INTRODUCTION The concepts of mechanical impedance and mobility are part of the design process and analysis in structural dynamics. Developed from electromechanical and electroacoustic analogies in the 1920’s , they define the relationship between force and velocity at a given location of a structure, both with respect to amplitude and phase. Point mobility measurement is crucial in determining the dynamic properties of mechanical structures to improve their noise and vibration characteristics, in addition to allowing the derivation of physical parameters of materials. For instance, the development of efficient, lightweight structures in transportation industries is of paramount concern, and expertise and experimental techniques are needed so that problems due to vibration, fatigue, or potential design flaws can be anticipated and brought under control. The point mobility characteristics of a mechanical structure, which is based on a collocated measurement of force and velocity, has broad and practical applications in: 1) modal testing to determine the natural frequencies, modal masses, mode shapes and damping of structures , 2) materials testing to determine certain specific properties of materials such as the loss factor and the modulus of elasticity , 3) structural inspection and monitoring in order to investigate non-uniformities, defect or tolerance deviation that can be detected by comparing the mobility function of a standard with a recently manufactured item . It is also used in system modeling since mobility measurement performed at many locations on a structure can be used as verification of mathematical models, or the response of subsystems or cascaded substructures may be measured and the properties of the assembled system be predicted before final assembly , similarly to the study of complex electrical circuits. In smart structures and vibration isolation, furthermore, it allows the dynamic interaction between electromechanical transducers and structure to be involved in the development process ; vibration isolator, proof-mass actuator and machinery mounting can then be optimized to minimize structural vibration or sound transmission . The measurement of mechanical impedance and mobility in general requires three devices : an actuator, a force sensor and a vibration sensor . In impact testing, the structure under test is excited by a sharp and clean force input of small time duration, generally by using an impact hammer equipped with a load cell, and the vibration response is typically sensed by using an accelerometer. In shaker testing, a vibrating-rod actuator is required to mechanically excite the structure according to the amplified input signal, usually a sine sweep or random noise. In both cases mechanical energy is supplied to the structure with a known frequency content, allowing the mobility function to be calculated from the force applied and the vibration response. Achieving collocated sensing is a challenging task and an imperfect colocation may result in significant errors in the measured mobility function . An alternative approach for the direct measurement of the driving point mobility of a structure has recently been proposed by Sands _et al._ . In this method, an electrodynamic inertial actuator combined with two accelerometers is attached to the structure directly so that the assembly can be used as a mobility measuring device. The two sensors are actually placed strategically on the actuator frame and the dynamic mass, allowing the mass-loading effect of the device on the structure to be removed. The feasibility of the method was validated in on a cantilever beam. Although this approach overcomes some practical limitations of impact testing and shaker testing, such as allowing better control over the frequency, amplitude and duration of the excitation applied to the structure while compensating the mass-loading effect of the actuator on the structure, it requires developing a specific device. On the other hand, several authors have studied how to use electromechanical transducers as both sensor and actuator simultaneously, a technique which is commonly referred to as _self-sensing actuation_ in the literature. The idea of self-sensing actuation was developed in the early nineties by Hagood _et al._ and Dosch _et al._ concurrently, where the piezoceramic elements were used as sensors and actuators simultaneously to reduce implementation, cost and complexity, by achieving truly collocated control. The principle was later implemented to achieve active control of noise and vibration , to implement self-sensing vibration damping using electrodynamic shaker actuators , for active acoustic attenuation and for biomedical applications using an electromagnetic actuator . More recently, the principle has been applied to an electrodynamic inertial exciter for active vibration control of a simply supported plate . In particular, it was shown in Ref. that the vibration of the structure at the actuator base can be accurately estimated from the actuator impedances and the current and voltage at the transducer’s terminals, paving the way for the development of a _sensorless_ mobility measuring device. The approach of estimating the mechanical impedance and mobility of a structure from the electrical impedance of an electromechanical transducer attached onto it is not new. In the mid 1990s, studies on the coupling of the electrical impedance of piezoceramic materials and the mechanical impedance of structure at the location of the piezo