THE subject of the analysis is system composed by

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1 MECHANICAL VIBRATION ASSIGNEMENT 1 On 3 DOF system identification Diego Zenari, , M.Sc Mechatronics engineering Abstract The present investigation carries out several analyses on a 3-DOF system. Nominal values for springs are given: by considering the system as linear, i.e. with ideal springs, dampers, masses, the main objective of the study is to estimate the value of the last two. The identification is based on measured data of the empirical response to various types of inputs. Moreover, modal shapes and resonant frequencies are evaluated: their importance is fundamental as they describe the vibrational behaviour of the mechanical system, essential for establishing correct operating conditions. Index Terms 3-DOF, static equilibrium, modal shape, parameters identification, transfer functions. 1 INTRODUCTION THE subject of the analysis is system composed by three masses interconnected by springs. Every mass is supposed to have simply 1 DOF since they only translate horizontally: vertical and transversal motion are considered negligible. The first mass is connected via a rack and pinion joint to an electric motor which could provide torque, thus, an equivalent horizontal force. Since the springs deformation remains in a little interval, their behaviour is approximated as linear (ideal). In conclusion, mass are slithering on the base on which they are lying: this energy dissipation is firstly approximated by a ideal dampers with constant modulus. In figure is shown the model beforehand described: The nominal values of the stiffness [K] matrix are given, while the damping [C] and mass [M] are to be evaluated. 2 STATIC ANALYSIS The first part of the investigation focuses on properties that could be verified considering the system stationary. This is the case of the step response: after a brief transitory the system reaches an equilibrium position, in which the three masses reach a still condition. The equations of motion are ( d2 x i dt = dxi 2 dt = 0): k 1 x 1 k 1 x 2 = f step k 1 x 1 + (k 1 + k 2 ) x 2 k 2 x 3 = 0 (2) k 2 x 2 + (k 2 + k 3 ) x 3 = 0 f step represents the intensity of the step applied to the first mass. We could use a relation based on the motor control settings to calculate the force, and consequently estimate the spring values. The first mass is considered rigidly connected to the motor which generates various shapes and intensities of force inputs. The linear equations of motion in matrices form: [M] ẍ1 + [C] ẋ1 + [K] ẍ 2 ẍ 3 [K] = [M] = [C] = ẋ 2 ẋ 3 [ m m m 3 [ c c c 3 x1 x 2 x 3 ] [ k1 k 1 0 k 1 k 1 + k 2 k 2 0 k 2 k 2 + k 3 = f 0 (1) 0 ] ] Master course of mechanical vibrations, Professor Daniele Bortoluzzi, academic year 2015/2016. See f step = K a K t K mp volt input = 2, 62 [N] (3) [ ] [ ] [ ] K a = 2, K t = 0.1, K mp = A V NA m 1 m

2 MECHANICAL VIBRATION ASSIGNEMENT 2 Where volt input is the voltage applied to the control system of the motor for generating the step force. k 1 k 2 k 3 f step Nominal data 800 [N/m] 800 [N/m] 400 [N/m] 2.62 [N] By analysing the data contained in "data_steps" file, the equilibrium positions of the three masses are derived, and consequently, by solving the linear system, an estimation of the springs modules is carried out: x 1eq x 2eq x 3eq k 1 k 2 k 3 Estimated data 15.3 ± 0.1 [mm] 11.6 ± 0.1 [mm] 5.6 ± 0.1 [mm] 701 ± 10 [N/m] 716 ± 42 [N/m] 374 ± 24 [N/m] Conversely, by using the nominal value of the springs and experimental equilibrium positions, it is possible to do an estimation of the step force applied. From the first equation of the linear system: f step = k 1 x 1 k 1 x 2 = 2.99 [N] (4) Consequently K a K t K mp 6 3 DYNAMIC ANALYSIS 3.1 System identification with constant damping Starting from the system of equations of motion (1), the laplace transform is computed. Considering initial conditions equal to zero, only forced motion survives: X i(s) = ([M] s 2 + [C] s + [K]) 1 F (s) 0 (5) 0 By using the nominal stiffness of springs, and the relation (3) for the force, the parameters to be identified are modules of masses and dampers (m 1, m 2, m 3, c 1, c 2, c 3 ): the last three are assumed as constant parameters. Therefore, by computing symbolically the transfer functions between displacement of masses 1,2,3 and the force applied, it is possible to simulate the response of the system to various force inputs. The difference between the empirical data and the simulated response is defined residual. The empirical data are obtained by giving the first mass an impulse. In practical terms the impulse is represented by a very brief step (50 ms) and 3 volt input. The system identification is achieved by square residuals minimization, that is, the optimal set of parameters should minimize the square residual function. The residual function is composed by the sum of three sub-residuals: each one derived from every mass position. In impulse_dat are stored 4 consecutive impulse responses of the system (impulse is given every 10 seconds): consequently, the values obtained in table "identification" are a mean of 4 different trials. The optimal values computed could be substituted in the transfer functions earlier defined: by analysing the poles (root locus) of the transfer functions we could estimate various properties of the system:

