A Revisit of Spinning Disk Models, Part II: Linear Transverse Vibrations

Applied Mathematical Modelling 25 (2001) 561±578

www.elsevier.nl/locate/apm

A revisit of spinning disk models.

Part II: linear transverse vibrations N. Baddour, J.W. Zu

*

Department of Mechanical Engineering and Industrial Engineering, University of Toronto, 5 King's College Road, Toronto, Ontario, Canada M5S 3G8 Received 1 November 1999; received in revised form 30 August 2000; accepted 5 October 2000

Abstract In this paper, two recently derived linear models for the transverse vibrations of a spinning plate are considered. The disk is modelled as a pure plate with no membrane e ects. Furthermore, the e ect of the rotary inertia of the plate is taken into account. The ®rst model is based on the assumption of linear (Kirchho ) strains. The second model is based on the assumption of non-linear (von Karman) strains. The merits of both models are considered and their predictions are compared with those of the traditional linear model. Ó 2001 Elsevier Science Inc. All rights reserved.

Keywords: Spinning disk; Linear vibrations

1. Introduction Spinning disks can be found in many engineering applications. Common industrial applications include circular sawblades, turbine rotors, brake systems, fans, ¯ywheels, gears, grinding wheels, precision gyroscopes and computer storage devices. Spinning disks may experience severe vibrations which could lead to fatigue failure of the system. Thus, the dynamics of spinning disks has attracted much research interest over the years.

The spinning disk has traditionally been modelled as either a spinning membrane, or as a spinning membrane with added bending sti ness [1]. A di erent approach to modelling the dynamics of spinning disks is considered in the ®rst part of this paper. The disk is modelled as a plate using the plate theories of Kirchho and von Karman. Rotary inertia is automatically included.

Two possible forms are presented for the equation of motion of linear transverse vibrations. One equation is the result of using the assumption of linear strains. Let us refer to this as the linearstrain-model (LSM). The second possible equation is linear in the transverse vibrations but is based on the assumption of non-linear (von Karman) strains. Let us refer to this as the nonlinear-strain-model (NLSM).

The LSM bears resemblance to the linear equation of the transverse vibrations of a plate, with an additional term that accounts for the rotation of the plate. This is to be expected since this latter equation is also based on the assumption of linear strains. In other words, the LSM is *

Corresponding author. Tel.: +1-416-978-0961; fax: +1-416-978-7753.

E-mail addresses: baddour@mie.utoronto.ca (N. Baddour), zu@mie.utoronto.ca (J.W. Zu).

0307-904X/01/$ - see front matter Ó 2001 Elsevier Science Inc. All rights reserved.

PII: S 0 3 0 7 - 9 0 4 X ( 0 0 ) 0 0 0 6 6 - 4

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essentially the linear (stationary) plate equation with a new term that is due to the rotation of the plate. The NLSM bears resemblance to the (linear) equation of motion of a spinning plate as derived by Lamb and Southwell [1], plus the additional term that was present in the LSM. Just as the Lamb equation contains the stationary plate equation as a special case, the NLSM contains the LSM model as a special case. The LSM model is obviously a much simpler equation to solve than the NLSM. We also know that for stationary plates the assumption of linear strains is a good one. In this paper, we seek to investigate whether the assumption of linear strains is equally useful for rotating plates or if non-linear strains must be used to obtain meaningful results.

Furthermore, the e ect of the new plate term that arises out the rotation of the plate is considered.

2. Equations of motion Consider the problem of small transverse vibrations of a thin spinning circular plate. The equations of motion describing these vibrations is derived in the ®rst part of the paper. The equation of motion that results from using the assumption of linear (Kirchho ) strains (LSM) is given by

o2 w h2 2 2 o2 2 Eh2 X r4 w 0:

r w r w

1 ot2 3 ot2 3q 1 m2 Recall that h represents the half-thickness of the plate. The rotary inertia of the plate may be ignored by dropping the o2 =ot2 r2 w term. Note that Eq. (1) is basically the equation for the transverse vibrations of a stationary plate with additional terms that account for the rotary inertia and the bending moment due to the centrifugal force. The latter term is the X2 r2 w term.

The boundary conditions for the problem are 1. Either ow=or is speci®ed or

o2 w m ow 1 o2 w

0: 2

or2 r or r oh2 2. Either w is speci®ed or

1 m2 q o3 w o 2 1 m o2 ow 2 ow X r w E orot2 or or r2 oh2 or

w : r

3

The equation of motion that results from the assumption of non-linear (von Karman) strains is given by

q 1 m2 o2 w h2 X2 2 h2 o2 2 rw

rw E ot2 3 3 ot2

h2 4 o2 w dueq ueq ow 1 m dueq d2 ueq 1 o2 w dueq ueq r w 2

2 2 m m

; or or r r oh 3 dr r dr dr2 dr r 4 where ueq r must ®rst be found from d2 ueq 1 dueq

r dr dr2

ueq q 1 m2 2

X r ueq 0: E r2

5

As has been shown in the ®rst part of the paper, Eq. (4) is the same as the Lamb and Southwell [1] model (TLM) with the exception of the terms due to the rotary inertia of the disk and the

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563

bending moment due to the centrifugal force. The boundary conditions to be used with Eq. (4) are given by Eqs. (2) and (3).

