted COR, e'''

silica glass
1.36 to 1.71

Alumina
0.45 to 1.63

silicon nitride
0.38 to 1.63

silicon carbide
0.47 to 1.31

highest amorphous metal
1.27

tungsten carbide
0.73 to 1.13

magnesium alloys
0.5 to 0.89

titanium alloy grade 5
0.84

aluminum alloy 7075T6
0.75

glass (sodalime)
0.69

glass (borosilicate)
0.66

nickel alloys
0.15 to 0.70

stainless steel alloys
0.23 to 0.62

zinc alloys
0.21 to 0.62

cast iron
0.3 to 0.6

copper alloys
0.15 to 0.55

titanium grade 2
0.46

tungsten
0.37

aluminum alloys 3003 6061, 70750
0.35

zinc
0.21

nickel
0.15

copper
0.15

aluminum
0.1

lead
0.08




}
Plastics and rubbers will give higher values than their actual values because they are not as ideally elastic as metals, glasses, and ceramics because of heating during compression. So the following is only a guide to ranking of polymers.
'''Polymers''' (overestimated compared to metals and ceramics):
* polybutadiene (golf balls shell) 11.8
* butyl rubber 6.24
* EVA 4.85
* silicone elastomers 2.80
* polycarbonate 1.46
* nylon 1.28
* polyethylene 1.24
* Teflon 1.21
* polypropylene 1.14
* ABS 1.12
* acrylic 1.06
* PET 0.95
* polystyrene 0.87
* PVC 0.86
For metals the range of speeds to which this theory can apply is about 5 to 100 m/s which is a drop of 1 to 500 meters, provided the sphere is small enough for Hertzian contact theory to apply (see page 366http://www.ewp.rpi.edu/hartford/~ernesto/S2015/FWLM/Books_Links/Books/JohnsonCONTACTMECHANICS.pdf) But the above rankings that it provides remain accurate.
Dropping hard spherical objects onto a softer surface (lower elastic modulus) which also has a lower coefficient of restitution will reduce the apparent coefficient of restitution of the dropped object. For example, most rubber and plastic balls have a lower coefficient than glass and some metal alloys, but when dropped on wood or cement, the softer material will bounce higher. This is because the harder objects distribute the impact energy over a much smaller contact area, losing energy to heat by exceeding the elastic range of the floor.
For metals, the theoretically perfect elastic range (the coefficient theoretically equals 1.0 and the above equation does not apply) is when the velocity is less than
$v\; =\; \backslash left(26\; \backslash frac\{S\_y\}\{\backslash rho\}\; \backslash left(\backslash frac\{S\_\{y\}\}\{E\text{'}\}\backslash right)^4\; \backslash right)^\{0.5\}$
which is less than 0.1 m/s.
==Predicting the coefficient from material properties==
When colliding objects do not have a center of gravity that is inline with their direction of motion and point of impact, energy that would have been available for the postcollision velocity difference will be lost to rotation and friction. This section will consider only spherical objects colliding directly either with other spherical objects or with a flat surface to avoid rotation and friction losses.
When soft objects strike hard objects, most of the energy available for for the postcollision velocity different will be stored in the soft object and the value of the COR will depend on how efficient the soft object is at storing the compression energy without losing it to heat and plastic deformation. A rubber ball will bounce a lot better off concrete than glass, but the COR of glassonglass is a lot better than rubberonrubber. So when wanting to know the COR of an object, it's good to impact it with an object that is much harder. For this reason the [[Leeb rebound hardness test]] impacts test samples with a tip of tungsten carbide, one of the hardest substances available. There is no perfectly hard material, and the COR depends on both objects, so for ideal testing and theory, determining the COR of a material depends on both objects being that same material.
