Mechanical Systems and Signal Processing

Transcription

1 Mechanical Systems and Signal Processing 29 (22) Contents lists available at SciVerse ScienceDirect Mechanical Systems and Signal Processing journal homepage: Theoretical analysis of the dynamic behavior of presliding rolling friction via skeleton technique T. Tjahjowidodo n School of Mechanical and Aerospace Engineering, Division Mechatronics and Design, Nanyang Technological University, 5 Nanyang Avenue, North Spine (N3) Level 2, Singapore , Singapore article info Article history: Received 27 January 2 Received in revised form October 2 Accepted 4 October 2 Available online 5 November 2 Keywords: Rolling friction Presliding regime Skeleton Hilbert Wavelet abstract Depending on the application, friction in mechanical systems can be a desirable or an undesirable thing. In order to optimally utilize or compensate for the frictional effect in the system, characterization of the frictional behavior is a crucial task. Unfortunately, friction exhibits strong (hysteretic) nonlinear relationship with sliding velocity, displacement and time, which makes the characterization to become a difficult task to fulfill. The classical Coulomb friction has been widely used, especially in the field of control engineering to compensate for the static in a system. However, despite its simplicity, the effectiveness of the Coulomb model in capturing the frictional behavior of friction in the presliding regime is very low. This paper deals with the derivation of equivalent modal parameters, namely the stiffness and damping elements, to represent the hysteresis friction element on mechanical systems. The analysis is carried out using the skeleton technique, which employs the instantaneous amplitude and frequency of the excitation input and the response of the system. Subsequently, the dynamic analysis is performed on the equivalent system and the result is compared to that of the original system. The results show good agreement between the equivalent system and the original system for wide range of excitation. & 2 Elsevier Ltd. All rights reserved.. Introduction Being a complex nonlinear phenomenon, friction is perhaps the most difficult part to be identified and compensated for in a mechanical system. It is the result of interaction between one body over or along another and is dependent on many parameters, such as contact geometry, topography, surface materials and presence and type of lubrication and relative motion. Depending on the type of application, friction can be a desirable or an undesirable thing in a system. Brakes, clutches, clamps and friction damping are few examples of the first situation, while the friction in slides, bearings and joints is an example of the second. The simple (but incomplete) representation of friction, i.e. the classical Coulomb friction model approximation, is adequate for the analysis of many, but not all, physical systems. It defines the friction for non-zero relative velocity in the sliding regime (va); when v¼ the characteristic simply sets the friction below the static value, where v represents the relative velocity. This makes the model quite complex to simulate, since it requires accurate detection of the n Tel.: þ ; fax: þ address: ttegoeh@ntu.edu.sg /$ - see front matter & 2 Elsevier Ltd. All rights reserved. doi:.6/j.ymssp.2..7

2 T. Tjahjowidodo / Mechanical Systems and Signal Processing 29 (22) velocity zero crossing. It is also impractical for friction compensation at motion stop and reversal, where the effect of stick slip motion arises. In fact, motion never starts or stops abruptly and micro-sliding displacements are actually observed []. Friction has been distinguished into two main regimes: the pre-sliding and sliding regimes. When a contacting body is sliding and moving away from a reversal point exceeding a pre-sliding limit this stage is commonly called as a sliding regime,the friction predominantly appears to be a function of velocity. On the contrary, when the velocity of the contacting body is decreased and the motion is reversed, the system is entering a pre-sliding regime, and the frictional effects of the mechanism are predetermined also by a function of displacement. At this stage, the friction behavior depends not only on the velocity, but also on the displacement, where particularly the relation between friction and displacement involves a so-called non-local memory hysteresis [ 5]. At small displacements, within the pre-sliding region, friction predominantly appears as a function of displacement instead of velocity (see [2,6]). This unique behavior attracts many researchers to thoroughly model the behavior of the contact between the two surfaces, which is realized using asperity junctions. The asperities can deform elastically or plastically depending on the load and on the displacement and/or the relative velocity of the surfaces as reported by Lampaert [7]. Regarding to the complexity of the hysteresis phenomena on frictional elements, there are hardly any instances in the literature of the dynamic behavior analysis of mechanical systems subjected to hysteresis elements. Among few of them, Seelecke [8] presented a relevant work on the dynamic analysis of shape memory alloys that are well-known to possess hysteresis properties. Capecchi and Vestroni [9] studied the steady-state dynamic analysis of hysteresis systems in geoengineering. A more relevant study is presented by Al-Bender et al. [6], in which they analyzed the dynamic behavior of mechanical systems comprising rolling elements that exhibit the pre-sliding friction phenomena. The analysis is carried out by performing the describing function technique [], where the nonlinear system is linearized by considering only the output (response) at the fundamental frequency component when the corresponding system is excited by a sinusoidal input. Consequently, the linearized components (in this case are the equivalent stiffness and equivalent damping) appear as functions of frequency. The approach offers a potential tool to appreciate the dynamic behavior of the corresponding systems. However, as the equivalent parameters (stiffness and damping) are presented as functions of frequency, a simulation of the response for any arbitrary input/excitation signals is rather tedious to perform. As an alternative tool to present the equivalent dynamic parameters of a nonlinear mechanical system, the skeleton technique can be performed utilizing the relationship between the (nonlinear) natural frequency (-ies) and instantaneous amplitudes of the system. Feldman and Braun [] reported that every typical nonlinear modal parameter in a system has a unique relationship between the natural frequency and the instantaneous response amplitude when the system is freely vibrating. The relationship is referred to as the skeleton curve. Furthermore, Feldman [2] proposed the method of FreeVib to construct the modal parameters from the skeleton curve using the Hilbert transform approach. Extensive studies and practical applications of this technique can be found in some papers [3 5]. In some cases, when the nonlinear mechanical systems comprise a high damping factor and the transient response decays in a very short duration, the FreeVib method is not applicable. In order to overcome this problem, Feldman extended the method to a d vibrating condition [6]. In this method, which is referred to as the ForceVib method, the system under evaluation is actuated by a prescribed excitation input and after some mathematical manipulation, the skeleton curve, and in turn the modal parameters, can be obtained. Despite its simplicity, the extraction of the instantaneous amplitude and frequency from the signals using the Hilbert transform is limited by one essential condition. The extraction is strictly exact only if the amplitudes of the signals are slowly varying compared to the phase variations. As a response to this problem, Staszewski [7] utilized the wavelet transform technique to extract the instantaneous amplitude and frequency. The practical application of the corresponding technique is reported and presented by Tjahjowidodo [8] on the dynamic identification of a mechanical system with backlash elements. Hitherto, it is obvious that these methods are applicable to predict the modal parameters of mechanical systems with geometric nonlinearities not to those with material nonlinearities. 2 This paper deals with the derivation of equivalent dynamic parameters, namely the stiffness and the damping, of mechanical systems with (hysteresis) frictional elements. In particular, this paper presents the theoretical dynamic analysis of a mechanical system with rolling friction and application of the skeleton technique to derive the equivalent modal parameters. Prior to the derivation of the equivalent dynamic parameters, a mechanical system subjected to a frictional surface will be simulated. A hysteresis model, namely the Maxwell-slip [9], which is proven to be able to capture the unique properties of the nonlocal memory hysteretic friction, is utilized to mimic the (rolling) frictional in the system. Subsequently, the displacement outputs of the system under non-stationary excitations are evaluated and the skeleton analysis is performed via the wavelet transform to construct the equivalent dynamic parameters of the system. The parameters are expected to give a geometric nonlinear equivalence to the original system. Subsequently, the analysis In a system with geometric nonlinearity, the change in geometry as the structure deforms causes a nonlinear change of the parameter in the system. 2 In a system with material nonlinearity, the material behavior depends on the current deformation. This type of nonlinearity is usually associated with the presence of hysteresis behavior in the dynamics of the system, e.g. (presliding) friction.

3 298 T. Tjahjowidodo / Mechanical Systems and Signal Processing 29 (22) is verified on a real mechanical system, namely a DC motor system, which is subjected to a frictional torque originated from rotating elements, i.e. shaft and bearing supports. In the following, Sections 2 and 3 discuss briefly the theoretical bases of the hysteresis and the skeleton method. Subsequently, Section 4 presents the mathematical model of the dynamic system used in this analysis. In Section 5, the dynamic simulation of the system and the skeleton analysis will be discussed. The equivalent modal parameters will be derived in this section. The resulting parameters are analyzed from the dynamic point of view in Section 6. Still in the same section, the dynamic behavior of the equivalent system will be compared with that of the original system with hysteresis friction models. Section 7 presents the analysis of the equivalent (geometric nonlinear) system of a DC motor system to validate the proposed approach. Finally, some appropriate conclusions are drawn in Section Rolling friction in a nutshell The relationship between the rolling friction and the displacement can be characterized by a hysteresis function exhibiting a non-local memory property. The key features of this behavior, as also illustrated in Fig., are: (i) the friction always follows a certain profile in a function of the displacement (referred to as the virgin curve), passing through all reversal points, which have to be memorized, but, (ii) every time a loop of the hysteresis curve is closed, the last reversal point can be forgotten. Al-Bender and Symens detailed the behavior of the corresponding rolling element friction in [2]. He also concluded that the behavior of a (rolling) friction essentially depends on the virgin curve function, y(t), and, as a reasonable but analytically tractable, the virgin curve is presented in the following form: " yðtþ¼h d # 2 ð2:þ dþx Despite its complexity, the aforementioned behavior can be modeled relatively simple using a parallel connection of Maxwell-slip elements also called Jenkin s elements (see e.g. [9]). The behavior of a Maxwell-slip element can be represented by two state equations. If the elementary model sticks, it behaves like a linear spring, thus the elementary friction, F i, can be modeled mathematically as df i dt ¼ k iv ðstickþ ð2:2þ with k i being the stiffness of each elementary model and v is the velocity. The elementary model will slip if the friction of each element reaches the maximum value of the, W i, that it can sustain. Beyond this point the elementary friction equals the maximum, W i, obtained by solving the following equation: df i dt ¼ ðslipþ ð2:3þ When the motion is reversed, the elementary model will stick, and it will behave again like a linear spring until the friction reaches the maximum, W i. Using a parallel connection of N Maxwell-slip elements, the total friction is equal to the sum of all elementary friction components F f ¼ X N F i ð2:4þ Fig. 2 shows the schematic of the corresponding Maxwell-slip model. 2 x t F m y(x) 3,3 -xm x m x -y(-x) 4 2,4 5 Fig.. Rolling friction behavior: the upper panel shows the displacement input, the lower panel depicts the function of friction vs. displacement. When a mass subjected on the friction is displaced, initially the friction follows an odd function of displacement, the so called virgin curve, y(t). If the motion is reversed at x m, the trajectory of the friction follows the flipped and double stretched of y(t) and the system memorizes the reversal point.

4 T. Tjahjowidodo / Mechanical Systems and Signal Processing 29 (22) W κ F W i W i κ i x κ i x κ N -W i W N Fig. 2. Parallel connection of N Maxwell-slip elements. The Maxwell-slip model is proven to be able to characterize the rolling (and presliding dry-) friction in a relatively simple way; however, one problem remains, the dynamic analysis of mechanical systems subjected to the aforementioned complex hysteresis phenomena is still a difficult task to perform. 3. Skeleton techniques According to Feldman [2], a large number of signals, including vibrations of systems with geometric nonlinearities, can be converted to analytic signals in complex time. Let us consider the following d vibration equation of a singledegree-of-freedom system: y þ2h ðaþ _y þo 2 ðaþy ¼ f =m ð3:þ where y is the response signal, f is the d excitation signal, m is the mass of the system, h and o are the symmetrical viscous damping and stiffness characteristic of the system, respectively, which depend on the amplitude (A). Feldman [6] shows that Eq. (3.) can be converted by the Hilbert transform to an analytic signal form: Y þ2h ðaþ Y _ þo 2 ðaþy ¼ F=m ð3:2þ where F¼F(t) is the analytic signal of the d excitation and Y is an analytic signal of the response of the system. The analytic signal, F(t), is expressed by FðtÞ¼fðtÞþj f ~ ðtþ, where the tilde notation represents the Hilbert transform of the corresponding real-value signal. While the analytic signal of the response of the system, Y(t), can be represented in the form of the combination of slow varying function called envelope, A(t), and instantaneous phase, j(t): Y ¼ YðtÞ¼yðtÞþj ~yðtþ¼aðtþe jjðtþ ð3:3þ Taking the derivatives of the analytic signal, Y(t), and solving the equations of the real and imaginary parts from Eq. (3.2), the modal parameters can be written in the expressions of the instantaneous amplitude and frequency: o 2 ðtþ¼o2 þ aðtþ m bðtþ _ A Aom A A þ 2 A _ 2 A 2 þ _ A _o Ao h ðtþ¼ bðtþ 2om A _ A _o ð3:5þ 2o where o is the time derivative of the instantaneous phase f, while a and b are the real and imaginary parts of input and output signals ratio ðfðtþ=yðtþþ ¼ aðtþþjbðtþ, respectively. Ruzzene et al. [2] reported that the envelope and instantaneous frequency extraction of a signal in Eq. (3.3) using the Hilbert-transform technique will introduce errors when the system is highly damped. As an alternative, Staszewski [7] proposed to implement the wavelet technique with the complex Morlet mother wavelet function [22]: qffiffiffiffiffiffiffi cðtþ¼ pf b e joct e t2 =f b where f b is the bandwidth parameter and f c ¼o c /2p is the center frequency. Since the wavelet analysis transforms a function to translation (t) and scale (s) domains, the extraction of the envelope and instantaneous frequency of the signal cannot be performed straightforward. The following equations are utilized to extract the corresponding parameters: qffiffiffiffiffiffiffi 9Wðs,tÞ9 ¼ p 2 f b 9s9AðtÞe f b 4 ðso n o cþ 2 +Wðs,tÞ¼otþy ð3:6þ Due to space limitation, interested readers on the derivation of Eq. (3.6) are suggested to refer to [7,8]. 4. A mechanical vibration system subjected to a frictional surface In this paper, the dynamic behavior of a mechanical vibration system subjected to a frictional will be evaluated by means of the skeleton technique. In the case of a mass supported by a linear spring and resting on a frictional surface, the schematic of the system can be illustrated in Fig. 3. ð3:4þ

5 3 T. Tjahjowidodo / Mechanical Systems and Signal Processing 29 (22) x k m F(t) f fr Fig. 3. Schematic drawing of a mechanical vibration system supported by a linear spring with stiffness of k and frictional, f fr. Table Parameter set of the simulated system. m k 5 h.64 d.656 Table 2 Discretized parameters of the Maxwell-slip model. W W 2 W 3 W 4 W 5 W 6 W 7 W k k 2 k 3 k 4 k 5 k 6 k 7 k W 9 W W W 2 W 3 W 4 W 5 W 6 SW k 9 k k k 2 k 3 k 4 k 5 k 6 Sk Thus, the equation of motion of the corresponding system can be written as m x þf fr ðx, _xþþkx ¼ FðtÞ ð4:þ where F(t) is the input, x is the displacement output of interest, m is the equivalent mass of the mechanical system and, thus, kx term is the restoring. The second term, f fr, represents the rolling friction generated from the interaction between two contact surfaces. The reason to support the mass m with a linear spring, k, is to avoid the mechanical drift of the mechanism that might occur because of the external, F(t). Table lists a set of the system parameters corresponding to Eq. (4.). It has to be underlined here that these parameters are chosen arbitrarily but based upon realistic values. Since the values are not based on physical measurements and only qualitative results are examined, no units are assigned to the numerical values of the different variables. Subsequently, for the purpose of performing the dynamic simulation of the corresponding system using the Maxwell-slip model, the virgin curve in Eq. (2.), where the parameters h and d are listed in Table, is discretized using 6 elementary models, resulting in 32 ( ¼2 6) Maxwell-slip parameters of W and k. Effective discretization values are evaluated by means of optimization process, i.e. minimizing the difference between the deterministic virgin-curve function and the discrete Maxwell-slip model using the Genetic Algorithm and the Nelder Mead Simplex algorithm, similar to the work presented in [23]. The result of the 32 parameters is tabulated in Table 2, where parameters W i are in the unit and k i in stiffness unit, while the discrete model is illustrated in Fig. 4 together with the original virgin curve. 5. Simulation and skeleton analysis To evaluate the equivalent modal parameters of the corresponding system, some simulations are performed on the Matlab/Simulink environment. In this simulation, the Maxwell-slip friction model is implemented in an S-function block written in C language to allow fast computation time. Regarding to the study of the ForceVib method, a chirp signal is chosen as the excitation to the system, since the frequency content of the signal is relatively easy to control [6].

6 T. Tjahjowidodo / Mechanical Systems and Signal Processing 29 (22) virgin curve, y(t) continuous virgin curve discretized virgin curve.5.5 displacement (x) Fig. 4. Continuous virgin curve and its discrete elementary model. 5-5 displacement (x) Hilbert wavelet time (s) Fig. 5. Input ( excitation) signal and output (displacement) signal of the system. The initial frequency of the signal is Hz and the frequency continues to increase at a linear rate, reaching 2 Hz in 2 s 3 with constant amplitude of G: FðtÞ¼G sinð:pt 2 Þ ð5:þ as illustrated in the upper panel of Fig. 5, where in the first simulation, the amplitude, G, is set to 5 unit. Simulations are taken at a fixed frequency sampling of khz with the Runge Kutta method as the explicit differential solver approach. The effect of different rates and amplitudes of the signal will be discussed further at the end of this section. The displacement response of the system due to the excitation is depicted in the lower panel of Fig. 5 together with the envelope extraction obtained from the Hilbert and wavelet analyses. The light line represents the envelope estimation based on the Hilbert transform, while the bold one is that from the wavelet analysis with bandwidth parameter of the mother function, f b, is.5 Hz, and the center frequency, f c, is.8 Hz. It is clearly seen from the figure that the wavelet analysis offers a smoother envelope extraction compared to that of the Hilbert transform as also reported by Ruzzene et al. [2]. The same conclusion is also inferred from the instantaneous frequency extraction of the response signal as shown in Fig. 6. For the Hilbert approach, the instantaneous frequency of the signal presented in the figure is obtained by taking the angle expressed in Eq. (3.3), while that of the wavelet approach is achieved by utilizing the second part of Eq. (3.6). It is 3 The signal is designed in such the frequency content covers the estimated (nonlinear) natural frequency of the system. Rough estimation of the natural frequency is obtained from the total stiffness (¼Sk i þke) and mass (¼), which results in o n ¼O.

7 32 T. Tjahjowidodo / Mechanical Systems and Signal Processing 29 (22) Frequency Extraction frequency (Hz) 5-5 wavelet extraction Hilbert extraction time (s) Fig. 6. Instantaneous frequency extraction of the output signal. Restoring Force displacement (x) Fig. 7. Estimation of the equivalent restoring of the rolling friction in the system under study based on the wavelet analysis. clearly seen that the wavelet approach offers smoother instantaneous frequency extraction compared to that of the Hilbert transform. Due to the superiority of the wavelet analysis for the parameters extraction, all of the envelope and instantaneous frequency extraction in the rest of this paper will be performed using the corresponding analysis. Utilizing Eqs. (3.4) (3.6), the equivalent modal parameters can be derived. Figs. 7 and 8 depict the restoring and the damping, respectively. Please note that the restoring plotted in Fig. 7 has been subtracted by the restoring of the linear spring, k. Thus the plot represents the equivalent restoring of the rolling friction solely. From the figure, we can see that the restoring implies the presence of a softening spring effect. This can be easily understood, since at a very small displacement range when the hysteresis forms a relatively small loop (see Fig. ), the hysteresis predominantly behaves as a linear spring as the state is sticking. In other words, from the Maxwell-slip modeling point of view, most of the elementary models are also in the sticking state. On the other hand, at a small displacement range, assuming a moderate frequency content of the excitation signal, the system will oscillate at a relatively low velocity. Therefore, intuitively, no damping effect will also appear at the vicinity of zero velocity. However, the result of the analysis of the equivalent damping does not cover the damping value at the corresponding velocity range as seen in Fig. 8. The lack of this information is attributed to the energy content in the excitation signal. In order to refine the results, the skeleton technique is re-implemented at varying amplitude of excitation, G. Five additional skeleton analyses are performed with different amplitude values, from G¼.2 until G¼2 and the resulting equivalent modal parameters are presented in Figs. 9 and. Different amplitude of the excitation signal, or equivalently, different energy of the excitation signal will manifest itself at different region in the equivalent dynamic parameters as shown in the figures. In turn, composing all of partial equivalent parameters, complete sets of those parameters are constructed. Fig. 9 depicts the complete equivalent restoring of the system under study. At low displacement, the equivalent spring exhibits a maximum stiffness ( E5) as implied by the local gradient of the restoring. The approximated value comes close to the theoretical stiffness at low displacement (¼Sk i ). However, an accurate local stiffness at very low

8 T. Tjahjowidodo / Mechanical Systems and Signal Processing 29 (22) Damping Force velocity Fig. 8. Damping estimation of the system based on the wavelet analysis. Restoring Force G = G = G = 5 G = -.8 G = 5 G = displacement (x) Fig. 9. Construction of the equivalent restoring of the rolling friction using various amplitude of excitation signals Damping Force G = G = 2 G = 5 - G = G = 5 G = velocity Fig.. Construction of the equivalent damping using various amplitude of excitation signals. Inset panel shows the magnification of the constructed equivalent damping in the vicinity of the origin.

9 34 T. Tjahjowidodo / Mechanical Systems and Signal Processing 29 (22) Equivalent Stiffness virgin curve displacement displacement is nearly impossible to evaluate (due to a reason that will be discussed in the next paragraph). The stiffness subsequently decreases until the restoring reaches its maximum at a point, where the virgin curve about to reach its saturation/sliding ( 7.4 displacement unit). As for the damping, from Fig., we can see that the is saturating more-or-less at the sliding of the virgin curve (7.5 unit). Interesting evidence appears at the vicinity of zero velocity, as can be seen in the inset plot of the same figure, where a dead-zone region is present within 7.5 range. At low velocity, that is to say, within a small distance after reversal points, the system is primarily characterized by the equivalent stiffness of the friction. Intuitively, the damping at low velocity, indeed, will be very low. However, the reason for having the dead-band region in the equivalent damping is due to the discretization in the Maxwell-slip model. According to Table 2, the highest element stiffness is attributed to the element with the smallest maximum allowable, W, namely the last element with W 6 ¼.8. Consequently, when the displacement response is damped and the amplitude is becoming less than or equal to the corresponding limit, W 6,the system is switched to an undamped vibration system with a constant stiffness (¼Sk i ¼5.) and no equivalent damping element acting on the system. Thus, the system is freely vibrating with certain (velocity) response amplitude, which corresponds to the dead-band limit of the equivalent damping parameter as shown in Fig.. This is also the reason, why the local stiffness on the equivalent stiffness parameter at very low displacement is also difficult to estimate. In particular, Fig. shows the positive part of the constructed equivalent stiffness together with the virgin curve of the rolling friction element. The figure confirms the previous proposition that immediately after its reversal point, the system is predominated by the total stiffness of the hysteresis elements. Subsequently, the (equivalent) damping is taking its part and the (equivalent) stiffness effect is becoming less compared to that of the virgin curve. 6. Dynamic analysis Fig.. Equivalent stiffness in comparison to the virgin curve. In order to analyze the dynamic behavior of the system and its equivalent modal parameters, the fundamental frequency response function and its higher order forms will be evaluated. Prior to analyzing the system, the equivalent modal parameters are fitted into continuous functions. Considering the behavior of the equivalent stiffness, a combination of exponential function and gamma function is proposed: F k ðxþ¼:38uð e 3:2x :4 Þþ ð6:þ Gð:3x :6 Þ where G represents the gamma function. Concurrently, the equivalent damping is fitted with an exponential function. However, in order to have a direct comparison with the discrete Maxwell-slip model, the dead-band region that appears in Fig. is also considered in the fitting function represented by a Heaviside function, HðÞ: F d ðxþ¼:5ð e :3ðx :5Þ ÞHðx :5Þ ð6:2þ The resulting functions are shown in Fig. 2 together with the analyzed data. Subsequently, the frequency response functions are evaluated by prescribing sinusoidal inputs at different frequencies and amplitudes. 4 To have a direct comparison with the original system, the simulation will also be carried out 4 The analyses are taken without using the linear spring as in the previous sections. The reason is to emphasize more the frictional effect on the dynamic results.

10 T. Tjahjowidodo / Mechanical Systems and Signal Processing 29 (22) Restoring Force.5 Damping Force displacement (x) velocity Fig. 2. Fitted functions of the equivalent modal parameters. on the original system with the Maxwell-slip model. In addition, a simplified model incorporating the Coulomb friction model represented by a signum function of velocity with.5 break-away is also simulated. The simulations consider range of frequency from. to Hz, in which the equivalent natural frequency is covered. 5 The first column of Fig. 3 shows the first three orders of the absolute frequency response function (AFRF) 6 obtained from the original system. The higher order AFRFs represent the harmonic response of the system on the corresponding order. For example, the second order AFRF at frequency f a represents the output of the system at the frequency 2 f a as a response of an excitation input at frequency f a. Similarly, the second and the third column depict the AFRFs for the Coulomb model and the proposed equivalent model, respectively. The plots show good agreement between the original case and the equivalent case, while the Coulomb case fails to capture the dynamic behavior of the original system; especially for the low amplitude excitation cases (please note the arrow in the figure that indicates the increasing amplitude of the excitation signal). Interestingly, the equivalent system is able to mimic the original system even for the low amplitude excitation level. 7. Experiment and skeleton analysis This section presents the validation of the geometric nonlinear equivalence of a system with hysteresis friction on a real mechanical system, namely a DC motor that comprises frictional due to the rotating parts and bearing supports. 7.. System apparatus and identification Experimental validation of the geometric nonlinear equivalent derivation was carried out on the ABB DC-servomotor with brushes, type M2AA, with maximum rated torque of.49 N m/amp and armature inertia of. kg m 2. To allow a straightforward position and velocity data collection of the motor, an incremental angular encoder was connected to the shaft through a timing belt. The pulleys give a reduction ratio from :3 to increase the sensitivity of the encoder, which has a resolution of 5 pulses per revolution. Owing to the concept that the torque in a DC motor is proportional to the current, the working torque on the DC motor is, therefore, estimated from the current measurement. Fig. 4 shows a schematic of the experimental setup. In order to derive the geometric nonlinear equivalence of the DC motor system, two distinct approaches can be utilized, i.e. the direct method and the indirect method. The direct method is performed by perturbing the real system and utilizing the observed instantaneous properties (envelope and frequency) of the displacement output to derive the equivalent dynamic parameters. In the indirect method, the system is identified and modeled based on the apriori knowledge of the system (e.g. a system with hysteresis nonlinear friction that can be represented as a mass with the Maxwell-slip friction model) and, subsequently, the equivalent dynamic parameters are derived using simulations, similar to those presented in Section 5 in this paper. In case of a high damping factor in the system, the direct method is rather difficult to perform, since the decay rate of the response signal is high. This reason motivates the analysis of the geometric nonlinear equivalence of the DC motor with friction element using the indirect method. The identification procedure was started by applying an input signal to the DC motor. The input command, which is composed of a uniformly distributed random signal with cut-off frequency of 4 Hz, is enveloped by an 5 The excitation amplitudes for all cases in the simulation are.,.2,.5,.8,,.5, 2, 5 and unit. 6 An AFRF presents the frequency content of the respective response without dividing them by the amplitude of the excitation. The reason to use the AFRF, instead of the popular FRF, is to show direct pictures of the output level with respect to the amplitude excitation level.

11 36 T. Tjahjowidodo / Mechanical Systems and Signal Processing 29 (22) Original Coulomb Equivalent AFRF increasing G AFRF AFRF Frequency (Hz) Frequency (Hz) -8 Frequency (Hz) Fig. 3. AFRFs for three different cases. The first column is for the original system, the second column is for the one with Coulomb friction, the third column is for the system with the equivalent modal parameters. Different rows show different orders of AFRF. Fig. 4. Schematic of the experimental setup. approximately-rectangular signal. The envelope signal consists of the three first non-zero terms of the Fourier series of a rectangular signal with an amplitude of 5 rad/s, an offset of rad/s and a period of s (as a result, the command signal is capped at 5 rad/s for s and at 5 rad/s for another s). The purpose of enveloping this input signal is to

12 T. Tjahjowidodo / Mechanical Systems and Signal Processing 29 (22) emphasize the behavior of pre-sliding friction in the experiment, which corresponds to the friction at low velocity and low displacement. The system is simply modeled as a moment of inertia subjected to a rotational friction and excited by an external torque as represented in the following equation: J y þf fr ðy, _ yþ¼tðtþ ð7:þ where the equivalent moment of inertia (the armature and equivalent of the belt pulley system) is.22 kg m 2 and the torque is measured from the current in the motor. Using the Genetic Algorithm and the Nelder Mead Simplex algorithm, the parameters of the Maxwell-slip model are optimized and the identification result is listed in Table 3. The identification and optimization processes are detailed in [23] and interested readers are suggested to refer to the paper. Repeating the same procedures in the previous sections, the dynamic responses of the system is simulated by prescribing a chirp signal that reaches 5 Hz in.25 s at varying amplitude, G (¼5,, 2, 3, 4 N m), and subsequently the equivalent dynamic properties are reconstructed as presented in Fig. 5. In the final step, the dynamic analysis of the DC motor system is evaluated by means of the AFRF analysis. Fig. 6a shows the measured AFRF of the system that was excited using stepped-sine signals at different torque amplitudes. Eight varying amplitudes were considered, i.e..4,.5,.7,.9,., 2., 4. and 6. N m. All the measurements were taken in a closed-loop current form to maintain controlled torque input to the system. At low torque amplitude, the system behaves like a mass spring damping system, while at higher amplitude only the mass-line is identified in the AFRF plot. The results show clearly that the measured transfer function depends strongly on the amplitude and type of the excitation signal. The nonlinear behavior apparent in the figure is attributed to the presence of friction [6]. Fig. 6b depicts the AFRFs for the simulated system with the Coulomb model, where the static is defined to be equal to the breakaway (SW¼.7566 N m) as presented in Table 3, while Fig. 6c shows those for the simulated system with the analyzed equivalent dynamic parameters (Fig. 5). The plots clearly show good agreement between the measurement and the equivalent geometric nonlinear system; while the Coulomb model fails to capture the dynamic behavior of the real system at low amplitude excitation cases (note the arrow in the figure that indicates the increasing torque input). The shortage of the performance of the model with Coulomb friction is attributed to the absence of (asperity) stiffness element in the simulated system. Table 3 Identified friction parameters of the DC-motor setup. W W 2 W 3 W 4 W 5 W 6 W 7 W k k 2 k 3 k 4 k 5 k 6 k 7 k W 9 W W W 2 W 3 W 4 W 5 W 6 SW k 9 k k k 2 k 3 k 4 k 5 k 6 Sk Restoring Torque (Nm) 2-2 Damping Torque (Nm) Displacement (rad) x Velocity (rad/sec) Fig. 5. Equivalent dynamic parameters of friction in DC motor.

13 38 T. Tjahjowidodo / Mechanical Systems and Signal Processing 29 (22) Nm Magnitude - rad (db) Nm Frequency (Hz) Frequency (Hz) Frequency (Hz) Fig. 6. AFRF s of the DC motor: a. direct measurement from the DC motor, b. simulated AFRF using Coulomb model, c. simulated AFRF using the equivalent geometric nonlinear system. 8. Conclusion This paper has evaluated and presented the derivation of the equivalent damping and stiffness parameters of a mechanical system comprising a hysteresis (rolling) friction element with respect to its dynamic behavior. After an appropriate study of the hysteresis phenomenon and the skeleton technique, it is shown that the equivalent (geometric nonlinear) modal parameters can be evaluated and constructed. As expected, the equivalent stiffness of the rolling element at very low displacement corresponds to the total stiffness elements of the hysteresis model. Subsequently, the equivalent stiffness value is decreasing until it reaches its maximum restoring, which corresponds to the point where the virgin curve reaches its sliding state. From this point onward, the stiffness of the equivalent spring continuously decreases until it loses its stiffness at high displacements. On the other hand, the equivalent damping is increasing from a relatively low value until it reaches its maximum that corresponds to the the friction element can sustain. One interesting thing occurs at the vicinity of zero velocity of the equivalent damping, where a dead-zone region appears. The effect manifests itself because of the discretization in the Maxwell-slip model that is utilized in the skeleton analysis. In order to overcome the problem, substituting the (finite) Maxwell-slip model with a continuous friction model 7 [24] seems to be a potential solution. However, the chirp excitation signal will result in open loop hystereses that might lead to a stack overflow problem. Based on the dynamic analyses of the original system, the simplified system with the Coulomb element and the presented equivalent system, it is shown that the proposed equivalent parameters offer a good correspondence with the original one. Slight discrepancies occur when the system is excited by low excitation signals, when the non-local memory property is exposed extensively. The simulation results are then verified on a DC motor system, where the equivalent (geometric nonlinear) dynamic parameters were constructed and the dynamic behavior of the equivalent system were compared and contrasted to that of the actual system in term of the AFRFs. The results confirm the correspondence of the equivalent system with the actual system for micro-level applications. In short, model simplification by utilizing a piecewise linear function of a damping as presented by the Coulomb friction function, sometimes is not sufficient to capture the dynamic behavior of the system satisfactorily. Utilization of a (nonlinear) restoring up to certain extent, sometimes, is necessary to capture the frictional behavior effectively. References [] B. Armstrong-Hélouvry, Control of Machines with Friction, Kluwer Academic Publishers, Massachusetts, 99. [2] T. Prajogo, Experimental Study of Pre-rolling Friction for Motion-Reversal Error Compensation on Machine Tool Drive Systems, Ph.D. Thesis, Department Werktuigkunde, Katholieke Universiteit Leuven, Belgium, 999. [3] C. Canudas de Wit, H. Olsson, K.J. Åström, P. Lischinsky, A new model for fontrol of systems with friction, IEEE Trans. Autom. Control 4 (3) (995) [4] J. Swevers, F. Al-Bender, C.G. Ganseman, T. Prajogo, An integrated friction model structure with improved pre-sliding behavior for accurate friction compensation, IEEE Trans. Autom. Control 45 (4) (2) [5] F. Al-Bender, V. Lampaert, J. Swevers, Modeling of dry sliding friction dynamics: from heuristic models to physically motivated models and back, Chaos: Interdiscip. J. Nonlinear Sci. 4 (2) (24) [6] F. Al-Bender, W. Symens, J. Swevers, H. Van Brussel, Theoretical analysis of the dynamic behavior of hysteresis elements in mechanical systems, Int. J. Non-Linear Mech. 39 (24) Continuous model of friction requires a memory stack technique to memorize every reversal point of the system. The memory of the reversal point can be released when a trajectory loop is closed.