patch were reported by Liang _et al._ . In the early 2000s, Ling _et al._ developed a technique for measuring the mechanical impedance of structures using the sensing capability of a piezoceramic inertial actuator . Experimental results for monitoring structural integrity of a clamped-clamped beam can be found in , showing that some structural defects can be detected by examining the changes of the electrical impedance of the actuator. In the footsteps of Ref. , Doutres _et al._ developed a methodology for determining the loss factor and the Young’s modulus of a porous sample from the measurement of the electrical impedance of an electrodynamic loudspeaker used to apply static and dynamic stress on it . This article intends to improve the mobility measuring device proposed by Sands _et al._ by jointly using the sensing capabilities of the inertial exciter when actuated, as suggested in Ref. . The objective is to demonstrate that the point mobility function of a structure can be measured accurately and reliably without the need for an external sensor, but directly from variations in the coil electrical impedance of the actuator attached onto it. This results in a versatile experimental tool with sensorless structural health monitoring and vibration control capabilities, that can also be used to evaluate certain characteristic parameters of materials. The remainder of the paper is organized as follows. Section 2 provides a lumped electromechanical parameter model for the electrodynamic actuator, including its operation for self-sensing actuation. In particular, the analytical derivation of the mobility function of the structure based on variations in the actuator electrical input impedance is thoroughly examined using an impedance-based approach. In Section 3, we present the methodology used to experimentally evaluate the electromechanical parameters of the actuator. The proof of concept is provided in Section 4 on the basis of a simply supported plate and experimental results are compared to standard shaker testing. As a conclusion, the benefits and applicability of this sensorless mobility measuring device (SMMD) is discussed, in particular as regards practical implementation and potential applications in structural dynamics. THE ELECTRODYNAMIC SENSORIACTUATOR This section provides the basic theoretical knowledge about the operation and the characteristics of the electrodynamic inertial exciter, and specifies how it can be employed for self-sensing actuation. A lumped parameter model is used in the following where complex numbers are used to represent sinusoidal functions of time, i.e., f(t)=Re[fexp(jωt)] where $=$ and ω is the angular frequency (in rads−1). All variables in the system are constrained to a fixed direction of action (without considering spatial dependency); therefore relationships between each quantity become scalar. It is assumed that all forces acting on the system are small enough so that the displacement remains proportional to applied forces (small-signal range). Otherwise, some of the model parameters need to be considered as time-varying nonlinear functions of the input variables . Model of the electrodynamic inertial exciter An electrodynamic inertial exciter is a voice coil actuator used to mechanically excite the structure to which it is attached. When actuated by ac power, the voice coil provides the motive force to the dynamic mass of the actuator by the reaction of a magnetic field to the current passing through it. Figure [schematic] gives a schematic representation of the actuator, where the current i is defined positive when flowing into the electrical port of the transducer and the velocities va and vs are defined positive when oriented towards the structure. When attached to a flexible structure, the equations of motion of the dynamic mass Ma and frame mass Mf of the actuator derive from the Newton’s second law and can be written as \omega M_av_a &= \left(R_a+K_a/\omega \right)({v}_{s} -{v}_{a})-F_{mag}\\ f_s &= -\left(R_a+K_a/\omega \right)({v}_{s} -{v}_{a})+F_{mag}-\omega M_f\,{v}_{s}\\ %\right. where va and vs are the axial velocities of the dynamic mass and base of the actuator, Ra and Ka are the mechanical resistance and stiffness of the mount, respectively, fs is the reaction force at the base of the actuator , and Fmag = Bl i is the force of electric origin resulting from the magnetic field acting on a free moving charge (current) where B represents the magnetic flux density (in T) and l the total length (in m) of the current-carrying wire. Moreover, it is assumed that the two velocities va and vs are collinear. The governing equation of the electrical dynamics is based on Kirchhoff’s circuit laws and can be written as u=\left(\omega L_{e}+R_{e}\right){i}+{\varepsilon} where u is the input voltage applied to the transducer terminals, i is the electrical current flowing through the coil, and ε = Bl(vs − va) is the back electromotive force (voltage) generated by the relative motion between the magnet and coil. The constant parameters Re and Le are the resistance and self inductance of the coil, respectively. Based on assumptions of linear dynamic coupling between the actuator and structure, Eqs. ([newton]) and ([kirchhoff]) may also be rewritten as \left\{ \omega M_av_a &= {Z}_{ms}(\omega)\left({v}_{s} - {v}_{a}\right)-Bl\,{i}\\ f_s &=-\omega M_a \,{v}_a-\omega M_f\,v_s\\ {u} &= {Z}_{e}(\omega)\,{i} + Bl\left({v}_{s}-{v}_{a}\right)\\ \right. where Ze(ω)=Re + jωLe is the blocked electrical impedance of the