Many materials are assumed to be perfectly elastic when their yield strength is not approached during impact. The impact energy is theoretically stored only in the springeffect of elastic compression and results in e=1. In practice a various stainless steels have a large variation well below e=1. Amorphous metals can achieve e = 0.95 or higher. The elastic range can be exceeded at low velocities because all the kinetic energy is concentrated at the point of impact. If the velocity is above 1 m/s, the yield strength of metals is usually exceeded in part of the contact area, losing energy to "plastic deformation" by not remaining in the elastic region. To account for this, the following method estimates the percent of the initial impact energy that did not get lost. Approximately, it divides how easy a volume of the material can store energy in compression ($1/\{\backslash text\{elastic\; modulus\}\}$) by how well it can stay in the elastic range ($1/\{\backslash text\{yield\; strength\}\}$):
$\%\; \backslash text\{impact\; energy\; available\; for\; restitution\}\; \backslash propto\; \backslash frac\{\backslash text\{yield\; strength\}\}\{\backslash text\{elastic\; modulus\}\}$
For a given material density and velocity this results in:
$\backslash text\{coefficient\; of\; restitution\}\; \backslash propto\; \backslash sqrt\{\backslash frac\{\backslash text\{yield\; strength\}\}\{\backslash text\{elastic\; modulus\}\}\; \}$
To be more precise, these and two more quantities can be shown to be important when predicting the COR at moderate velocities. A high ''yield strength'' allows the material to stay in the elastic region at higher energies. A lower ''elastic modulus'' allows a larger surface area of contact during impact so the energy is distributed to a larger volume at the contact point which helps prevent the yield strength from being exceeded. A ''lower velocity'' increases the coefficient by needing less energy to be absorbed. A ''lower density'' also means less initial energy needs to be absorbed. The density instead of mass is used because the volume of the sphere cancels out with the volume of the affected volume at the contact area.
Combining these four variables, a theoretical estimation of the coefficient of restitution can be made when a ball is dropped onto a surface of the same material.http://itzhak.green.gatech.edu/rotordynamics/Predicting%20the%20coefficient%20of%20restitution%20of%20impacting%20spheres.pdf
* e = coefficient of restitution
* S_{y} = dynamic yield strength (dynamic "elastic limit")
* E' = effective elastic modulus
* ρ = density
* v = velocity at impact
* μ = Poisson's ratio
$e\; =\; 3.1\; \backslash left(\backslash frac\{S\_\{y\}\}\{1\}\backslash right)^\backslash frac\{5\}\{8\}\; \backslash left(\backslash frac\{1\}\{E\text{'}\}\backslash right)^\backslash frac\{1\}\{2\}\; \backslash left(\backslash frac\{1\}\{v\}\backslash right)^\backslash frac\{1\}\{4\}\; \backslash left(\backslash frac\{1\}\{\backslash rho\}\backslash right)^\backslash frac\{1\}\{8\}$
$E\text{'}\; =\; \backslash frac\{E\}\{1\backslash mu^2\}$
This applies for a direct impact and when:
$0.001\; <\; \backslash frac\{\backslash rho\; v^2\}\{S\_y\}\; <\; 0.1$
Although the accuracy of this equation is not good, it is easy to calculate and accurately predicts the relative coefficient for many materials, even at velocities above and below its intended range.
Theoretical coefficient of restitution solid spheres dropped 1 meter (v= 4.5 m/s). Values > 1.0 indicates the equation indicates the equation has errors.http://wwwmdp.eng.cam.ac.uk/web/library/enginfo/cueddatabooks/materials.pdf Yield strength instead of dynamic yield strength was used.
{
'''Metals and Ceramics:'''
'''Predicted COR, e'''

