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On the direct control of follower vibrations in cam–follower mechanisms
Mechanism and Machine Theory 45 (2010) 23–35
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On the direct control of follower vibrations in cam–follower mechanisms Gianluca Gatti *, Domenico Mundo Department of Mechanical Engineering, University of Calabria, 87036 Arcavacata di Rende (CS), Italy
a r t i c l e
i n f o
Article history: Received 6 October 2008 Received in revised form 16 June 2009 Accepted 22 July 2009 Available online 18 August 2009 Keywords: Vibration control Follower vibration Cam–follower mechanism Vibration absorber
a b s t r a c t This paper addresses a preliminary study on the control of follower vibrations in cam–follower mechanisms when acting directly onto the follower. Due to the elasticity of the mechanical components, the follower motion is affected by undesired dynamics, which may compromise its accuracy. Different techniques are commonly used to reduce such undesired vibrations, but they approach the problem in an indirect way. The aim of this paper is to investigate the feasibility of controlling follower motion by applying a secondary force directly onto it. As far as the authors are aware, there is not a clear indication in literature of why such a direct control has not been considered by researchers so far. This paper would thus like to address the limitations of such a control approach giving the reader a general and broader view on this issue. Simple active and passive control strategies are investigated and compared. Simulations are performed based on ‘purely mechanical’ considerations only and these bring, alone, to draw some important conclusion about their effectiveness or practical feasibility. Ó 2009 Elsevier Ltd. All rights reserved.
1. Introduction Cam–follower mechanisms are widely used in modern machineries thanks to several attractive features, such as design simplicity, manufacturing cheapness and employment versatility. In high-speed cam-driven mechanisms, motion accuracy of the follower is a key factor, since it affects the whole machine performance. Several researchers investigated the effects of cam profile inaccuracy and system flexibility on the output motion. Rothbart [1] proposed a method to predict how the follower response is affected by cam surface irregularities, while Koster [2] used a four degree-of-freedom (d.o.f.) model to simulate the effects of backlash and camshaft deflections on the output of the mechanism. Kim and Newcombe [3] and Grewal and Newcombe [4] investigated the combined effects of three sources of errors: geometric inaccuracy, kinematic errors and dynamic effects. Norton [5,6] investigated the effect of manufacturing tolerances on the dynamic performance of eccentric and double-dwell cams by means of experimental analyses. Recently, Wu and Chang [7] proposed an analytical method for the analysis of mechanical errors in disc-cam mechanisms. Conversely, several approaches have been proposed to improve the dynamic behaviour of cam–follower mechanisms by reducing follower undesired vibrations. Those may be classified into three main classes: (1) optimal selection of design parameters, (2) optimal design of cam profile or (3) optimal control of the mechanism input-speed. The first category aims at determining the optimal values of all significant design parameters (i.e. cam base-circle radius, follower roller radius, cam thickness, return-spring stiffness and so on) to limit undesired dynamics [8]. A sensitivity analysis is often performed to avoid that unacceptable design results may derive from variations of system dynamic parameters [9]. * Corresponding author. Tel.: +39 (0)984 494157; fax: +39 (0)984 494673. E-mail address: [email protected] (G. Gatti). 0094-114X/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.mechmachtheory.2009.07.010
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The second class of optimization methods aims at improving the kinematic behaviour [10] or the dynamic response of the follower by modifying the cam profile. Ting et al. [11] proposed the use of the Bezier technique to synthesize polynomial and other curves. Srinivasan and Ge [12] used Bernstein–Bezier harmonic curves to design dynamically compensated and robust cam profiles. As a result of the design process, residual vibrations are minimized at the design speed and made insensitive to speed variations. Recently, Andresen and Singhose [13] developed a procedure for minimizing unwanted vibrations in high-speed cam systems by using input shaping to modifying cam profile, while Qiu et al. [14] proposed a universal optimal approach to cam curve design, based on a B-spline description of the profile. More recently, Flocker [15] presented a modification of the cam profile by analyzing the frequency content of inertia forces. The third category of optimization methods is based on the control of the cam motion. By providing the camshaft with a proper input-speed function, both kinematic and dynamic performances of the follower can be improved. In 1956, Rothbart [1] proposed the use of a Withworth quick-return mechanism to provide a cam with a variable input-speed, thus reducing cam dimensions and pressure angle. Later, Tesar and Matthew [16] derived motion equations for the analysis of variable input-speed cam–follower mechanisms. The rapid development of servomotors suggested researchers to design servo-integrated mechanisms, characterized by a computer-controlled input-speed. Yan et al. [17,18] studied how the kinematic characteristics of followers are related to the cam speed curve. Furthermore, they proposed the theory of ‘Motion Control’ of cam mechanisms [19] by developing a method to design the optimal computer-controlled input-speed function to improve the kinematic performances. Kim et al. [20] presented a method for the optimal synthesis of a cam with non-constant angular velocity, based on the dynamic model of a complete spring-actuated cam mechanism. This paper would like to address the problem of controlling follower motion in a direct way by applying a secondary force on it, illustrating the limitations of such an approach and giving the reader a general view to this issue, which seems not to have been dealt in literature so far. Starting from a particular study-case application, which is given in [4] as a typical highspeed cam mechanism example, the basic and simplest possible control strategies are applied for follower vibration reduction, and some general considerations are inferred. Active and passive control strategies are applied to improve the dynamic performance of cam–follower systems, by directly connecting actuators onto the follower (active case) or through an attached tuned mass damper (passive case). Dynamic simulations are performed based on ‘purely mechanical’ models, i.e. without considering the effect of the control electronics, such as sampling frequency, time-delay, and electro-mechanical coupling in the secondary actuator. On the other hand, this is not a limitation of the present work since, based on those considerations only, important conclusions are drawn on the feasibility of those strategies. 2. Model of the cam–follower system 2.1. Kinematic model In this work, and without loss of generality, the follower lift function has been selected to follow a dwell-rise–dwellreturn–dwell motion and cycloidal function has been adopted for the rise and return phases. In particular, the follower lift function equation is reported below, and geometric parameters are listed in Table 1
8 0 > > h i > > > hc h1 1 > h 21p sin 2p hc h > max b b > < yðhc Þ ¼ hmax h i > > > h h h h > > hmax 4 b c 21p sin 2p 4 b c > > > : 0
for 0 < hc h1 ; for h1 < hc h2 ; for h2 < hc h3 ; for h3 < hc h4 ; for h4 < hc 2p;
where hc denotes the cam rotation angle. Fig. 1 illustrates the follower kinematic motion in terms of displacement, velocity and acceleration, whose values are normalized to their respective maxima. 2.2. Dynamic model The physical simulation of the cam–follower mechanism dynamics has been implemented considering a four d.o.f dynamic model, which is a simplified version of a more complete seven d.o.f. model discussed by Grewal and Newcombe [4]. In this paper, a two-dimensional lumped model, with one rotational and three translational d.o.f., is adopted to approximately represent the structural flexibility that exists in a typical cam–follower system, as shown in Fig. 2. The model takes into account the important effects due to the torsional and translational flexibility and damping of the cam and camshaft, the contact compliance at the cam/roller interface, the return spring and preload, and all significant Table 1 Geometric system parameters. Parameter Value Unit
Rb 0.06 m
Rr 0.015 m
hmax 0.03 m
h1 p/4 rad
h2 3p/4 rad
h3 5p/4 rad
h4 7p/4 rad
b
p/2 rad
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Fig. 1. Normalized kinematic motion of the follower. Displacement (dashed line), velocity (dotted line) and acceleration (solid line).