3. Frequency analysis of the equations of motion We ®rst consider the frequencies of vibration as predicted by the NLSM of Eq. (4). In this regard, we will follow the approach taken by Lamb and Southwell in their historic paper [1]. The same formula as proposed by Lamb and Southwell [1] is employed. Speci®cally, in their paper they write ``If the restoring forces which control the vibrations of an elastic system can be separated into two or more groups which a ect the potential energy independently, and if the gravest frequency of vibration be found on the assumption that each group acts independently (the inertia being unchanged), then the sum of the squares of the frequencies thus found is less than the square of the greatest frequency which can occur when all the groups act simultaneously''.

In particular, suppose that rotary inertia is ignored, then as an approximation to ®nding the exact frequency, the following formula is proposed: p2 p12 p22 ;

6

where p1 is the natural frequency of vibration of Eq. (4) when all h2 terms are dropped.

Remark that the resulting equation (when all h2 terms are dropped) is identical to the equation for a spinning membrane as introduced by Lamb and Southwell [1] and Simmonds [2]. This was shown in the ®rst part of this paper. These frequencies have been found by Lamb and Southwell in their historic paper [1]. Similar to their approach, p2 is taken as the natural frequencies of Eq. (1) with the rotary inertia dropped. Why has the e ect of rotary inertia been disregarded? This has been done so that we may apply the aforementioned result which requires the inertia of the two subsystems to be the same. Hence, the primary di erence between this consideration and the work of Lamb and Southwell is the inclusion of the X2 r2 w term that represents the bending moment due to the centrifugal force. It was shown in the ®rst part of this paper that this term actually arises from the kinetic energy of the system. In particular, it is a consequence of the variation of the rw rw term which only depends on space derivatives. Hence, although rw rw is technically part of the kinetic energy, from a purely mathematical point of view, it could be easily lumped in with the potential energy of the system (with an appropriate change in sign). This would permit the use of the aforementioned result which requires the inertias of the two subsystems to be the same.

Hence, to summarize, we shall use Eq. (6) to analyze the frequency predictions of the NLSM. Our p1 is identical to the p1 of Lamb and Southwell [1]. Our p2 will be the natural frequency of Eq. (1). In comparison, for Lamb and Southwell, p2 were the natural frequencies of the stationary disk equation. In other words, to compare the natural frequency predictions of the NLSM to the predictions of the traditional linear model (TLM) of Lamb and Southwell, it su ces to compare the two values of p2 . Thus, the primary di erence between the NLSM and the Lamb and Southwell model (TLM) is in the calculation of p2 . In the TLM, p2 are the frequencies of vibration of the stationary plate while in the NLSM, p2 are the frequencies of the rotating plate of the LSM. Based on this logic, a large portion of the ensuing work will focus on the analysis of the LSM, Eq. (1) and its comparison to the stationary plate equation.

The complete solution and analysis of the LSM will now be undertaken. It is the natural frequencies of vibration and the modes of vibration that are of interest here. To ®nd the frequencies of vibration, assume that w takes the form w r; h; t eikt X r; h ;

7

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This form of w is substituted into Eq. (1). The remaining equation to be solved for the spatial dependence of w takes the form r4 X br2 X cX 0;

8

where b

m2

q 1 E

X2 k2 ;

9

3q 1 m2 2 k: 10 Eh2 Note that b P 0 and c 6 0. Furthermore, if the rotary inertia is ignored, all occurrences of X2 k2 are replaced with X2 . Eq. (8) for the spatial dependence of w demonstrates the power of the new formulation of the problem. This equation is much simpler to solve exactly then the equations resulting from the TLM or the NLSM. Furthermore, the inclusion of the rotary inertia is a new development in the vibration of spinning disks. However, as can be seen from Eq. (8), the inclusion of the rotary inertia does not present much of an additional complication since the form of the equation remains unchanged whether it is included or not. That is, b is non-zero for a spinning plate regardless of the inclusion of the rotary inertia of the plate.

For stationary (non-spinning) plates, the omission of the rotary inertia simpli®es the form of the equation considerably since now b 0. For low-frequency vibrations, this is the approach traditionally taken for stationary plate problems. We see that for spinning plates, the primary traditional motivation to ignoring the rotary inertia is not present. Furthermore, from the analysis of stationary plates, it is known that for high-frequency vibrations, the rotary inertia can no longer be ignored without introducing signi®cant errors. From previous works, it is known that spinning a plate raises its natural frequencies of vibration. This fact also provides further incentive to preserve the rotary inertia term in the ensuing analysis.

If r2 is considered as an operator, then the form of Eq. (8) suggests a quadratic in r2 . This observation can be used to ``factor'' Eq. (8) in the same manner that one would factor a quadratic.

Thus in place of solving Eq. (8), i.e. one fourth-order PDE, two second-order PDEs must be solved instead c

r2 X a2 X 0;

11

b2 X 0;

r2 X where

p b2 4c b ; 2 12 p b b2 4c 2 : a 2 For future reference, the following relationships between a and b follow as a consequence of their de®nition: b

2

a2

b2

a2 b2

m2

q 1 E

X2 k2 ;

3q 1 m2 2 k: Eh2

13 14

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565

The simplicity of this formulation becomes apparent in Eq. (11). First, the problem has been reduced to solving two second-order equations, which is a much simpler task than solving a fourth-order equation. Second, these second-order equations happen to be Helmholtz equations.

In polar coordinates, the solution of these equations will involve the use of Bessel functions and modi®ed Bessel functions.

