4.4 Dynamic Accuracy 153 FIGURE 4.48 Example of a mechanism described by linear equations. When designing, the engineer (if it is important) must use these approaches (choos- ing the assumptions as required) and numerically solve the appropriate equations. It must be mentioned that dynamic errors (q v q 2 , etc.) are often serious obstacles in the effort to increase the efficiency, accuracy, quality, and/or productivity of newly designed equipment. Let us make a short digression and consider an example relating to the recent history of typewriters and the essence of their dynamics. The classical structure of the type- writer included a carriage that holds the paper and moves it along the typed line, pro- viding the correct intervals between the characters. The typebars are fastened onto specially shaped levers that are actuated manually or electrically. The wider the paper sheet or the more copies being typed simultaneously, the larger must the mass of the carriage and the dynamic effort of the mechanism be. To compensate, it was neces- sary to limit the typing speed and the dimensions of the parts. These limitations were overcome by the introduction of the IBM concept, where the carriage does not move along the lines and thus no inertia! forces occur. The line of characters is typed by a small, moving "golfball" element. It is made of light plastic, and therefore its mass is much less than that of the carriage in the old concept. The dynamic efforts are thus considerably reduced and do not depend on the paper width and the number of copies. The speed of manual typing is not limited by this concept. However, problems can (and certainly do) appear when this kind of typewriter is attached to a computer, which can type much faster. Going back to our subject, let us now consider an example of dynamic distur- bances in an industrial machine. Figure 4.49 shows an indexing table drive (we will deal with these devices in more detail in Chapter 6). This drive consists of a one- revolution mechanism like the one discussed earlier and shown in Figure 4.21. In this case, one-revolution mechanism 1 (when actuated) drives spatial cam 2 (into which the one-revolution mechanism is built): the cam is engaged with a row of rollers 3 fastened around the perimeter of rotating table 4. The mechanism is driven by motor 5 and trans- mission 6. The wheel 11 is the permanently rotating part of the mechanism. Key 7 is kept in the disengagement position by "teeth" 10 on cylinder 8. The latter is actuated by electromagnet 9 whose movement rotates cylinder 8, disconnecting "teeth" 10 from TEAM LRN 154 Kinematics and Control of Automatic Machines FIGURE 4.49 Layout of an indexing table as an example of a source of dynamic errors. key 7, thus engaging cam 2 with one-revolution mechanism 1 and thus causing rota- tion of the cam. During this revolution the magnet must be switched off and "teeth" 10 are put in the way of the rotating cam with key 7 attached to it, leading to disconnec- tion of cam 2 from mechanism 1. During this one revolution of cam 2 it causes rollers 3 to move, rotating table 4 for a corresponding angle (one pitch between the rollers). The profile of the cam is designed in such a way as to optimize the dynamic behav- ior of the table during its rotation. The main aim is to accelerate the process of index- ing; that is, to rotate for one pitch as quickly as possible, and to shorten transient processes such as parasitic oscillations as much as possible. The shorter the indexing time, the better the device. The best devices existing on the market complete this process in 0.25-0.3 seconds. By experimentally analyzing the mechanism in Figure 4.49 and measuring the dynamics of its behavior, we obtained the graphs shown in Figure 4.50a) and b), for the rotational speed and acceleration, respectively, of the cam when actuated by the one-revolution mechanism. These graphs imply that, instead of uniform rotation of the cam (for which it was calculated and designed), the rotation is essentially nonuniform, especially at the beginning of the revolution. How can we predict such behavior before the mechanism is built? What happens at the beginning of the engagement between the cam and the drive of the one-revolution mechanism? When "teeth" 10 (see Figure 4.49) free the key 7, it collides with the surface of the half-circular slot on the permanently rotating body of the mechanism (see Figure 4.21). As a result of this collision, the cam with the key and the rotating body rebound. From this moment these parts move independently until a new collision takes place. Thus, two modes of operation occur at the time of engagement. In the first, the two rotating bodies move together, connected to each other through the elastic key (Figure 4.51a)); and in the second, after the rebound, when the bodies rotate independently (Figure 4.51b)). TEAM