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Non-rotating beams isospectral to a given rotating uniform beam
International Journal of Mechanical Sciences 66 (2013) 12–21
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International Journal of Mechanical Sciences journal homepage: www.elsevier.com/locate/ijmecsci
Non-rotating beams isospectral to a given rotating uniform beam Sandilya Kambampati, Ranjan Ganguli n, V. Mani Department of Aerospace Engineering, Indian Institute of Science, Bangalore 560012, India
a r t i c l e i n f o
a b s t r a c t
Article history: Received 23 July 2012 Received in revised form 9 October 2012 Accepted 17 October 2012 Available online 30 October 2012
In this paper, we seek to find non-rotating beams with continuous mass and flexural stiffness distributions, that are isospectral to a given uniform rotating beam. The Barcilon–Gottlieb transformation is used to convert the fourth order governing equation of a non-rotating beam, to a canonical fourth order eigenvalue problem. If the coefficients in this canonical equation match with the coefficients of the uniform rotating beam equation, then the non-rotating beam is isospectral to the given rotating beam. The conditions on matching the coefficients leads to a pair of coupled differential equations. We solve these coupled differential equations for a particular case, and thereby obtain a class of non-rotating beams that are isospectral to a uniform rotating beam. However, to obtain isospectral beams, the transformation must leave the boundary conditions invariant. We show that the clamped end boundary condition is always invariant, and for the free end boundary condition to be invariant, we impose certain conditions on the beam characteristics. We also verify numerically that the frequencies of the non-rotating beam obtained using the finite element method (FEM) are the exact frequencies of the uniform rotating beam. Finally, the example of beams having a rectangular cross-section is presented to show the application of our analysis. Since experimental determination of rotating beam frequencies is a difficult task, experiments can be easily conducted on these rectangular non-rotating beams, to calculate the frequencies of the rotating beam. & 2012 Elsevier Ltd. All rights reserved.
Keywords: Free vibration Rotating beam Isospectral Modal analysis Finite element Eigenvalue
1. Introduction An undamped vibrating system has a set of natural frequencies, forming a spectrum, at which it vibrates freely without external forces. An important class of problems for such vibrating systems can be broadly classified into inverse problems and isospectral problems. In inverse problems [2,3], one tries to determine the material properties of a system for a given frequency spectrum. In general, more than one frequency spectrum is required for a reconstruction procedure to determine the material properties of the system [4]. Systems, that have the same vibrating frequencies, but have different material properties are called isospectral systems. Isospectral systems are of great interest in mechanics as they yield alternative usable designs. The existence of isospectral systems also proves that a system cannot be uniquely identified from its spectrum. 1.1. Isospectral spring–mass systems Examples of discrete isospectral systems are in-line spring– mass systems. A schematic of a 2-DOF spring mass system is n
Corresponding author. E-mail address: [email protected] (R. Ganguli).
0020-7403/$ - see front matter & 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijmecsci.2012.10.004
shown in Fig. 1 , where m1 ,m2 are the masses and k1 ,k2 are the spring constants of the springs. Let us consider one such 2-DOF system (System-A) [1] where the values of masses m1 and m2, and spring constants k1 and k2, and the natural frequencies of SystemA (o1 and o2 ) are tabulated in Table 1. Similarly, let us consider one more system (System-A0 ) where m01 and m02 are the masses, k01 and k02 , are the spring constants and o01 and o02 are the natural frequencies, whose values are tabulated in Table 1. From Table 1, we can see that both System-A and System-A0 have the same frequency spectrum as o1 ¼ o01 and o2 ¼ o02 . Therefore, System-A and System-A0 are isospectral. Gladwell [5] described four ways to form in-line spring mass systems isospectral to a given one. Gladwell [6] also considered a discrete model of a vibrating cantilever beam, and presented two procedures for finding families of such beams, isospectral to a given one. Gottlieb [7,8] studied isospectral vibrating strings with discontinuous coefficients. All these studies addressed discrete models of vibrating beams. Borg [9] studied isospectral systems corresponding to a vibrating string with continuous coefficients (second order governing equation). Gottlieb [10] analyzed the non-uniform Euler–Bernoulli beam equation and gave seven classes of non-uniform beams isospectral to the given uniform beam with different boundary conditions. Subramanian and Raman [11] generalized Gottlieb’s method for tapered beams. However, studies on isospectral systems have not
S. Kambampati et al. / International Journal of Mechanical Sciences 66 (2013) 12–21
Fig. 1. Schematic of a 2DOF spring–mass system.