Solutions of Helmholtz equations via separation of variables is well documented in the literature. Note that the function X will be the sum of the solutions to the two Helmholtz equation (11). Thus, X is given by 1 X An Jn ar Bn Yn ar Cn In br Dn Kn br En cos nh Fn sin nh ; X r; h

15

n 0

where An ; Bn ; Cn ; Dn ; En ; Fn are constants, Jn and In are Bessel functions and modi®ed Bessel function of the ®rst kind, respectively, while Yn and Kn are Bessel functions and modi®ed Bessel functions of the second kind, respectively.

3.1. Solid plate with free boundary Substitution of w eikt X r; h into the boundary conditions as given by Eqs. (2) and (3) leads to the boundary conditions for a free boundary in terms of X as

o2 X m oX 1 o2 X 0; 16

or2 r or r oh2

1 m2 q 2 o 2 1 m o2 oX X 2 oX k X rX 0: 17 E or or r2 oh2 or r De®ne Xa and Xb such that they satisfy r2 Xa a2 Xa 0; r2 X b

b2 Xb 0:

18

In other words, X r; h Xa r; h Xb r; h Ra r Rb r Hn h :

19

From the de®nition of the Laplacian in polar coordinates, we may write o2 X r2 X or2

1 oX r or

1 o2 X r2 oh2

20

Note that o2 X =oh2 n2 X : This follows from the separation of variables of the Helmholtz equation in polar coordinates. Furthermore, n must be an integer for the solution to be singlevalued. From the above de®nition of Xa and Xb , it follows that r2 X r2 Xa r2 Xb b2 Xb a2 Xa . The above facts can be combined to simplify boundary condition (16) to the following expression

2 2

n n 1 m oRa oRb 2 2 1 m b Rb 2 1 m a Ra

: 21 r r2 r or or

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Boundary condition (17) can be similarly simpli®ed to

1 m n2 1 m2 q 2 1 m n2 2 R

R

X k a b E r3 r2

1 m2 q 2 1 m n2 oRb

b2 k X2 0: E r2 or

a

2

oRa or 22

3.2. Frequency equation Consider the free transverse vibrations of a spinning solid plate with a free outer radius. The Bessel and modi®ed Bessel functions of the second kind, Yn ar and Kn br are not bounded at r 0. They are thus discarded from the general solution. That is, Bn Dn 0 for all n. Furthermore, Ra r An Jn ar and Rb r Cn In br , where n 0; 1; 2; . . .. These expressions are subsequently substituted into Eqs. (21) and (22) and evaluated at the outer radius of the plate, a.

This yields two equations for the constants An and Cn : 2

n 1 m Pa2 Jn Pa 1 m Pa Jn0 Pa An

n2 1 m Pb2 In Pb 1 m Pb In0 Pb Cn 0; 23

1 m n2 Jn Pa Pb2 1

1 m n2 In Pb Pa2

m n2 Pa Jn0 Pa An

1 m n2 Pb In0 Pb Cn 0;

24

where Pa aa; Pb ba and a is the radius of the plate. Note that the de®nitions of a and b have been used to simplify Eqs. (21) and (22).

Eqs. (23) and (24) represent two homogeneous equations in terms of the two unknowns An and Cn . For non-trivial solutions the determinant is required to vanish. This yields the frequency equation of the system: Pa2 Jn Pa 1 Pb2 In Pb 1

m Pa Jn0 Pa m Pb In0 Pb

n2 Jn Pa Pb2 Pa Jn0 Pa 1

n2 In Pb Pa2 Pb In0 Pb 1

m n2 Pa Jn0 Pa Jn Pa m n2 Pb In0 Pb In Pb

25

Note that the new model has yielded a frequency equation in closed form for the spinning disk problem. A closed-form expression for the spinning disk problem with bending sti ness taken into account was hitherto unavailable in the literature. While this is a considerable simpli®cation, we must still investigate the validity of the linear strain assumption which leads to this equation. Also note that the above frequency equation has the exact same form as that for a stationary circular plate (also based on the linear strain assumption). In fact, it can be shown that the frequency equation for a stationary circular plate can be obtained as a special case from Eq. (25), as would be expected.

3.3. Mode shapes The mode shapes for a solid disk can be found from Eq. (15). For reference, these are given here by r C r n In Pb ; 26 Jn Pa a a An where Cn n2 1 m Pa2 Jn Pa 1 m Pa Jn0 Pa :

n2 1 m Pb2 In Pb 1 m Pb In0 Pb An Naturally, Pa and Pb in the above expressions are those satisfying the frequency equation.

27

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567

3.4. Nodal circles Now consider ®nding the radius of the nodal circles, r , associated with a particular mode of vibration. For the radius of a particular nodal circle, it follows that R r An Jn ar Cn In br 0:

28

From the de®nition of Pa and Pb and using the frequency equation, the above equation can be rewritten as Jn Pa r =a Pa2 Jn Pa 1 m Pa Jn0 Pa n2 Jn Pa

: In Pb r =a Pb2 In Pb 1 m Pb In0 Pb n2 In Pb

29

Thus, for particular values of Pa and Pb which correspond to a particular mode of vibration, Eq. (29) can be solved for r =a or r .