silica glass
1.36 to 1.71

Alumina
0.45 to 1.63

silicon nitride
0.38 to 1.63

silicon carbide
0.47 to 1.31

highest amorphous metal
1.27

tungsten carbide
0.73 to 1.13

magnesium alloys
0.5 to 0.89

titanium alloy grade 5
0.84

aluminum alloy 7075T6
0.75

glass (sodalime)
0.69

glass (borosilicate)
0.66

nickel alloys
0.15 to 0.70

stainless steel alloys
0.23 to 0.62

zinc alloys
0.21 to 0.62

cast iron
0.3 to 0.6

copper alloys
0.15 to 0.55

titanium grade 2
0.46

tungsten
0.37

aluminum alloys 3003 6061, 70750
0.35

zinc
0.21

nickel
0.15

copper
0.15

aluminum
0.1

lead
0.08




}
Plastics and rubbers will give higher values than their actual values because they are not as ideally elastic as metals, glasses, and ceramics because of heating during compression. So the following is only a guide to ranking of polymers.
'''Polymers''' (overestimated compared to metals and ceramics):
* polybutadiene (golf balls shell) 11.8
* butyl rubber 6.24
* EVA 4.85
* silicone elastomers 2.80
* polycarbonate 1.46
* nylon 1.28
* polyethylene 1.24
* Teflon 1.21
* polypropylene 1.14
* ABS 1.12
* acrylic 1.06
* PET 0.95
* polystyrene 0.87
* PVC 0.86
For metals the range of speeds to which this theory can apply is about 5 to 100 m/s which is a drop of 1 to 500 meters, provided the sphere is small enough for Hertzian contact theory to apply (see page 366http://www.ewp.rpi.edu/hartford/~ernesto/S2015/FWLM/Books_Links/Books/JohnsonCONTACTMECHANICS.pdf) But the above rankings that it provides remain accurate.
Dropping hard spherical objects onto a softer surface (lower elastic modulus) which also has a lower coefficient of restitution will reduce the apparent coefficient of restitution of the dropped object. For example, most rubber and plastic balls have a lower coefficient than glass and some metal alloys, but when dropped on wood or cement, the softer material will bounce higher. This is because the harder objects distribute the impact energy over a much smaller contact area, losing energy to heat by exceeding the elastic range of the floor.
For metals, the theoretically perfect elastic range (the coefficient theoretically equals 1.0 and the above equation does not apply) is when the velocity is less than
$v\; =\; \backslash left(26\; \backslash frac\{S\_y\}\{\backslash rho\}\; \backslash left(\backslash frac\{S\_\{y\}\}\{E\text{'}\}\backslash right)^4\; \backslash right)^\{0.5\}$
which is less than 0.1 m/s.
Friday, February 3, 2017
Wikipedia edit to coefficient of restitution
=== COR variation due to object shape and offcenter collisions ===
When colliding objects do not have a direction of motion that is inline with their centers of gravity and point of impact, or if their contact surfaces at that point are not perpendicular to that line, some energy that would have been available for the postcollision velocity difference will be lost to rotation and friction. Energy losses to vibration and the resulting sound are usually negligible.
=== Colliding different materials and practical measurement ===
When a soft object strikes a harder object, most of the energy available for the postcollision velocity will be stored in the soft object. The COR will depend on how efficient the soft object is at storing the energy in compression without losing it to heat and plastic deformation. A rubber ball will bounce better off concrete than a glass ball, but the COR of glassonglass is a lot higher than rubberonrubber because some of the energy in rubber is lost to heat when it is compressed. When a rubber ball collides with a glass ball, the COR will depend entirely on the rubber. For this reason, determining the COR of a material when there is not identical material for collision is best done by using a much harder material.
Since there is no perfectly rigid material, hard materials such as metals and ceramics have their COR theoretically determined by considering the collision between identical spheres. In practice, a 2ball [[Newton's cradle]] may be employed but such a set up is not conducive to quickly testing samples.
The [[Leeb rebound hardness test]] is the only commonlyavailable test related to determining the COR. It uses a tip of tungsten carbide, one of the hardest substances available, dropped onto test samples from a specific height. But the shape of the tip, the velocity of impact, and the tungsten carbide are all variables that affect the result that is expressed in terms of 1000*COR. It does not give an objective COR for the material that is independent from the test.