3 MECHANICAL VIBRATION ASSIGNEMENT 3 parameters identified c ± 0.99 [Ns/m] c ± 0.51 [Ns/m] c ± 0.55 [Ns/m] m ± 0.03 [kg] m ± 0.03 [kg] m ± 0.02 [kg] identified response rms error x ± 1.3 [%] x ± 1.7 [%] x ± 1.6 [%] H 1 { resonant poles frequencies 8.26 [rad/s] [rad/s] [rad/s] damping ratios ξ [ ] ξ [ ] ξ [ ] zeros frequencies w z [rad/s] w z [rad/s] H 2{ w z [rad/s] H 3{ cally coincident: U11 U21 U31 U 12 U 22 U 32 = (7) U 13 U 23 U By looking at the bode plot and the pole-zero table, it is possible to notice that there is a zero-pole quasicancellation (around 47 rad/s) in the transfer function of the first mass. resonant poles frequencies - numerical methods 8.29 ±10 5 [rad/s] ±10 5 [rad/s] ±10 5 [rad/s] The general relation for root: s = ξw n ± w n 1 ξ 2 (6) Where s is the Laplace variable. All the roots are couples of complex conjugate numbers: consequently the transfer function is factorizable with second order terms. This implies that every root is accompanied by a phase change of π radiants. The resonant frequencies of the undamped and damped system are practically coincident since the damping ratios obtained are very small. The resonant frequencies and modal shapes could be calculated by variuos methods: Eigenvalue problem: by solving the eigenvalue problem ([K] wi 2 [M]){U} = 0. Ryleigh s quotient: by computing the gradient of Rayleigh s ratio R = 0, R = U T [K]U U T [M]U. Stationary points coincide with the modal shapes. Matrix iteration: by defining the dynamical matrix [D 1 ] = [K] 1 [M], and a guess {U 10 }, the product [D 1 ] {U 10 } is normalized and iterated until convergence. In such way {U 1 } is achieved. The process is identical for every mode of vibration, except for the dynamical matrix, which is "deflated" respected to the previous mode. 1 The results produced by the three methods are practi- 1. generic deflation : [D n+1 ] = [D n] {U n}{u n} T [M], where {U n} satisfies the normalization: {U n} T [M]{U n} = Conclusions A first identification of the system has been achieved. However, it shows some weaknesses: No friction modelling: in the real system masses are slithering on a plain surface. Friction effect could be seen from the impulse response plot: motion unexpectedly stops as it approaches the equilibrium. It also causes poor reliability to the estimated springs and force values in the static section. Constant damping modulues: the system identification process has lead to dampers values with an high uncertainty. Owing to this, it is reasonable to affirm that the linear model with constant damping could be improved. The mass m 1 found represents an equivalent mass, composed by slithering mass, rack and pinion. Consequently, the real slithering mass m s is: m eq = m s + m r + J θ /l 2 Where m r is the connecting rack mass, J θ /l 2 is the pinion rotational inertia over its radius. 3.2 System identification with linear damping The damping matrix is now substituted by a linear function: [C] = α[m] + β[k] (8) Where α and β are two scalar values. This expresses how the damping is distributed. We expect that, since the original system configuration contemplates only absolute motion damping thus generating a diagonal matrix,

4 MECHANICAL VIBRATION ASSIGNEMENT 4 the β parameter will be very low. In addition, since the new set of 5 parameters used for the minimization α, β, m 1, m 2, m 3, is smaller respect to the constant damping set, the minimization will probably not improve. c c c 3 αm 1 + βk 1 βk 1 0 = βk 1 β(k 1 + k 2 ) + αm 2 βk 2 (9) 0 βk 2 β(k 2 + k 3 ) + αm 3 As before the resonant frequencies and modal shapes masses, α and β m ± 0.05 [kg] m ± 0.04 [kg] m ± 0.03 [kg] α 1.34 ± 0.15 [ ] β 1.1 ± [ ] resonant poles frequencies 8.31 [rad/s] [rad/s] [rad/s] { H 1 zeros frequencies w z [rad/s] w z [rad/s] H 2{ w z [rad/s] H 3{ modal damping ratios ξ [ ] ξ [ ] ξ [ ] proportinal damp. rms response error x ± 1.3 [%] x ± 1.7 [%] x ± 1.5 [%] could be calculated: U11 U21 U31 U 12 U 22 U 32 = (10) U 13 U 23 U resonant poles frequencies - numerical methods 8.31 ±10 5 [rad/s] ±10 5 [rad/s] ±10 5 [rad/s] The system could be decoupled in three equations, each one dependent on a modal coordinate (q 1, q 2, q 3 ) and a modal force. This is allowed only in few cases of damping, including proportional damping. The response of the system is generated by superposition: x1(t) x 2(t) = x 3(t) U11 U 12 U 13 q 1(t) + U21 U 22 U 23 q 2(t) + U31 U 32 U 33 q 3(t) (11) Where every {U i } satisfies the normalization: {U n } T [M]{U n } = 1 q i (t) + (α + βwni) 2 q i (t) + wni 2 q i (t) = {U i } T {F } (12) Where {U i } T {F } is called modal force, since it is likely the contribution of the external force to the modal shape Conclusions As for the constant damping, this new minimizations hasn t achieved significant improvements. But there are a few aspects deserving some attention The minimization converged to a result very similar to the one obtained with constant damping, without achieving better advancements. The minimization process is more stable, the dispersion on minimization parameters is significantly lower. As last remark, the linear damping achieves parameters more stable results with a lower uncertainty. 4 SINE SWEEP ANALYSIS Sine sweep is a frequency domain technique for estimating transfer functions. It could be applied in fields where it is possible to generate harmonic input signals, e.g. mechanical, electrical. The technique is based on a fixed amplitude and frequency varying input: the output signal produced is continuously measured. Consequently, by estimating the phase delay and amplitude respect