Eq. (29) can further be rewritten to match the expression derived by Colwell and Hardy [3] for stationary plates: Jn Pa r =a

Pa2 Jn Pa 1 m Pa Jn0 Pa n2 Jn Pa

: Pb2 In Pb =In Pb r =a d 1 m Pb In0 Pb =In Pb r =a n2 In Pb =In Pb r =a 30

Following the argument given in [3], the modi®ed Bessel function In is like hyperbolic function in that it starts at small values and increases rapidly to in®nity. For values of r not near the edge, r =a 1, it follows that In Pb In Pb r =a and In0 Pb In Pb r =a . Thus, the right-hand side of Eq. (30) becomes vanishingly small for small r . In other words, for high frequencies and for nodal circles not close to the edge, the nodal circles may be approximated by Jn Pa r =a 0:

31

That is, the values of the nodal circles may be approximated from the zeros of the Bessel function.

Note that the e ects of rotary inertia and of the spin do not enter into the argument. It must be emphasized once again that the above approximation is for high frequencies and for small r .

4. The frequency equation for p2 The frequency equation (25) represents an equation in terms of two variables, Pa and Pb . Of course, these two variables are not independent, but are related through Eq. (13). The actual frequency, k may consequently be found from Eq. (14). In fact, Eq. (25) is the frequency equation for four possible circular plate cases and it is the relationship between Pa and Pb that determines precisely which case is being considered. In its present form, Eq. (25) depends on the parameters n and m, as well as the two variables Pa and Pb . Note that thus far the spin rate and/or the plate thickness-to-diameter ratio do not appear directly in Eq. (25). In turns out that these factors will appear when the appropriate relationship between Pa and Pb is formulated, thus entering the frequency equation indirectly through this second equation. The aforementioned four cases are for a spinning or non-spinning (stationary) plate with or without rotary inertia taken into account.

De®ne the variable k such that k2

q 1

m2 X2 : E

32

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Note that k represents the e ect of the spin. For a stationary (non-spinning) plate, X 0 and thus k 0 as well. Also de®ne the bending sti ness of the plate as D

2Eh3 ; 3 1 m2

33

where h is the half-thickness of the plate. Recall that the following relationships hold between a and b as a consequence of their de®nition a2

b2 b

2 2

ab

q 1

m2 X2 k2 E

3q 1 m2 k2 c : Eh2

34 35

Each case will now be considered separately.

1. Stationary plate, no rotary inertia In this case, X 0 and rotary inertia is neglected. Thus, b 0 and Eq. (34) yields Pa Pb :

36

2. Stationary plate, rotary inertia included Here X 0 as before, but now b is non-zero. In fact, b q 1 m2 k2 =E . However, Eq. (35) can be solved for k in terms of Pa and Pb . This procedure yields the required relationship between Pa and Pb s 3Pa2 : 37 Pb 3 Pa2 h=a 2 Note that the ratio of plate-thickness-to-diameter, h=a now enters the frequency equation through this relationship. Since rotary inertia is being taken into account, this seems a reasonable result.

3. Spinning plate, no rotary inertia In this case, b is nonzero. However, b k 2 since rotary inertia is being neglected. Thus the relationship between Pa and Pb is given by q Pb Pa2 ka 2 : 38 Note that in this case, the variable k has entered into the frequency equation. As previously mentioned, k represents the e ect of the plate spin and thus it is to be expected that it appears in the calculation of natural frequencies.

4. Spinning plate, rotary inertia included Based on the results of the previous cases, it would be expected at this point that both the thickness-to-diameter ratio of the plate and the e ect of rotation appear in the relationship between Pa and Pb . This is indeed the case. Eq. (35) can be solved for k and the resulting expression substituted into Eq. (34). This procedure yields the following relationship s 3 Pa2 ka 2 : 39 Pb 3 Pa2 h=a 2 Thus to ®nd the natural frequencies of vibration of a circular plate for one of the above four cases, it su ces to solve Eq. (25) in conjunction with one of Eqs. (36)±(39).

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Once the appropriate values of Pa and Pb have been found, the frequency itself may be found from (14) s Pa2 Pb2 D : 40 k 2qha4 Again, note that this is merely a generalized version of the classical relationship found in many texts for stationary circular plates. Since the classical case corresponds to the no-spin, no rotary inertia case, it may be obtained by substituting Pa Pb into the above equation.

The frequency (25) can be rewritten in a simpler form as A11 Jn Pa In Pb A21 Jn0 Pa In Pb A12 Jn Pa In0 Pb A22 Jn0 Pa In0 Pb 0;

41

where A11 1 m n2 Pa2 Pb2 ; i h 2 2 A21 Pa 1 m n2 n2 1 m Pb2 ; i h 2 2 2 2 2 A12 Pb n 1 m Pa 1 m n ; A22 1

m Pa Pb Pa2 Pb2 :

As previously mentioned, this equation must be solved in conjunction with one of Eqs. (36)±(38) or (39). For numerical simulations, the easiest approach is to substitute one of Eqs. (36)±(38) or (39) into the frequency equation. This will yield an equation in Pa only, which can then be solved numerically. For each value of n, there will be an in®nite number of solutions. However, there is another way of viewing the above problem which lends itself to insightful observations. Instead of substituting one of Eqs. (36)±(38) or (39) into the frequency equation, it is more interesting to consider the problem of solving the frequency equation and one of Eqs. (36)±(38) or (39) simultaneouslya. In this case, the frequency equation expresses a relationship between Pa and Pb and is thus a curve in the Pa ±Pb plane. Similarly, Eqs. (36)±(39) are relationships between Pa and Pb and are thus also curves in the Pa ±Pb plane. Hence the values of Pa and Pb that satisfy both the frequency equation and one of Eqs. (36)±(38) or (39) are intersections of the two curves in the Pa ±Pb plane. By sketching these curves, numerical simulations are not required to obtain insights into the e ect of certain parameters on the natural frequencies. Note that from their de®nition, interest is con®ned to the case where both Pa and Pb are positive. That is, only the ®rst quadrant of the Pa ±Pb plane will be considered.

