Toplu ve yayılı parametreli sistemlerde titreşim gücünün dağılımı : bir kalite kontrol sistemine uygulanması

Abstract

Yapılarda titreşim gücü dağılımı ve titreşim gücünün yapı içerisinde izlediği yol, titreşimsiz ürün tasarımı, yapılarda titreşim kontrolü ve ürün kalite kontrolü gibi konularda gün geçtikçe önem kazanmaktadır. Titreşim gücünü etkileyen parametrelerin belirlenmesi, söz konusu konularda yapılan çalışmalara oldukça büyük ölçüde katkı sağlayacaktır. Bu çalışmada, öncelikle liner yapılarda titreşim gücünün farklı analiz (Sonlu Elemanlar ve Modal Analiz) yöntemleriyle formüle edilmesi ve modellenmesi sağlanmıştır. Elde edilen modeller, endüstride üretim hattında elektrik motorundan akan aktif titreşim gücünü temel alan bir kalite konrol sisteminin ölçüm düzeneği üzerine uygulanmıştır. Göz önüne alman ölçüm düzeneğinin yapısal özellikleri deneysel çalışmalar sonucu elde edilmiştir. Ayrıca ölçüm düzeneğini belirleyen özellikler, farklı yazılım ortamlarında geliştirilen modellere girilmiştir. Yapılan çalışmalar sonucunda, liner bir yapıya girilen titreşim gücünün; yapısal özelliklere, sınır şartlarına, uygulanan kuvvetin genliğine, frekansına ve faz farkına bağlı olarak değiştiği tespit edilmiştir. Elde edilen bu parametrelerin değişmesi durumunda, yapı üzerindeki titreşim gücü dağılımının nasıl bir dağılım göstereceğine dair bilgiler elde edilmiştir. Titreşim gücünü etkileyen özellikler göz önüne alınarak, kalite kontrol amacıyla kullanılacak en uygun ölçüm düzeneğine yönelik tasarım parametreleri elde edilmiştir. Ayrıca çalışma sonucu elde edilen bilgiler, titreşimsiz ürün tasarımında ve titreşim kontrolünde de kullanılabilecektir. Kalite kontrol amacıyla titreşim gücü sinyalinin kullanılması konusunda yapılan ilk çalışmalardan olması nedeniyle oldukça önemlidir.

The purpose of this study is to understand the dynamic characteristics of the mechanical power flow in linear structures so that the parameters influencing the power flow can better be understood. The ultimate goal of the study is, to obtain the parameters of the optimum measurement set-up which is used for quality control of the electrical motors of the washing machine. In order to get the design parameters of the optimum measurement set-up, the model of the active vibration power flowing from the product to the measurement set-up should be established. On the production line in industry, the functionality expected from the measurement set-up which is developed for quality control purposes, is to support the static load of the product and to maximize the active vibration power flowing through the power flow transducer which is located above the energy absorber, Figure 1. Figure. 1. The Measurement Set-up Designed For Measuring Active Power Flowing From The Electrical Motor. XVU The proposed measurement set-up contains; T bone plate located on three air-springs so that the motor can be supported, the energy absorber which is positioned in the middle of the plate, and the power flow transducer located above the absorber in order to measure the active power flowing from the product. The sketch of the measurement set-up is shown in Figure. 1. The power flow transducer consists of two parts, first part is the accelerometer and the other part is the force transducer. These two transducers are put together and the power flow transducer is obtained. The cross correlation of the signal coming from the accelerometer and the force transducer gives the mechanical power signal at the measurement point in the structure. The cross correlation signal is used for quality control purpose. Supplied mechanical power in linear structures, using analysis methods (Modal Analysis, Finite Element Methods), is discussed theoretically. The formulation of the supplied mechanical power in lumped and continuous systems is given in Section 2. Experimental studies are performed to get the structural properties of the components of the measurement set-up in Section.3. Using the formulation derived in Section 2, the models of the supplied mechanical power in linear structures are established in Section 4. The evaluation of these models are made, and then the models are applied on the measurement set-up in Section 5. During the studies conducted, the parameters influencing the measurement technique are revealed. Therefore the design parameters of the optimum measurement set-up for a quality control system, using the mechanical power signal coming from the product, are obtained. With the three different points of view, three different models of mechanical power for linear structures are developed. Taking the measurement set-up into account as a lumped parameter system, the average value of the supplied mechanical power to the structure is obtained. The purpose of using a lumped model is to see the dependence of the power flow on the forcing frequency variation and the rigid body motion of the structure. In the second model, the modal model of the structure is formed to obtain the mechanical power at a point defined in the as function of the frequency. Using this approach the effects of the elastic body motion of the structure on the mechanical power spectrum can be obtained. In the third model the Finite Element approach is used and the mechanical power distribution in the structure is obtained at a specific frequency. Finite Element model of the mechanical power distribution in the structure is conducted for a harmonic force input. The theoretical background of three different models of mechanical power in a linear structure are discussed in Section.2. The supplied mechanical power is simply formulated as a one degree-of-freedom linear system in Section.2. 