DESIGN OF MECHANICAL SYSTEMS HAVING MAXIMALLY FLAT RESPONSE AT LOW FREQUENCIES

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1 DESIGN OF MECHANICAL SYSTEMS HAVING MAXIMALLY FLAT RESPONSE AT LOW FREQUENCIES V.Raachran, Ravi P.Raachran C.S.Gargour Departent of Electrical Coputer Engineering, Concordia University, Montreal, QC, CANADA, HG M8. Departent of Electrical Coputer Engineering, Rowan University, Glassboro, New ersey, U.S.A., 8. École de Technologie Supérieure, University of Quebec, Montreal, CANADA, HC. Abstract This paper discusses a ethod of designing translational rotational echanical systes having axially flat response at the lower end of the frequency spectru. Introduction One of the ethods of analyzing a syste is to obtain an analogous electrical networ then use nown analysis techniques to obtain the required inforation []. The syste ay be echanical (translational rotational, fluid, theral or of any other type. The analysis can be carried out either in the tie-doain or in the frequency-doain. Also, Laplacetransfor techniques can be conveniently used in all cases, when the syste considered is linear []. It is the purpose of this paper to show that the sae concept can be applied in the design of such systes. Specifically, since a considerable aount of literature exists in the design of electrical low-pass filters, it can be ade use of in the design of siilar echanical systes also. Magnitude of the Transfer function One starts with the agnitude of the transfer function H ( j V out Ω ( V n in Ω where V in is the input velocity, V out is the output velocity, Ω is the noralized frequency ω, ω c ω is the operating frequency, ω c is the cut-off frequency, n is the order of the syste. Since the velocity is an acrossvariable, the analogous variable in the electrical syste is the voltage therefore Eq.( can be considered to be a voltage transfer function. This perits one to use the theory of electrical filters to study the properties of Eq.( its design. Soe iportant properties of the agnitude transfer function considered. Property : At zero frequency, that is, at Ω, the response is always unity. Property : At unity noralized frequency, that is, at Ω, (which is the cut-off frequency, the response is

2 always, irrespective of the order of the syste. Property : It is readily verified that the first (n- derivatives at Ω will always be zero. This gives rise to the axially flat response in the neighbourhood of Ω. These three properties do not give any inforation regarding the selection of order n. Property : The agnitude response is onotonically decreasing. Property 5 : It can be shown that d H ( jω dω Ω -.55n ( This gives the slope of the agnitude response at the cut-off frequency. This is a possibility for the specification of the syste. Note that n has to be an integer. If n coes out to be a fraction, it is converted into the next higher integer. Alternatively, the order n could be deterined by specifying the agnitude response at a frequency Ω >. In this case also, if n coes out to be a fraction, it is converted into the next higher integer. Table I gives the agnitude of the transfer functions for different orders for Ω.5 Ω.. Deterination of the transfer function After the deterination of n, the order of the syste, the next step is to generate the appropriate transfer function. This is carried out with the help of the coplex frequency given by S σ jω ( S can also be considered as a Laplace-transfor variable. This perits one to write Table I Magnitudes of the transfer function for different orders at Ω.5 Ω. radians. Order Ω.5 Ω. Magnitude in db Magnitude in db H j Ω H(S. H(-S ( Ω -S ( ( S n It is evident that when the denoinator is equated to zero, it contains n roots S, S,...., S n-, S n, each one having agnitude of unity. In other words, all of the lie on the unit circle in the coplex-plane S. Since the function has to be stable, H(S shall contain all its poles in the left-half of the S -plane, or H ( S (5 D( S where D(S is a strictly Hurwitz polynoial. It is readily shown that the roots of the equation (- n S n (6a are given by π (- j n S e,,,,n (6b Fro these roots, D(S a a S a S a S.. (6c can be readily obtained these are given upto n 6 in Table. For details, please see []

3 Table Coefficients of the denoinator polynoial of the transfer function upto order 6. Order a a a a, a. a a a a a a a a.66 a. 5 a 5 a a a.668 a a a 6 a a 5 a.867 a a.9785 a 9.6 Ipleentation of the generated Transfer Function We are ainly interested in a echanical syste, which results in the generated transfer function. One possibility is to design a passive electrical networ then obtain the corresponding echanical syste by appropriate odeling. One such networ is shown in Fig.. Vin LC - networ R L Vout ' ' Fig.: The passive electrical networ considered. The LC-networ consists of only inductors capacitors the utual inductances are absent. (There are other possibilities they are not considered here. It can be readily shown that ( H S V - y out (7 Vin y R L where y input adittance at - with - short-circuited R L reoved, -y feedbac transfer adittance between - - with R L reoved - shortcircuited. It can be shown that y y are the ratio of even to odd polynoials [], because they are obtained for LCnetwors only. Writing D(S M(S N(S (8 where M(S Even part of D(S, N(S Odd part of D(S. It can be shown that -y N( S (9 M(S -y ( N( S with R L Ω (taen without any loss of generality. M(S Exping around S N( S into a continued fraction expansion, the required LC-networ is obtained. Once the electrical networ is realized, the corresponding echanical syste is obtained by replacing the resistor by a daper, the inductor by a spring the capacitor by a ass (for translational systes a flywheel (for rotational systes. This is illustrated by the following two exaples where the orders considered are n n respectively. Exaple Let the order n be. The transfer function will be