K rs
Crs
yf
Ccb
Mf Kf
Fs
Cf yr
Mr Kh
Cb yc
θc
K sf
Ic , M c
θm K vs
Cvs
Csf Fig. 2. Schematic representation of the cam–follower dynamic model.
damping and stiffness as well as mass and inertia of the mechanism components. With reference to Fig. 2, the equation for the rotational motion is written as
Ic €hc ¼ C sf ðh_ c h_ m Þ K sf ðhc hm Þ C b h_ c F c cosð/Þ
ðy_ r y_ c Þ ; h_ c
ð1Þ
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where
Fc ¼
Fp þ Fh cosð/Þ
and / ¼ a tan
ðy_ r y_ c Þ 1 _hc Rb þ Rr þ yr yc
are the total contact force at the cam/roller interface and the pressure angle, respectively. Fp is the preload and Fh is the elastic force due to Hertzian contact, which may be calculated as [4]
F h ¼ K h ðy þ yc yr Þ cosð/Þ: The preload is computed by assuming that the return spring is assembled in compression, with an initial displacement drs. If K eq is the equivalent stiffness due to Kf, Kh and Kvs in series, the preload force can be expressed as
F p ¼ drs
K eq K rs ; K eq þ K rs
where K eq ¼
1 1 1 þ þ K vs K f K h
1 :
The contact stiffness Kh is due to the Hertzian contact between the cam and the roller, and it is considered to be constant, as assumed in [4]. The equations for translational motion may be written as follows
€c ¼ C sv y_ c K sv yc F h cosð/Þ; Mc y €r ¼ C f ðy_ r y_ f Þ K f ðyr yf Þ þ F h cosð/Þ; Mr y
ð2Þ ð3Þ
€f ¼ C f ðy_ f y_ r Þ K f ðyf yr Þ C rs y_ f K rs yf F w F cb F s : Mf y
ð4Þ
Table 2 Dynamic system parameters. Parameter
Value
Units
Parameter
Value
Units
Ic Mc Mr Mf Ksf Kvs Kh Kf
0.000664 0.42365 0.02276 0.340 2.26 104 2.60 109 1.182 108 1.751 108
kg m2 kg kg kg N/m rad N/m N/m N/m
Krs Csf Cvs Cb nf Crs
2.10 104 0.0135 752.9 0.113 0.1 0.7558 0.08 0.0127
N/m N m s/rad N s/m N m s/rad – N s/m – m
lcb drs
Fig. 3. Follower acceleration: normalized difference between kinematic and dynamic case.
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The structural damping of the follower Cf and the friction force Fcb at the follower/guide interface are described by simple models, i.e. a viscous-type damping and a force proportional to the normal load by the Coulomb friction coefficient lcb, respectively, as
C f ¼ 2ff
qffiffiffiffiffiffiffiffiffiffiffiffi K f Mf ;
F cb ¼ lcb jF c sinð/Þj signðy_ f Þ:
A ‘‘jump criterion” has been also formulated according to [4]. Without loss of generality, the external work load Fw has been set to zero and system parameters have been adjusted in order to prevent jumping of the follower. 2.3. Dynamic system response The nonlinear system of Eqs. (1)–(4) is implemented in Simulink, given an input-speed of h_ m ¼ 1500 rpm and the dynamic parameters as listed in Table 2. Fig. 3 shows the algebraic difference between the follower acceleration in the pure kinematic case and the one in the dynamic case, normalized with respect to the maximum kinematic value of 1885 m/s2. Results are shown for the third cam cycle only, after which a steady state is achieved. As can be appreciated from Fig. 3, the maximum deviation in acceleration from the desired kinematic reference case is in the order of magnitude of a few percent.
3. Active control of follower dynamics In this section, the limitation of an active control strategy, which relies on the use of an external actuator, is investigated through the use of the inverse dynamics of the system. A very basic and simple control approach is then applied to reduce follower vibrations. 3.1. Fundamental limitation of active control In order to investigate the feasibility to supply a secondary force on the follower to suppress its undesired dynamics, a particular case of feed-forward strategy is simulated at first. In the ideal situation, if an accurate dynamic model of the system is available and the spinning frequency of the camshaft motor is known, feed-forward control can completely eliminate the effect of the primary disturbance whenever a signal correlated to it is available [21]. In the actual case, the follower kinematic motion could be taken as the reference and the inverse dynamic model can be solved to evaluate the secondary force signal to be provided to the follower for a complete vibration cancellation. The nonlinear system of Eqs. (1)–(4) is then rewritten as follows
8 _ _ Ic €hc ¼ C sf ðh_ c h_ m Þ K sf ðhc hm Þ C b h_ c F c cosð/Þ ðyrh_ yc Þ ; > > c > > €r ¼ C f ðy_ r y_ f Þ K f ðyr yf Þ þ F h cosð/Þ; > Mr y > > : €f C f ðy_ f y_ r Þ K f ðyf yr Þ C rs y_ f K rs yf F w F cb ; F s ¼ M f y
ð5a-dÞ
where symbols with a ‘star’ superscript denote the kinematic reference quantities, as depicted in Fig. 1. Fig. 4a shows the result of simulation in terms of the difference between the desired and actual follower acceleration with and without control. As can be noticed, in the ideal situation, where dynamic parameters are exactly known, and system and measurement are free of errors, feed-forward control strategy can completely suppress all unwanted dynamics. In Fig. 4b, the solid line represents the secondary force that would be required for the cancellation of the unwanted dynamics, while the dotted line represents the force required to exactly drive the follower according to the reference signal. In fact, due to the elasticity of the system, the total lift of the follower would be less than the kinematic value hmax by a static deflection df, which for the actual system (in the dwell phase after the rise) is given as
df ¼ hmax
K rs 9:2 106 006m: K rs þ K eq
This constant deflection compensation is the reason for the presence of a DC force component in the dotted line of Fig. 4b during the dwell phase. On the other hand, if the aim is, as usual, to reduce only the oscillatory motion of the follower, the required secondary force would be given by the solid line in Fig. 4b. One should note the relatively high force that would be required by the auxiliary actuating system in implementing such a strategy, and this is due to the fact that Eq. (5d), which provides the secondary force to obtain the desired follower motion, is decoupled from Eqs. (5a–c), which govern the cam and roller motion. This means that, no matter this latter motion, the follower is actuated by the external secondary force, which does not play a supporting role in reducing vibrations, but rather is the main source of input power. Even if this results in being a limiting case for follower dynamics control, it would be interesting to examine the detriment in performance due to parameters uncertainties. To illustrate this with an example, Fig. 5 shows the difference between the
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Fig. 4. Feed-forward control performance: (a) normalized difference between the desired and actual follower acceleration, without (dotted line) and with (solid line) control; (b) force signal for the secondary actuator, with (dotted line) and without (solid line) compensation for static deflection.
desired and actual follower acceleration if the controller is based on a wrong estimate of the dynamic parameters listed in Table 2, which are varied by a random amount between ±1% and ±5% of their nominal values. 3.2. Secondary force application In order to apply whatever secondary force to the system depicted in Fig. 2, some considerations are due. Generally speaking, it could be a direct force or the reaction force due to an inertial actuator. Fig. 6a and b illustrates two possible arrangements, where in either cases a force Fa is provided by a magnetic coil. The magnetic actuator is introduced here just as an example to show one of the possible physical implementations. Since the approach of the paper is based only on mechanical considerations, as discussed in the introduction, the particular electro-mechanical coupling, which is an actuator-dependent
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Fig. 5. Feed-forward control performance with parameters variation: normalized difference between the desired and actual follower acceleration, without (dotted line) and with (solid line) control.
