Rotational Mechanical Systems. Unit 2: Modeling in the Frequency Domain Part 6: Modeling Rotational Mechanical Systems

Transcription

1 Unit 2: Modeling in the Frequency Domain Part 6: Modeling Rotational mechanical systems are modelled in almost the same way as translational systems except that... We replace displacement, x(t) with angular displacement θ(t); Angular velocity is ω(t) We replace force with torque Engineering 5821: Control Systems I Faculty of Engineering & Applied Science Memorial University of Newfoundland January 22, 2010 For a force F acting on a body at point P, torque is defined as, T = FR sin φ where R is the distance from P to the body s axis of rotation and φ is the angle the force makes to the ray from the axis of rotation to P. Hence, if the force is perpendicular to the axis of rotation then, T = FR A rotating body can be considered a system of particles with masses m1, m2, m3,.... The moment of inertia is defined as, J = m1 R12 + m2 R22 + m3 R32 + The total kinetic energy is, 1 K = Jω 2 2 Recall that the kinetic energy for a translational system is 12 mv 2. So J is analagous to mass in translational motion. Also, similar to the equation F = ma in translational systems, we can relate torque and angular acceleration, T (t) = J dω d 2θ =J 2 dt dt We define the components of our rotational system as springs, viscous dampers, and rotating masses.

2 e.g. Find the transfer function θ2(s)/t (s) for the following system: We model the system in (a) as consisting of two sections joined by a spring. We draw a free-body diagram of J1: Sum the torques in (c) (J1s 2 + D1s + K)θ1(s) Kθ2(s) = T (s) Now consider J2 and apply the same analysis... (a) J1 rotating, J2 held still (b) J2 rotating, J1 held still (c) All torques on J1 Kθ1(s) + ( J2s 2 + D2s + K ) θ2(s) = 0 We can easily solve these two linear equations for the transfer function θ2(s)/t (s). Rarely do we see mechanical systems without gear trains. allow us to trade-off speed for torque. We will assume that connected gears fit perfectly together. However, in reality gears exhibit backlash where one gear will move through a small angle before its teeth meet those of the other gear. This is a non-linear effect that we will not model analytically.

3 The input gear on the left has radius r1 and teeth. It is rotated by θ1(t) due to a torque T1(t). What is the relationship between the rotation of the input gear and that of the output gear, θ2(t)? Although, the angles will differ, the arc length through which both gears turn will be the same: r1θ1 = r2θ2 Therefore the relation between angles is as follows, θ2 = r1 θ1 r2 Since the number of teeth is proportional to the radius, then the following also holds, θ2 = θ1 We can relate T1 and T2 through energy considerations. The amount of work done by the rotation of gear 1 is T θ. We are assuming that no energy is lost, therefore T1θ1 = T2θ2 = T2 T1 = The relationships between gears are pictured as transfer functions below: Reflecting Impedances Consider the following system: Now assume we are interested in the relation between T1 and θ1. Assume that we are interested in the relationship between the imposed torque T1 and θ2. The two rotating members are not linearly independent. So there is really only one rotating member to consider. We can reflect the imposed torque T1 to the rotating member J so that the gears are removed from the schematic: The torque due to each of the impedances on shaft 2 can be reflected to an equivalent torque on shaft 1. Consider the torque due to the damper on shaft 2: TD2 = Dsθ2(s) = Ds θ1(s) The relationship between the torques due to the damper is, TD1 = TD2 This is valid since T2 = T1. ( ) 2 = TD1 = Ds θ1(s) = Dsθ1(s)

4 The general pattern for the reflectance of impedances is as follows: ( ) 2 Ndest Tdest = ZMθdest Nsrc In this manner we can reflect all impedances on shaft 2 to shaft 1: This system can be modelled as follows, ( ( ) 2 ( ) 2 ( ) ) 2 Js 2 + Ds + K θ1(s) = T1(s) e.g. Find the transfer function θ2(s)/t1(s) for the following system, We can now write the equation of motion: Since the output is defined as θ2(s) we should reflect the impedances from shaft 1 onto shaft 2: where, Therefore (Je + De + K2) θ2(s) = T1(s) ( ) 2 Je = J1 + J2 ( ) 2 De = D1 + D2 G(s) = θ2(s) T1(s) = / Jes 2 + Des + K2

5 If we allow our gears to be large enough we can obtain any desired gear ratio. However, it is usually impractical to allow gears with large radii. Instead, gear trains are employed. e.g. Reflect all impedances in the abbreviated schematic below onto the input shaft: Solution: The equivalent gear ratio is the product of gear ratios for pairs of meshed gears. Notice that we are assuming the rotations of all other shafts are directly tied through the gear train to θ1.