4 H ( S S S S This yields y S S S y S S y has to be exped into continued fractions this yields y ( S S S S This results in the electrical networ shown in Fig.(a..5H.5H F ' ' Fig.(a: A third-order low-pass filter having axially flat response around ω. Ω The corresponding noralized echanical translation syste is shown in Fig.(b the noralized echanical rotational syste is shown in Fig.(c. n n n b where n, n, n b. Fig.(b: A third-order noralized echanical translational syste corresponding to the electrical networ of Fig.(a. n n n B where n, n, n B. Fig.(c: A third-order noralized echanical rotational syste corresponding to the electrical networ of Fig.(a. Exaple Let the order n be. The transfer function will be H ( S S a S a S a S where a a.66, a.. This yields - y.6s.66s S.S y.6s.66s The continued fraction of y around S yields y C S L S C S LS where L.57, C.5776, L.89, C.868. This results in the electrical networ shown in Fig.(a.

5 H L H L C F ' ' Fig.(a: A fourth-order low-pass filter having axially flat response around ω. F C Ω The corresponding noralized echanical translational syste is shown in Fig.(b the noralized echanical rotational syste is shown in Fig.(c. n n n n b where n.98797, n.658, n.86, n.5776, b. Fig.(b: A fourth-order noralized echanical translational syste corresponding to the electrical networ of Fig.(a. n n n n B where n.988, n.658, n.86, n.5776, B. Fig.(c: A fourth-order noralized echanical rotational syste corresponding to the electrical networ of Fig.(a. Scaling Having obtained the noralized values in the design, one has to denoralize the in order to obtain the actual values. This process is nown as scaling. There are two types of scaling, naely (a Ipedance scaling (b Frequency scaling. Both these can be carried out independently or siultaneously, depending on the requireents. They are discussed below: (a Ipedance scaling : This is necessitated by the fact that b or B is taen to be unity initially. By ipedance scaling, the ipedance of every coponent has to be scaled by the sae quantity. However, the transfer function reains unaltered. Let be the ipedance scaling factor. It is readily seen that b B becoe b B S respectively. The ipedances S becoes S S respectively. These yield the results that the spring constants becoes respectively. Considering the asses, the ipedances S becoe respectively, S S S giving us the new values as after ipedance scaling. These new values ay be further subjected to frequency scaling discussed below. (b Frequency scaling : This is required because the operating frequency has been noralized to unity. We can s readily put S, where β is the β frequency scaling factor. The quantities b B reain unchanged. By a siilar treatent as given in the case of 5

6 Ipedance scaling, it can be shown that the quantities,, becoe β, β, β β respectively. (c Cobined Ipedance Frequency scaling : Both the scaling operations can be cobined the final results are given in Table III. The quantities in colun are given after they are subjected to frequency scaling only. The quantities in colun can be obtained fro those in colun are obtained by putting those in colun are obtained by putting β in those given in colun. Table Effect of scaling on the coponent values Quantity b B Ipedance scaling Frequency scaling b b B B β β β β Cobined scalings b B β β β β translational syste considered in Fig.(a with b N.s/ a cutoff frequency of 5 radians/second. This gives β 5. The various denoralized coponent values coe out to be: n N/, n 9 8 N/ n g. If the fourth 9 order syste considered in Fig.(a is considered with the sae scaling factors, the various denoralized will be n 5.98 N/, n 8.88 N/, n.559 g n.5 g. Exaple In this exaple, we shall consider the denoralization of the rotational syste considered in Fig.(c. Let B N..s the cut-off frequency be 5 radians/second. The various coponent values coe out to be n N./radians, n N./radians, n 6 g.. When the fourth order rotational syste given in Fig.(c is considered, the various denoralized coponent values coe out to be n.796 N./radians, n.776 N./radians, n.5 g. n 7.86 g.. The responses for the orders are shown in Fig.. By using these scalings, we shall now deterine the actual values to be used for the two exaples considered above. Exaple In this exaple, we shall consider the scaling of a third order 6

7 Fig.: Magnitude responses for n n. Conclusions In this paper, it is shown that the design ethod used in electrical wave filters can be effectively used in the design of echanical low-pass systes. Specifically, axially-flat response near the zero frequency has been considered the corresponding echanical syste has been designed. Though lower order systes are considered, the sae concept can be used to design higher order systes as well. It is envisaged that the sae concept can be used when other types of responses are considered. References [].F.Lindsay V.Raachran, Modeling Analysis of Linear Physical Systes, Weber systes Inc., 99. [] V.Raachran Ravi P.Raachran, Tellegan s Theore applied to Mechanical, Fluid Theral Systes, Annual Conference Exposition of Aerican Society for Engineering Education, New Mexico, Session 79, une -7,. [] V.Raachran, Course notes for Modeling Analysis of Linear Physical Systes, Concordia University, Montreal, QC, CANADA,