LRN 4.4 Dynamic Accuracy 155 FIGURE 4.50 Records of a) speed and b) acceleration of the cam in the mechanism shown in Figure 4.49. Here, T = intervals of common drive-cam motion; T 2 , T 4 = intervals of independent drive-cam motion. Here we use the simplest model for the inertial behavior of bodies possessing moments of inertia, respectively J c and/ rf . The equations for the first mode of opera- Here, The initial conditions for t= 0 are <j) di = <p ci , § di = co di , and <j) ci = a> ci , where / = 0,2,4, , 2n for n = 0,1,2, the number of time intervals (here the collision intervals are considered). For the second mode of motion we have the following equations: FIGURE 4.51 Models of two modes of move- ment of the parts of a one-revolution mecha- nism: a) The rotating masses function as one system; b) The rotating masses move independently. TEAM LRN 156 Kinematics and Control of Automatic Machines The initial conditions for t = 0 are faj = <f) cj , Q dj = co dj and 4/ = o) cj where; = 1,3,5, , 2n -1, for n = 1,2, 3, Here b c , b d are the lumped coefficients of viscous friction of the cam and drive, respectively. Omitting transformations, we obtain the following solutions for the system of Equa- tions (4.36): where and We obtain the following formula for the durations T: The stiffness c is assumed to be the stiffness of the key 7 supported as shown in Figure 4.52. Thus FIGURE 4.52 The key of the mechanism as an elastic beam and its cross sections. A model for stiffness calculation. TEAM LRN 4.5 Damping of Harmful Vibrations 157 Here E is Young's modulus; /j and 1 2 are clear from Figure 4.52, and /j and 7 2 are iner- tial moments of the cross-sections I and II, respectively. The solution of the independent motion of the cam from Equation (4.37) is as follows: where n = b c /J c . Equations (4.38) and (4.44) are presented graphically in Figure 4.53. Here it is clear how the proposed model causes the changes in rotational speed and results in motion disturbances or errors. The intervals r of common motion and T 2 , r 4 , and T I are deter- mined here according to the modes of independent motion of the drive and cam. Figure 4.54 presents a comparison of the calculated and measured processes. The solid line represents the measured cam speed during its engagement with the drive, while the dashed line represents the process calculated according to Expressions (4.38) and (4.44). Considering the roughness of some assumptions made here, the curves do not look bad. 4.5 Damping of Harmful Vibrations The dynamic errors or disturbances we discussed in the previous section are obvi- ously not useful. They cause noise, fatigue, wear of the links, and inaccuracy in their movements, and result in decreased quality and productivity of the whole mechanism. The most obvious way to prevent such vibrations is to get rid of the forces that cause them. However, this is usually impossible or inconvenient. It may also be possible to move away from resonance conditions, by changing the mass (or the moment of inertia) of the vibrating element or the elasticity of a spring. There are several approaches to this, which we will discuss later. However, even the method of moving away from res- onance conditions does not always work. FIGURE 4.53 Calculated speeds of the drive and cam during acceleration of the indexing table. TEAM LRN 158 Kinematics and Control of Automatic Machines FIGURE 4.54 Comparison between the experimental (solid line) and the calculated (dashed line) speed of the cam of the one-revolution mechanism during operation. The third possibility is to try to use a dynamic vibration damper or dynamic vibra- tion absorber such as that shown schematically in Figure 4.55. This design is based on the finding that when the natural frequency equals the frequency co of the excitation force P = P 0 sin cot, the main mass M will not oscillate, because of the vibrations of the added absorber's mass m. Thus, the force k 2 (x 2 - xj of the stretched spring will balance the force P acting on the main mass M. The equations describing the behavior of this system (assuming that friction is negli- gible) are as follows: The solutions of this system are where for the amplitudes a\ and a 2 we obtain TEAM LRN 4.5 Damping of Harmful Vibrations 159 FIGURE 4.55 Model of a dynamic damper. Equation (4.47) proves what was stated earlier, i.e., that when co a = (o the numera- tor of this expression, and obviously also the amplitude a lt equals 0. Let us consider a model of free vibration. The model shown in Figure 4.55 is free to oscillate when no excitation force is applied, i.e., P = 0. For this case the equations for the mass movements are The solutions are in the form To find the natural frequencies we must solve the following biquadratic equation: The amplitudes are related as follows: From the latter expression we obtain the condition for the