1. The XVlll equation of motion of one degree-of-freedom viscos damped system is considered in Eq.l. mx(t) + cx(t) + kx(t) = F(t) (1) Using harmonic motion assumption, the second order differential equation is solved. F(t) = Feiû5t (2) x(t) = Vei(ût (3) where F = the amplitude of the force [N], V = the amplitude of the velocity [m/s], t = time [s], co = frequency [rad/s]. If the Eq.2 and Eq.3 are put into the Eq.l, and the time averaged values of the force and the velocity are considered, the mobility of one degree of freedom system can be obtained as in Eq.4. V ceo 2 - ico (k - co m) ? = |_(k-co2m)2+c2co2 where i denotes the imaginary part. (4) The mechanical power is defined as the multiplication of the time averaged values of the force and the velocity : n = t=^{FV*} (5) where XIX (...), = time averaged values, = complex conjugate, IT = mechanical power [W]. IfEq.4 is substituded into Eq. 5 then Eq.6 is obtained. ceo2 + ico(k-©2m) n-Ip (k-co m)2 +c2co: (6) In Eq.6, the real and imaginary parts are defined as the active and reactive power respectively. These two definitions are given in Eq.7 and Eq.8. 1 (k~co m) +c^co' Pn = -F2co2 "_,,2, 2 (7) Qrr lxfl (k~ro m) vn = -Fco- y--2 j-j (8) 2 (k-co m) +c2© where p n = the active power [W], Q IT = the reactive power [W]. In a multi-degree of freedom system, similiar formulation is valid, but in this case, active and reactive components of the power are expressed in terms of matrix quantities. Using the formulation derived so far, the lumped parameter model of the structure is obtained. In order to govern the modal model of the structure, the vibration modes of the structure are obtained by calculating the eigen-values and the eigen-vectors. The superposition of the vibration modes of the structure gives the modal model of the structure. Modal properties of the structure are obtained from spatial model of the structure,by using the orthogonality principle or by performing experimental modal analysis on the structure and using the curve fitting method. While the modal model of the structure giving the mobility characteristics for one degree of freedom system is given by Eq. 4, the eigen-values of the modal model is calculated in Eq. 9 and Eq. 10. The eigen-values of the modal model are complex quantities since the elements of the modal model are complex. XX ^ (k.-^m.V+^c? (9) AQi= *» -*2*i (10) 1 ', (ki-û)2mi)2+©2c? X? = the real part of eigen values of modal model in i * mode, X? = the imaginary part of eigen values of modal model in ift mode. Hence, using the modal model of the structure, the mechanical power is obtained. The formulation of the mechanical power in an elastic linear structure is defined in the continuum mechanics by Eq.ll, using the continuity equation and assuming that the damping mechanism in the material is modelled by linear viscoelastic model of Kelvin type. r1 = -25ljYj=Pri+iQri (i= 1,2,3,j= 1,2,3 ) (11) where çTj. = stress tensor [N/rm], T; = mechanical power flowing unit area in i direction [W/rm], p rf = active power flowing unit area in i direction [W/rm], Q Tj ' = reactive power flowing unit area in i direction [W/rm], v j = velocity tensor [m/s]. The velocity and stress values which are defined as tensors in the continuum mechanic formulation, in Finite Element approach become matrices due to discretization. The Finite Element model of the mechanical power is developed in Section.2.3, in this derivation Eq. 11 is used. Using the harmonic forcing, at a specific frequency the distribution of the active and reactive power in the structure for three directions, is calculated. In the Finite Element approach in order to create the solid model of the structure, an eight node cotinuum element is used. This element has displacements in the three direction as the degrees-of-freedom at the nodes and capable of yielding the stresses in three directions. In order to understand the dynamic behaviour of the measurement set-up the test motor is placed on the T-bone plate and started and "Operational Deflection XXI Shape" study is performed. This study is explained in Section.3.1. As a result of this study, it is seen that there is a substantial amount of motion not only in the vertical direction but also in the horizontal plane. Experimental Modal Analysis study is performed on the T bone plate with free free boundary conditions. Later, the analysis is repeated in Section.3.2 on the. assembled measurement set-up so that the effects of the other components on T bone plate can be identified. Modal properties of the all measurement set-up is obtained from Experimental Modal Analysis study in order to add them into the modal model of the measurement set-up. Mode shapes of the T bone plate in free free boundary condition are also calculated in Finite Element environment in Section. 3. 