4.1. Case 1: stationary plate, no rotary inertia In this case, we have Pa Pb . In the Pa ±Pb plane, this is straight line with unit slope passing through the origin. This is the classic problem whose solution is well documented. The solutions Pa of the frequency equation depend only on the value of m and can be found in the literature [4,5].

As a ®rst approximation, Kirchho has given the value p Pa n 2s ; 2

n; s 0; 1; 2; . . . ;

42

where s is the number of nodal circles and n is the number of nodal diameters [3,4]. This value can be used as an initial guess when using a numerical method to solve the frequency equation.

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4.2. Case 2: stationary plate, rotary inertia included This case is illustrated in Fig. 1, with case 1 included for comparison. As can be seen from the ®gure, the Pa ±Pb curve for case 2 resembles a parabola opening up onto the Pa axis. For small case values of Pa , it matches the Pa Pb straight line of p 1 exactly. However, as Pa increases, the curve approaches the horizontal asymptote of Pb 3a=h. From this, it can be concluded that at low frequencies, rotary inertia may be safely ignored, since there is virtually no di erence between the Pa ±Pb curves for the two di erent cases. However, at high frequencies, there is a marked di erence between the two cases and rotary inertia should be taken into account. The thinner the plate, the smaller the thickness-to-diameter ratio h=a and thus the two curves become di erent at higher and higher frequencies. Furthermore, it may be seen that compared to case 1, the values of Pb that satisfy both the frequency equation and the Pa ±Pb relation are much smaller. From Eq. (40), it can be seen then that the net e ect of the inclusion of the rotary inertia is to lower the natural frequencies of vibration.

By graphing the Pa ±Pb curves for cases 1 and 2 in this manner, the frequency at which rotary inertia must be included, can be deduced. This occurs wherever the Pa ±Pb curves for the two cases start to diverge. The fact that the rotary inertia can be ignored at low frequencies for thin plates and that this becomes less valid as the thickness of the plate or the frequency of vibration increase is well known. While these conclusions are not new, the ease with which they were obtained is once again emphasized.

4.3. Case 3: rotating plate, no rotary inertia This case is illustrated in Fig. 2. From the form of the equation or from the ®gure, it may be seen that the Pa ±Pb relation for case 3 is a hyperbola. The straight line Pa ±Pb corresponding to case 1 is an asymptote to the curve, and furthermore, Pa P ka. From this, we may deduce that the rotation a ects mostly the lower frequencies where the two curves are di erent. At high

Fig. 1. Graphs of Pa vs Pb for cases 1 and 2.

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571

Fig. 2. Graphs of Pa vs Pb for cases 1 and 3.

frequencies, the natural frequencies of a rotating plate and those of a stationary plate will be very similar, although they would not necessarily correspond to the same mode of vibration. Thus, di erent modes may correspond to the same frequencies. This is an interesting result since if the (high) natural frequencies are of interest without regard to which mode they belong to, those of a stationary plate may be used instead. This fact is useful since these values are readily available in the literature and are easier to measure experimentally.

Furthermore, the e ect of the rotation of the plate is readily apparent. In the low frequency range, the values of Pa that satisfy both the frequency equation and the Pa ±Pb relation are higher than in the ®rst case, while the values of Pb may be comparable. This follows since Pa cannot be less than ka. From Eq. (40), it can be deduced that the e ect of the rotation generally is to raise the natural frequency of vibration. The frequency of vibration can also be similarly raised by increasing D, the sti ness of the plate. In other words, the rotation has the general e ect of sti ening the plate. This sti ening has been previously observed in the ®eld of rotating structures.

The frequency equation may once again be rearranged Jn Pa Pa Pb4 2n2 1 m Pb2 n2 n2 1 1 m 2 fIn Pb =In0 Pb g 1 m Pa Pb Pa2 Pb2 :

Jn0 Pa Pb Pa4 2n2 1 m Pa2 n2 n2 1 1 m 2 1 m n2 Pa2 Pb2 fIn Pb =In0 Pb g 43 Now consider the low frequencies for the case where the spin rate, X, is very large. This implies that Pa is large and Pa Pb . From Eq. (43), if Pa dominates, then the right-hand side is very small.

In other words, for Pa Pb , Eq. (43) becomes Jn Pa 0: Since Pa is assumed to be large, the asymptotic expression for Jn may be used r

2 np p cos x : Jn x px 2 4

44

45

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Thus we have

Jn Pa cos Pa

np 2

p 0: 4

46

The zeros of the cosine function are the odd multiples of p=2. Furthermore, recall that Pa must be at least as large as ka and that Pa is assumed to be large. Thus the above argument yields as a ®rst approximation p 47 Pa n 2m ka; 2 where n is the number of nodal diameters, n 0; 1; 2; . . . and m 0; 1; 2; . . .. While n is the number of nodal diameters as in the stationary plate case, m does not necessarily correspond to the number of nodal circles as it would for the stationary plate case. Note how this approximation reduces to Kirchho 's ®rst approximation when k 0, as it should. Furthermore, it implies the raising of the natural frequencies due to the e ect of the spin, so it captures the aforementioned sti ening e ect of the rotation. This is a crude approximation but it may be used as an initial guess in numerical solutions.