minimum value of 02 in the form Obviously, if the models discussed here represent rotational vibrations, the mass characteristic must be replaced by the moments of inertia and the springs must be described by their angular stiffness. Rotational vibrations have very important effects on indexing tables (see Chapter 7), which require some time to come to a complete rest after every step. An example is shown in Figure 4.56 of a pneumatically driven indexing table. In case a) the table, which has moment of inertia/! +J 2 when stopped, comes to rest as illustrated by the acceleration recording shown below. This process TEAM LRN 160 Kinematics and Control of Automatic Machines FIGURE 4.56 Indexing table (pneumatic drive) and its acceleration at the end of each rotation: a) Undamped; b) Dynamically damped. takes about 200 milliseconds. In case b) the masses are separated by a rod with an angular stiffness k 2 . The acceleration recording shows considerably lower acceleration (and thus also lower impact forces) and a shorter resting time. Now we must consider a situation where, in the layout given in Figure 4.55, an energy absorber is installed parallel to spring k 2 . Then the equations describing the motion of the masses will be as follows (c-damping coefficient): Using the method of complex numbers we obtain, for mass M (which is the point of interest), the following expression: where 7= V-l. To put this answer in the form of real numbers, we must express the complex numbers in terms of their absolute values. Thus, for the amplitude of vibrations of mass M we derive, from Equation (4.55), By substituting c = 0 into Equation (4.56) we obtain the result shown earlier in (4.47). For the opposite case, i.e., when c —»°°, the model under consideration becomes a system with one degree of freedom where the oscillating mass is M+ m. In both cases, when c = 0 and c -> °°, no energy is consumed in the absorber. This brings us to the TEAM LRN 4.5 Damping of Harmful Vibrations 161 conclusion that there must be some value of c at which a maximum amount of energy is absorbed during the relative movement of the vibrating masses, and at which the vibrational amplitude of mass Mis thus minimized. Denoting a> a /Q.=f, it can be proved that the minimum vibrational amplitude of the main mass M is reached when For this condition the vibrational amplitude flj of the main mass M can be estimated from the expression The last specific case applied to the model shown in Figure 4.55 that we will consider here describes the situation where c * 0, and k 2 = 0; that is, when masses M and m are connected only by means of the absorber. Such an absorber is called a Lanchester damper; it is based on viscous friction. The implementation of such a damper for indus- trial purposes can have various forms. An example is shown in Figure 4.57. This design consists of two discs 1 freely rotating on bearings 2. The latter are mounted on bushing 3 fastened onto the vibrating shaft 9. This bushing has disc 4 with friction gaskets 5. Springs 6 can be tuned by bolt 7 and nut 8 to develop the required factional torque. The optimum amplitude (the minimum vibrational amplitude) in this case is esti- mated by the Formula (4.58). (Obviously, for rotational vibration the masses and stiff- nesses must be replaced by the appropriate concepts: masses by moments of inertia, FIGURE 4.57 Lanchester damper—cross section. TEAM LRN 162 Kinematics and Control of Automatic Machines stiffnesses by angular stiffnesses, linear displacements by angles.) The optimum ratio ra/Mis 0.08-0.1. It must be mentioned that absorbers are less effective as vibration dampers than are dynamic dampers. The extinction of vibrations by dynamic means is more effec- tive. However, as follows from the above discussion, damping can be achieved when the parameters of the system are tuned accurately for all methods of damping. Unfor- tunately, accurate tuning is often almost impossible to achieve, for various reasons. For instance, the moments of inertia can change during the action of the manipula- tor, and the stiffnesses can also change because of changes in the effective lengths of the shaft when a sliding wheel moves along it. Vibration dampers that rely on friction are especially inaccurate. This brings us to the idea of using automatically controlled vibration damping. The next section is devoted to this question and describes some ideas on adaptive systems that can automatically tune themselves so as to minimize harmful vibrations. More detailed information on this subject can be found in the excellent book MechanicalVibrations by J. P. DenHartog (McGraw-Hill, New York, 1956). 