transducer and Zms(ω)=Ra + Ka/jω is the mechanical impedance of the spring-dashpot mounting system. Note that the voice coil actuator is an anti-reciprocal system that can alternatively be represented as a two-port network (see [ABCDmatrix]), involving the voltage u and current i at the input (electrical) port, and the force fs and velocity vs at the output (mechanical) port. When attached to a flexible structure, the mobility function (in ms−1N−1) of the structure at the location of the actuator is given by {Y}(\omega) = }{{f_s}} The block diagram and the analogous electrical circuit of the electrodynamic inertial actuator, derived from Eqs. ([characteristic]) and ([mobility]), is given in Fig. [blockdiagram]. (a) (b) Electrical input impedance The electrical input impedance of an electromechanical actuator is the complex ratio of the voltage to the current in the electrical circuit of the transducer. It determines the electrical characteristics of the actuator ’seen’ by the electronic drive source or any electrical network connected across its input terminals. It usually serves three purposes: (1) providing information for impedance matching between the actuator and electronic drive source, (2) determining the actuator efficiency and driving voltage from current response (or vice versa), and (3) predicting analytically the performance of the actuator. Although it is measured electrically, it also depends on the mechanical characteristics of the actuator. When attached to a rigid foundation, the closed form expression of the actuator electrical input impedance (in Ω) can be obtained from Eq. ([characteristic]) by setting vs = 0, as {Z}_{in0}(\omega)=}{{i}}\Big|_{v_s=0}={Z}_{e}(\omega) + {{Z}_{ms}(\omega)+\omega M_a} As can be seen in Eq. ([Zin0]), it is basically the sum of the blocked electrical impedance and motional impedance of the actuator. Substituting now Eq. ([mobility]) into Eqs. ([characteristic]), the closed form expression of the actuator electrical input impedance can therefore be expressed as {Z}_{in}(\omega)=}{{i}}={Z}_{e}(\omega) + {{Z}_{ms}(\omega)+\omega M_a\omega M_f{Y}(\omega)}{1+\omega (M_a+M_f)\,{Y}(\omega)}}} Eq. ([Zin]) clearly states that the motional impedance in Zin is a function of the mobility characteristics of the structure under study. The actuator electromechanical response By driving a current through the voice coil, the transfer function between the input current and the reaction force at the base of the actuator can be derived from Eqs. ([characteristic]) and ([mobility]), as H_i(\omega) = {Bli} = \omega M_a}{Z_{ma}(\omega) + \omega M_a\left(Z_{ms}(\omega)+{M_a}Z_{ma}(\omega)\right)Y(\omega)} where Zma(ω)=Zms(ω)+jωMa. As shown in Eq. ([Hi]), the reaction force due to the acceleration of the dynamic mass is in turn exerted to the structure at the actuator base. Similarly, the open loop response function (in ms−1A−1) of the actuator when attached to a flexible structure can be written from Eqs. ([characteristic]-[Zin0]) and ([Zin]), as {G}_{i}(\omega)=_{s}}{{i}}=_{ma}(\omega)}{{\omega M_aBl}} \big({Z}_{in}(\omega)-{Z}_{in0}(\omega)\big)} Eq. ([Gi]) clearly shows that the dynamic response of the structure at the attachement point can be expressed as a function of the actuator impedances. Derivation of mobility function from the actuator impedances Combining Eqs. ([mobility]) and ([Hi]-[Gi]), the mobility function of the structure at the attachment point can be rewritten as {Y}(\omega)=_{ma}(\omega)}{\omega M_a}\,{Bl-\left(Z_{ms}(\omega)+{M_a}Z_{ma}(\omega)\right){G}_i(\omega)} As can be seen in Eqs. ([Gi]-[mobilityest]), the mobility function of the structure and the open loop response function of the actuator attached onto it can be readily obtained without external sensor, provided that the actuator electrical input impedance and the fundamental parameters Ma, Ka, Ra, Mf and Bl are known. More details on the analytical derivation of Eqs. ([Hi]-[mobilityest]) are given in [actuatormodel]. Method to achieve a sensorless mobility measuring device In summary, the proposed method to experimentally achieve the mobility characteristics of a mechanical structure using only an inertial electrodynamic exciter requires 1) measuring the electromechanical parameters Ma, Ka, Ra, Bl, Mf obtained from the transmissibility response and the input electrical impedance of the actuator when attached to a rigid foundation (see Eqs.([trans]) and ([Zin0]), respectively), and 2) measuring the electrical input impedance of the actuator when attached to the structure under study, as indicated in Eq. ([Zin]). The mobility function of the structure can therefore be calculated from the frequency response function of the input electrical impedance of the actuator when mounted to the structure according to Eq. ([mobilityest]). ÊÊÊ EXPERIMENTAL ASSESSMENT OF ELECTROMECHANICAL PARAMETERS This section presents a methodology to accurately measure the fundamental parameters of the electrodynamic actuator described in Section [sensoriactuator], from the velocity transmissibility responses and the measurement of the actuator electrical input impedance. The experimental setup is illustrated in Fig. [setup1]. (a) Velocity transmissibility test (b) Electrical impedance test Velocity transmissibility The velocity transmissibility for a single-degree-of-freedom (SDOF) system is the complex ratio between the output and the input velocities . For instance, the closed form expression of the velocity transmissibility between the dynamic mass Ma and the base of the