4.4. Case 4: rotating plate, rotary inertia included As would be expected, the individual features of cases 2 and 3 are now both in play. The Pa ±Pb curves in Fig. 3 compare cases 1 and 4 where the equivalent curves are shown; case 1 without the e ect of rotation or rotary inertia and case 4 including both. The Pa ±Pb curves seen in Fig. 4 compare the equivalent curves with or without the e ect of rotation only since both curves include the e ect of rotary inertia. As previously observed, for plates with larger thickness-to-diameter ratios at high frequencies, the e ect of rotary inertia should not be disregarded. Note that when the spin rate X is high, the `low' frequencies of vibration are actually high compared to those of a stationary plate. In other words, the e ect of rotary inertia should be included even at low

Fig. 3. Graphs of Pa vs Pb for cases 1 and 4.

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573

Fig. 4. Graphs of Pa vs Pb for cases 2 and 4.

frequencies of vibration of a plate spinning at a high spin-rate. This is true even if the plate is thin.

Again, to reiterate, the e ect of rotary inertia may be safely ignored at low frequencies of vibration of a thin stationary plate, but not for a thin plate that is rotating very rapidly. Note that this conclusion could not have come out of the analysis of a stationary plate. This serves to point out that some of the approximations that are well-accepted for stationary plates may not be applied to spinning plates as they may not necessarily be as valid for the spinning problem.

5. Numerical simulations Numerical simulations were carried out to solve the frequency equation for various cases. For all cases, the values of E 2 1011 N/m, m 0:25; q 7:8 103 kg=m3 , plate radius a 0:6 m, and plate half-thickness h 0:01 m were used. These values correspond exactly to those used by Lamb and Southwell [1] in their pioneering paper.

5.1. Frequencies of vibration The ®rst case simulated was for X 0, that is, no spin. In this case Pa Pb and the solutions to the frequency equation should be those for a stationary plate. These solutions are widely available in the literature, thus providing a way of verifying the accuracy of the developed equations and code. The resulting values of Pa are given in Table 1.

Next, a stationary plate (X 0) with rotary inertia was considered. The resulting solutions Pa of the frequency equation are given in Table 2. Note that taking the e ect of rotary inertia into account results in smaller values of Pa . In addition, the lower frequencies do not di er much from their values without rotary inertia taken into account. It is at high frequencies that the presence of rotary inertia becomes apparent. In other words, the e ect of rotary inertia on the frequencies of vibration of a stationary disk can be ignored at low frequencies. However, at high

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Table 1

Solutions of the frequency equation for a stationary disk without rotary inertia, ka 0 Pa for a stationary disk, ka 0 n

s 0

s 1

s 2

s 3

s 4

s 5

0 1 2 3 4 5 6

± ± 2.3476 3.5699 4.7236 5.8446 6.9459

2.9816 4.5177 5.9405 7.2906 8.5916 9.8572 11.0960

6.1916 7.7286 9.1842 10.5837 11.9416 13.2672 14.5669

9.3619 10.9028 12.3801 13.8102 15.2034 16.5668 17.9056

12.5186 14.0635 15.5559 17.0071 18.4252 19.8157 21.1828

15.6694 17.2177 18.7210 20.1879 21.6245 23.0356 24.4247

Table 2

Solutions of the frequency equation for a stationary disk with rotary inertia, ka 0 Pa for a stationary disk with RI, ka 0 n

s 0

s 1

s 2

s 3

s 4

s 5

0 1 2 3 4 5 6

± ± 2.3473 3.5693 4.7226 5.8432 6.9439

2.9809 4.5161 5.9380 7.2870 8.5869 9.8513 11.0888

6.1887 7.7243 9.1782 10.5759 11.9319 13.2556 14.5532

9.3556 10.8944 12.3693 13.7969 15.1875 16.5482 17.8842

12.5075 14.0496 15.5390 16.9871 18.4019 19.7890 21.1527

15.6522 17.1970 18.6967 20.1598 21.5926 22.9997 24.3846

frequencies of vibration, it should be taken into account. Again, this is a well-known result and serves to verify the validity of the equations developed here.

Now consider higher values of the dimensionless spin parameter ka, without the e ect of rotary inertia. It appears that certain modes of vibration of the stationary plate get cut o . For example, the resulting values of Pa corresponding to ka 1:65 are given in Table 3. For ka 1:65, the n 2, s 0 mode is cut o . That is, our model predicts that this mode should not be observed at all in a spinning plate. The n 2, s 0 mode would have been the mode with the lowest frequency of vibration had it not been cut o . Now that this mode is cut o , the mode with the lowest frequency of vibration is the n 0, s 1 mode. However, the n 0, s 1 mode corresponds to Pa 2:5912 for ka 1:65, compared with Pa 2:9816 for ka 0. In other words, the e ect of the spin is to lower the frequency of vibration of a particular mode even though the frequencies of vibration are overall now higher due to the absence of particular modes.