Table 1 Material and spectral properties of System-A and System-A0 . System-A0
System-A m1 m2 k1 k2
o1 o2
6.25 kg 4 kg 92.5 kN/m 20 kN/m 62.2 rad/s 138.3 rad/s
m01 m02 k01 k02
o01 o02
1.286 kg 1 kg 15.857 kN/m 6 kN/m 62.2 rad/s 138.3 rad/s
addressed rotating beams. In this study, we seek to find nonuniform beams that are isospectral to a given rotating beam. Such beams, if they exist, have important applications in the dynamics of rotating systems. Rotating beams serve as a useful mathematical model to simulate vibration of helicopter blades, wind turbines, long flexible rotating space booms, turbo-machinery blades, etc. A rotating mechanical system can suffer from high vibration, if its natural frequencies coincide with multiples of the rotation speed. Therefore, an accurate determination of the frequencies is an important aspect in rotating blade design. The natural frequencies and mode shapes can be determined using an approximation scheme such as the Rayleigh– Ritz method [12], Galerkin method [13], finite element method [14–19], differential transform method [20,21] or the dynamic stiffness method [22]. Hodges and Rutkowski [23] analyzed the out of plane vibrations of a rotating beam using a finite element method of variable order. Wright et al. [24] presented an accurate solution for the mode shapes of a beam attached to a rotating hub using the method of Frobenius. Storti and Aboelnaga [25] listed the classes of beams which admit hypergeometric solutions to the mode shape equation. Naguleswaran [26] solved the mode shape equation using the method of Frobenius. Low [27] developed an algorithm for solving frequency equations for a cantilever double-span non-rotating beam and a uniform rotating beam. Thus, we see that considerable amount of research has been done on the models of a rotating beam. Experimental determination of rotating beam frequencies can be difficult. For example, Senatore [28] experimentally determined the frequencies of a rotating beam, using lumped parameter axial loads on a uniform non-rotating beam, in order to simulate an approximation of the centrifugal force field, acting on the rotating beam. However, this method could not predict exactly the first natural frequency and mode shape due to the lumped axial loads. Hence, it is interesting to see, if we can find any non-rotating beam which is isospectral to the uniform rotating beam. It is important to take into account the exact centrifugal force acting on the rotating beam. The reasons for obtaining isospectral beams are the following. (a) It is difficult to conduct experiments on rotating beams to obtain its natural frequencies. If we find an isospectral non-rotating beam, one can easily conduct experiments on the non-uniform beam, to obtain the natural frequencies. (b) Such isospectral beams can provide insight by predicting the stiffening effect of the centrifugal force. (c) Provide benchmark problems for finite element analysis. In this paper, we find non-rotating beams with continuous mass and flexural stiffness distributions, that are isospectral to a given uniform rotating beam. The mass and stiffness functions of non-rotating beams, isospectral to a uniform beam rotating at different speeds are derived. We note that for high rotating speeds, the derived mass and stiffness functions of the
13
non-rotating beam are not physically realizable, owing to stiffening effect of the centrifugal force. In such situations, if we attach a torsional spring, of a spring constant KR, at the free end of the non-rotating beam, the obtained mass and stiffness functions become physically realizable. We also show numerically that the frequencies of the non-rotating beam obtained using the finite element method (FEM) are the exact frequencies of the uniform rotating beam. This confirms numerically that the non-uniform beam obtained in this method is isospectral to the given uniform rotating beam. We also provide an example of realistic beams having a rectangular cross-section to show a physically realizable application of our analysis.