actuator can be written as T(\omega) = {v_s}=(\omega)}{Z_{ms}(\omega)+\omega M_a} where Zms(ω)=Ra + Ka/jω is the impedance of the spring-dashpot mounting system. This result is obtained from Eqs. ([characteristic]) by setting i = 0, i.e. considering that the mass Ma is moving as a result of an external force acting on the actuator. Experimental setup In practice, the velocity transmissibility responses of the SDOF (see Eq. [trans]) are measured by mechanically driving the actuator using a shaker. As shown in Fig. [setup1](a), a K2007E01 Mini SmartShaker (5 lbf rms) with integrated power amplifier is used to mechanically drive the actuator and a VibroMet 500V laser Doppler velocimeter (sensitivity 5 mV/(ms−1)) is used to sense the axial velocity va of the dynamic mass Ma. A PCB model 288D01 impedance head (sensitivity 22.4 mV/N for the force sensor and 10.3 mV/(ms−2) for the accelerometer) is connected between the shaker and the mounting system to which the actuator is attached, allowing the force and acceleration transmitted by the shaker to be monitored. The accelerometer signal is processed by a time integrator, thus determining the axial velocity vs at the base of the inertial actuator. The magnitude and phase of the transmissibility responses obtained from experimental testing are shown in Fig. [transmissibility]. The actuator electrical input impedance is determined by applying a sinusoidal current to the transducer terminals while monitoring the induced voltage. As shown in Fig. [setup1](b), the measurement is performed when the actuator base is attached to a rigid foundation, allowing the electrical input impedance to be evaluated according to Eq. ([Zin0]). Similar to the procedure for the determination of electromechanical parameters of a loudspeaker system, measurement is performed at small input level around the resonance frequency of the dynamic mass on its suspension so that the mechanical behavior of the drive unit is effectively linear, i.e. proportional to the input  . (a) (b) (c) Derivation of the electromechanical parameters • Dynamic mass Ma The dynamic mass Ma is determined in two steps by measuring the velocity transmissibility between the dynamic mass and base when the actuator is attached to the shaker at its base (see Fig. [setup](a)), and then combined with a mass m added to the dynamic mass (see Fig. [setup](b)). Denoting fm and fn the resonance frequencies (in Hz) of the actuator with and without added mass, respectively, the dynamic mass Ma can therefore be derived after M_a = ^2}{{f_n}^2-{f_m}^2}m Figure [transmissibility] shows the frequency response functions (FRFs) of the measured velocity transmissibility. As can be seen in Fig. [transmissibility] with an added mass m = 0.0305 kg (case B), the measured resonance frequency of the actuator at fn = 75 Hz (case A) is found to be shifted at fm = 62 Hz, thus giving a dynamic mass Ma = 0.108 kg. • Effective stiffness Ka The effective stiffness Ka of the suspension can be derived once the dynamic mass has been identified, as K_a = \left(2\pi f_n\right)^2M_a which gives an effective stiffness Ka ≃ 24000 Nm−1. (a) (b) • Mass of the frame Mf The mass Mf of the actuator base can be determined from the transmissibility response when the actuator is mounted upside down (see Fig. [setup](c)) so that the dynamic mass Ma is not allowed to vibrate freely, as M_f = {\left(2\pi f_f\right)^2} where ff is the resonance frequency when the actuator is mounted upside down on the shaker. As can be seen in Fig. ([transmissibility])(a), the measured resonance frequency in this case equals ff = 222 Hz, thus giving the mass of the actuator base Mf = 0.0134 kg. • Effective mechanical resistance Ra The effective mechanical resistance Ra can be derived from the peak-transmissibility which is the maximum amplitude of the transmitted velocity for a given amplitude of the input velocity. For a linear system the peak-transmissibility is |T(ω)|max ≃ 1/(2ζ), where $\zeta = R_a/(2)$ is the damping ratio of the SDOF. The effective mechanical resistance is therefore given by R_a = 2\zeta As can be seen in Fig. ([transmissibility])(a), the peak-transmissibility equals 9.56 for all configurations tested, thus giving the effective mechanical resistance of the actuator Ra = 5.32 Nsm−1. • Transduction coefficient Bl The transduction coefficient Bl of the voice coil can be derived from the actuator electrical input impedance measured when only the dynamic mass Ma is permitted to vibrate, provided that the effective resistance Ra is known, as Bl = -R_e\right)} where Re is the resistance of the voice coil and Rem is the electrical _motional_ resistance measured at the natural resonance fn of the actuator. As can be seen in Fig. [transmissibility](b), the resistance Re = 6.65 Ω and the maximum resistance at the actuator mechanical resonance Rem = 10.63 Ω, thus giving the electrodynamic transduction coefficient Bl = 4.6 NA−1. The velocity transmissibility test and electrical impedance measurement are therefore helpful to obtain the actuator physical parameters required for self-sensing actuation. The values of the physical parameters of the actuator used in this study are summarized in Tab.