Reaction mass
Coil
Magnetic core
Ma
Coil
Magnetic core
Fa
Fa
K rs
Crs
yf
Ccb
Mf
Ka
ya Ca
Fs K rs
Crs
yf
Ccb
(a) - Case 1
Mf
Fs
(b) - Case 2 Fig. 6. Two possible arrangements to apply a secondary force to the follower: (a) direct application; (b) by means of an inertial actuator.
issue, is out of the scope of this work. On the other hand, this does not affect the results that will be drawn as discussed in the conclusions. For Case 1, is
Fa ¼ Fs; while for Case 2 is
F a ¼ F s þ C a ðy_ a y_ f Þ þ K a ðya yf Þ: Of course, the use of an inertial actuator would turn the system into a five d.o.f. one. If the dynamic equations of the system are updated accounting for the additional mass Ma and its connection to the mechanism, it is possible to calculate the force Fa to be supplied to the magnetic coil and the reaction mass motion. Table 3 gives a list of the inertial actuator
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Table 3 Inertial actuator parameters. Parameter
Value
Units
Ka Ma na
96 0.032 0.4
N/m kg –
parameters that has been adopted, which refer to a commercially available small, lightweight actuator (Micromega-Dynamics, model IA-01). For a complete cancellation of follower vibrations, using feed-forward control strategy, the simulation gives the results illustrated in Fig. 7a and b, which show, respectively, the control force to the actuator and the reaction mass
Fig. 7. Feed-forward control performance in Case 2 arrangement: (a) control force and (b) reaction mass normalized displacement.
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Reaction mass
31
Ma
Magnetic core y
Fa Coil
Ka
Ca
Fs
Fa = M a y + Ca y + K a y Fs = − Fa + Ca y + K a y = −M a y
Frequency domain
T=
Fs ω2 = 2 Fa −ω + 2 jξaωaω + ωa2
Fig. 8. Inertial actuator dynamics and frequency response function.
Fig. 9. Feedback control performance: (a) normalized difference between the desired and actual follower acceleration, without (dotted line) and with (solid line) control; (b) control force; (c) reaction mass normalized displacement.
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displacement, normalized to the maximum kinematic displacement of the follower, hmax. It is interesting to note that both the control force and the reaction mass displacement are not periodic, but increasing with time. At the third cycle, for instance, an ideally unconstrained reaction mass would reach a displacement hundred times grater than the follower displacement. Such an arrangement for the actuator turns thus to be unfeasible. 3.3. Simple application of active control A simple case of active control is applied here, which is based on a feedback approach, where the difference between the actual and desired output signal is passed into a compensator and applied back to the system to control its output. This strategy differs from a classic feedback control in the fact that a model of the system is needed to provide the reference signal. The method can be thus realized having a cam mechanism dynamic model and with guaranteed stability, provided that the sensor and actuator are collocated, i.e. the actuator exerts its force in the location where the sensor observes the motion [21]. In this paper, relative velocity feedback control is investigated and applied as the simplest possible control strategy. In such a control, the force applied by the actuator is proportional to the difference in velocity between the desired and actual follower motion. The higher the feedback gain, the higher the damping force and the better the controller performance. The final effect is to add damping to the system by the use of a control force that opposes to any deviation from the reference, no matter the source of such a deviation. Either the actuator arrangements depicted in Fig. 6 can be adopted, and results in terms of performance and control force showed to be the same for both cases. The inertial actuator is, in fact, performing pffiffiffiffiffiffiffiffiffiffiffiffiffiffi as a perfect force generator for frequencies above its suspension frequency, xa ¼ K a =Ma ¼ 8:7 Hz, as illustrated in Fig. 8. To give the reader a qualitative understanding of the controller behaviour, Fig. 9a shows the effect of velocity feedback control on the system performance when a gain of 1000 N s/m is adopted. For this gain, the control force and the reaction mass displacement are periodic and illustrated in Fig. 9b and c, respectively. In this case, for instance, the reaction mass displacement is of the same order of magnitude than the follower displacement. It is worth to emphasize here that no practical implementation is considered in this paper; hence the particular value adopted for the gain has only the aim to give an illustrative example. 4. Passive control of follower dynamics In this section, the tuned vibration absorber (TVA) is investigated as the simplest passive vibration control device to attenuate follower undesired dynamics [22]. In its simplest configuration, the vibration absorber, also referred to as mass damper, consists of an auxiliary mass Ma attached to the host structure through a spring Ka and a damper Ca. The schematic representation depicted in Fig. 6b may be also adopted in this case as a reference, where now no coil or magnetic core is used.
Fig. 10. Normalized frequency response function of the follower acceleration, without (dotted line) and with (solid line) damping.
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When the primary system is excited by a harmonic force, its vibration can be suppressed by attaching a vibration absorber, which must be tuned to attenuate a particular frequency or a frequency band [23]. Since the actual mechanism is described by a nonlinear system of equations – mainly due to the pressure angle variation, which varies significantly within a cycle – a simple and approximate approach is followed here for tuning the absorber. This is based on a frequency analysis of the three d.o.f. translational linear system, corresponding to a dwell phase of the mechanism motion, i.e. when / = 0. In this case, the dynamics of the system is described by the following equation
ðMx2 þ Cx þ KÞ Y ¼ F;
ð6Þ
where Y = [yf, yr, yc], F is the vector of the applied harmonic force amplitudes and M, C, K are the mass, damping and stiffness matrices respectively, given by
Fig. 11. Tuned vibration absorber performance: (a) normalized difference between the desired and actual follower acceleration, without (dotted line) and with (solid line) absorber; (b) absorber mass normalized displacement.