=== Predicting from material properties ===
The COR is not a material property because it changes with the shape of the material and the specifics of the collision, but it can be predicted from material properties and the velocity of impact when the specifics of the collision are simplified. To avoid the complications of rotational and frictional losses, we can consider the ideal case of an identical pair of spherical objects, colliding so that their centers of mass and relative velocity are all inline.
Many materials like metals and ceramics (but not rubbers and plastics) are assumed to be perfectly elastic when their yield strength is not approached during impact. The impact energy is theoretically stored only in the springeffect of elastic compression and results in ''e'' = 1. But this applies only at velocities less than about 0.1 m/s to 1 m/s. The elastic range can be exceeded at higher velocities because all the kinetic energy is concentrated at the point of impact. Specifically, the yield strength is usually exceeded in part of the contact area, losing energy to plastic deformation by not remaining in the elastic region. To account for this, the following estimates the COR by estimating the percent of the initial impact energy that did not get lost to plastic deformation. Approximately, it divides how easy a volume of the material can store energy in compression (<math>1/{\text{elastic modulus}}</math>) by how well it can stay in the elastic range (<math>1/{\text{yield strength}}</math>):
<math>% \text{impact energy available for restitution} \propto \frac{\text{yield strength}}{\text{elastic modulus}} </math>
For a given material density and velocity this results in:
<math>\text{coefficient of restitution} \propto \sqrt{\frac{\text{yield strength}}{\text{elastic modulus}} }</math>
A high yield strength allows more of the "contact volume" of the material to stay in the elastic region at higher energies. A lower elastic modulus allows a larger contact area to develop during impact so the energy is distributed to a larger volume beneath the surface at the contact point. This helps prevent the yield strength from being exceeded.
A more precise theoretical development<ref>http://wwwmdp.eng.cam.ac.uk/web/library/enginfo/cueddatabooks/materials.pdf</ref> shows the velocity and density of the material to also be important when predicting the COR at moderate velocities faster than elastic collision (greater than 0.1 m/s for metals) and slower than large permanent plastic deformation (less than 100 m/s). A lower velocity increases the coefficient by needing less energy to be absorbed. A lower density also means less initial energy needs to be absorbed. The density instead of mass is used because the volume of the sphere cancels out with the volume of the affected volume at the contact area. In this way, the radius of the sphere does not affect the coefficient. A pair of colliding spheres of different sizes but of the same material have the same coefficient as below, but multiplied by <math>\left(\frac{R_1}{R_2}\right)^{\frac{3}{8}}</math>
Combining these four variables, a theoretical estimation of the coefficient of restitution can be made when a ball is dropped onto a surface of the same material.<ref>http://itzhak.green.gatech.edu/rotordynamics/Predicting%20the%20coefficient%20of%20restitution%20of%20impacting%20spheres.pdf</ref>
* ''e'' = coefficient of restitution
* ''S''<sub>y</sub> = dynamic yield strength (dynamic "elastic limit")
* ''E''′ = effective elastic modulus
* ''ρ'' = density
* ''v'' = velocity at impact
* ''μ'' = Poisson's ratio
<math>e = 3.1 \left(\frac{S_\text{y}}{1}\right)^\frac{5}{8} \left(\frac{1}{E'}\right)^\frac{1}{2} \left(\frac{1}{v}\right)^\frac{1}{4} \left(\frac{1}{\rho}\right)^\frac{1}{8} </math>
<math>E' = \frac{E}{1\mu^2}</math>
This equation overestimates the actual COR. For metals, it applies when v is approximately between 0.1 m/s and 100 m/s and in general when:
<math>0.001 < \frac{\rho v^2}{S_\text{y}} < 0.1</math>
At slower velocities the COR is higher than the above equation predicts, theoretically reaching e=1 when the above fraction is less than <math>10^{6}</math> m/s. It gives the the following theoretical coefficient of restitution for solid spheres dropped 1 meter (''v'' = 4.5 m/s). Values greater than 1.0 indicate that the equation has errors. Yield strength instead of dynamic yield strength was used.
{
'''Metals and Ceramics:'''
'''Predicted COR, ''e'''''