[VISATONEX60S]. PARAMETER NOTATION VALUE UNIT -------------------------- ---------- -------- ------- Coil resistance Re 6.65 Ω Coil inductance Le 0.05 mH Transduction coefficient Bl 4.6 NA−1 Dynamic mass Ma 0.108 kg Frame mass Mf 0.0134 kg Mechanical resistance Ra 5.32 Nsm−1 Suspension stiffness Ka 24×10³ Nm−1 Resonance frequency fn 75 Hz : Physical parameters of the VISATON EX 60 S electrodynamic exciter measured in small-signal range. EXPERIMENTAL PROOF OF CONCEPT This section is intended to provide experimental evidence that the electrodynamic inertial actuator examined in Section [sensoriactuator] can be used for self-sensing actuation, allowing the mobility function of the host structure to be assessed without the need for an external sensor. For the purpose of illustration, the results presented below relate specifically to a simply supported plate, the dynamic model of which is given in [platemodel]. Experimental setup The simply supported plate used for the measurements is shown in Fig. [shaker]. As can be seen in Fig. [shaker](a), a Brüel and Kjaer Type 4810 electrodynamic mini-shaker (2.25 lbf pk) is employed to measure the reference mobility function. A PCB model 208B04 force sensor, the sensitivity of which is 1.174 mV/N, is connected between the plate and the shaker using a stinger. As illustrated in Fig. [shaker](b), the electrically-excited inertial actuator is attached in the same place in order to be compared to the measurements obtained with the shaker. Unlike the shaker, the inertial actuator will exert a reaction force on the structure which is due to acceleration of the dynamic mass, without the need of a grounded support. Not shown in Fig. [shaker], a VibroMet 500V laser Doppler velocimeter (sensitivity 5 mV/(ms−1)) is also used to sense the vibration response on the back side of the measurement point location. Signal acquisition is done with 18 bit precision analog inputs at a sampling frequency of 20 kHz. The digital signal processing is performed using a SpeedgoatPerformance Real-Time Target Machine running on Simulink. The physical parameters of the VISATON EX 60 S electrodynamic exciter used in this study can be found in Tab. [VISATONEX60S] and the methodology employed to assess the key parameters is detailed in Section[lumped]. A picture of the inertial actuator used to perform the experiments is given in Fig. [electronic](a). System calibration Before making a valid measurement, the electronic circuit used for both controlling the actuator and detect the voltage and current at its input terminals is subject to calibration. A known precision resistor is substituted for the electrical load impedance to be measured and a scaling factor (complex gain factor Gc(ω)) is calculated for subsequent measurements. The complex gain actually provides information on how the electronic circuit will affect the measured electrical impedance in terms of amplitude and phase. Once calibration is complete, the actuator is substituted for the calibration resistor. The valid electrical impedances can therefore be calculated by multiplying the measured data by the complex gain Gc(ω). A picture of the electronic circuit used to perform the experiments is given in Fig. [electronic](b). (a) (b) Electrical input impedance A convenient way to get the electrical motional response of the voice coil actuator is to attach it to a rigid foundation, so that only the dynamic mass Ma is permitted to vibrate. Figure [FRF_ZIN](a) illustrates the frequency response function of the actuator electrical input impedance when the base is prevented from vibrating, i.e. for vs = 0 as given in Eq. ([Zin0]). As can be seen in Fig. [FRF_ZIN](a), there is a good agreement between the measured and computed data. The actuator electrical input impedance is mainly resistive at low and middle frequencies, except around the mechanical resonance at 75 Hz where there is also a large increase in the actuator impedance, and becomes inductive as the frequency increases due to the coil inductance. (a) (b) When the actuator is attached to a flexible structure, the mass of the base Mf is free to vibrate which will affect the actuator electrical input impedance , as shown in Eq. ([Zin]). Figure [FRF_ZIN](b) illustrates the frequency response function of the input impedance of the actuator when fixed near a corner of the plate (see Fig. [shaker](b)). As can be seen in Fig. [FRF_ZIN](b), there is a good agreement between the measured and computed data. Compared to Fig. [FRF_ZIN](a), it can be noted that the transducer’s mechanical resonance, which is clearly seen at 75 Hz in Fig. [FRF_ZIN](b), strongly interacts with the first structural resonance at 77 Hz for the plate under study. The following peaks in the frequency response are therefore clearly connected to the structural resonances which modify the actuator electrical input impedance. The question now is to _extract_ the structural response from the overall electrical input impedance. Dynamic response of the structure from variations in the actuator input impedance As shown in Eqs. ([Gi]-[mobilityest]), the mobility function of the structure and the open loop response function of the actuator attached onto it can be obtained with no need for external sensor. In order to investigate the accuracy and reliability of the method, and to provide a defined degree of confidence in the mobility measurement, we will first look at determining the frequency range over which it is possible to identify the dynamic response of the structure based on variations in the actuator electrical input impedance. This can be calculated mathematically by subtracting Zin(ω)−Zin0(ω) in accordance with Eq. ([Gi]). Figure [FRF_DIFF] illustrates the computed and measured frequency response function Zin(ω)−Zin0(ω), showing the dynamic response of the structure from variations in the actuator