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2
Mf
0
6 M¼4 0
Mr
0
0
0
3
7 0 5; Mc
2
C rs þ C f
6 C ¼ 4 C f 0
C f Cf 0
3
2
7 0 5; Cvs
6 K¼4
0
K rs þ K f
K f
0
K f
Kh þ Kf
K h
0
K h
K h þ K vs
3 7 5:
By solving Eq. (6) for F = [1, 0, 0], the frequency response function of the follower acceleration, due to a unit secondary force, is obtained and given in Fig. 10. The undamped natural frequencies of the system are easily calculated by solving the standard eigenvalue problem |M1K xnI| = 0, which gives the following values: xn1 = 2375 Hz, xn2 = 12925 Hz and xn3 = 19530 Hz. As can be seen from Fig. 10, the system is dominated by its first resonance frequency, manly due to the higher damping of the last two. Following this observations, a mass damper attached to the follower may be possibly tuned to this dominant resonance frequency. The tuning conditions are roughly selected according to the H1 optimization for an absorber attached to a one d.o.f. system [23]. Those are given below as
l¼
Ma ; Mf
b¼
sffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffi Ka 1 1 ¼ ; 1þl M a xp
nopt ¼
1 2
sffiffiffiffiffiffiffiffiffiffiffiffi 3l ; 2þl
C a ¼ 2Ma
sffiffiffiffiffiffiffi Ka n : M a opt
For a mass ratio l = 0.05, the optimum damping ratio is estimated as nopt 0.135. The effect of the tuned mass damper is illustrated in Fig. 11a and b, which show, respectively, the follower acceleration attenuation and the normalized damper displacement. 5. Discussion and conclusion In this paper, a theoretical discussion on the vibration control of cam–follower mechanisms by applying a direct secondary force on the follower has been presented, on the base of purely mechanical considerations only. This issue seems not to be present in literature and has driven the authors to give some general motivations for that and explore some limitations that may be of help to the interested reader. Due to the elasticity of the mechanical elements in the system, the follower motion is affected by undesired dynamics, which may possibly compromise the accuracy of the final motion when applied to high-speed automated machine. A four degree-of-freedom model is used to evaluate the mechanism performance when different basic control strategies are applied. A study-case is taken from literature and analyzed as a typical application of high-speed cam–follower mechanism. From this example some more general considerations are drawn. The kinematic acceleration of the follower is taken as a reference and performances are presented in terms of the normalized difference between dynamic and kinematic follower acceleration. The following general considerations may be drawn. A secondary force may be applied directly on the follower using an actuator attached to the frame. However, such a configuration would require a relatively high force and quick response. In fact, simulation results show that there is a fundamental limitation on applying active control to exactly compensate for follower unwanted dynamics. This is due to the fact that the equation which solves for the secondary control force is decoupled from the equation of motion of the cam and roller. This results in a secondary force which actually ‘actuates’ the follower instead of the cam mechanism. A simple application of active control is then illustrated and implemented in simulations adopting a feedback approach. An inertial actuator may be attached to the follower and if properly selected, it would perform as a pure force generator, requiring a force which may be orders of magnitude smaller than that of the limiting case. This is obtained by implementing a velocity feedback control and using the system model to generate the reference signal. The overall effect is to add damping to the system and thus reducing the undesired oscillation. The reaction mass of the actuator would achieve displacements which may be feasible with a practical implementation. An accelerometer must be attached to the follower and the difference to the kinematic reference signal must be fed to a simple controller, i.e. an integrator. A potential limit to this strategy would be the need to achieve the required value for the feedback gain. This, in fact, is practically limited by the electronic equipment and the sensor and actuator transfer functions. Another limitation would be the synchronization of the actual cam acceleration (provided by the sensor) to the kinematic reference one (provided by the model implemented into the control electronics). A comparison to a classic passive control strategy is performed. Such a control may be relatively easy to realize by attaching a vibration absorber to the follower which must be properly tuned to achieve effective vibration attenuation. A simple tuning of the absorber is realized and simulation results show that this control strategy could be feasible and qualitatively performing as active control without the need of external actuators. The absorber mass would pretty much follow the follower displacement, allowing for practical implementation. As a final consideration, active control directly on the follower is seen to be challenging in the sense that simple strategies seem not to be competitive compared to simple passive ones, and further investigations could be significant. From a mechanical point of view, control effort and actuator performance seem to be the major issues. References [1] H.A. Rothbart, Cams: Design, Dynamics, and Accuracy, Wiley, New York, 1956.
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[2] M.P. Koster, Vibrations of Cam Mechanism, Philips Technical Library, MacMillan, New York, 1974. [3] H.R. Kim, W.R. Newcombe, The effect of cam profile errors and system flexibility on cam mechanism output, Mechanism and Machine Theory 17 (1982) 57–72. [4] P.S. Grewal, W.R. Newcombe, Dynamic performance of high-speed semi-rigid follower cam systems – effects of cam profile errors, Mechanism and Machine Theory 23 (1988) 121–133. [5] R.L. Norton, Effect of manufacturing method on dynamic performance of cams – an experimental study. Part I – Eccentric cams, Mechanism and Machine Theory 3 (1988) 191–199. [6] R.L. Norton, Effect of manufacturing method on dynamic performance of cams – an experimental study. Part II – Double dwell cams, Mechanism and Machine Theory 23 (1988) 201–208. [7] L.I. Wu, W.T. Chang, Analysis of mechanical errors in disc cam mechanisms, Proceedings of the Institution of Mechanical Engineers, Part C, Journal of Mechanical Engineering Science 219 (2) (2005) 209–224. [8] J. K Mills, L. Notash, R.G. Fenton, Optimal design and sensitivity analysis of flexible cam mechanisms, Mechanism and Machine Theory 28 (4) (1993) 563–581. [9] A.C. Rao, Minimum flexibility error and optimum sensitivity synthesis of cam mechanisms, Mechanism and Machine Theory 14 (3) (1979) 209–214. [10] W.M. Hwang, C.Z. Yu, Optimal synthesis of the adjustable knock-out cam-follower mechanism of a bolt former, Proceedings of the Institution of Mechanical Engineers, Part C, Journal of Mechanical Engineering Science 219 (8) (2005) 767–774. [11] K.-L. Ting, N.L. Lee, G.H. Brandan, Synthesis of polynomial and other curves with the Bezier technique, Mechanism and Machine Theory 29 (6) (1994) 887–903. [12] L.N. Srinivasan, Q.J. Ge, Designing dynamically compensated and robust cam profiles with Bernstein–Bezier harmonic curves, Transactions of the ASME, Journal of Mechanical Design 120 (1) (1998) 40–45. [13] U. Andresen, W.A. Singhose, A simple procedure for modifying high-speed cam profiles for vibration reduction, Transactions of the ASME, Journal of Mechanical Design 126 (6) (2004) 1105–1108. [14] H. Qiu, C.J. Lin, Z.Y. Li, H. Ozaki, J. Wang, Y. Yue, A universal optimal approach to cam curve design and its applications, Mechanism and Machine Theory 40 (6) (2005) 669–692. [15] W.F. Flocker, Controlling the frequency content of inertia forces in dwelling ca-follower systems, Transactions of the ASME, Journal of Mechanical Design 129 (5) (2007) 546–552. [16] D. Tesar, F.K. Matthew, The Dynamics Synthesis, Analysis and Design of Modeled Cam Systems, Lexington Books, Lexington, MA, 1976. [17] H.S. Yan, M.C. Tsai, M.H. Hsu, An experimental study of the effects of cam speeds on cam–follower systems, Mechanism and Machine Theory 31 (4) (1993) 397–412. [18] H.S. Yan, M.C. Tsai, M.H. Hsu, A variable-speed method for improving motion characteristics of cam–follower systems, Transactions of the ASME, Journal of Mechanical Design 118 (2) (1996) 250–258. [19] Y. Yao, C. Zhang, H.S. Yan, Motion control of cam mechanisms, Mechanism and Machine Theory 35 (2000) 593–607. [20] J.H. Kim, K.Y. Ahn, S.H. Kim, Optimal synthesis of a spring-actuated cam mechanism using a cubic spline, Proceedings of the Institution of Mechanical Engineers, Part C, Journal of Mechanical Engineering Science 216 (9) (2002) 875–884. [21] R. Alkhatib, M.F. Golnaraghi, Active structural vibration control: a review, The Shock and Vibration Digest 35 (2003) 367–383. [22] J. Ormondroyd, J.P. Den Hartog, The theory of the dynamic vibration absorber, Transactions of the ASME, Journal of Applied Mechanics 50 (7) (1928) 9– 22. [23] T. Asami, O. Nishihara, Closed-form exact solution to H1 optimization of dynamic vibration absorbers (application to different transfer functions and damping systems), Transactions of the ASME, Journal of Vibration and Acoustics 125 (2003) 398–411.