2. Mathematical analysis In this section, the mathematical formulation of the problem is presented. The governing differential equation for the transverse free vibrations V(Z) of a uniform rotating beam (Fig. 2) of length L, stiffness EI0 and mass M0 rotating with an angular speed O is given in [23] as: " # " # ! 4 2 2 d V d dV 2 L Z M EI0 O ð1Þ o2 M0 V ¼ 0, 0 rZ r L 0 dZ dZ 2 dZ 4 We introduce a non-dimensional variable z ¼ Z=L so that the above equation can be rewritten as 4 2 d V d dV 2 1z Z2 V ¼ 0, 0 r z r1 l ð2Þ dz 2 dz4 dz where Z is the non-dimensional frequency given by qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z ¼ o M0 L4 =EI0
ð3Þ
and l is the non-dimensional rotation speed given by qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi l ¼ O M0 L4 =EI0
ð4Þ
Similarly, the governing equation for the out of plane free vibrations YðXÞ of a non-uniform non-rotating Euler–Bernoulli beam (Fig. 3), which is isospectral to the rotating beam is given by [29] as " # 2 2 d d Y EIðXÞ ð5Þ o2 MðXÞY ¼ 0, 0 rX rL dX 2 dX 2 where EI(X) is the flexural stiffness , M(X) is the mass/length, L is the length of the beam, which is same as that of the uniform rotating beam (L ¼ LÞ. Now, we introduce non-dimensional variables f , m and x as f ðxÞ ¼
EIðXÞ , EI0
mðxÞ ¼
MðXÞ , M0
x¼
X L
ð6Þ
Eq. (5) can be rewritten as ðf ðxÞY 00 Þ00 ¼ Z2 mðxÞY,
0 r x r1
ð7Þ
Here, the notation Y 0 and Y 00 represent the first and the second derivatives respectively of Y w.r.t x. The transformation, which
Fig. 2. Schematic of a uniform rotating beam.
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S. Kambampati et al. / International Journal of Mechanical Sciences 66 (2013) 12–21
Here, the notation q(z) means the value of q at the coordinate z. The first, second, third and the fourth derivatives, respectively of q w.r.t z are denoted by qz, qzz, qzzz and qzzzz. Therefore, we can conclude the above analysis by establishing the following. The governing differential equation of the non-uniform nonrotating beam is " # 2 2 d d Y f ð20Þ Z2 mY ¼ 0, 0 r x r 1 dx2 dx2
Fig. 3. Schematic of a non-uniform non-rotating beam.
converts the above non-rotating non-uniform beam equation (5) to the canonical fourth order equation for the function U(z) 4 d U d dU A þ BU ¼ Z2 U, 0 rz r 1 þ ð8Þ dz dz dz4 2
is proposed by Barcilon [30]. When A ¼ l ð1z2 Þ=2 and B ¼0, U corresponds to V in Eq. (2). In the Barcilon–Gottlieb transformation [10,30], two auxiliary variables p and q are defined as 1=4 m 1 1=8 p¼ , q¼ ð9Þ 3 f m f which also means that f ¼ p3 q2 ,
m ¼ pq2
ð10Þ
The transformation of the independent variable (x) is given implicitly as Z x z¼ pðzðxÞÞ dx ð11Þ 0
which implies that Z z dz x¼ 0 pðzÞ
ð12Þ
d d 1 dx ¼ pðzÞ ) ¼ dx dz pðzÞ dz
ð13Þ
We note that x¼0 3 z¼0
ð14Þ
Also, we have x¼1 3 z¼1
YðxÞ ¼ qðzÞUðzÞ
ð16Þ
where 2
