Equation of Energy Loss by Friction Clutch During Engagement
Read: What is a Clutch? - Types of Clutches
Consider a plate or disc clutch
Let
IA = mass moment inertia of rotors attached to shaft A
IB = mass moment inertia of rotors attached to shaft B
ωA = Angular speed of shat A before engagement
ωB = Angular speed of shat B before engagement
ω = Common angular speed of shat A and Shaft B after engagement
According to the principle of conservation of momentum, Total momentum before clutch engage is equal to the total momentum of the clutch after clutch disc engagement.
IA ωA + IB ωB = (IA + IB)ω
Common angular speed after engagement of clutch pressure plate
Total Kinetic energy before friction clutch engagement
Kinetic energy after clutch engagement
Put the value of ω into above equation,
Now the loss of energy during clutch engagement, E= E1-E2
Consider a plate or disc clutch
Let
IA = mass moment inertia of rotors attached to shaft A
IB = mass moment inertia of rotors attached to shaft B
ωA = Angular speed of shat A before engagement
ωB = Angular speed of shat B before engagement
ω = Common angular speed of shat A and Shaft B after engagement
According to the principle of conservation of momentum, Total momentum before clutch engage is equal to the total momentum of the clutch after clutch disc engagement.
IA ωA + IB ωB = (IA + IB)ω
Common angular speed after engagement of clutch pressure plate
![ω= (I_A ω_A+I_B ω_B)/(I_A+I_B ) common angular speed](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj7IYPzJRljNy3Y4wmTDPOj2gDmqJ2xxxEAiYXKJ6Xas4IHy31rTU9kmsnGoxGhmFgUNC5B8CkIfhkiipaWDY1QEdJA8aqQXs3QoXGQoco6sFwxokk4LZU5sXFD7MZwb4gtSBbwH2x-iZQB/s1600/1+angular+speed.png)
Total Kinetic energy before friction clutch engagement
![E_1=1/2 I_A 〖ω_A〗^2+ 1/2 I_B 〖ω_B〗^2 E_1=1/2 〖(I〗_A 〖ω_A〗^2+I_B 〖ω_B〗^2) Energy equation before engagement](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiCMSUatVgEeTJfpAN9xJT7cgPrGfytCa9vmLo9yHhbOtbvIVDBhmbZmQEElwzG5NvUPYUKfi5DVrsZcd64Cafu56CnJCiEKkvDtjzS6DqALAgqyJuJH64l6BjkwlXYNeEqBVqWZbVvMhRh/s1600/2+Kinetic+energy.png)
Kinetic energy after clutch engagement
![E_2=1/2 〖(I〗_A+I_b)ω^2 energy equation after engagement](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg0eye2LvPQ3jwVu4L25rykGoz6PMgJU-6VK-U0Q8cYLrNtkFjNrSW6TYoic4DnEv237vUuyxQZbvUBa1TQz8oL1_RWPQwXxeeH5QBjr2lcxyrzc0DFjM2GnqgzaC5FFG_c5NV7Vnyfu9ds/s1600/3+Kinetic+energy.png)
Put the value of ω into above equation,
![E_2=1/2 〖(I〗_A+I_b)〖((I_A ω_A+I_B ω_B)/(I_A+I_B ))〗^2 = 〖〖(I〗_A ω_A+I_B ω_B)〗^2/(2(I_A+I_B)) energy equation after engagement of clutch](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhVuzHUpfdiWDleUUQkZK8NEtVt4X3mn9r7BQlE68fUJkFXXKaBNFmTijHZ9jbmsrfuoo6YdI1sLYNu2-sacqmsOSP5Z1rChSPeyLnQ1XaO1apsDWSacfnM_99EVUnH9B9WqDHfIoexgurI/s1600/4+Kinetic+energy.png)
Now the loss of energy during clutch engagement, E= E1-E2
![E= 1/2 〖(I〗_A 〖ω_A〗^2+I_B 〖ω_B〗^2)-〖〖(I〗_A ω_A+I_B ω_B)〗^2/(2(I_A+I_B)) E= (I_A I_B 〖(ω_A-ω_B)〗^2)/(2(I_A+I_B)) energy loss by friction during clutch engagement](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi5Mpo6pFB7zT0Y1ISq6sDuTTNcElEjHwMD-1dMVSPrzs3-G6cr_MxreE7zy9HfkKiNm3W1sTNUoOqwgDzlH_4W4SrM3c3kznOCNEu1Q-dKnH8dscrt37t-9X2GQnNXMnP8ysqKtRM04sP0/s1600/5+energy+loss%25281%2529.png)
Apply Different condition for above equation
Condition I - The rotor attached and hence the shaft B at rest ωB = 0
Put these condition in equation (a), and equation (b) we get
Common angular speed after the clutch engagement,
![ω= (I_A ω_A)/(I_A+I_B ) coomon angular speed](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgk-PnWD2fDErAtBDBYYiJ6AnvbiJgeNLBW1S2z5viPGu3iLlaOTRqTr6T3A243MrwkO1ZbVx8Pu07goM7reShvwZR707B3RgpgwUcIwDEBmjbTSFeaRCLFhGdONBB33EzOjK8wPRhtB5GM/s1600/6+angular+speed.png)
Loss of kinetic energy
![E= (I_A I_B 〖ω_A〗^2)/(2(I_A+I_B)) energy loss](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgCJ4smxZtuZ_XhIw0zC-aCHxfeo5AfiwHRP7Y68HyUsOktgmm5m7HgfaWrBrM7tpzltUJ8v2SfIhyphenhyphenAGIb4MlnMhPICfZp1UmMVfdX3NchJQWYE3pzl_9KNB5Llch-aF3KYFWzKHiBNbLS7/s1600/7+energy+loss.png)
Condition II - If rotor B at rest (ωB = 0) and IB is very small when compared to IA
Common angular speed after the clutch engagement,
![ω= (I_A ω_A)/(I_A+I_B )= ω_A common angular speed](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhuNvlqpDh7U9hrttFuZLW7eFaANiwBePuFx9mAc4lRTe5Epmj5DUe2HTC_lBLj4y4nApD6_djFz77OphLSMRqiiX9LiHUCd4VNsEMQY2Ha5COKIqVVfGDAwrbDVz-jgFX62KdEJuEOVkHm/s1600/8+angular+speed.png)
Kinetic energy loss,
![E= (I_A I_B 〖ω_A〗^2)/(2(I_A+I_B))=(I_A I_B ω^2)/(2I_A ) ,here (ω=ω_A) E=(I_B ω^2)/2 kinetic energy loss](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj-a5r3ZdKP06kE-zXuOjCsaaD428IdpHiAB6JUY1s9XlAQHKsuziDMIFw_2fn4RJ4Jn8moljFh-FXbYxMMkz8aRoPgZAe0A8Xv6mY6_yolei3hwyWrP0FdVtClbOblEx4WSaUwWv0BaN_s/s1600/9+energy+loss.png)