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Mechanism and Machine Theory journal homepage: www.elsevier.com/locate/mechmt
On the direct control of follower vibrations in cam–follower mechanisms Gianluca Gatti *, Domenico Mundo Department of Mechanical Engineering, University of Calabria, 87036 Arcavacata di Rende (CS), Italy
a r t i c l e
i n f o
Article history: Received 6 October 2008 Received in revised form 16 June 2009 Accepted 22 July 2009 Available online 18 August 2009 Keywords: Vibration control Follower vibration Cam–follower mechanism Vibration absorber
a b s t r a c t This paper addresses a preliminary study on the control of follower vibrations in cam–follower mechanisms when acting directly onto the follower. Due to the elasticity of the mechanical components, the follower motion is affected by undesired dynamics, which may compromise its accuracy. Different techniques are commonly used to reduce such undesired vibrations, but they approach the problem in an indirect way. The aim of this paper is to investigate the feasibility of controlling follower motion by applying a secondary force directly onto it. As far as the authors are aware, there is not a clear indication in literature of why such a direct control has not been considered by researchers so far. This paper would thus like to address the limitations of such a control approach giving the reader a general and broader view on this issue. Simple active and passive control strategies are investigated and compared. Simulations are performed based on ‘purely mechanical’ considerations only and these bring, alone, to draw some important conclusion about their effectiveness or practical feasibility. Ó 2009 Elsevier Ltd. All rights reserved.
1. Introduction Cam–follower mechanisms are widely used in modern machineries thanks to several attractive features, such as design simplicity, manufacturing cheapness and employment versatility. In high-speed cam-driven mechanisms, motion accuracy of the follower is a key factor, since it affects the whole machine performance. Several researchers investigated the effects of cam profile inaccuracy and system flexibility on the output motion. Rothbart [1] proposed a method to predict how the follower response is affected by cam surface irregularities, while Koster [2] used a four degree-of-freedom (d.o.f.) model to simulate the effects of backlash and camshaft deflections on the output of the mechanism. Kim and Newcombe [3] and Grewal and Newcombe [4] investigated the combined effects of three sources of errors: geometric inaccuracy, kinematic errors and dynamic effects. Norton [5,6] investigated the effect of manufacturing tolerances on the dynamic performance of eccentric and double-dwell cams by means of experimental analyses. Recently, Wu and Chang [7] proposed an analytical method for the analysis of mechanical errors in disc-cam mechanisms. Conversely, several approaches have been proposed to improve the dynamic behaviour of cam–follower mechanisms by reducing follower undesired vibrations. Those may be classified into three main classes: (1) optimal selection of design parameters, (2) optimal design of cam profile or (3) optimal control of the mechanism input-speed. The first category aims at determining the optimal values of all significant design parameters (i.e. cam base-circle radius, follower roller radius, cam thickness, return-spring stiffness and so on) to limit undesired dynamics [8]. A sensitivity analysis is often performed to avoid that unacceptable design results may derive from variations of system dynamic parameters [9]. * Corresponding author. Tel.: +39 (0)984 494157; fax: +39 (0)984 494673. E-mail address: [email protected] (G. Gatti). 0094-114X/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.mechmachtheory.2009.07.010
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The second class of optimization methods aims at improving the kinematic behaviour [10] or the dynamic response of the follower by modifying the cam profile. Ting et al. [11] proposed the use of the Bezier technique to synthesize polynomial and other curves. Srinivasan and Ge [12] used Bernstein–Bezier harmonic curves to design dynamically compensated and robust cam profiles. As a result of the design process, residual vibrations are minimized at the design speed and made insensitive to speed variations. Recently, Andresen and Singhose [13] developed a procedure for minimizing unwanted vibrations in high-speed cam systems by using input shaping to modifying cam profile, while Qiu et al. [14] proposed a universal optimal approach to cam curve design, based on a B-spline description of the profile. More recently, Flocker [15] presented a modification of the cam profile by analyzing the frequency content of inertia forces. The third category of optimization methods is based on the control of the cam motion. By providing the camshaft with a proper input-speed function, both kinematic and dynamic performances of the follower can be improved. In 1956, Rothbart [1] proposed the use of a Withworth quick-return mechanism to provide a cam with a variable input-speed, thus reducing cam dimensions and pressure angle. Later, Tesar and Matthew [16] derived motion equations for the analysis of variable input-speed cam–follower mechanisms. The rapid development of servomotors suggested researchers to design servo-integrated mechanisms, characterized by a computer-controlled input-speed. Yan et al. [17,18] studied how the kinematic characteristics of followers are related to the cam speed curve. Furthermore, they proposed the theory of ‘Motion Control’ of cam mechanisms [19] by developing a method to design the optimal computer-controlled input-speed function to improve the kinematic performances. Kim et al. [20] presented a method for the optimal synthesis of a cam with non-constant angular velocity, based on the dynamic model of a complete spring-actuated cam mechanism. This paper would like to address the problem of controlling follower motion in a direct way by applying a secondary force on it, illustrating the limitations of such an approach and giving the reader a general view to this issue, which seems not to have been dealt in literature so far. Starting from a particular study-case application, which is given in [4] as a typical highspeed cam mechanism example, the basic and simplest possible control strategies are applied for follower vibration reduction, and some general considerations are inferred. Active and passive control strategies are applied to improve the dynamic performance of cam–follower systems, by directly connecting actuators onto the follower (active case) or through an attached tuned mass damper (passive case). Dynamic simulations are performed based on ‘purely mechanical’ models, i.e. without considering the effect of the control electronics, such as sampling frequency, time-delay, and electro-mechanical coupling in the secondary actuator. On the other hand, this is not a limitation of the present work since, based on those considerations only, important conclusions are drawn on the feasibility of those strategies. 2. Model of the cam–follower system 2.1. Kinematic model In this work, and without loss of generality, the follower lift function has been selected to follow a dwell-rise–dwellreturn–dwell motion and cycloidal function has been adopted for the rise and return phases. In particular, the follower lift function equation is reported below, and geometric parameters are listed in Table 1
8 0 > > h i > > > hc h1 1 > h 21p sin 2p hc h > max b b > < yðhc Þ ¼ hmax h i > > > h h h h > > hmax 4 b c 21p sin 2p 4 b c > > > : 0
for 0 < hc h1 ; for h1 < hc h2 ; for h2 < hc h3 ; for h3 < hc h4 ; for h4 < hc 2p;
where hc denotes the cam rotation angle. Fig. 1 illustrates the follower kinematic motion in terms of displacement, velocity and acceleration, whose values are normalized to their respective maxima. 2.2. Dynamic model The physical simulation of the cam–follower mechanism dynamics has been implemented considering a four d.o.f dynamic model, which is a simplified version of a more complete seven d.o.f. model discussed by Grewal and Newcombe [4]. In this paper, a two-dimensional lumped model, with one rotational and three translational d.o.f., is adopted to approximately represent the structural flexibility that exists in a typical cam–follower system, as shown in Fig. 2. The model takes into account the important effects due to the torsional and translational flexibility and damping of the cam and camshaft, the contact compliance at the cam/roller interface, the return spring and preload, and all significant Table 1 Geometric system parameters. Parameter Value Unit
Rb 0.06 m
Rr 0.015 m
hmax 0.03 m
h1 p/4 rad
h2 3p/4 rad
h3 5p/4 rad
h4 7p/4 rad
b
p/2 rad
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Fig. 1. Normalized kinematic motion of the follower. Displacement (dashed line), velocity (dotted line) and acceleration (solid line).