5 MECHANICAL VIBRATION ASSIGNEMENT 5 to the input, it is possible to reconstruct the transfer function point by point. Two data sets are provided: the difference consists in the sampling frequency: 100 Hz for the first, 200 Hz for the second. As expected, better results are achieved as the sampling time becomes smaller (greater modules of resonance peaks). However the difference is not remarkable: the transfer function estimation doesn t suffers of this discrepancy between sampling times. In addition, the "slow" sine sweep uses a lower acceleration of frequency: this could help to express system dynamics, and owing to this, it is taken as reference. The plot of the frequency spectrum is obtained through fourier transform of the output signal. It indicates which are the harmonics that the system exploits to exchange energy: the grater their amplitude, the greater the energy exchanged. The three black vertical lines indicate the three resonant frequencies identified before: 8, 27, 42 rad/s. The first two are well coinciding with amplitude peaks. However, this is not strongly evident for the third frequency. From 80 rad/s and further the amplitude stabilizes at with "tfest" Through the command tfest matlab provides an estimation of the transfer function between given input and output: in this case force and displacement. Additive informations are provided: the number of poles and zeros of every transfer function. In addition tfest filters the data with a default normalized cut frequency of 0.1. resonant poles frequencies 8.30 [rad/s] [rad/s] [rad/s] Conclusion The estimation with tfest carried out a partially correct result: the first two poles, around 8 and 27 rad/s are discretely identified, conversely the pole at 42 rad/s is not present. The highest resonance frequency is also not clearly identified in the frequency spectrum plot: probably, it is due to the dynamics of the system, the

6 MECHANICAL VIBRATION ASSIGNEMENT 6 inertias are tuned such that a complete expression of the modal shape corresponding to the resonant frequency is not possible. In this sense, it could be possible to follow the path of signal amplification with Fourier transform ratio Now, recalling the fourier transform of the input and output data, their ratio is computed. Then, the transfer function is filtered by means of a Butterworth filter. In such way the third pole, which was not visible beforehand, is now localized. The approximation is valid for ξ < 0.4. Alternatively the bandwidth amplitude method could be also used if ξ < 0.05: the 1-DOF system is excitated with a sine sweep input in order to estimate its transfer function. Then it is possible to find some interesting points: G max = G(jw max amp ) 1 2ξ (15) where G max corresponds to the maximum amplitude of output that could be produced. Then we find the "half power points", i.e. take w such that: which results G(jw) = G max = ξ (16) w 2 = w 2 n (1 ± 2ξ 2 ) (17) and in conclusion ξ = w 2 2 w2 1 2w 2 n w 2 w 1 2w n (18) where w 1 and w 2 are the half power pulsations Butterworth polynomial filter 6 th order w cut 70 [rad/s] The cut frequency is sufficiently high since the poles frequencies are close around 8, 27, 42 rad/s, consequently, they coincide with previous estimations, and filtering does not introduces dynamic delay. APPENDIX A EMPIRICAL ALTERNATIVE METHOD We suppose to stop every mass around its equilibrium point, then, we assume to excite only one mass, keeping the other still. Consequently, we produced a 1-DOF oscillator. Considering the oscillator as described by a linear equation of motion, it could be possible to estimate the parameters c,m by means of logarithmic decrement. This parameter allows to evaluate how quick the system attenuates the amplitude of free oscillations. We could deal with free oscillations since the impulse is equivalent to a velocity initial condition. APPENDIX B NOTES ON ERRORS static section: errors are obtained trough linear propagation. The method is itself correct in this case, but only the equilibrium points error are considered, while force error is neglected. Moreover, the preceding calculation is correct only if the original errors are not correlated, that is, the error is generated by casual factors. This hypothesis is weak since there are some systematic errors not evaluated, in particular friction. dynamic section: the minimization is conducted on a group of 4 samples. Even if this population is very small, variance is computed in order to give an idea of the dispersion of the different minimization trails. Generally, some values are not accompanied by an error for two reasons: either it is neglected or the calculations for error propagation where difficult or lead to not significant results. REFERENCES [1] Mechanical vibrations, Sinigresu S. Rao, Prentice Hall, [2] Valutazione della capacita dissipativa di un sistema strutturale, F. Ponzo, R. Di Tommaso, δ = log U i U i+1 (13) δ evaluates the ratio between consecutive peaks. Under the hypotesis of exponential governed oscillation: δ = 2πξ 2πξ (14) 1 ξ 2