electrical input impedance. As can be seen in Fig. [FRF_DIFF], there is a good agreement between simulation and measurement over the frequency range 50 Hz – 1000 Hz. For higher frequencies, however, the phase response deviates and it is no longer between ±90 degrees as predicted by the theory. Therefore, results presented below will be restricted to 1 kHz which is the frequency range wherein the measurements are accurate and reliable. Outside the working frequency range, it may not be possible to prevent the phase response from rolling off due to the non-ideal behavior of the coil inductance . Velocity-current open loop response Figure [FRF_Gi] shows the comparison between the velocity-current open loop frequency response function obtained using a laser Doppler velocimeter, the beam of which is focused on the plate on the opposite side of the actuator, and that derived from Eq. ([Gi]). As can be seen in Fig. [FRF_Gi], there is a good agreement between reference (true) and estimate, showing that the electrodynamic actuator detailed in Section [sensoriactuator] can be used for self-sensing actuation. Mobility function of the plate Figure [FRF_Y] illustrates the comparison between the mobility function of the plate measured by shaker testing and that obtained from Eq. ([mobilityest]) using the SMMD. As can be seen in Fig. [FRF_Y], both show that there is a good agreement in the working frequency range. The most notable differences are due to experimental conditions, especially the location of the two types of transducers on the plate and their own dynamics that may interact differently with the structure. As can be seen in Figs. [FRF_DIFF]-[FRF_Y], the electrical input impedance of the inertial actuator mounted on a structure is a sensitive function of the mobility characteristics of the structure at the attachment point, due to the electrodynamic coupling within the voice coil. This result provides experimental evidence that the mobility function of a mechanical structure can be readily obtained using an electrodynamic inertial exciter without the need for an external sensor. DISCUSSION As demonstrated experimentally in Section 4, the electrodynamic sensoriactuator developed in Section 2 can be used to measure the mobility of a mechanical structure. Upon actuation by an ac power source, the inertial exciter generates a reaction force on the structure to which it is attached, and the mobility of the structure concurrently affects the actuator electrical input impedance due to the electrodynamic interaction occurring in the voice coil. Once the physical parameters of the actuator are fully identified, the point mobility characteristics of the structure under study can be calculated from the measured electrical input impedance of the actuator via simple mathematical operations. In comparison to the experimental technique proposed by Ling _et al._ for detecting mechanical impedance of structures using the sensing capabilities of a piezoceramic inertial actuator , the electromechanical parameters are easier to measure in the case of the electrodynamic inertial actuator and with better accuracy. Compared to the concept proposed by Sands _et al._ , this sensorless mobility measuring device (SMMD) is based on an actuator commercially available and does not require extra sensor. In the procedure suggested by Sands _et al._, moreover, a frequency-dependent settling time function is actually needed, in the order of a few tens of milliseconds for each frequency, to allow stabilization for the mobility calculation. With the SMMD, the mobility function of the structure can be evaluated by a single measurement for the whole frequency range of interest, using a sine sweep or random test signal. The SMMD also has advantages in relation to the state of the art measurement techniques and can overcome some common problems encountered in practice. Impact testing has the advantage of being non-intrusive with a low mass-loading effect exerted upon the structure, providing the mobility characteristics directly. It is therefore convenient for small lightweight structures. However, some practical problems related to the input force applied to the structure or the recording of the measurement signals may limit its use in some cases . For example, multiple impacts, misalignment of impacts in both position and direction over the averaging period, for example, will affect the measured frequency response functions (FRFs). Furthermore, due to the short duration of the impact force high peak levels but low RMS levels are likely to be produced. This may lead to a poor signal-to-noise ratio of the measured signals and possibly overload the input channel of the multichannel analyzer, while supplying little energy into the structure. The duration of the contact time also influences the frequency content of the force, with a larger contact time causing a smaller range of bandwidth. As a result, different force transducers have to be used according to the force demands  and impact hammers equipped with different masses may be required to cover the frequency range of interest. Note that the hammer can be equipped with an impedance head combining a force sensor and accelerometer, but this device is not free of measurement error . On the other hand, shaker testing can supply more energy to the structure over a longer period of time and hence provide better control of the frequency content, amplitude and duration of the mechanical excitation. A grounded mounting support is necessary, however, which