K rs
Crs
yf
Ccb
Mf Kf
Fs
Cf yr
Mr Kh
Cb yc
θc
K sf
Ic , M c
θm K vs
Cvs
Csf Fig. 2. Schematic representation of the cam–follower dynamic model.
damping and stiffness as well as mass and inertia of the mechanism components. With reference to Fig. 2, the equation for the rotational motion is written as
Ic €hc ¼ C sf ðh_ c h_ m Þ K sf ðhc hm Þ C b h_ c F c cosð/Þ
ðy_ r y_ c Þ ; h_ c
ð1Þ
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where
Fc ¼
Fp þ Fh cosð/Þ
and / ¼ a tan
ðy_ r y_ c Þ 1 _hc Rb þ Rr þ yr yc
are the total contact force at the cam/roller interface and the pressure angle, respectively. Fp is the preload and Fh is the elastic force due to Hertzian contact, which may be calculated as [4]
F h ¼ K h ðy þ yc yr Þ cosð/Þ: The preload is computed by assuming that the return spring is assembled in compression, with an initial displacement drs. If K eq is the equivalent stiffness due to Kf, Kh and Kvs in series, the preload force can be expressed as
F p ¼ drs
K eq K rs ; K eq þ K rs
where K eq ¼
1 1 1 þ þ K vs K f K h
1 :
The contact stiffness Kh is due to the Hertzian contact between the cam and the roller, and it is considered to be constant, as assumed in [4]. The equations for translational motion may be written as follows
€c ¼ C sv y_ c K sv yc F h cosð/Þ; Mc y €r ¼ C f ðy_ r y_ f Þ K f ðyr yf Þ þ F h cosð/Þ; Mr y
ð2Þ ð3Þ
€f ¼ C f ðy_ f y_ r Þ K f ðyf yr Þ C rs y_ f K rs yf F w F cb F s : Mf y
ð4Þ
Table 2 Dynamic system parameters. Parameter
Value
Units
Parameter
Value
Units
Ic Mc Mr Mf Ksf Kvs Kh Kf
0.000664 0.42365 0.02276 0.340 2.26 104 2.60 109 1.182 108 1.751 108
kg m2 kg kg kg N/m rad N/m N/m N/m
Krs Csf Cvs Cb nf Crs
2.10 104 0.0135 752.9 0.113 0.1 0.7558 0.08 0.0127
N/m N m s/rad N s/m N m s/rad – N s/m – m
lcb drs
Fig. 3. Follower acceleration: normalized difference between kinematic and dynamic case.
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The structural damping of the follower Cf and the friction force Fcb at the follower/guide interface are described by simple models, i.e. a viscous-type damping and a force proportional to the normal load by the Coulomb friction coefficient lcb, respectively, as
C f ¼ 2ff
qffiffiffiffiffiffiffiffiffiffiffiffi K f Mf ;
F cb ¼ lcb jF c sinð/Þj signðy_ f Þ:
A ‘‘jump criterion” has been also formulated according to [4]. Without loss of generality, the external work load Fw has been set to zero and system parameters have been adjusted in order to prevent jumping of the follower. 2.3. Dynamic system response The nonlinear system of Eqs. (1)–(4) is implemented in Simulink, given an input-speed of h_ m ¼ 1500 rpm and the dynamic parameters as listed in Table 2. Fig. 3 shows the algebraic difference between the follower acceleration in the pure kinematic case and the one in the dynamic case, normalized with respect to the maximum kinematic value of 1885 m/s2. Results are shown for the third cam cycle only, after which a steady state is achieved. As can be appreciated from Fig. 3, the maximum deviation in acceleration from the desired kinematic reference case is in the order of magnitude of a few percent.
3. Active control of follower dynamics In this section, the limitation of an active control strategy, which relies on the use of an external actuator, is investigated through the use of the inverse dynamics of the system. A very basic and simple control approach is then applied to reduce follower vibrations. 3.1. Fundamental limitation of active control In order to investigate the feasibility to supply a secondary force on the follower to suppress its undesired dynamics, a particular case of feed-forward strategy is simulated at first. In the ideal situation, if an accurate dynamic model of the system is available and the spinning frequency of the camshaft motor is known, feed-forward control can completely eliminate the effect of the primary disturbance whenever a signal correlated to it is available [21]. In the actual case, the follower kinematic motion could be taken as the reference and the inverse dynamic model can be solved to evaluate the secondary force signal to be provided to the follower for a complete vibration cancellation. The nonlinear system of Eqs. (1)–(4) is then rewritten as follows
8 _ _ Ic €hc ¼ C sf ðh_ c h_ m Þ K sf ðhc hm Þ C b h_ c F c cosð/Þ ðyrh_ yc Þ ; > > c > >
ð5a-dÞ
where symbols with a ‘star’ superscript denote the kinematic reference quantities, as depicted in Fig. 1. Fig. 4a shows the result of simulation in terms of the difference between the desired and actual follower acceleration with and without control. As can be noticed, in the ideal situation, where dynamic parameters are exactly known, and system and measurement are free of errors, feed-forward control strategy can completely suppress all unwanted dynamics. In Fig. 4b, the solid line represents the secondary force that would be required for the cancellation of the unwanted dynamics, while the dotted line represents the force required to exactly drive the follower according to the reference signal. In fact, due to the elasticity of the system, the total lift of the follower would be less than the kinematic value hmax by a static deflection df, which for the actual system (in the dwell phase after the rise) is given as
df ¼ hmax
K rs 9:2 106 006m: K rs þ K eq
This constant deflection compensation is the reason for the presence of a DC force component in the dotted line of Fig. 4b during the dwell phase. On the other hand, if the aim is, as usual, to reduce only the oscillatory motion of the follower, the required secondary force would be given by the solid line in Fig. 4b. One should note the relatively high force that would be required by the auxiliary actuating system in implementing such a strategy, and this is due to the fact that Eq. (5d), which provides the secondary force to obtain the desired follower motion, is decoupled from Eqs. (5a–c), which govern the cam and roller motion. This means that, no matter this latter motion, the follower is actuated by the external secondary force, which does not play a supporting role in reducing vibrations, but rather is the main source of input power. Even if this results in being a limiting case for follower dynamics control, it would be interesting to examine the detriment in performance due to parameters uncertainties. To illustrate this with an example, Fig. 5 shows the difference between the
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Fig. 4. Feed-forward control performance: (a) normalized difference between the desired and actual follower acceleration, without (dotted line) and with (solid line) control; (b) force signal for the secondary actuator, with (dotted line) and without (solid line) compensation for static deflection.
desired and actual follower acceleration if the controller is based on a wrong estimate of the dynamic parameters listed in Table 2, which are varied by a random amount between ±1% and ±5% of their nominal values. 3.2. Secondary force application In order to apply whatever secondary force to the system depicted in Fig. 2, some considerations are due. Generally speaking, it could be a direct force or the reaction force due to an inertial actuator. Fig. 6a and b illustrates two possible arrangements, where in either cases a force Fa is provided by a magnetic coil. The magnetic actuator is introduced here just as an example to show one of the possible physical implementations. Since the approach of the paper is based only on mechanical considerations, as discussed in the introduction, the particular electro-mechanical coupling, which is an actuator-dependent
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Fig. 5. Feed-forward control performance with parameters variation: normalized difference between the desired and actual follower acceleration, without (dotted line) and with (solid line) control.