4.6 Automatic Vibration Damping We consider here some ideas for dynamic damping that can be useful for auto- matic vibration control under changing conditions. As a first example we discuss a rapidly rotating shaft. As is known, a shaft with a concentrated mass on it, as in Figure 4.58, vibrates according to the solid curve shown in Figure 4.59. The amplitudes A are described by the following formula FIGURE 4.58 Dynamically damped rapidly rotating shaft. TEAM LRN [...]... force developed by the electromagnet, which is a function of the DC current in the coils; 6 = the air gap between the damper's mass and the magnet; and x = the displacement of the damper's mass during vibration Another approach to this problem is to apply an active force to the vibrating mass, thus creating an Active Damper (AD) The AD device generates a variable force P applied to the oscillating mass... Armature FIGURE 4.72 Comparison of the free oscillation of mass M (computation) without (damping takes about 20 sec) and with actuation of the AD (damping takes about 10 sec) force is applied to the mass M Obviously, the bigger the mass of the armature 4, the bigger the force The core 4 is fastened to the arm of the manipulator (or any other object) An example of a comparison of the vibrations damping processes... pulses Information is carried encoded as the amount of pulses (say, the higher the number of pulses, the larger the measured dimension), as the frequency of pulses, or as some other pulse-duration parameter The amplitude of the pulses usually has no importance in information transmission 5.1 Linear and Angular Displacement Sensors The most common task of a feedback is to gather information about the real... computers The disadvantage of this damper is that it is essentially nonlinear and therefore when the vibrational amplitude of the vibrating base changes, the natural frequency of the damper must be retuned An analytical approximation of the nonlinear stiffness k of this damper is: Here (see Figure 4 .67 ), k = the constant stiffness of the mechanical spring or elastic element of the damper; P = the force... tried, for instance, to produce the minimum vibrational amplitudes in beam A while the rotation speed of masses 8 slowly changed Figure 4 .64 shows the results of two independent experiments The upper record in both cases shows the vibration amplitude (in volts) of beam A before damping was attempted (indicating the frequencies in the excitation force) The lower record shows the damped vibrations To the. .. (b) 4 .6 Automatic Vibration Damping 165 FIGURE 4 .61 Dynamically damped cam mechanism so that mass 4 on rods 5 will have the required parameters as a dynamic damper The disturbances (or errors) q change their frequency at different cam speeds; therefore, the damper must be tuned accordingly Figure 4 .62 a) shows the usual (without damping) acceleration of the follower at two cam speeds, and Figure 4 .62 b)... means The DD designed in accordance with the patent mentioned above is shown in Figure 4 .67 and consists of a base 1, a spring system 2, a mass 3, a magnet core 4, and a coil 5 By changing the voltage (about 15-20 millivolts) in the coil 5, we control the restoring force developed by the flat springs 2, or, in other words, the stiffness of the springs, and, as a result, the natural frequency of the DD... nonlinear nature) The vibrations of the beams are measured by strain gauges 10 glued to the beams close to the fastening points The positions of masses 3 are measured by potentiometers 11 The information about the deflections of the beams and the positions of masses 3 is processed by a computer so as to move masses 3 into the proper positions for minimizing vibrations of beam A (or any other beam) Several... manufacturing machines The main advantage of this idea is the possibility to interact between the mechanics and the control electronics or computer This kind of interaction recently has been given the name mechatronics Exercise 4E-1 For the mechanisms shown in Figure 4E-1 a) and b), write the motion functions y = n(.x) and yf = n'(jt), respectively For case a) calculate the speed y and the acceleration y... amplitudes at a certain value of the frequency of the applied force P(t) The force generator proposed in this case is an electromagnet fastened to the vibrating mass M (say, the arm of the robot) The magnet consists of a core 2, a coil 3, and an armature 4 An elastic layer (not shown in Figure 4.71) is placed between 3 and 4 When energized, the magnet develops a force P, pulling the armature As a result, . the AD (damping takes about 10 sec). force is applied to the mass M. Obviously, the bigger the mass of the armature 4, the bigger the force. The core 4 is fastened to the. excitation force is applied, i.e., P = 0. For this case the equations for the mass movements are The solutions are in the form To find the natural frequencies we must solve the . vibrations of the added absorber's mass m. Thus, the force k 2 (x 2 - xj of the stretched spring will balance the force P acting on the main mass M. The equations describing the