can complicate the implementation of the shaker in some cases. In addition, the shaker can significantly influence the dynamic characteristics of the structure under test. Moreover, the stinger has to be axially stiff but soft in bending in order to reduce the transmission of bending moments to the armature of the shaker on one hand and the force transducer and structure under test on the other hand . Since the SMMD eliminates vibration and force sensors and mounting support, concerns such as the alignment directions of the sensors or access to tight locations no longer exist. Different types of voice coil linear actuators are commercially available with dynamic masses ranging from a few grams to several kilograms, which might be adapted to a wide range of mechanical structures. Their compact size compared to a conventional electrodynamic shaker allows them to fit into small spaces. As suggested in Sands _et al._, furthermore, the mass-loading effect exerted upon the structure by the actuator can be removed mathematically. The SMMD operating range, however, is limited in terms of frequency bandwidth. As shown in Figs. [FRF_DIFF] to [FRF_Y], the mobility characteristics of the plate can be obtained with a high degree of confidence up to about 1 kHz with the actuator used for this study. Beyond this cutoff frequency, it can be seen that the phase roll-off may occur with increasing frequency. This undesirable phenomenon is probably caused by the non-ideal inductive behaviour of the coil which is likely to vary between two successive measurements of the electrical impedance (on the rigid foundation and the structure under test). Notwithstanding these comments, the SMMD is a convenient and cost effective experimental tool that can be used in the analysis of dynamic structures and materials testing. As revealed above, the electrical input impedance of the electrodynamic inertial actuator fixed to a flexible structure is a sensitive function of the mechanical impedance of the structure at the attachment point. Variations of the mechanical impedance directly indicates changes of structural properties such as stiffness changes due to defects, mass changes due to wearing, or changes of boundary conditions due to malfunction of a coupled mechanical element . Therefore, any change of structural conditions due to problems of failure, changes of supporting conditions, etc., can be easily detected by comparing consecutive measurements of electrical impedance. Accurate knowledge of the mechanical impedance or mobility characteristics is also crucial in active vibration control. For example, the maximum feedback gain that can be implemented for a specific application can be predicted more reliably, and hence the stability of the feedback loop where the control unit is mounted . Another major strength of the SMMD in the context of smart structures is the ability to achieve structural monitoring and active vibration control from the same device. This allows for instance obtaining the driving point mobility of the structure precisely at the location of the actuator without the need to remove it, and therefore detect any sign of failure that may occur over time. In short, the sensorless mobility measuring device may provide a practical tool for diagnosing structural problems, determining the modal parameters of a mechanical structure or evaluating certain physical parameters of materials. CONCLUSION This paper presented a stand-alone experimental tool for measuring the mobility characteristics of a mechanical structure using only an electrodynamic inertial exciter. Unlike the state of the art techniques such as impact testing and shaker testing, it is shown that mobility measurement can be performed accurately and reliably without involving external sensors such as accelerometers, load cells or impedance head, but directly from variations in the electrical input impedance of the actuator fixed onto it. This sensorless mobility measuring device can therefore provide valuable information about the dynamic properties of mechanical structures and evaluate some specific characteristics of materials quickly and inexpensively, which are then used to improve noise and vibration characteristics. The electrical input impedance is available when switching on the device or can be assessed in operation, allowing early detection of inhomogeneities, defects or other changes. This results in a versatile experimental tool which can also be employed for sensorless control methods. It is believed that the formulations presented in this paper can be used to investigate new designs of electrodynamic actuator in order to optimize the performance properties of the type of sensoriactuator depending on the structure to be characterized. ACKNOWLEDGMENT This work was supported by the Swiss National Science Foundation (SNSF) under grant PBELP2-146567, the National Sciences and Engineering Research of Canada (NSERC) and Bombardier Aerospace. TRANSMISSION MATRIX OF THE ELECTRODYNAMIC INERTIAL ACTUATOR The electrodynamic inertial exciter is a linear power device that can be modeled as a two-port network. As illustrated in Fig. [two-port], the complex magnitudes of effort (voltage) and flow (current) at the input (electrical) port are related to the complex magnitudes of effort (force) fs and flow (velocity) vs at the output (mechanical) port. The electromechanical transducer equations can alternatively be represented in transmission matrix by \left[ {c} u \\ i \right] = \left[ {cc} T_{11}(\omega)& T_{12}(\omega) \\ T_{21}(\omega) & T_{22} (\omega) \right] \left[ {c} f_s \\ v_s \right] where the frequency dependent parameters $T_{11}(\omega)={f_s}\left|_{v_s=0}\right.