Reaction mass
Coil
Magnetic core
Ma
Coil
Magnetic core
Fa
Fa
K rs
Crs
yf
Ccb
Mf
Ka
ya Ca
Fs K rs
Crs
yf
Ccb
(a) - Case 1
Mf
Fs
(b) - Case 2 Fig. 6. Two possible arrangements to apply a secondary force to the follower: (a) direct application; (b) by means of an inertial actuator.
issue, is out of the scope of this work. On the other hand, this does not affect the results that will be drawn as discussed in the conclusions. For Case 1, is
Fa ¼ Fs; while for Case 2 is
F a ¼ F s þ C a ðy_ a y_ f Þ þ K a ðya yf Þ: Of course, the use of an inertial actuator would turn the system into a five d.o.f. one. If the dynamic equations of the system are updated accounting for the additional mass Ma and its connection to the mechanism, it is possible to calculate the force Fa to be supplied to the magnetic coil and the reaction mass motion. Table 3 gives a list of the inertial actuator
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Table 3 Inertial actuator parameters. Parameter
Value
Units
Ka Ma na
96 0.032 0.4
N/m kg –
parameters that has been adopted, which refer to a commercially available small, lightweight actuator (Micromega-Dynamics, model IA-01). For a complete cancellation of follower vibrations, using feed-forward control strategy, the simulation gives the results illustrated in Fig. 7a and b, which show, respectively, the control force to the actuator and the reaction mass
Fig. 7. Feed-forward control performance in Case 2 arrangement: (a) control force and (b) reaction mass normalized displacement.
G. Gatti, D. Mundo / Mechanism and Machine Theory 45 (2010) 23–35
Reaction mass
31
Ma
Magnetic core y
Fa Coil
Ka
Ca
Fs
Fa = M a y + Ca y + K a y Fs = − Fa + Ca y + K a y = −M a y
Frequency domain
T=
Fs ω2 = 2 Fa −ω + 2 jξaωaω + ωa2
Fig. 8. Inertial actuator dynamics and frequency response function.
Fig. 9. Feedback control performance: (a) normalized difference between the desired and actual follower acceleration, without (dotted line) and with (solid line) control; (b) control force; (c) reaction mass normalized displacement.
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displacement, normalized to the maximum kinematic displacement of the follower, hmax. It is interesting to note that both the control force and the reaction mass displacement are not periodic, but increasing with time. At the third cycle, for instance, an ideally unconstrained reaction mass would reach a displacement hundred times grater than the follower displacement. Such an arrangement for the actuator turns thus to be unfeasible. 3.3. Simple application of active control A simple case of active control is applied here, which is based on a feedback approach, where the difference between the actual and desired output signal is passed into a compensator and applied back to the system to control its output. This strategy differs from a classic feedback control in the fact that a model of the system is needed to provide the reference signal. The method can be thus realized having a cam mechanism dynamic model and with guaranteed stability, provided that the sensor and actuator are collocated, i.e. the actuator exerts its force in the location where the sensor observes the motion [21]. In this paper, relative velocity feedback control is investigated and applied as the simplest possible control strategy. In such a control, the force applied by the actuator is proportional to the difference in velocity between the desired and actual follower motion. The higher the feedback gain, the higher the damping force and the better the controller performance. The final effect is to add damping to the system by the use of a control force that opposes to any deviation from the reference, no matter the source of such a deviation. Either the actuator arrangements depicted in Fig. 6 can be adopted, and results in terms of performance and control force showed to be the same for both cases. The inertial actuator is, in fact, performing pffiffiffiffiffiffiffiffiffiffiffiffiffiffi as a perfect force generator for frequencies above its suspension frequency, xa ¼ K a =Ma ¼ 8:7 Hz, as illustrated in Fig. 8. To give the reader a qualitative understanding of the controller behaviour, Fig. 9a shows the effect of velocity feedback control on the system performance when a gain of 1000 N s/m is adopted. For this gain, the control force and the reaction mass displacement are periodic and illustrated in Fig. 9b and c, respectively. In this case, for instance, the reaction mass displacement is of the same order of magnitude than the follower displacement. It is worth to emphasize here that no practical implementation is considered in this paper; hence the particular value adopted for the gain has only the aim to give an illustrative example. 4. Passive control of follower dynamics In this section, the tuned vibration absorber (TVA) is investigated as the simplest passive vibration control device to attenuate follower undesired dynamics [22]. In its simplest configuration, the vibration absorber, also referred to as mass damper, consists of an auxiliary mass Ma attached to the host structure through a spring Ka and a damper Ca. The schematic representation depicted in Fig. 6b may be also adopted in this case as a reference, where now no coil or magnetic core is used.
Fig. 10. Normalized frequency response function of the follower acceleration, without (dotted line) and with (solid line) damping.
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When the primary system is excited by a harmonic force, its vibration can be suppressed by attaching a vibration absorber, which must be tuned to attenuate a particular frequency or a frequency band [23]. Since the actual mechanism is described by a nonlinear system of equations – mainly due to the pressure angle variation, which varies significantly within a cycle – a simple and approximate approach is followed here for tuning the absorber. This is based on a frequency analysis of the three d.o.f. translational linear system, corresponding to a dwell phase of the mechanism motion, i.e. when / = 0. In this case, the dynamics of the system is described by the following equation
ðMx2 þ Cx þ KÞ Y ¼ F;
ð6Þ
where Y = [yf, yr, yc], F is the vector of the applied harmonic force amplitudes and M, C, K are the mass, damping and stiffness matrices respectively, given by
Fig. 11. Tuned vibration absorber performance: (a) normalized difference between the desired and actual follower acceleration, without (dotted line) and with (solid line) absorber; (b) absorber mass normalized displacement.