silicon
1.79

Alumina
0.45 to 1.63

silicon nitride
0.38 to 1.63

silicon carbide
0.47 to 1.31

highest amorphous metal
1.27

tungsten carbide
0.73 to 1.13

stainless steel
0.63 to 0.93

magnesium alloys
0.5 to 0.89

titanium alloy grade 5
0.84

aluminum alloy 7075T6
0.75

glass (sodalime)
0.69

glass (borosilicate)
0.66

nickel alloys
0.15 to 0.70

zinc alloys
0.21 to 0.62

cast iron
0.3 to 0.6

copper alloys
0.15 to 0.55

titanium grade 2
0.46

tungsten
0.37

aluminum alloys 3003 6061, 70750
0.35

zinc
0.21

nickel
0.15

copper
0.15

aluminum
0.1

lead
0.08




}
Plastics and rubbers will give higher values than their actual values because they are not as ideally elastic as metals, glasses, and ceramics because of heating during compression. So the following is only a guide to ranking of polymers.
'''Polymers''' (overestimated compared to metals and ceramics):
* polybutadiene (golf balls shell) 11.8
* butyl rubber 6.24
* EVA 4.85
* silicone elastomers 2.80
* polycarbonate 1.46
* nylon 1.28
* polyethylene 1.24
* Teflon 1.21
* polypropylene 1.14
* ABS 1.12
* acrylic 1.06
* PET 0.95
* polystyrene 0.87
* PVC 0.86
For metals the range of speeds to which this theory can apply is about 0.1 to 5 m/s which is a drop of 0.5 mm to 1.25 meters (page 366<ref>http://www.ewp.rpi.edu/hartford/~ernesto/S2015/FWLM/Books_Links/Books/JohnsonCONTACTMECHANICS.pdf</ref>).
When colliding objects do not have a direction of motion that is inline with their centers of gravity and point of impact, or if their contact surfaces at that point are not perpendicular to that line, some energy that would have been available for the postcollision velocity difference will be lost to rotation and friction. Energy losses to vibration and the resulting sound are usually negligible.
=== Colliding different materials and practical measurement ===
When a soft object strikes a harder object, most of the energy available for the postcollision velocity will be stored in the soft object. The COR will depend on how efficient the soft object is at storing the energy in compression without losing it to heat and plastic deformation. A rubber ball will bounce better off concrete than a glass ball, but the COR of glassonglass is a lot higher than rubberonrubber because some of the energy in rubber is lost to heat when it is compressed. When a rubber ball collides with a glass ball, the COR will depend entirely on the rubber. For this reason, determining the COR of a material when there is not identical material for collision is best done by using a much harder material.
Since there is no perfectly rigid material, hard materials such as metals and ceramics have their COR theoretically determined by considering the collision between identical spheres. In practice, a 2ball [[Newton's cradle]] may be employed but such a set up is not conducive to quickly testing samples.
The [[Leeb rebound hardness test]] is the only commonlyavailable test related to determining the COR. It uses a tip of tungsten carbide, one of the hardest substances available, dropped onto test samples from a specific height. But the shape of the tip, the velocity of impact, and the tungsten carbide are all variables that affect the result that is expressed in terms of 1000*COR. It does not give an objective COR for the material that is independent from the test.
=== Predicting from material properties ===
The COR is not a material property because it changes with the shape of the material and the specifics of the collision, but it can be predicted from material properties and the velocity of impact when the specifics of the collision are simplified. To avoid the complications of rotational and frictional losses, we can consider the ideal case of an identical pair of spherical objects, colliding so that their centers of mass and relative velocity are all inline.
Many materials like metals and ceramics (but not rubbers and plastics) are assumed to be perfectly elastic when their yield strength is not approached during impact. The impact energy is theoretically stored only in the springeffect of elastic compression and results in ''e'' = 1. But this applies only at velocities less than about 0.1 m/s to 1 m/s. The elastic range can be exceeded at higher velocities because all the kinetic energy is concentrated at the point of impact. Specifically, the yield strength is usually exceeded in part of the contact area, losing energy to plastic deformation by not remaining in the elastic region. To account for this, the following estimates the COR by estimating the percent of the initial impact energy that did not get lost to plastic deformation. Approximately, it divides how easy a volume of the material can store energy in compression (<math>1/{\text{elastic modulus}}</math>) by how well it can stay in the elastic range (<math>1/{\text{yield strength}}</math>):
<math>% \text{impact energy available for restitution} \propto \frac{\text{yield strength}}{\text{elastic modulus}} </math>
For a given material density and velocity this results in:
<math>\text{coefficient of restitution} \propto \sqrt{\frac{\text{yield strength}}{\text{elastic modulus}} }</math>
A high yield strength allows more of the "contact volume" of the material to stay in the elastic region at higher energies. A lower elastic modulus allows a larger contact area to develop during impact so the energy is distributed to a larger volume beneath the surface at the contact point. This helps prevent the yield strength from being exceeded.
A more precise theoretical development<ref>http://wwwmdp.eng.cam.ac.uk/web/library/enginfo/cueddatabooks/materials.pdf</ref> shows the velocity and density of the material to also be important when predicting the COR at moderate velocities faster than elastic collision (greater than 0.1 m/s for metals) and slower than large permanent plastic deformation (less than 100 m/s). A lower velocity increases the coefficient by needing less energy to be absorbed. A lower density also means less initial energy needs to be absorbed. The density instead of mass is used because the volume of the sphere cancels out with the volume of the affected volume at the contact area. In this way, the radius of the sphere does not affect the coefficient. A pair of colliding spheres of different sizes but of the same material have the same coefficient as below, but multiplied by <math>\left(\frac{R_1}{R_2}\right)^{\frac{3}{8}}</math>
Combining these four variables, a theoretical estimation of the coefficient of restitution can be made when a ball is dropped onto a surface of the same material.<ref>http://itzhak.green.gatech.edu/rotordynamics/Predicting%20the%20coefficient%20of%20restitution%20of%20impacting%20spheres.pdf</ref>
* ''e'' = coefficient of restitution
* ''S''<sub>y</sub> = dynamic yield strength (dynamic "elastic limit")
* ''E''′ = effective elastic modulus
* ''ρ'' = density
* ''v'' = velocity at impact
* ''μ'' = Poisson's ratio
<math>e = 3.1 \left(\frac{S_\text{y}}{1}\right)^\frac{5}{8} \left(\frac{1}{E'}\right)^\frac{1}{2} \left(\frac{1}{v}\right)^\frac{1}{4} \left(\frac{1}{\rho}\right)^\frac{1}{8} </math>
<math>E' = \frac{E}{1\mu^2}</math>
This equation overestimates the actual COR. For metals, it applies when v is approximately between 0.1 m/s and 100 m/s and in general when:
<math>0.001 < \frac{\rho v^2}{S_\text{y}} < 0.1</math>
At slower velocities the COR is higher than the above equation predicts, theoretically reaching e=1 when the above fraction is less than <math>10^{6}</math> m/s. It gives the the following theoretical coefficient of restitution for solid spheres dropped 1 meter (''v'' = 4.5 m/s). Values greater than 1.0 indicate that the equation has errors. Yield strength instead of dynamic yield strength was used.
{
'''Metals and Ceramics:'''
'''Predicted COR, ''e'''''