Since Pa cannot be less than ka, we can now see why certain modes get cut o resulting in overall higher vibration frequencies. In essence, the e ect of the spin is to lower the frequency of Table 3

Solutions of the frequency equation for a rotating disk without rotary inertia, ka 1:65 Pa for a rotating disk, ka 1:65 n

s 0

s 1

s 2

s 3

s 4

s 5

0 1 2 3 4 5 6

± ± ± 3.4394 4.6591 5.8053 6.9191

2.5912 4.3955 5.8778 7.2520 8.5653 9.8382 11.0816

6.1319 7.6918 9.1588 10.5651 11.9273 13.2559 14.5578

9.3372 10.8848 12.3664 13.7992 15.1945 16.5594 17.8993

12.5050 14.0529 15.5472 16.9999 18.4191 19.8105 21.1783

15.6609 17.2106 18.7150 20.1828 21.6201 23.0317 24.4212

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vibration of a particular mode until it becomes disallowed as Pa must always be greater than or equal to ka. This points to the existence of critical speeds of rotation. At some particular value of the spin rate X, some modes may cease to exist. Again, note that despite the cutting o of certain modes and the lowering of the frequencies of neighbouring modes, the higher modes of vibration (those not near the cuto point) are hardly a ected by the e ect of spin. This con®rms the predictions made by investigating the shapes of the Pa ±Pb curves. For values of Pa much larger than ka, the Pa ±Pb curves for a spinning plate approach asymptotically those of a stationary plate. The prediction that the modes and frequencies of vibration of a stationary plate can be used in place of those of a spinning plate for the higher modes has thus been veri®ed numerically.

The same trend is observed for higher values of dimensionless spin parameter ka. For higher values of ka such as ka 3, even more modes are cut o . The frequencies of modes adjacent to the cuto modes are lower compared to the corresponding values for a stationary plate, while those of modes far from the cuto modes are once again hardly di erent (although still lower) from the corresponding values for a stationary plate. The resulting values of Pa can be found in Table 4.

Now consider the e ect of both rotary inertia and the spin of the disk. The solutions to the frequency equation Pa for ka 3 with the e ect of the rotary inertia included are given in Table 5.

From the table, it is easy to deduce that the presence of rotary inertia a ects a spinning plate the same way as it a ects a stationary plate. In other words, rotary inertia tends to lower the frequencies of vibration and mostly a ects the higher frequencies of vibration.

The next case simulated was one for which X 100p, without the e ect of rotary inertia taken into account. This is the same value used by Lamb and Southwell [1] and corresponds to ka 0:036. The resulting values of Pa can be found in Table 6. Note that the smallest non-zero value of Pa that is a solution to the frequency equation for a stationary plate is Pa 2:3476. Since Pa , Pb and ka are dimensionless quantities, the small value of ka for this case leads us to expect that the spin will not have much e ect on the resulting solutions to the frequency equation. This is indeed the case, as can be seen from Table 6. Only the smaller values of Pa are a ected, and even so only at the fourth decimal place. The higher frequencies do not appear to be a ected at all by the spin of the plate. The value of ka is not large enough to cut o any of the stationary plate modes. In other words, for the spin rate used by Lamb and Southwell, the values of p2 predicted by the LSM and the TLM are nearly identical.

5.2. Observations from simulations 5.2.1. Some stationary plate modes get cut o After performing many more simulations with various values of the spin parameter, a general trend starts to become apparent. It is the relationship between the value of dimensionless spin parameter ka and dimensionless natural frequencies Pa of the stationary plate that determines the Table 4

Solutions of the frequency equation for a rotating disk without rotary inertia, ka 3:0 Pa for a rotating disk, ka 3:0 n

s 0

s 1

s 2

s 3

s 4

s 5

0 1 2 3 4 5 6

± ± ± ± 4.4401 5.6919 6.8474

± 3.9348 5.6949 7.1490 8.4983 9.7909 11.0464

5.9678 7.5972 9.0958 10.5196 11.8929 13.2289 14.5360

9.2763 10.8414 12.3334 13.7733 15.1734 16.5420 17.8847

12.4727 14.0276 15.5268 16.9830 18.4049 19.7983 21.1678

15.6407 17.1941 18.7011 20.1709 21.6098 23.0227 24.4133

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Table 5

Solutions of the frequency equation for a rotating disk with rotary inertia, ka 3:0 Pa for a rotating disk with RI, ka 3:0 n

s 0

s 1

s 2

s 3

s 4

s 5

0 1 2 3 4 5 6

± ± ± ± 4.4392 5.6904 6.8453

± 3.9339 5.6925 7.1454 8.4935 9.7848 11.0389

5.9652 7.5930 9.0899 10.5119 11.8832 13.2171 14.5221

9.2701 10.8330 12.3227 13.7600 15.1575 16.5233 17.8632

12.4616 14.0137 15.5099 16.9630 18.3816 19.7716 21.1375

15.6235 17.1734 18.6769 20.1429 21.5779 22.9868 24.3732

Table 6

Solutions of the frequency equation for a rotating disk without rotary inertia, ka 0:036 Pa for a rotating disk, ka 0:036. n

s 0

s 1

s 2

s 3

s 4

s 5

0 1 2 3 4 5 6

± ± 2.3475 3.5699 4.7235 5.8446 6.9459

2.9815 4.5176 5.9405 7.2906 8.5916 9.8572 11.0960

6.1915 7.7286 9.1842 10.5837 11.9416 13.2672 14.5669

9.3619 10.9028 12.3801 13.8102 15.2034 16.5668 17.9056

12.5186 14.0635 15.5559 17.0071 18.4252 19.8157 21.1828

15.6694 17.2177 18.7210 20.1879 21.6245 23.0356 24.4247

results for the rotating plate. If the dimensionless frequency Pa of a particular stationary plate mode is less than ka, then that mode will be absent for the rotating plate. If the stationary plate value of Pa is larger than ka but close in value to ka, that mode may or may not exist for the rotating plate. Recall that the spin will lower the frequencies of a particular mode and thus it is not clear if a particular frequency will be lowered to less than ka or not. Finally, if the stationary plate value of Pa is much greater than ka, then the frequency and mode shape of that particular mode are hardly a ected by the rotation of the plate.

