THE SCIENCE AND APPLICATIONS OF ACOUSTICS - H. H. Arnold ...

14.11 Other Mechanical Drive Elements 383Figure 14.10. Roller chain.For transmission of high loads at high speeds, inverted-tooth (or “silent”) chainsare often applied. Because the inverted-tooth chain engages with the sprocketswith less impact force, it generally operates more quietly than roller chain.The links are of finite length in either roller chain or inverted-tooth chain. Whilethe chain links engage a sprocket, the pitch line (i.e., the centerline of the link pins)bobbles up and down due to the chordal action. In Figure 14.10, the center of alink pin is located at R max from the center of the sprocket over which it moves.R max is the maximum center of the pitch line from the sprocket center. As the linkengages further, the pitch line moves nearer to the sprocket center and both pinsof the link coincide with the pitch line. The new position is given by( 180◦)R min = R max cos(14.31)N Swhere N S indicates the number of sprocket teeth. Roller chains usually have aminimum of 8 teeth and inverted-tooth chains, 17 teeth.Consider a drive sprocket rotating at constant speed n s . The pitch line velocityof the chain will change in direct proportion to its distance from the center of thesprocket, i.e., V = R × (2πn s ). The fractional velocity change V /V is given byVV= R max − R meanR max=1 − 1 + cos(180◦ /N S )21 + cos(180 ◦ /N S )2= 1 − cos(180◦ /N s )1 + cos(180 ◦ /N s )(14.32)The angular velocity (2πn s ) is canceled itself out in the ratio of Equation (14.32),but the speed ratio of a chain drive is given byn 2= N 1(14.33)n 1 N 2where n refers to the rotational speed (rpm) of the sprocket and N the numberof sprocket teeth. Subscripts 1 and 2 denote the driver sprocket and the drivensprocket, respectively. Fundamental frequencies of noise and vibration can be dueto the following: