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Introduction to heavy flavour physics

HFAG-charm


Micky Rakotondrabe

Yassine Haddab

Philippe Lutz

High-Stroke motion modelling and voltage/frequency

proportional control of a stick-slip microsystem.

High-Stroke Motion Modelling and Voltage/Frequency

Proportional Control of a Stick-Slip Microsystem

Micky Rakotondrabe, Yassine Haddab and Philippe Lutz

Abstract-A new control type for stick-slip microsystems is proposed in this paper: the voltage/frequency (U/f ) proportional control. It gives a best resolution relatively to the classical control algorithm. It is also an englobalization of three classical controllers: the sign controller, the classical proportional controller and the frequency proportional controller. A high stroke model of a stick-slip microsystem is first given. Then, we theoretically analyze the performances of the closed loop process with the U/f controller. Finally, we give some experimental results obtained with different values of the proportional gains.

I. Introduction

Among the stepping principles for micropositioning, stick-slip [START_REF] Breguet | Actionneurs 'Stick and Slip' pour micromanipulateurs[END_REF] is the most used : examples are given in [START_REF] Matsuda | Micro-step XY-stage using piezoelectric tube actuator[END_REF][3][4] [START_REF] Fatikow | A piezoelectric mobile robot for handling of microobjects[END_REF]. Stick-slip actuators, generally based on piezoelectric materials, are characterized by their simplicity, rapidity, low cost and possibility of batch fabrication. Two modes of motion can be obtained with a stickslip micropositioner [START_REF] Bergander | Micropositioners for microscopy applications based on the stick-slip effect[END_REF] : the stepping mode and the scanning mode. The stepping mode consists in applying a sawtooth voltages to the micropositioner and let it move step by step, in high range and with a high velocity on the workspace (Fig. 1-a, b andc). The resolution in this mode is limited to one step. When the difference between the target position and the current position becomes less than the value of a step, the legs (piezoelectric actuators) are bent slowly until the final position is reached (Fig. 1d). This is the scanning mode and the obtained resolution can be very high. It is possible to have a similar resolution in the stepping mode by reducing the amplitude U of the sawtooth but the vibrations occuring in each steps [START_REF] Bergander | Performance improvements for stick-slip positioners[END_REF] may influence the performances.

Two modes of control for stick-slip micropositioners are possible: control in stepping mode for high stroke positioning and control in scanning mode for fine positioning. Each of them may use an open-loop or a closedloop structure. As the fine positioning may be controlled in closed loop with standard controllers (P ID, optimal controllers, robust controllers, etc.), this paper deals with the high stroke positioning control.

The first section is a brief presentation of existing control methods. Then, we present the high stroke characteristics and model our stick-slip microsystem for ex-Laboratoire d'Automatique de Besançon -CNRS UMR6596, ENSMM -Université de Franche-Comté, 25000 Besançon -France {mrakoton,yhaddab,plutz}@ens2m.fr periments. After that, we detail a new type of control : the voltage/frequency proportional control. It inferes good resolution of the motion relative to the existing methods. On the other hand, the vibrations possibilities are reduced in this approach. Finally, we present the experimental and simulation results.

II. Existing methods

Open-loop control based on step counting as in stepper motors are not applicable for stick-slip microsystems because:

• the steps are never identical along a displacement due to the nonlinearities phenomena coming from the friction and the piezoelectric materials, • the step is very low relative to the distance, then the number of steps of the displacement is very high. Thus, small counting error may generate high static error, • the step errors (step errors = step estimated -step real ) accumulate along the motion unlike those of in stepper motors, • the external disturbances (adhesion forces, thermal effect, vibrations of the workspace etc.) influence the accuracy.

Thus, the classical employed closed-loop controller is the following algorithm [START_REF] Breguet | Piezoactuators for motion control from centimeter to nanometer[END_REF] except for a sign:

W hile |x c -x| ≥ step Do apply one step EndW hile (1)
where x c is the target position and x is the current position. When the accuracy of the sensor is worse than the value of a step, the term step in the precedent algorithm is replaced by n × step which gives the limit of the sensor accuracy.

Another method based on the hybrid modeling of the stick-slip systems has been proposed in [START_REF] Sedghi | Control design of hybrid systems via dehybridization[END_REF]. The hybrid model is approximated into a continuous one (called dehybridization) and a continuous controller is applyed (P D). This method gives the possibility to reach the target position without using two separate control modes (stepping and scanning). However, the dehybridization requires a hybrid controller in series with the stickslip microsystem which makes the whole algorithm more complex than the precedent algorithms.

Finally, Breguet and Clavel proposed a numerical frequency proportional controller in [START_REF] Breguet | Stick and slip actuators: design, control, performances and applications[END_REF]. In this method, the frequency is proportional to the error while the amplitude U of the voltage still stays constant. Thereby, as the step is constant along the motion, the resolution is also constant.

III. Characteristics and modelling of the microsystem Fig. 2-a shows the stick-slip microsystem [START_REF] Rakotondrabe | Design, devbelopment and experiments of a high stroke-precision 2Dof (linear-angular) microsystem[END_REF] used for experiments in this paper. Fig. 2-b shows how the microactuators are spread out inside the microsystem. It has two degrees of freedom (2DoF: linear and angular) but our test will only be performed in the linear motion. The maximal step value of the microsystem is about 200nm and the speed can reach 2mm/s. These performances are obtained using a sawtooth input voltage with an amplitude of ±150V and a frequency of 10kHz.

When applying a continuous input voltage U , the microsystem works in scanning mode. Hypothesizing that the scanning displacement x sc is dynamically linear reative to the voltage, we can write the following function:

X sc (s) = G sc (s).U (s) ( 2
)
Where s is the Laplace variable and G sc is the transfer function connecting the amplitude U and the scanning position x sc . When the voltage is abruptly removed, the resulting step x step is smaller than the corresponding amplitude x U sc so that [START_REF] Driesen | Energy consumption of piezoelectric actuators for inertial drives[END_REF](Fig. 3-a): Hypothesizing that the flashback ∆ back is dynamically linear reative to the amplitude U , the step can be written as follow:

x step = x U sc -∆ back (3)
X step (s) = G step (s).U (s) (4)
When the sequence is repeated (stepping mode) with a frequency f = 1/T , the dynamic of the continous part G step is not visible and the microsystem works with a quasi-static manner. Thus, the step can be approximated by:

x step = α.U (5)
where α > 0 is the static gain of G step . From the Fig. 3-b and the (Eq. 5), we infer the bilinearity of the speed: When the amplitude U is below a value U 0 , the torque is not sufficient and the microsystem can't move. Thus, an offset is introduced to the (Eq. 5) and the final model becomes:

v = ẋ = α.U.f (6) t [s] T T 2T 3T x [µm] (a) (b) ∆back x x x sc sc U step t [s] x [mm] v [mm/s]
v = 0 if |U | ≤ U 0 v = α.f. (U -sgn(U ).U 0 ) if |U | > U 0 (7)
Fig. 4 summarize the speed performances of the microsystem. Until f = 10kHz, the speed is approximately linear versus f (Fig. 4-a). Above this frequency value, a saturation and a fluctuation appear. The identified parameters are: α = 15.652173 × 10 -7 and U 0 = 35V (Fig. 4-b).

IV. U/f proportional control A. Principle scheme

The principle scheme of U/f proportional control is shown in Fig. 5. The saturations avoid the overvoltages and limit the microsystem work inside the linear zone of the frequency. The proportional gains K U > 0 and K f > 0 are to be adjusted like in a classical proportional controller.

B. Equations and analysis

In this section, we express the speed according to the values of K U and K f . Let U s and f s indicate the saturations respectively used for the voltage and for the frequency. a) if K U . |x c -x| > U s and K f . |x c -x| > f s From the Fig. 5 and the bi-afine expression in (Eq. 7), we obtain:

ẋ = α.f s . (U s -U 0 ) .sgn (x c -x) (8)
This case is equivalent to a sign control (Fig. 6-a).

When the microsystem is at the neighborhood of the final position, there are oscillations in sign control.

The frequency and the amplitude of these oscillations depend on the response time T r of the process, on the refreshing time T s (Fig. 6-b) of the controller and on the frequency saturation f s . It is the way that realtime feedback systems give the best performances.

b) if U 0 > K U . |x c -x| ∀ f = K f . |x c -x| we have: ẋ = 0 (9)
c) if U s ≥ K U . |x c -x| ≥ U 0 and K f . |x c -x| > f s
Here, the closed loop process is equivalent to a voltage proportional control (Fig. 7), also known as proportional control in the control theory.

The equation of the closed loop is:

ẋ = α.f s . (K U . (x c -x) -sgn (x c -x) .U 0 ) (10)
For simplification but without loss of generality, let us take a positive target position x c and an initial value

x(t = 0) equal to zero, we obtain the following Laplace transformation:

X = 1 1 + 1 α.fs.K U .s .X c - 1 K U 1 + 1 α.fs.K U .s .U 0 ( 11
)
where s is the Laplace variable. The (Eq. 11) means that the closed loop process is a first order dynamic system with a static gain equal to unity and a disturbance U 0 . We can infer that, the more K U is great, the more a voltage proportional control of a stick-slip system is accurate. The voltage proportional control always gives stability because it is a first order.

d) if K U . |x c -x| > U s and f s ≥ K f . |x c -x|
This case is equivalent to a frequency proportional control (Fig. 8). The following equation is easily obtained:

ẋ = α.K f . |x c -x| . (U s -U 0 ) .sgn (x c -x) (12)
The expression (Eq. 12) is equivalent to the following transfer function:

X X c = 1 1 + 1 α.K f .(Us-U0) .s (13)
The (Eq. 13) differently means that in this case, the closed loop process is a first order system with a static gain equal to unity. The stability is always ensured.

e) if U s ≥ K U . |x c -x| ≥ U 0 and f s ≥ K f . |x c -x|
The frequency and the voltage are both proportional to the error ε. From the Fig. 5 and the formula (Eq. 7), we have the following expression:

ẋ = α.K f . |x c -x| . (K U . (x c -x) -sgn (x c -x) .U 0 ) (14)
The expression (Eq. 14) is equivalent to:

dx dt = (α.K f .U 0 -α.K f .K U . |x c -x|) .x + (-α.K f .U 0 + α.K f .K U . |x c -x|) .x c ( 15
)
which is in the form of:

dx dt = A (x c , x) .x + B (x c , x) .x c (16)
Where A and B are the state and the input coefficients.

Here, the closed loop system has a 1st order pseudo-linear behavior.

C. Stability

According to the values of K U , K f and the error (x c -x), all the above cases may appear during a displacement. Without loss of generality, let us suppose that x c = 0 and x(t = 0) > 0 for the stability analysis. Let us divide the displacement into two phases (Fig. 9):

• phase-1; the error (x c -x) is initially high so that the voltage U and the frequency f are both in their saturations (case a)). The speed is constant. • phase-2; the error becomes smaller and the speed

is not yet constant (equivalent to the other cases). As the equations are quasi-static, ie. there is no acceleration, one case does not influence the succeeding case. So, the phase-2 may be studied independently from the phase-1. In the phase-2, two sub-phases happen:

• phase-2.1: either the frequency is in saturation but not the voltage (case c)) or the voltage is in saturation but not the frequency (case d)), • phase-2.2: both the frequency and the voltage are not in saturation (case e)). Once again, the phase-2.1 does not influence the phase-2.2. As the possible two cases inside the phase-2.1 have a 1st order linear behaviors, the displacement from the initial position to current position has not yet an overshoot. On the other hand, when the voltage becomes smaller than U 0 (case b)), the stick-slip microsystem stops independently to the precendent phases. Thus, the stability analysis of the microsystem may be done only with the phase-2.2 (case e)). For that, we use the direct method of Lyapunov.

A dynamic system dx dt = f (x, e, t), where e is the input vector, is Lyapunov stable if a Lyapunov function V (x) exists such as:

V (x = 0) = 0 (17)

V (x) > 0 ∀ x = 0 (18) dV (x) dt ≤ 0 ∀ x = 0 (19)
From the (Eq. 15) and the conditions x(t = 0) > 0 and x c = 0, we obtain:

dx dt = -α.K f .x. (K U .x -U 0 ) (20)
We use the following quadratical functional V (x):

V (x) = γ.x 2 (21)
where γ is a positive constant. The conditions (Eq. 17) and (Eq. 18) are filled. From (Eq. 20) and (Eq. 21), we have:

dV (x) dt = -2.γ.α.K f .x 2 . (K U .x -U 0 ) (22)
As (K U .x -U 0 ) > 0, the condition (Eq. 19) is filled and all the trajectory is asymptotically (Lyapunov) stable. When (K U .x -U 0 ) < 0, the microsystem stops, the stability is obvious and the static error is given.

V. Experiments

The experimental setup is composed of a computer, an amplifier, the microsystem and a laser sensor (0.5nm resolution, 10m accuracy). The computer has no real-time operating system (Windows-XP) and we use LabView software for the implementation of the U/f controller.

The choice of K U is a compromise. If K U is very low, the static error is high. If K U is very high, oscillations may occur (case a) when the refreshing time T s and the response time T r are not negligible. For all the experiments, the target point x c = 10mm and the initial point x(0) = 0mm.

The first experiment are done with high values of K u and K f . They have been chosen so that the phase-2 never happen. It is equivalent to the (case a). The Fig. 10-a gives the experimental and the simulation curves. Due to the fact that the computer has a non real-time operating system, the experimental results have oscillations (Fig. 10-in solid plot).

Then, we use a low K U and a high K f . The frequency stays in saturation while the voltage becomes non saturated when x c -x is inferior to a given value x U S (Fig. 10b) such as x U S = U s /K U . In that case, the behavior is equivalent to (Eq. 11) of the case c), ie. a voltage proportional control. As shown in the figure, there is a static error. Its value is equal to ε stat = U 0 /K U . After that, we apply a high K U but a low K f (Fig. 11-a). The leaving saturation speed is obtained at (x c -x) = x f S = f s /K f . From this point, the closed loop system has the frequenty proportional controller behavior and there is no static error (case d)).

Finally, we test moderated values of K U and K f (Fig. 11-b). First, the frequency leaves the saturation at x f s = f s /K f while the voltage stays saturated. When x U S = U s /K U is reached, the voltage leaves the satuation. According to the values of K U and K f , the inverse circumstance can happen. The static error is given by ε stat = U 0 /K U .

VI. Discussion

The U/f proportional controller encompasses some classical controllers: the sign controller (see case a), the classical proportional controller (see case c) and the frequency proportional controller (see case d). On the other hand, in comparison with the classical algorithm of (Eq. 1) and with the frequantial proportional controller, the U/f proportional controller seems to have a high resolution. The resolution obtained with the two formers is one step (respectively n × step) while the one obtained with the U/f controller is better than one step, within the limits of the sensor accuracy. In fact, the diminution of the applied voltage U = K U .ε reduces the value of a step. In addition, the diminution of the frequency f = K f .ε reduces the number of the steps and then the vibrations. A test has been performed with an interferometer sensor (resolution = 1.24nm) to compare the static precision obtained with the algotrithm in (Eq. 1) and the U/f proportional controller. The setup used is composed of materials with high values of T s and T r : the interferometer datagate and a LabView software under Windows-XP. Thus, we use a low frequency (1Hz) for the classical controller in order to avoid oscillations. Fig. 12 shows the results with a setpoint (reference) of 10m. The zoom (Fig. 12-b) indicates that the static error (accuracy) is about 130nm. As we can see, the resolution before the stop is the half value of a step instead of a step because the applied voltage is a peak-to-peak signal (±150V ), then when stopping it, the half only is cut out (+150V → 0V ).

Fig. 13 gives the results obtained with the U/f controller when applying a setpoint of 1mm. The zoom (Fig. 13-b) indicates that the frequency and the step amplitude become increasingly small. The medium static error also becomes increasingly small. However, the flashback ∆back becomes increasingly important. That is due to the diminution of the amplitude U so that there will not be enough torque to move the microsystem during the slip phase. The proposed method appears to converge to the reference position much faster than the classical controller. That is due to the low frequency used for this latter. Increasing the frequency increases the speed of the classical controller but oscillations may appear.

VII. Conclusion

A new controller type for high stroke displacement of stick-slip microsystems was proposed: voltage/frequency proportional controller. Here, the frequency f and an amplitude U of a sawtooth signal are proportional to the error. To analyze the performances given by such controller, we have first modeled the high stroke displacement of the microsystem. Then, we have analysed the stability according to the value of the proportional gains K U and K f . Finally, we have presented experiments and simulations of different cases. The U/f proportional control is a globalization of existing some existing controllers, the sign controller, the classical proportional controller and the frequency proportional controller, and it gives a better resolution relative to existing controllers for stick-slip microsystems.

Fig. 1 .

 1 Fig. 1. Stick-slip principle [6]. a, b and c : stepping mode. d : scanning mode.

Fig. 2 .

 2 Fig. 2. The stick-slip microsystem.

Fig. 3 .

 3 Fig. 3. a: motion of a stick-slip system. b: speed approximation.

Fig. 4 .

 4 Fig. 4. Speed performances of the microsystem. a: speed versus the frequency f . b: speed versus the amplitude U .

Fig. 5 .

 5 Fig. 5. Principle scheme of the U/f proportional control.

Fig. 6 .

 6 Fig. 6. a: sign control. b: oscillations in sign control.

Fig. 7 .

 7 Fig. 7. Voltage proportional control.

Fig. 8 .

 8 Fig. 8. Frequency proportional control.

Fig. 9 .

 9 Fig.9. The displacement may be divided into two phases.

Fig. 10 .

 10 Fig. 10. a: high values of K U and K f : (case a). b: low K U and high K f .

Fig. 11

 11 Fig. 11. a: high K U and low K f . b: decent values of K U and K f .

Fig. 12 .

 12 Fig. 12. Classical controller using an interferometer sensor. a: high stroke motion. b: zoom of the final position.

Fig

  Fig. 13. U/f controller using an interferometer sensor. a: high stroke motion. b: zoom of the final position.

Acknowledgment

Particular thanks to Dr Jean-Marc Breguet and his team from the LSRO -EPFL (Lausanne Switzerland) for accepting us to use their interferometer sensor. This work is partially supported by the European Project EUPASS (http://www.eupass.org/)


A MIXED SHOOTING -HARMONIC BALANCE METHOD FOR UNILATERALLY CONSTRAINED MECHANICAL SYSTEMS

Keywords: Shooting method, Harmonic Balance Method, local nonlinearities, periodic solutions, nonsmooth dynamics

In this paper we present a mixed shooting -harmonic balance method for large linear mechanical systems on which local nonlinearities are imposed. The standard harmonic balance method (HBM), which approximates the periodic solution in frequency domain, is very popular as it is well suited for large systems with many degrees of freedom. However, it suffers from the fact that local nonlinearities cannot be evaluated directly in the frequency domain. The standard HBM performs an inverse Fourier transform, then calculates the nonlinear force in time domain and subsequently the Fourier coefficients of the nonlinear force. The disadvantage of the HBM is that strong nonlinearities are poorly represented by a truncated Fourier series. In contrast, the shooting method operates in time-domain and relies on numerical time-simulation. Set-valued force laws such as dry friction or other strong nonlinearities can be dealt with if an appropriate numerical integrator is available. The shooting method, however, becomes infeasible if the system has many states. The proposed mixed shooting-HBM approach combines the best of both worlds.

Introduction

Finding periodic solutions of mechanical systems is an important task in the design process of machines and mechanical devices. For instance, knowledge of the response of the system to harmonic excitation is essential to obtain information about high cycle fatigue behaviour. In engineering systems local nonlinearities are present due to contact or coupling elements. These local nonlinearities can have a strong impact on the global system behaviour. Therefore, the nonlinearities have to be considered in the design process and must be modelled accurately as well as in a computationally efficient way.

The most popular methods to find periodic steady-state responses of nonlinear differential equations are the Harmonic Balance Method (HBM) [START_REF] Magnusand | Schwingungen (Vibrations)[END_REF][START_REF] Nayfehand | Nonlinear Oscillations[END_REF][START_REF] Von Grolland | The harmonic balance method with arc-length continuation in rotor/stator contact problems[END_REF][START_REF] Cardona | A multiharmonic method for non-linear vibration analysis[END_REF] and the shooting method [START_REF] Ascher | Numerical solution of boundary value problems for ordinary differential equations[END_REF]. The standard HBM approximates the periodic solution in frequency domain and is very popular as it is well suited for large systems with many states. Local nonlinearities cannot be evaluated directly in the frequency domain. The standard HBM performs an inverse Fourier transformation, and then calculates the nonlinear force in time domain and subsequently the Fourier coefficients of the nonlinear force. This procedure is often denoted as the Alternating Frequency Time Method (AFT) [START_REF] Cameron | An alternating frequency/time domain method for cal-culating the steady-state response of nonlinear dynamic systems[END_REF]. The disadvantage of the HBM is that strong nonlinearities are poorly represented by a truncated Fourier series. Additionally, the restriction of the AFT method to single-valued functions only allows the use of smooth contact laws. In contrast, the shooting method operates in time-domain and relies on numerical time-simulation. Set-valued force laws such as dry friction or other strong nonlinearities can be dealt with if an appropriate numerical integrator is available. The shooting method, however, becomes infeasible if the system has many states. The proposed mixed shooting-HBM approach combines the efficiency of HBM and the accuracy of the shooting method and has therefore many advantages.

In this paper the mixed shooting-HBM (MS-HBM) approach is introduced as a novel method to calculate periodic solutions of forced mechanical systems. Two different variants of the mixed shooting-HBM approach are presented in Section 3.2 and 3.3 respectively. Depending on the position of the local nonlinearities within the mechanical system, the one or the other is better suitable. The more general method of Section 3.2 is tested on a multi-mass oscillator at the end of the paper and is compared to the full HBM and full shooting method. As local nonlinearities, dry friction as well as a hard unilateral constraint are investigated.

Fourier Transformation

In this paper the Fourier transformation is used and therefore its nomenclature is introduced in this section for an arbitrary time signal x(t) ∈ R n . The Fourier coefficients

x = FFT(x(t)) = 2 T T 0 JV(t)x(t)dt (1)
of the time signal are obtained using the Fast Fourier Transformation (FFT). Hereto,

V(t) = ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ I cos(ωt)I sin(ωt)I . . . cos(n H ωt)I sin(n H ωt)I ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ , J = ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ 1 2 I 0 0 0 I 0 . . . 0 0 I . . . . . . ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ (2)
are defined where ω = 2π T and n H denote the number of considered harmonics and I is the n-dimensional identity matrix. The harmonic oscillation which is described by the Fourier coefficients can be calculated using the matrix multiplications

x h (t) = IFFT( x) = V(t) T x (3)
and its time-derivative by

ẋh (t) = V(t) T x = V(t) T Ω x (4)
where

Ω = diag(Ω 1 , Ω 2 , . . . , Ω n H ), Ω k = ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ 0 0 0 0 0 kωI 0 -kωI 0 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ . (5)
Equation ( 3) is often referred to as the inverse Fourier transformation. For the understanding of the following sections note that IFFT(FFT(x(t))) = x(t) only holds for a harmonic signal of which all contained harmonics are considered in the ansatz function V(t).

Mixed shooting-HBM approach

The mixed shooting-HBM approach exploits the local character of the nonlinearities to find periodic solutions of mechanical systems efficiently. The system must therefore be divided into linear and nonlinear subsystems. This can be done with or without an intermediate subsystem leading to the methods of Section 3.2 and 3.3. First the system description is given and subsequently both methods are discussed.

System description

We consider a Lagrangian system of the form M q(t) + C q(t) + K q(t) = f ex (t) + f nl (q(t), q(t)), [START_REF] Cameron | An alternating frequency/time domain method for cal-culating the steady-state response of nonlinear dynamic systems[END_REF] where f nl contains the nonlinear forces and f ex (t) = f ex (t + T ) is the periodic forcing. We assume that the system consists of a linear and a nonlinear subsystem with the generalized coordinates

q = ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ q N q L ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ , (7)
that the nonlinear forces only act on the nonlinear subsystem, and that the system matrices M, C and K have the following structure

M = ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ M NN M NL M LN M NN ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ , f nl (q, q) = ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ f nl,N (q N , qN ) 0 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ . ( 8
)
The nonlinear subsystem is subjected to nonlinear forces, which only depend on its own positions and velocities e.g. the three DOF oscillator shown in Fig. 1.

MS-HBM with a linear and a nonlinear subsystem

The reduction into two subsystems can reduce the computational effort for systems for which the relationship dim(q L ) dim(q N ) between the dimensions of the subsystems holds. The linear subsystem which is described by

M LL qL + C LL qL + K LL q L + f c = f ex,L (9)
with the dynamic coupling force between both subsystem

f c = M LN qN + C LN qN + K LN q N , (10)
is approximated using a Ritz-Galerkin Method. The ansatz for the displacement field is chosen as

U n H = {q L (t) ∈ U | q L (t) = V L (t) T qL }, ( 11
)
where U is the space of absolutely continuous functions. In other words, due to the Galerkin approach, the motion of the linear subsystem is approximated by a truncated Fourier series

q L (t) = q0 L + n H k=1 qc,k L cos(kωt) + qs,k L sin(kωt) = V L (t) T qL . ( 12
)
The equality (12) can be envisaged as a constraint on the system and will be referred to as the harmonic constraint on the linear subsystem.

According to the Galerkin method the corresponding virtual displacement field

V n H = {δq L (t)| δq L (t) = V L (t) T δ qL } (13)
is described using the same ansatz functions. To obtain the harmonically constrained subsystem one requires that the virtual work of the linear subsystem

δW = T 0 δq T L M LL qL + C LL qL + K LL q L + f c -f ex,L dt = 0, (14)
vanishes for all virtual displacements δq L ∈ V n H and q L ∈ U n H . From ( 14) with ( 11) and (13) the virtual work can be described as

δW = δ qT L T 0 V L M LL Ω 2 V T L + V L C LL ΩV T L + V L K LL V T L dt qL + + δ qT L T 0 (V L f c )dt -δ qT L T 0 V L f ex,L dt = 0 ∀δ qL . (15)
Using ( 1) and by pre-multiplying (15) with 2 T J, the last two force terms in (15) can be interpreted as the Fourier coefficients fc and fex,L of their respective time signals. The dynamics of the linear subsystem can therefore be described in frequency domain as

H LL qL + fc -fex,L = 0, (16)
where qL represent the Fourier coefficients of the linear subsystem and

H LL = 2 T T 0 JV L M LL Ω 2 V T L + JV L C LL ΩV T L + JV L K LL V T L dt (17)
is the dynamic stiffness matrix. The Fourier coefficients of the coupling force resulting from the nonlinear system are given by fc

= 2 T T 0 JV L f c (t)dt = = 2 T T 0 JV L f c,h (t)dt (18)
where f c,h (t) is the harmonic part of the coupling force f c (t). The harmonic part f c,h (t) only depends on the harmonic part of the motion of the nonlinear subsystem which can be expressed in the Fourier coefficients qN , i.e.

f c,h (t) = M LN qN,h + C LN qN,h + K LN q N,h = = M LN Ω 2 V T N qN + C LN ΩV T N qN + K LN V T N qN . (19)
Hence, the Fourier coefficients of the coupling force can be expressed as

fc = 2 T T 0 JV L M LN Ω 2 V T N + JV L C LN ΩV T N + JV L K LN V T N dt qN = = H LN qN (20)
The harmonic constraint can be viewed as a Galerkin projection of the dynamics of the linear subsystem on the chosen harmonic basis (which is of course an approximation of the dynamics). The nonlinear subsystem, which is not constrained to oscillate harmonically, acts as external forcing f c (t) on the linear subsystem. The inharmonic part of this forcing, e.g. f c (t)-f c,h (t), is counterbalanced by the perfect constraint force which enforces the harmonic constraint (12). Similarly, the inharmonic part of the excitation force f ex,L (t) is absorbed by the harmonic constraint and its contribution to (16) is given by

fex,L = 2 T T 0 JV L f ex,L dt. ( 21
)
Using ( 16), we obtain the linear relation

qL = H -1 LL ( fex,L -H LN qN ) (22)
between the Fourier coefficients of the linear subsystem qL and the Fourier coefficients of the nonlinear subsystem qN . The dynamic stiffness matrices H i j result from Equation ( 17) and ( 20)

H i j = diag(H i j,0 , H i j,1 , . . . H i j,n H ) ( 23
)
where the diagonal entries

H i j,k = ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ -M i j (kω) 2 + K i j C i j kω -C i j kω -M i j (kω) 2 + K i j ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ (24)
are given for each considered harmonic oscillation. For given qN the equation of motion of the linear subsystem is therefore completely described by ( 22). The equation of motion of the nonlinear subsystem

M NN qN + C NN qN + K NN q N = = -M NL qL -C NL qL -K NL q L -f ex,N + f nl,N (25)
is described in time domain. If qN is known, then the time-evolution q L (t) and its derivatives are given using (22) together with (12). The influence of the linear subsystem can therefore be considered as external forcing. To ensure that both subsystems oscillate correspondingly a connectivity condition must be considered in the residuum function. Hence, only the nonlinear subsystem must be described as differential equation, which has to be solved for q N (t) using numerical time integration. In particular, if the nonlinear force f nl,N is a dry friction force or, more generally, described by a set-valued force law, then dedicated time-integration schemes such as timestepping methods [START_REF] Acary | Numerical Methods for Nonsmooth Dynamical Systems: Applications in Mechanics and Electronics[END_REF][START_REF] Leine | Dynamics and Bifurcations of Nonsmooth Mechanical Systems[END_REF] have to be used. Here it should be noted, that the system (6) and consequently (25) turns into a differential inclusion if a set-valued force law is considered. A periodic solution of the complete system can be represented by the trajectory q N (t) on the interval 0 ≤ t ≤ T and by the Fourier coefficients qN , as qL is expressed by (22). The initial condition q N (0) and qN (0) together with q L (t) = V L (t) T qL allow to construct q N (t) over one period. The vector of unknowns

x = ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ qN q N (0) qN (0) ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ (26)
therefore fully represents a periodic solution of the system. Similar to a shooting method, we require for the nonlinear subsystem the periodicity conditions q N (T )-q N (0) = 0 and qN (T )-qN (0) = 0, where the state at t = T is obtained through numerical time-integration of (25). The periodicity condition of the linear subsystem is given in frequency domain by (22). Hence, we seek a periodic solution by finding a zero of the nonlinear function

f R (x) = ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ qN -FFT(q N (t)) q N (T ) -q N (0) qN (T ) -qN (0) ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ . ( 27
)
Note that FFT(q N (t)) is the Fourier transformation

FFT(q N (t)) = 2 T T 0 JV N q N (t)dt. ( 28
)
of the solution of the differential equation ( 25) and qN are the Fourier coefficients which represent the dynamical behaviour of the linear subsystem in the vector of unknowns x through (22). If the connectivity condition qN -FFT(q N (t)) = 0 holds, then the linear subsystem is oscillating in correspondence to the movement of the nonlinear subsystem. Obviously, due to the harmonic approximation, the connectivity condition between both subsystems leads to an approximated solution of the linear and the nonlinear subsystem. The periodic solution converges to the exact solution for an increasing number of considered harmonic oscillations. The zeros of f R (x) can be solved with a Newton-type method by iterating

x i+1 = x i - ∂ f R ∂x -1 f R (x i ), ( 29
)
where the Jacobian is obtained through finite differences. The calculation procedure of this method is depicted in Fig. 2.

MS-HBM using an intermediate subsystem

For the second approach System (6) must be divided into three subsystems. The first subsystem is subjected to the nonlinear force and is coupled through the linear Subsystem 2 to the third subsystem which has to be linear as well. For the second approach the system matrices must therefore exist in the form

M = ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ M 11 M 12 0 M 21 M 22 M 23 0 M 32 M 33 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ , f nl (q, q) = ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ f nl,N (q 1 , q1 ) 0 0 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ , q = ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ q 1 q 2 q 3 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ . ( 30
)
This method is suitable if the relation between the dimensions of the subsystems dim(q 3 ) dim(q 1 ) > dim(q 2 ) (31) holds. Using the first approach Subsystem 1 and 3 do not have to be uncoupled since the system is not restricted to condition (31). The first approach is therefore more general than the method described in this section. If the relation (31) between the dimensions is fulfilled, then the approach using an intermediate subsystem can be beneficial. This may for instance be the case for an extension of the system depicted in Fig. 1 to a chain of n masses, where the first n 1 masses are subjected to friction and the mass n 1 + 1 acts as intermediate subsystem.

For Subsystem 2 and 3 we use the Galerkin approach, as before for the linear subsystem, and impose perfect constraints on the system which force the response to be harmonic of the form

q 2 (t) = q0 2 + n H k=1 qc,k 2 cos kωt + qs,k 2 sin kωt = V 2 (t) T q2 , q 3 (t) = V 3 (t) T q3 . ( 32
)
The motion q 1 (t) of Subsystem 1 is described in time domain and is not constrained to be harmonic. The Fourier coefficients of the generalized coordinates q 1 (t) are obtained from

q1 = 2 T T 0 JV 1 (t)q 1 (t)dt. ( 33
)
The equations of motion of Subsystem 2 and 3 can therefore be expressed in frequency domain as

H 21 q1 + H 22 q2 + H 23 q3 = fex2 , H 32 q2 + H 33 q3 = fex3 , (34)
where H i j are the dynamic stiffness matrices. Using (34) the Fourier coefficients q3 can be expressed in q2 as

q3 = H -1 33 ( fex3 -H 32 q2 ) (35)
and can therefore be eliminated from the equations of motion in frequency domain, i.e.

H 21 q1 + (H 22 -H 23 H -1 33 H 32 ) q2 = fex2 -H 23 H -1 33 fex3 . ( 36
)
The equations of motion of Subsystem 1 are nonlinear and are simulated in time-domain. For known q2 one can calculate its time-domain representation q 2 (t) and its derivatives and solve the differential equation for q 1 (t)

M 11 q1 (t) + C 11 q1 (t) + K 11 q 1 (t) = -(M 12 q2 (t) + + C 12 q2 (t) + K 12 q 2 (t)) + f ex1 (t) + f nl1 (q 1 (t), q1 (t)) (37)
using numerical integration techniques. A periodic solution is completely described by the vector of unknowns

x = ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ q2 q 1 (0) q1 (0) ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ . ( 38
)
We require for Subsystem 1 the periodicity conditions q 1 (T ) -q 1 (0) = 0 and q1 (T ) -q1 (0) = 0, where the state at t = T is obtained through numerical time-integration of (37). The periodicity conditions of Subsystems 2 and 3 are given in frequency domain by ( 36) and ( 35). The residuum function to find a periodic solution

f R (x) = ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ H 21 q1 +(H 22 -H 23 H -1 33 H 32 ) q2 -fex2 +H 23 H -1 33 fex3 q 1 (T ) -q 1 (0) q1 (T ) -q1 (0) ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ (39)
is defined by (36) and the periodicity condition of Subsystem 1. The zeros of f R (x) can be solved with a Newton-type method by iterating

x i+1 = x i - ∂ f R ∂x -1 f R (x i ). ( 40
)

Numerical examples

The three DOF-oscillator (Fig. 1) is used as a numerical benchmark to compare the MS-HBM approach (method of Section 3.2) with the full shooting method and the full HBM, in both computation effort as well as accuracy. The system is inertially decoupled due to its diagonal mass matrix. It is important to note, that the MS-HBM is not restricted to inertially decoupled systems. However, the presence of inertial coupling may necessitate to use a large number n H of harmonics to arrive at a sufficiently accurate approximation of the periodic solution. Another way to improve the accuracy could be to expand the nonlinear subsystem in order to describe a larger part of the system in time domain. Since the full shooting method and the MS-HBM approach solve the nonlinear subsystem as a nonlinear differential inclusion, modern time-stepping methods with a set-valued force law are used for both methods. In contrast to the full shooting method and MS-HBM, the standard HBM with alternating frequency time approach only calculates the nonlinear force in time domain which makes it impossible to use the same contact model. Two types of contacts are considered separately in this work to compare the different methods for a system which is subjected to friction or to a completely elastic unilateral constraint.

System with friction

First, the different methods are investigated for a system under influence of dry friction. Using the MS-HBM or the shooting approach a set-valued force law can be used within the concept of (measure) differential inclusions. The friction force is expressed by the set-valued relationship

-λ T ∈ μF N Sign(γ T ) = ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ μF N , γ T > 0, -1, 1 μF N , γ T = 0, -μF N , γ T < 0. (41)
The parameters μ and F N are the friction coefficient and normal load, respectively. This friction model cannot be used for the HBM. To compare the methods in a most suitable way, the friction force for the HBM is approximated using an arctangent function

-λ smooth T = μF N 2 π arctan(κγ T ), (42)
being a smoothed approximation of (41). The approximation (42) tends to the set-valued force law (41) for large values of the smoothing parameter κ.

In Fig. 3 the displacements of the system calculated with all three methods for the period T = 10 s are shown. During this period the first mass shows a pronounced stick-slip behaviour. Though for the HBM 20 harmonics and for the MS-HBM only 3 harmonics are considered, the mixed method approximates much better the results of the full shooting method. The smoothing parameter is chosen preferably high (κ = 800). The mixed and full shooting method employ the set-valued description (41) of the friction law and can therefore describe stiction precisely. The HBM, however, not only uses the smoothed friction law (42) but also uses harmonic shape functions to approximate the friction force which leads to a poor description of this force.

In contrast, the MS-HBM describes the nonlinear subsystem in time domain and approximates only the coupling between both subsystems with harmonic shape functions. The MS-HBM approach becomes more advantageous than the full shooting method if the dimension of the linear subsystem is much larger than that of the nonlinear subsystem. To demonstrate this, the linear subsystem is extended with additional masses. This expanded model is used to compare the full HBM, the full shooting and the mixed approach. The excitation force is chosen as f exi = 0 for i = 1 . . . n -1 and f exn = 5 cos(ωt). The methods are compared for one excitation frequency in computation effort and accuracy. To start the calculation for a specific excitation frequency, a starting guess for the first iteration is needed. However, the methods iterate in different unknowns and the same starting guess can therefore not be given. The HBM needs an initial guess for the Fourier coefficients of the nonlinear subsystem, whereas the starting guess for the MS-HBM consists of the state of the nonlinear subsystem and Fourier coefficients which describe the motion of the linear subsystem through (22). However, to provide comparable starting guesses, solutions for an excitation frequency close to the actual frequency are used as starting vectors for the iterative loops of the respective approximation methods. This choice seems reasonable, since this is done as well if a frequency response diagram is calculated.

In Fig. 4 the relative error of the amplitude of the first and nth mass and the calculation effort is shown for different numbers of considered harmonics n H . Both ratios are with respect to the full shooting method, which is chosen as reference as it is almost exact.

The results show that the computation effort for a moderate accuracy can be reduced drastically by using the MS-HBM approach. Compared to the HBM, the mixed approach shows for all values of n H more accurate results. The horizontal plateau of the relative error of the mixed method can be explained by the limited resolution of the used Fourier transformation and the integration schemes. Therefore, an increase in the number of harmonics can only decrease the error down to a problem dependent minimum value. The used parameters for the calculations of the system with friction are summarized in Table 1.

System with unilateral constraint

In the second example, the friction force on the first mass is replaced by a unilateral constraint. The unilateral constraint is modelled within the concept of measure differential inclusions using the hard contact law

0 ≤ g N ⊥ λ N ≥ 0, (43)
where g N is the contact distance (g N = g N,0q 1 ) and λ N represents the contact force. The Newtonian impact law is expressed through the inequality complementarity

0 ≤ γ + N + e N γ - N ⊥ Λ N ≥ 0 with 0 ≤ e N ≤ 1, (44)
with the post-and pre-impact relative velocities γ + N and γ - N , the contact impulse Λ N and the restitution coefficient e N . For a more detailed description of the contact law and impact law see e.g. [START_REF] Leine | Dynamics and Bifurcations of Nonsmooth Mechanical Systems[END_REF]. The concept of measure differential inclusions with set-valued contact and impact law can only be used for the MS-HBM and full shooting method. As discussed at the beginning of Section 4, the HBM only allows a single-valued contact law. Therefore, the contact for the HBM is modelled using a one-sided springdamper element

-λ penalty N = ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ k c g N + d c γ N g N ≤ 0 0 g N > 0, (45)
which is in fact a nonsmooth contact law for d c > 0 as the contact force jumps at collision time instants. The equivalent restitution coefficient e N for a specific one-sided spring-damper element can be calculated following Brogliato [START_REF] Brogliato | Nonsmooth Mechanics: Models, Dynamics and Control[END_REF]. Since only a non-dissipative elastic contact (e N = 1) is used in this work, d c is taken as zero and the model tends to the hard contact if k c is tending to infinity. In Fig. 5, the displacements of a five DOF oscillator with a contact distance (g N,0 = 0.1) at the first mass for the HBM, MS-HBM and the full shooting method are depicted. The used parameters are summarized in Table 2. Figure 6 shows the velocity of the first mass. Like before for the system with dry friction, the HBM has difficulties to approximate the jump in the velocity of the first mass at the collision time-instant (t = 0.55), although the contact stiffness k c is chosen relatively high. The mixed and full shooting method show a true velocity jump whereas the HBM only gives a rough approximation of this phenomenon.

To demonstrate the efficiency of the MS-HBM for the system with unilateral constraint, it is extended to the size of 50 masses. The investigated

Concluding Remarks

The presented mixed shooting-HBM approach shows good characteristics in accuracy as well as in calculation effort, at least for the investigated benchmark system with dry friction or unilateral constraint. Depending on the system size and the nonlinear characteristics the method can be a good alternative to the commonly used methods like HBM and shooting. It should be noted, that the numerical efficiency of the methods are hard to compare and that there exist alternative Harmonic Balance Methods to compute periodic solutions of systems with dry friction and impact. Further research will focus on providing a better comparison of the mixed shooting-HBM method with existing methods [START_REF] Nacivet | A dynamic lagrangian frequency-time method for the vibration of dry-friction-damped systems[END_REF].

Fig. 1 .

 1 Fig. 1. Three DOF oscillator with dry friction and unilateral constraint

Fig. 2 .

 2 Fig. 2. Calculation scheme of MS-HBM with linear and nonlinear subsystem

Fig. 3 .

 3 Fig. 3. Displacement and friction force for a periodic solution with period time T = 10 s of the three DOF oscillator with dry friction

Fig. 4 .

 4 Fig. 4. Work-precision-diagram of the HBM and the mixed shooting-HBM approach in relation to full shooting for a system of n = 30 masses with friction and different numbers of considered harmonics (n H = 1, 3, . . . , 25)

Fig. 5 .Fig. 6 .

 56 Fig. 5. Displacements of a five DOF oscillator with impact calculated with the different methods

Fig. 7 .

 7 Fig. 7. Phase diagram of the nonlinear subsystem of the 50-mass oscillator calculated with 3 and 9 considered harmonics using the HBM (left), MS-HBM (right), as reference the full shooting approach is depicted in both diagrams as black curve

Table 1 .

 1 Selected parameters for the system with friction

	parameter m i k i c i	μ	ω	f ex,30
	value	1	1	0 0.8	1 5	π 5 cos(ωt)


Wilhelm Klat
email: wilhelm.klat@uni-bielefeld.de

Christian Stummer
email: christian.stummer@uni-bielefeld.de

Reinhold Decker
email: rdecker@uni-bielefeld.de

From Web Analytics to Product Analytics: The Internet of Things as a New Data Source for Enterprise Information Systems

Keywords: Internet of things, Product analytics, Enterprise information systems

The internet of things (IoT) paves the way for a new generation of consumer products that collect and exchange data, constituting a new data source for enterprise information systems (EIS). These IoTready products use built-in sensors and wireless communication technologies to capture and share data about product usage and the environment in which the products are used. The dissemination of the internet into the physical world of everyday products thus establishes new opportunities to apply methods well-established in web analytics to IoT-products, allowing enterprises to tap into a new and rich source of consumer data. In this paper we examine technical challenges of enabling everyday products to generate consumer data for EIS and discuss the application of web analytics methods to IoT-ready consumer products.

Introduction

A growing number of recently introduced new consumer products are able to sense their environment and share data with users, other products, and companies via the internet of things (IoT). Examples are LG's refrigerator "Smart ThingQ", VW 's smart minivan "BUDD-e", Sleep Number 's mattress "It Bed", or Verbund 's energy monitoring and controlling system "Eco-Home". These products have in common that they extend the basic functionalities of regular products with the ability to collect and share data [START_REF] Bauer | After you open the box: making smart products more usable, useful, and desirable through interactive technology[END_REF][START_REF] Körling | Smart products: why adding a digital side to a toothbrush could make a lot of sense[END_REF][START_REF] Mayer | Economic aspects of smartproducts[END_REF][START_REF] Meyer | Intelligent products for enhancing the utilization of tracking technology in transportation[END_REF]. We call this new category of data-collecting and -sharing products IoT-ready products (or for short, IoT-products). The IoT was not established at a specific point in time; rather, it has emerged in a continuous, and still ongoing, process. It can be interpreted as "a global network infrastructure, linking physical and virtual objects through the exploitation of data capture and communication capabilities" [START_REF]RFID and the inclusive model for the internet of things[END_REF]. Accordingly, for the purpose of this paper we define an IoT-product as a consumer product that autonomously collects and exchanges consumer data.

The IoT is a potential game changer for nearly every business and it has received attention by both academia and practitioners around the world (for a recent overview of research trends and challenges see [START_REF] Whitmore | The internet of things: a survey of topics and trends[END_REF][START_REF] Xu | Internet of things in industries: a survey[END_REF]). From the perspective of enterprise information systems (EIS), the IoT will be the key to unlocking the full potential of EIS (see, for example, [START_REF] Romero | Enterprise information systems state of the art: past, present and future trends[END_REF]). Conceptual designs for corresponding EIS have already been outlined in prior work (e.g., [START_REF] Ma | Enterprise information management system integration based on internet of things technology[END_REF][START_REF] Wei | Framework model on enterprise information system based on internet of things[END_REF]). Our approach differs from these approaches by focusing on consumer data that is collected by the physical IoT-products themselves in the consumer's environment. While websites, social networks, search engines, and other traditional sources for data generation in sales market research are built on the traditional internet and the World Wide Web, IoT-products provide enterprises with access to a new kind of data. With traditional products, enterprises lose direct access to their products at the point of sale [START_REF] Mayer | Economic aspects of smartproducts[END_REF]. With IoT-products, however, enterprises have the opportunity to gain insights into the actual interaction of customers with their products as well as collecting additional data from the customer's environment, which provides these enterprises with access to large amounts of longitudinal data in a way that has not been previously available [START_REF] Porter | How smart, connected products are transforming competition[END_REF]. The data collected by IoT-products primarily supports decision making in marketing and sales, but other enterprise divisions and EIS-subsystems may also benefit from this data (see Fig. 1). The car manufacturer Tesla, for instance, has already captured data about 780 million miles of driving from its IoT-ready cars, using the data to improve the technical functionality of the autopilot and to optimize the user interface [START_REF] Coren | Tesla has 780 million miles of driving data, and adds another million every 10 hours[END_REF]. The new opportunities for data collection and analysis from connected everyday products emerge from a concept we refer to as "product analytics" (see also [START_REF] Klat | Marketing management in the era of the internet of things[END_REF]). Product analytics is the IoT-equivalent of web analytics and aims at the autonomous collection and analysis of usage data from the customers' environment. IoT-products thus constitute a promising platform for the application of analytics technologies that have already proven their effectiveness on the web and on mobile phones. The research contribution of this paper therefore lies in the exploration and discussion of the new data collection opportunities embodied by IoT-products for enterprise information systems.

Accordingly, the remainder of the paper is organized as follows: Section 2 deals with technical aspects of IoT-products and outlines major challenges of transforming traditional products into IoT-ready ones with respect to technologies for data collection, data exchange, and energy supply. Section 3 then focuses on the vital aspect of data collection and discusses the application of established methods from web analytics to IoT-products. The paper concludes with a summary and suggestions for promising research directions in Section 4.

Technical Foundation of IoT-Products

The application of product analytics to physical consumer products requires them to be ready for the IoT. The core components for this purpose are (i) sensors and processors for data collection and processing, (ii) transceivers for wireless data exchange, and (iii) energy supply (see Fig. 2).

Fig. 2. Components of IoT-products necessary for product analytics

Integrating these IoT-components seems rather trivial for large-sized products that provide sufficient space, can draw on a genuine energy supply, and already have built-in sensors or communication technologies (e.g., in-home health care devices, as described in [START_REF] Pang | Design of a terminal solution for integration of in-home health care devices and services towards the internet-of-things[END_REF]). Readying everyday products such as shoes, watches, or flower pots for the IoT, though, still constitutes a technical challenge. However, such small everyday IoT-products in particular exhibit a huge potential for generating rich consumer data [START_REF] Akyildiz | Wireless sensor networks: a survey[END_REF]. Tracking a consumer's number of daily footsteps may serve as an illustrative example. A smartphone infers this data by tracking movement with GPS, accelerometers, and other built-in sensors. IoT-ready shoes, in contrast, are much closer to the event of interest and therefore can capture footsteps directly and more precisely including the actual number of steps, the pressure applied, the weight distribution between the feet, and much more. The rich data from IoT-ready shoes therefore offer enterprises from industries such as footwear manufacturing to healthcare a highly valuable opportunity to better understand individual usage patterns and to offer databased services. The IoT-pioneer Orpyx, which offers IoT-ready shoe inserts that transform traditional shoes into IoT-shoes, for instance, already uses the data generated by these inserts to remotely diagnose certain diseases at an early stage.

The continuous progress in computer miniaturization, processing speed, and storage will pave the way to integrating IoT-components into small products even if they so far lack any electronic support [START_REF] Weiser | The computer for the 21st century[END_REF]. In principle, IoT-products can autonomously acquire data from three sources, namely, internally from builtin sensors, externally from other IoT-products in close proximity, and from the EIS itself. Moreover, enterprises may also indirectly access data generated by third-party IoT-products that has been mutually exchanged with their own IoTproduct.

Small, everyday products can typically host only one, if any, wireless communication technology. Each technology has its advantages and drawbacks and is suitable for specific fields of application (for an overview, see Fig. 3). In the recent past the development of wireless communication technologies has focused on personal and body area networks as well as on wireless sensor networks. Bluetooth Low Energy, Wi-Fi Direct, and Near Field Communication are particularly promising technologies for small and mobile IoT-products. In general, the assessment of communication technologies for a specific IoT-product should take into account communication distance, required data throughput, latency, reliability, practicability for customers, and technology dissemination [START_REF] Chong | Sensor networks: evolution, opportunities, and challenges[END_REF].

Especially for miniaturized and mobile IoT-products, energy consumption and supply constitutes a significant technical challenge. Although batteries may be an inexpensive solution, the increasing number of IoT-products will render periodic recharging and replacement of numerous batteries difficult [START_REF] Shebli | Minimizing energy consumption within wireless sensor networks using optimal transmission range between nodes[END_REF][START_REF] Sudevalayam | Energy harvesting sensor nodes: survey and implications[END_REF]. A promising approach to address this issue is energy harvesting. Energyharvesting-enabled IoT-products are capable of autonomously harnessing energy from the environment and converting it to electrical energy. Most energyharvesting methods for (miniaturized) IoT-products rely on solar [START_REF] Chirap | A study on light energy harvesting from indoor environment: the autonomous sensor nodes[END_REF][START_REF] Raghunathan | Design considerations for solar energy harvesting wireless embedded systems[END_REF], wind [START_REF] Hsiao | Study on pyroelectric harvesters integrating solar radiation with wind power[END_REF][START_REF] Kamalinejad | Wireless energy harvesting for the internet of things[END_REF], differences in temperatures (e.g., body heat) [START_REF] Hoang | Thermal energy harvesting from human warmth for wireless body area network in medical healthcare system[END_REF][START_REF] Lu | Thermal energy harvesting for WSNs[END_REF], movement (e.g., body movement) [START_REF] Gorlatova | Movers and shakers: kinetic energy harvesting for the internet of things[END_REF][START_REF] Khaligh | Kinetic energy harvesting using piezoelectric and electromagnetic technologies: state of the art[END_REF][START_REF] Mitcheson | Energy harvesting from human and machine motion for wireless electronic devices[END_REF], vibrations [START_REF] Beeby | Energy harvesting vibration sources for microsystems applications[END_REF], radio frequency [START_REF] Lu | Wireless networks with RF energy harvesting: a contemporary survey[END_REF][START_REF] Mishra | Smart RF energy harvesting communications: challenges and opportunities[END_REF][START_REF] Piñuela | Ambient RF energy harvesting in urban and semi-urban environments[END_REF], or microbial activity [START_REF] Yang | Energy-neutral communication protocol for very low power microbial fuel cell based wireless sensor network[END_REF]. Piezo-electric materials embedded in IoT-ready shoes or light switches, for instance, generate sufficient electrical energy from controllable mechanical deformation to send radio signals over short distances to smartphones or IoT-ready light bulbs [START_REF] Paradiso | A compact, wireless, self-powered pushbutton controller[END_REF][START_REF] Shenck | Energy scavenging with shoe-mounted piezoelectrics[END_REF]. For a more in-depth overview of energy-harvesting techniques in wireless sensor networks, we refer to [START_REF] Dewan | Alternative power sources for remote sensors: a review[END_REF][START_REF] Sudevalayam | Energy harvesting sensor nodes: survey and implications[END_REF][START_REF] Vullers | Energy harvesting for autonomous wireless sensor networks[END_REF]. Until now, enterprises have generally lost direct access to their products at the point of sale [START_REF] Mayer | Economic aspects of smartproducts[END_REF]. In order to gain deeper insights into actual product usage, product condition, and other information of interest, enterprises had to conduct resource-intense market studies and maintain extensive consumer dialogs, which leave little room for trials and feedback on a large scale [START_REF] Körling | Smart products: why adding a digital side to a toothbrush could make a lot of sense[END_REF]. For the purpose of collecting data for their EIS, enterprises more often than not resort to the internet, which has thus become a popular source for data mining, including online consumer ratings, reviews, discussions, subscriptions, and other consumer-related actions. Modern web technologies also reduce the common information asymmetry between consumers and enterprises [START_REF] Bohn | Living in a world of smart everyday objects: social, economic, and ethical implications[END_REF][START_REF] Filipova-Neumann | Reducing asymmetric information in insurance markets: cars with black boxes[END_REF]. Still, the internet has undergone development from the initial stationary internet, to the current mobile internet stage, and on to the already emerging internet of things (for an overview, see Fig. 4).

The stationary internet and web analytics have allowed capturing of consumer behavior in a novel way through company websites. Google, comScore, Adobe, and Mixpanel are currently the worldwide leading providers of well-Fig. 4. The evolution of the internet of things and product analytics established web analytics technologies. Although the web has significantly shortened the distance between enterprises and consumers, both physically and emotionally, to just a click [START_REF] Jansen | Understanding user-web interactions via web analytics[END_REF], web analytics still requires consumers to generate feedback data actively on the web. The dissemination of smartphones and the mobile internet have further increased the time consumers are close to enterprises, because, in contrast to a personal computer, smartphones can be used almost anytime and anywhere (for an application example, see [START_REF] Weber | Panel surveys go mobile[END_REF]). Thus, mobile analytics have provided new opportunities. With regard to the mobile internet, Google Analytics, Flurry Analytics, Crashlytics, and HockeyApp are popular examples of tools that capture the consumer's interaction with mobile apps and mobile web browsers. Following these predecessors, the current evolution stage of the internet, IoT, is about to establish a permanent connection between consumers and enterprises. In the future, consumers will be surrounded by IoT-products that permanently capture the consumer's interactions with IoT-products, everyday objects, and other persons. With the expansion of the internet from computers and smartphones to everyday things, the application field of tools used in web and mobile analytics likewise expands [START_REF] Bohn | Living in a world of smart everyday objects: social, economic, and ethical implications[END_REF][START_REF] Filipova-Neumann | Reducing asymmetric information in insurance markets: cars with black boxes[END_REF]. Thus, IoT-products constitute a promising platform to collect data about the actual product usage.

We refer to the IoT-counterpart of web and mobile analytics as "product analytics" because the medium for customer interaction is not a web browser but the physical product itself and, if available, a corresponding mobile app that functions as a remote control for the IoT-product. Although a wide range of IoT-ready products already exists on the market, the application of product analytics is still in an infant stage; standardized tools and best practices for product analytics are likely to emerge in the upcoming years. In the remainder of this section we explore established methods from web analytics and discuss their application to IoT-products.

Log Files

Web servers store a log of page requests including data such as the originating internet protocol (IP) address, date and time of the request, referrer, and some information about the device from which the pages are requested [START_REF] Jansen | Understanding user-web interactions via web analytics[END_REF]. These log files serve as a data foundation for the most common metrics such as the frequency and duration of visits, visitor paths, demographic and geographic visitor information, referring websites and keywords, operating system statistics, and so forth. Data collection with log files can be executed on both the server and the client level [START_REF] Srivastava | Web usage mining: discovery and applications of usage patterns from web data[END_REF]. The collected data helps enterprises to understand consumer behavior such as how consumers react to product information, how they make purchase decisions, which consumer segment is most likely to make product purchases, and reasons for consumers to bounce from (fail to follow through with) the purchase process. The insights extracted from log files are used to support decision making in various enterprise divisions and levels including communication or pricing decisions in marketing, product feature decisions in product development, training of sales representatives and technicians, etc.

IoT-products offer an attractive platform for the autonomous collection of product usage data in log files that are automatically pushed to EIS at intervals or on request. The data from built-in sensors can be analyzed in order to determine the frequency and intensity of product usage, usage patterns and triggers of product abandonment, product malfunctions, incorrect product usage, and much more. Through an EIS, these insights can be utilized to predict product maintenance and the topic of inbound calls in call centers [START_REF] Porter | How smart, connected products are transforming competition[END_REF], analyze the sequence of events on the customer journey [START_REF] Norton | Using the customer journey to road test and refine the business model[END_REF], maintain direct relationships with consumers throughout the whole product life cycle [START_REF] Konana | Physical product reengineering with embedded information technology[END_REF], capture longitudinal data for customer relationship management [START_REF] Jayachandran | The role of relational information processes and technology use in customer relationship management[END_REF][START_REF] Rigby | Avoid the four perils of CRM[END_REF], or apply novel business models that consider usage behavior [START_REF] Dover | Worldwide SaaS enterprise applications 2014-2018 forecast and 2013 vendor shares[END_REF]. Amazon, for instance, uses log files to offer ebook authors per-page payouts. Another promising field of application for information being derived from log files is product optimization. Log files are an established source for capturing the behavior of website visitors in so-called A/B tests. In these experiments, the performance of two operational versions of a website that differ in only one single variable such as a logo or a button is compared, which ultimately allows testing of hypotheses with respect to conversion rates and other visitor actions of interest. When applied to IoT-products, experimental tests of variations in digital user interfaces and product features can deliver information for EIS about consumer preferences, latent needs, cognitive capabilities of using the product, and so forth. Log file data from IoT-products can also provide insights about the consumer's environment by applying association rules to identify complementary products that consumers frequently use alongside their own IoT-products. Such association rules are currently used in the web to relate pages or objects that are frequently referenced together in a single session [START_REF] Srivastava | Web usage mining: discovery and applications of usage patterns from web data[END_REF]. A common field of application is product recommendations in web shops, referring to products that "other customers also bought" using association rules to identify products that are complementary to items in the shopping cart of the consumer. In the IoT context, insights on products in the consumer's environment can also serve as a foundation to determine cross-and upselling opportunities and, moreover, they can support decision making about technical interfaces to ensure compatibility with complementary products.

Tagging

Tagging is a more active method of capturing IoT-product usages. In the traditional internet, tagging describes the placement of invisible images or, more recently, pixels, in specific sections of a website or email that trigger actions when they are loaded by web browsers or email clients [START_REF] Jansen | Understanding user-web interactions via web analytics[END_REF]. Unlike the rather continuous and passive data collection in log files, tagging data is pushed to the EIS in real-time if a predefined user action is tracked. When applied in IoT-products tagging is a promising method for enterprises to collect additional real-time data that can automatically trigger actions in different enterprise divisions through the EIS. As an example, IoT-product tags may indicate continued usage difficulties on the consumer's side and notify customer service to proactively contact the consumer for assistance. Real-time IoT-tagging can also be used to remotely personalize IoT-products with respect to the consumer's usage patterns, similar to website personalization [START_REF] Pierrakos | Web usage mining as a tool for personalization: a survey[END_REF]. Some vehicle insurance enterprises such as Metromile already offer "pay how you drive (PHYD)" policies wherein the driving behavior of consumers is taken into account in calculating premiums. A small box installed in the car of the consumer tracks the driving behavior, including driving speed, braking behavior, and turns, and sends this data to the insurance providers. Thus, tagging can instantly reveal infringements such as driving through a red traffic light, allowing the enterprise to aline premiums with actual risk.

Conclusions

The internet of things gives birth to a novel category of consumer products that can both collect data and share data. These so-called IoT-products constitute a new source of real-time consumer data for EIS. They use built-in sensors and wireless communication technologies to capture actual product usage and push data to the EIS. The current expansion of the internet from the web to physical products provides an opportunity to apply well-established methods from web analytics to IoT-products. This paper contributes to the field of research in the intersection between the IoT and EIS by highlighting new data collection opportunities for EIS. By integrating microsensors, processors, wireless communication technologies, and energy supplies into traditional products, enterprises may gain real-time access to data beyond the point of sale. In particular, everyday consumer products may serve as a platform for product analytics, as consumers use them frequently and such products are typically close to the event of interest. Tracking and analyzing actual product usage can be implemented passively with log files or actively with tagging. Both approaches provide enterprises with the means to extract insights from rich product usage data with methods such as A/B testing and association rules.

While our work has focused on marketing and sales, future research should also analyze the opportunities of product analytics for other enterprise divisions. A second promising direction for further research in IoT and EIS lies in the development of frameworks that allow easy cross-platform application of product analytics in IoT-products (see, for example, [START_REF] Kryvinska | Conceptual model of business service availability vs. interoperability on collaborative IoT-enabled eBusiness platforms[END_REF][START_REF] Tiwana | Platform evolution: coevolution of platform architecture, governance, and environmental dynamics[END_REF]). Such platforms will ultimately be necessary for product analytics to become as successful and popular as its counterpart in the web. Finally, IoT-products pose security and privacy challenges (see, for example, [START_REF] Körling | Smart products: why adding a digital side to a toothbrush could make a lot of sense[END_REF][START_REF] Strazdins | Open security and privacy challenges for the internet of things[END_REF][START_REF] Weber | Internet of things: new security and privacy challenges[END_REF]). In the context of product analytics, further works on data survivability, intrusion detection, and data authentication seem to be particularly worthwhile [START_REF] Di Pietro | Security in wireless ad-hoc networks: a survey[END_REF].

Fig. 1 .

 1 Fig. 1. IoT-products as a new data source for Enterprise Information Systems (see, for example, [32]). DSS = Decision Support Systems, MIS = Management Information System, ESS = Executive Support Systems, TPS = Transaction Processing Systems, KWS = Knowledge Work Systems

Fig. 3 .

 3 Fig. 3. Wireless communication technologies categorized by their typical field of application. WBAN = Wireless Body Area Network, WPAN = Wireless Personal Area Network, WLAN = Wireless Local Area Network, WMAN = Wireless Metropolitan Area Network, WWAN = Wireless Wide Area Network. Year of market introduction in brackets