silicon
1.79

Alumina
0.45 to 1.63

silicon nitride
0.38 to 1.63

silicon carbide
0.47 to 1.31

highest amorphous metal
1.27

tungsten carbide
0.73 to 1.13

stainless steel
0.63 to 0.93

magnesium alloys
0.5 to 0.89

titanium alloy grade 5
0.84

aluminum alloy 7075T6
0.75

glass (sodalime)
0.69

glass (borosilicate)
0.66

nickel alloys
0.15 to 0.70

zinc alloys
0.21 to 0.62

cast iron
0.3 to 0.6

copper alloys
0.15 to 0.55

titanium grade 2
0.46

tungsten
0.37

aluminum alloys 3003 6061, 70750
0.35

zinc
0.21

nickel
0.15

copper
0.15

aluminum
0.1

lead
0.08




}
Plastics and rubbers will give higher values than their actual values because they are not as ideally elastic as metals, glasses, and ceramics because of heating during compression. So the following is only a guide to ranking of polymers.
'''Polymers''' (overestimated compared to metals and ceramics):
* polybutadiene (golf balls shell) 11.8
* butyl rubber 6.24
* EVA 4.85
* silicone elastomers 2.80
* polycarbonate 1.46
* nylon 1.28
* polyethylene 1.24
* Teflon 1.21
* polypropylene 1.14
* ABS 1.12
* acrylic 1.06
* PET 0.95
* polystyrene 0.87
* PVC 0.86
For metals the range of speeds to which this theory can apply is about 0.1 to 5 m/s which is a drop of 0.5 mm to 1.25 meters (page 366<ref>http://www.ewp.rpi.edu/hartford/~ernesto/S2015/FWLM/Books_Links/Books/JohnsonCONTACTMECHANICS.pdf</ref>).
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