14 T. Tjahjowidodo / Mechanical Systems and Signal Processing 29 (22) [7] V. Lampaert, Modelling and Control of Dry Sliding Friction in Mechanical Systems, Ph.D. Thesis, Department Werktuigkunde, Katholieke Universiteit Leuven, Belgium, 23. [8] S. Seelecke, Modeling the dynamic behavior of shape memory alloys, Int. J. Non-Linear Mech. 37 (22) [9] D. Capecchi, F. Vestroni, Steady-state dynamic analysis of hysteretic systems, J. Eng. Mech. (2) (985) [] G.J. Taylor, R.G. Brown, Analysis and Design of Feedback Control System, second ed., McGraw-Hill, New York, 96. [] M. Feldman, S. Braun, Analysis of typical nonlinear vibration systems by using the Hilbert transform, in: Proceedings of the XI International Modal Analysis Conference, Kissimmee, Florida, 993, pp [2] M. Feldman, Non-linear system vibration analysis using Hilbert transform I. Free vibration analysis method FreeVib, Mech. Syst. Signal Process. 8 (2) (994) [3] S. Braun, M. Feldman, Time frequency characteristics of non-linear systems, Mech. Syst. Signal Process. (4) (997) [4] M. Feldman, Non-linear free vibration identification via the Hilbert transform, J. Sound Vib. 28 (3) (997) [5] M. Feldman, S. Seibold, Damage diagnosis of rotors: application of Hilbert transform and multi-hypothesis testing, J. Vib. Control 5 (999) [6] M. Feldman, Non-linear system vibration analysis using Hilbert transform II. Forced vibration analysis method ForceVib, Mech. Syst. Signal Process. 8 (3) (994) [7] W.J. Staszewski, Identification of nonlinear systems using multi-scale ridges and skeletons of the wavelet transform, J. Sound Vib. 24 (4) (998) [8] T. Tjahjowidodo, F. Al-Bender, H. Van Brussel, Experimental dynamic identification of backlash using skeleton methods, Mech. Syst. Signal Process. 2 (27) [9] W.D. Iwan, A distributed-element model for hysteresis and its steady-state dynamic response, J. Appl. Mech. 33 (4) (966) [2] F. Al-Bender, W. Symens, Characterization of frictional hysteresis in ball-bearing guideways, Wear 258 (25) [2] M. Ruzzene, L. Fasana, L. Garibaldi, B. Piombo, Natural frequencies and dampings identification using wavelet transform: application to real data, Mech. Syst. Signal Process. (2) (997) [22] M. Misiti, Y. Misiti, G. Oppenheim, J.M. Poggi, Wavelet Toolbox TM 4, User s Guide, Available online: / wavelet/wavelet_ug.pdfs (last accessed: 2). [23] T. Tjahjowidodo, F. Al-Bender, H. Van Brussel, W. Symens, Friction characterization and compensation in electro-mechanical systems, J. Sound Vib. 38 (27) [24] J. Swevers, F. Al-Bender, C. Ganseman, T. Prajogo, An integrated friction model structure with improved presliding behavior for accurate friction compensation, IEEE Trans. Autom. Control 45 (4) (2)