3 in order to compare the mode shapes obtained from Experimental Modal Analysis with the mode shapes obtained from Finite Element Analysis. In order to add the internal damping properties of the T bone plate in the Finite Element model, two different methods are used. The loss factor of the T bone plate in first mode is obtained in Section.3.4 by using the resonant and non-resonant methods.The average value of the two results is considered in the Finite Element model of the measurement set-up. The structural properties of the other components of the set-up, excluding the T bone plate, are provided by assuming these components as one degree of freedom elements individually and by performing mobility measurements on them. As a result of these mobility measurements, stiffness, damping and mass properties of the air spring, absorber and transducer with coupling stinger are obtained. These values are used in the three different models created for the measurement set-up in Section.5. The lumped parameter model of the structure is used for the two degree of freedom system to get the input active and reactive power at one point. Inputs of the model are, structural properties of the two degree of freedom system, degree of freedom where the force is applied, degree of freedom where the response is taken, amplitude and the frequency of the force. Outputs of the model are, mobility spectrum, active and reactive power spectrums at a point where the force is applied. First, this model is applied to a case study given in reference [8], and the model is validated, then the model is applied on the current measurement set-up. The measurement set-up is considered as the two degree of freedom system. First degree of freedom is associated to the mass of the T-bone plate and this mass is connected to the ground by the air springs, second degree of freedom is related to the mass of the absorber and this mass is grounded by the spring and dashpot of the absorber. xxn The connection properties between these two degree of freedom are taken from the stiffness and damping of the transducer coupled with the stinger. Modal model of the measurement set-up is governed, using the mode shapes and modal properties of the measurement set-up in 0-10 kHz frequency range. In order to get the active and reactive power spectrum at a point where the force is applied, the degree of freedom where the force is applied and the degree of freedom where the response is taken, should be added to the model. An exciting force which has the unit amplitude over the frequency bands of interest is applied on all the degrees of freedom of the measurement set-up, in order to determine the point which is the most suitable for the supplied active power to the structure to be maximum. Finite Element model of the structure is formed by using the geometry, material properties, boundary condition of the structure, point where the force is applied, amplitude and the frequency of the force. Outputs of the model corresponds to these inputs are, active and reactive power distribution for three direction in the structure at the excitation frequency. First, the model is applied on an already tested structure for validation [20], then the model is applied on the test set-up. In this approach the Finite Element model of the measurement set-up is established by using the solid model of the T bone plate, the mode shapes of the plate in free free boundary condition, structural properties of the air springs and absorber, the loss factor of the T bone plate, material properties of the plate, boundary condition, the point where the force is applied, the frequency and amplitude of the force. In the excitation condition, active and reactive power distribution is calculated for the three directions in the structure at the excitation frequency. Consequently, the factors influencing the mechanical power are determined as the structural properties, boundary conditions, application point, frequency, amplitude of the force, and also the phase angle between the excitation forces. Added to this, supplied mechanical power at one point in the structure is related to the mobility properties of the structure at that point. The real part of the mobility spectrum is proportional to the active power spectrum at same point. The imaginary part of the mobility is proportional to the reactive power spectrum at same point. There is a peak in the natural frequency region of the real part of the mobility spectrum while the phase angle between the velocity and the force signal is zero at resonance. At resonance region, the supplied active power to the structure is maximum, but the reactive power is minimum. XXlll Mechanical power spectrum can be modified by changing the structural properties this changes the mobility spectrum as well. The modification of the mass, stiffness and the boundary condition of the structure affect the natural frequency of the structure, and hence modify the active and reactive power spectrum of the structure. The change in the damping properties of the structure affects the amplitude of the mechanical power. If a quality control system is to be designed based on the measurement of active power signal flowing from the product in one direction at one point, then the structural properties on both ends of the point should be considered carefully because it dramatically affects the active power spectrum at this point. If it is desired that the effects of the structural properties are minimized, the natural frequency of the structure (the measurement set-up) should be away from the frequency range of interest. On the other hand, the maximum active power flowing into the structure is observed at the natural frequencies. If the natural frequencies of the structure and the excitation frequencies of the source coincide then the supplied active power flowing into the structure will be maximum, reactive power flowing in structure will be minimum at the excitation frequencies. If it is not possible to change the excitation frequencies of the source, then changing the properties of the structure and the designing a suitable mobility spectrum is necessary, depending on the excitation frequencies of the source. However, it is considerably difficult to design the mobility spectrum given the excitation frequencies.