$ and $T_{21}(\omega)={f_s}\left|_{v_s=0}\right.$ are the transduction function of voltage to force and the transduction function of current to force, respectively, when the actuator base is fixed to a rigid foundation; $T_{12}(\omega)={v_s}\left|_{f_s=0}\right.$ and $T_{22}(\omega)={v_s}\left|_{f_s=0}\right.$ are the transduction function of voltage to velocity and the transduction function of current to velocity, respectively, when the actuator base is free to vibrate. From Eqs. ([characteristic]), the four frequency dependent parameters T₁₁(ω),T₂₁(ω),T₁₂(ω) and T₂₂(ω) can be derived as T_{11}(\omega) = {f_s}\Big|_{v_s=0} =& {\omega M_a Bl}\left(Z_{e}(\omega)Z_{ma}(\omega)+(Bl)^2\right)\\ T_{21}(\omega) = {f_s}\Big|_{v_s=0} =& (\omega)}{\omega M_a Bl}\\ T_{12}(\omega) = {v_s}\Big|_{f_s=0} =& {Bl}\left({M_a}Z_{ma}(\omega)+Z_{ms}(\omega)\right)+Bl\left(1+{M_a}\right)\\ T_{22}(\omega) = {v_s}\Big|_{f_s=0} =& {Bl}\left({M_a}Z_{ma}(\omega)+Z_{ms}(\omega)\right) where Zma(ω)=Zms(ω)+jωMa. From Eqs. ([T11]-[T22]) and after some further manipulations, the determinant of the transmission matrix yields T = T_{11}(\omega)T_{22}(\omega)-T_{12}(\omega)T_{21}(\omega)=-1 showing that the electrodynamic inertial actuator is an anti-reciprocal electromechanical transducer which follows directly from Maxwell-FaradayÕs law and Lenz’s law . ANALYTICAL DERIVATION OF THE SENSORIACTUATOR TRANSFER FUNCTIONS From Eqs. ([characteristic]), the velocity of the dynamic mass Ma may be expressed as v_a = -{Z_{ma}(\omega) }i+ (\omega)}{Z_{ma}(\omega) }v_s and the reaction force at the base of the actuator can be written as f_s = {Z_{ma}(\omega)}Bl\,i - \left(\omega M_a(\omega)}{Z_{ma}(\omega) }+\omega M_f\right)v_s where Zma(ω)=Zms(ω)+jωMa. Substituting Eq. ([mobility]) into Eq. ([fs]), the transfer function between the magnetic force generated upon actuation and the reaction force can therefore be written as H_i(\omega) = {Bli} = \omega M_a}{Z_{ma}(\omega) + \omega M_a\left(Z_{ms}(\omega)+{M_a}Z_{ma}(\omega)\right)Y(\omega)} Note that Eq. ([HI]) is equivalent to Eq. ([Hi]). Substituting now Eq. ([va]) into Eq. ([kirchhoff]), the voltage generated at the transducer terminals upon actuation using a current drive source can be written as u = \left(Z_e(\omega)+{Z_{ma}(\omega)}\right)i+Bl\left(1-(\omega)}{Z_{ma}(\omega)}\right)v_s and after some further manipulations, the output velocity at the actuator base can be rewritten in terms of the input current and the induced voltage, as v_s = (\omega)}{\omega M_a Bl}\left(u-\left(Z_e(\omega)+{Z_{ma}(\omega)}\right)i\right) Finally, the transfer function between the input current and the output velocity of the structure at the attachment point can be derived from Eqs. ([vs2]) and ([Zin0]) as %G_i(\omega) ={i}= {Z_{ms}(\omega){j\omega M_a Y(\omega)}} + j\omega M_f} {G}_{i}(\omega)=_{s}}{{i}}=_{ma}(\omega)}{{\omega M_aBl}} \big({Z}_{in}(\omega)-{Z}_{in0}(\omega)\big)} where Zin(ω) is the electrical input impedance of the actuator when attached to the structure under study. Note that Eq. ([GI]) is equivalent to Eq. ([Gi]). From Eq. ([mobility]), the mobility function of the structure at the attachment point can also be expressed as {Y}(\omega)=_{s}}{f_s}={Bl}{H_i(\omega)} Substituting now Eqs. ([HI]) into Eq. ([Y1]) yields {Y}(\omega)={Bl}(\omega)}{\omega M_a} +{Bl} \left(Z_{ms}(\omega)+{M_a}Z_{ma}(\omega)\right){Y}(\omega) and after some further manipulations, Eq. ([Y2]) can be expressed as a function of Eq. ([GI]) and the impedances of the actuator, as {Y}(\omega)=_{ma}(\omega)}{\omega M_a}\,{Bl-\left(Z_{ms}(\omega)+{M_a}Z_{ma}(\omega)\right){G}_i(\omega)} Note that Eq. ([Y3]) is equivalent to Eq. ([mobilityest]). DYNAMIC MODEL OF A SIMPLY SUPPORTED PLATE Consider a flexural simply supported plate. Under pure bending assumption, the transverse velocity response vs(t)=Re[vsexp(jωt)] of the plate at a location (x, y) can be expressed using the modal summation formulation   of the complex velocity, as {v}_s(x,y) =\omega ^{\infty}^{\infty}\,{q}_{mn} \,(x,y) where ω is the angular frequency, $=$, q mn is the complex modal amplitude and ϕmn is the m, n-th mode shape function, which is defined for a simply supported Lx × Ly rectangular thin plate, as (x,y) = \sin\left(\gamma_m \,x\right)\sin\left(\gamma_n\, y\right) where γm = mπ/Lx and γn = nπ/Ly. Assuming the primary force disturbance to be harmonic, with time dependence of the form f(t)=Re[F exp(jωt)] where F  is the complex phasor of a transverse force, the complex modal amplitude can be written as {q}_{mn} ={\mu L_xL_y}\,_{mn}}{{}^2-\omega^2}} where Fmn is the corresponding modal force that depends on the nature of the excitation, μ is the mass per unit area (in kgm−2), and ωmn is the natural frequency (in rads−1) of the m, n-th mode, given by =h^3}{12(1-\nu^2)\mu}}\left({\gamma_m}^2+{\gamma_n}^2\right) where $=E(1+\eta)$ is the Young’s modulus (in Pa) including the structural damping factor η, h is the thickness (in m) and ν is the Poisson’s ratio of the plate. Assuming an ideal point force input F at the location (x, y), the transverse velocity response of the plate at the same location is given by {v}_{s}(x,y) =\omega {\mu L_xL_y}\,^{\infty}^{\infty}\,}^2(x,y)\;{F}}{{}^2-\omega^2}} and the driving point mobility function (in ms−1N−1) of the plate can be derived from Eq. 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