5.2.2. Stationary plate modes can be used for higher modes For values of Pa greater than but close to ka, a certain amount of care must be taken. In fact, it is these modes that will have the lowest frequencies of vibration and are thus of the greatest concern. These modes will also be the most di erent from their stationary plate values. For values of Pa not too close to ka, the stationary plate values of Pa may be used as starting points in the numerical solution of the rotating plate values. In fact, the stationary plate values may also be used as an approximation to the rotating plate values for Pa ka since it is known that they do not di er much.

5.2.3. Rotary inertia The question of whether or not the rotary inertia must be included can also be addressed in a similar fashion. From the stationary plate problem, it is known that for thin plates rotary inertia will tend to decrease the values of Pa that satisfy the frequency equation. Furthermore, it is known that the e ect of rotary inertia only a ects the higher values of Pa . Thus, for a given plate, it is usually known how large Pa has to be before the rotary inertia of the plate plays a role in the calculation of the natural frequencies. So, let us suppose that for Pa > Pa the rotary inertia must be included to correctly calculate the frequency. This critical value of Pa is still applicable to the

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rotating disk. Furthermore, if ka > Pa , then the e ect of rotary inertia must be taken into account even at the lowest frequencies of the rotating plate. Note that this will correspond to high angular speeds of the disk and the assumption of small in-plane displacements inherent in the model may cease to be valid.

6. Discussion Now that the frequency predictions of the LSM have been thoroughly analyzed, we may return to compare the LSM, NLSM, TLM and discussing the validity of the linear strain assumption. It is now obvious that the linear strain assumption for rotating plates is NOT a good assumption.

First, we note that for the spin rate taken by Lamb and Southwell, p2 for the stationary plate and the rotating plate of the LSM are identical. If the LSM model alone is usedp to predict frequency, 2 2 we would have p p2 . However, for the TLM of Lamb and Southwell, p p1 p2 . Unless p1 is insigni®cant, the LSM will underpredict the frequency. Furthermore, we note that the predictions of the NLSM will match those of the TLM since the e ect of p1 is included in this model.

Based on the foregoing analysis of the LSM, the frequencies predicted by the NLSM will be similar to but slightly smaller than the predictions of the TLM. For the angular velocity used by Lamb and Southwell, the two models are in complete agreement. At higher angular velocities, the bending moment due to the centrifugal force will become more apparent and the NLSM and TLM will begin to di er in the predictions of the frequencies of the lower modes of vibration.

Since it was observed from the simulations that the stationary plate frequencies can be used for the higher frequencies of the LSM, at high frequencies the NLSM and TLM will once again agree.

The ease and simplicity-of-use of the LSM comes at a price; it is evidently not valid for angular velocities large enough to render the assumption of linear strains inadequate. The LSM can be used to accurately re¯ect the dynamics of a rotating plate only for angular velocities small enough to make p1 negligible.

7. Conclusions Two linear models of spinning disks were considered, one arising from the use of linear strains and the other from the use of von Karman strains. The linear equation of motion for small transverse vibrations of a spinning disk modelled as a spinning plate with linear strains has been solved and analyzed and the natural frequencies of vibration were obtained. In contrast with the traditional model of a spinning disk, the frequency equation was found in closed form. This frequency equation is a relationship in two variables. The four cases of stationary and rotating plate with and without rotary inertia have the same frequency equation. The particular case being considered depends on the relation between the two variables in the frequency equation. Thus, ®nding the natural frequencies of vibration amounts to solving two equations in two variables. It turns out that the solution of the stationary or rotating plate problems are quite similar and the well-known results for the stationary plate can be used as a starting point for the rotating disk problem. This new model predicts that certain modes corresponding to the stationary plate will be cut o if the plate rotates at particular angular speeds. The net e ect of having some of the lower modes cut o is the raising of the lowest frequencies of vibration. However, comparison with existing work makes it evident that this equation cannot accurately represent the spinning disk at angular velocities that are not extremely small. For angular velocities that are typical of spinning disks, the second equation based on nonlinear strains must be used. This second equation is closer

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in form to the traditional model, yet it contains a few terms neglected in the traditional model. It was shown that this second equation is in agreement with the traditional model for the angular velocities used. At higher angular velocities, the bending moment due to the centrifugal force should be taken into consideration and thus the NLSM and TLM may begin to di er appreciably in their predictions of the lower modes. The most subtle conclusion is that despite of its success for stationary plates, the assumption of linear strains quickly becomes inadequate in the realm of spinning bodies.

Acknowledgements This research was ®nancially supported by the National Science and Engineering Research Council of Canada.

References [1] H. Lamb, R.V. Southwell, The vibrations of a spinning disk, Proc. R. Soc. Lond., Ser. A 99 (1921) 272±280.

[2] J.G. Simmonds, The transverse vibrations of a ¯at spinning membrane, J. Aeronaut. Sci. 29 (1962) 16±18.

[3] R.C. Colwell, H.C. Hardy, The frequencies and nodal systems of circular plates, Philos. Mag., Ser. 7 24 (165) (1937) 1041±1055.

[4] A. Leissa, Vibrations of Plates. NASA SP-160, Washington, DC, 1969.

[5] L. Raleigh, Theory of Sound, Macmillan, London, 1937.