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2
Mf
0
6 M¼4 0
Mr
0
0
0
3
7 0 5; Mc
2
C rs þ C f
6 C ¼ 4 C f 0
C f Cf 0
3
2
7 0 5; Cvs
6 K¼4
0
K rs þ K f
K f
0
K f
Kh þ Kf
K h
0
K h
K h þ K vs
3 7 5:
By solving Eq. (6) for F = [1, 0, 0], the frequency response function of the follower acceleration, due to a unit secondary force, is obtained and given in Fig. 10. The undamped natural frequencies of the system are easily calculated by solving the standard eigenvalue problem |M1K xnI| = 0, which gives the following values: xn1 = 2375 Hz, xn2 = 12925 Hz and xn3 = 19530 Hz. As can be seen from Fig. 10, the system is dominated by its first resonance frequency, manly due to the higher damping of the last two. Following this observations, a mass damper attached to the follower may be possibly tuned to this dominant resonance frequency. The tuning conditions are roughly selected according to the H1 optimization for an absorber attached to a one d.o.f. system [23]. Those are given below as
l¼
Ma ; Mf
b¼
sffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffi Ka 1 1 ¼ ; 1þl M a xp
nopt ¼
1 2
sffiffiffiffiffiffiffiffiffiffiffiffi 3l ; 2þl
C a ¼ 2Ma
sffiffiffiffiffiffiffi Ka n : M a opt
For a mass ratio l = 0.05, the optimum damping ratio is estimated as nopt 0.135. The effect of the tuned mass damper is illustrated in Fig. 11a and b, which show, respectively, the follower acceleration attenuation and the normalized damper displacement. 5. Discussion and conclusion In this paper, a theoretical discussion on the vibration control of cam–follower mechanisms by applying a direct secondary force on the follower has been presented, on the base of purely mechanical considerations only. This issue seems not to be present in literature and has driven the authors to give some general motivations for that and explore some limitations that may be of help to the interested reader. Due to the elasticity of the mechanical elements in the system, the follower motion is affected by undesired dynamics, which may possibly compromise the accuracy of the final motion when applied to high-speed automated machine. A four degree-of-freedom model is used to evaluate the mechanism performance when different basic control strategies are applied. A study-case is taken from literature and analyzed as a typical application of high-speed cam–follower mechanism. From this example some more general considerations are drawn. The kinematic acceleration of the follower is taken as a reference and performances are presented in terms of the normalized difference between dynamic and kinematic follower acceleration. The following general considerations may be drawn. A secondary force may be applied directly on the follower using an actuator attached to the frame. However, such a configuration would require a relatively high force and quick response. In fact, simulation results show that there is a fundamental limitation on applying active control to exactly compensate for follower unwanted dynamics. This is due to the fact that the equation which solves for the secondary control force is decoupled from the equation of motion of the cam and roller. This results in a secondary force which actually ‘actuates’ the follower instead of the cam mechanism. A simple application of active control is then illustrated and implemented in simulations adopting a feedback approach. An inertial actuator may be attached to the follower and if properly selected, it would perform as a pure force generator, requiring a force which may be orders of magnitude smaller than that of the limiting case. This is obtained by implementing a velocity feedback control and using the system model to generate the reference signal. The overall effect is to add damping to the system and thus reducing the undesired oscillation. The reaction mass of the actuator would achieve displacements which may be feasible with a practical implementation. An accelerometer must be attached to the follower and the difference to the kinematic reference signal must be fed to a simple controller, i.e. an integrator. A potential limit to this strategy would be the need to achieve the required value for the feedback gain. This, in fact, is practically limited by the electronic equipment and the sensor and actuator transfer functions. Another limitation would be the synchronization of the actual cam acceleration (provided by the sensor) to the kinematic reference one (provided by the model implemented into the control electronics). A comparison to a classic passive control strategy is performed. Such a control may be relatively easy to realize by attaching a vibration absorber to the follower which must be properly tuned to achieve effective vibration attenuation. A simple tuning of the absorber is realized and simulation results show that this control strategy could be feasible and qualitatively performing as active control without the need of external actuators. The absorber mass would pretty much follow the follower displacement, allowing for practical implementation. As a final consideration, active control directly on the follower is seen to be challenging in the sense that simple strategies seem not to be competitive compared to simple passive ones, and further investigations could be significant. From a mechanical point of view, control effort and actuator performance seem to be the major issues. References [1] H.A. Rothbart, Cams: Design, Dynamics, and Accuracy, Wiley, New York, 1956.
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[2] M.P. Koster, Vibrations of Cam Mechanism, Philips Technical Library, MacMillan, New York, 1974. [3] H.R. Kim, W.R. Newcombe, The effect of cam profile errors and system flexibility on cam mechanism output, Mechanism and Machine Theory 17 (1982) 57–72. [4] P.S. Grewal, W.R. Newcombe, Dynamic performance of high-speed semi-rigid follower cam systems – effects of cam profile errors, Mechanism and Machine Theory 23 (1988) 121–133. [5] R.L. Norton, Effect of manufacturing method on dynamic performance of cams – an experimental study. Part I – Eccentric cams, Mechanism and Machine Theory 3 (1988) 191–199. [6] R.L. Norton, Effect of manufacturing method on dynamic performance of cams – an experimental study. Part II – Double dwell cams, Mechanism and Machine Theory 23 (1988) 201–208. [7] L.I. Wu, W.T. Chang, Analysis of mechanical errors in disc cam mechanisms, Proceedings of the Institution of Mechanical Engineers, Part C, Journal of Mechanical Engineering Science 219 (2) (2005) 209–224. [8] J. K Mills, L. Notash, R.G. Fenton, Optimal design and sensitivity analysis of flexible cam mechanisms, Mechanism and Machine Theory 28 (4) (1993) 563–581. [9] A.C. Rao, Minimum flexibility error and optimum sensitivity synthesis of cam mechanisms, Mechanism and Machine Theory 14 (3) (1979) 209–214. [10] W.M. Hwang, C.Z. Yu, Optimal synthesis of the adjustable knock-out cam-follower mechanism of a bolt former, Proceedings of the Institution of Mechanical Engineers, Part C, Journal of Mechanical Engineering Science 219 (8) (2005) 767–774. [11] K.-L. Ting, N.L. Lee, G.H. Brandan, Synthesis of polynomial and other curves with the Bezier technique, Mechanism and Machine Theory 29 (6) (1994) 887–903. [12] L.N. Srinivasan, Q.J. Ge, Designing dynamically compensated and robust cam profiles with Bernstein–Bezier harmonic curves, Transactions of the ASME, Journal of Mechanical Design 120 (1) (1998) 40–45. [13] U. Andresen, W.A. Singhose, A simple procedure for modifying high-speed cam profiles for vibration reduction, Transactions of the ASME, Journal of Mechanical Design 126 (6) (2004) 1105–1108. [14] H. Qiu, C.J. Lin, Z.Y. Li, H. Ozaki, J. Wang, Y. Yue, A universal optimal approach to cam curve design and its applications, Mechanism and Machine Theory 40 (6) (2005) 669–692. [15] W.F. Flocker, Controlling the frequency content of inertia forces in dwelling ca-follower systems, Transactions of the ASME, Journal of Mechanical Design 129 (5) (2007) 546–552. [16] D. Tesar, F.K. Matthew, The Dynamics Synthesis, Analysis and Design of Modeled Cam Systems, Lexington Books, Lexington, MA, 1976. [17] H.S. Yan, M.C. Tsai, M.H. Hsu, An experimental study of the effects of cam speeds on cam–follower systems, Mechanism and Machine Theory 31 (4) (1993) 397–412. [18] H.S. Yan, M.C. Tsai, M.H. Hsu, A variable-speed method for improving motion characteristics of cam–follower systems, Transactions of the ASME, Journal of Mechanical Design 118 (2) (1996) 250–258. [19] Y. Yao, C. Zhang, H.S. Yan, Motion control of cam mechanisms, Mechanism and Machine Theory 35 (2000) 593–607. [20] J.H. Kim, K.Y. Ahn, S.H. Kim, Optimal synthesis of a spring-actuated cam mechanism using a cubic spline, Proceedings of the Institution of Mechanical Engineers, Part C, Journal of Mechanical Engineering Science 216 (9) (2002) 875–884. [21] R. Alkhatib, M.F. Golnaraghi, Active structural vibration control: a review, The Shock and Vibration Digest 35 (2003) 367–383. [22] J. Ormondroyd, J.P. Den Hartog, The theory of the dynamic vibration absorber, Transactions of the ASME, Journal of Applied Mechanics 50 (7) (1928) 9– 22. [23] T. Asami, O. Nishihara, Closed-form exact solution to H1 optimization of dynamic vibration absorbers (application to different transfer functions and damping systems), Transactions of the ASME, Journal of Vibration and Acoustics 125 (2003) 398–411.