If the coefficients A(z) and B(z) of the non-rotating beam in Eq. (21) match with the coefficients of the uniform rotating beam equation (22), then the non-rotating beam is isospectral to the 2 given rotating beam; i.e, AðzÞ ¼ l ð1z2 Þ=2 and BðzÞ ¼ 0. Hence, a non-uniform non-rotating beam isospectral to a given rotating uniform beam is obtained when U corresponds to V. In other words, Eqs. (20) and (22) have the same natural frequencies if and only if the auxiliary functions p(z) and q(z) satisfy the following conditions: 2
AðzÞ ¼ l ð1z2 Þ=2
ð23Þ
BðzÞ ¼ 0
ð24Þ
We need to solve the above equations to obtain non-rotating beams isospectral to a given uniform rotating beam. These equations are nonlinear fourth order coupled differential equations which are difficult to solve analytically for a general q(z). However, for qðzÞ ¼ q0 which is a constant, it is possible to obtain the results analytically. Also, note that if q ¼ q0 , then m and f are 3 simply related throughout by f ¼ q8 . When qðzÞ ¼ q0 , we can 0 m see that BðzÞ ¼ 0 as the derivatives qz ,qzz ,qzzz and qzzzz vanish. 2 When AðzÞ ¼ l ð1z2 Þ=2, then from Eq. (18) we have
We see that Eqs. (12) and (16) transform the variables x and Y into z and U, respectively. This completes the Barcilon–Gottlieb transformation. Using this transformation, we get " # " # 2 2 4 d d Y d dU 2 1 d U 2 A þBUZ U f þ ð17Þ Z mY ¼ pq dz dz dx2 dz4 dx2
A¼
The expressions A(z) and B(z), for this non-rotating beam, are given in Eqs. (18) and (19). The governing differential equation of the uniform rotating beam which is rotating with a non-dimensional speed l is from Eq. (1), " # 4 d V d l2 2 dV Z2 V ¼ 0, 0 rz r 1 ð1z þ Þ ð22Þ dz dz 2 dz4
ð15Þ
The transformation of the dependent variable Y is given as
q q2 4 zz 6 z2 q q
The above equation can be transformed into a canonical form using Barcilon–Gottlieb transformation as 4 d U d dU AðzÞ þBðzÞUZ2 U ¼ 0, 0 r z r 1 þ ð21Þ dz dz dz4
qz pz þ qp
pzz p2 2 z2 p p
qzzzz qq q2 q2 q qp q p q2 p 4 z 2zzz 2 zz þ6 z 3zz þ z zzz þ zz zz 4 z 2 zz 2 q qp qp q q q q p qz pz pzz qz qzz pz q3z pz q2z p2z qz p3z qzz p2z þ 6 3 þ 6 2 2 þ4 4 2 5 qp2 q2 p q p q p qp3 qp2
ð18Þ
l2 2
ð1z2 Þ ¼
pzz 2p2z 2 p p
ð25Þ
The above equation is divided throughout by p and rewritten as 2
2
d p^ l ð1z2 Þp^ ¼ 0 2 dz2
ð26Þ
where p^ is defined using Eq. (13) as p^ ¼ p1 ¼
dx dz
ð27Þ
Note that at z¼1, from Eq. (26), we have " # " # 2 d p^ l2 2 ^ ð1z ¼ Þ p ¼0 2 dz2 z¼1
ð28Þ
z¼1
Also, using qðzÞ ¼ q0 in Eq. (16) and using Eq. (13), we obtain the following:
B¼
ð19Þ
These expressions for A and B are originally given by Barcilon [30] and later corrected by Gottlieb [10].
Y ¼ q0 U
ð29Þ
Y x ¼ q0 pU z
ð30Þ
Y xx ¼ q0 pðpz U z þ pU zz Þ
ð31Þ
S. Kambampati et al. / International Journal of Mechanical Sciences 66 (2013) 12–21
Y xxx ¼ q0 ppz ðpz U z þpU zz Þ þ q0 p2 ðpzz U z þ 2pz U zz þpU zzz Þ
ð32Þ
The initial conditions that are to be imposed on Eq. (26) depend on the physical boundary conditions (clamped, free, hinged, etc.). In the next section, we solve this equation for cantilever rotating and non-rotating beams. 2.1. Cantilever rotating and non-rotating beams In this section, we look for a non-uniform non-rotating cantilever beam which is isospectral to a given uniform rotating cantilever beam. The rotating and the non-rotating beams are clamped at the root (x ¼ 0 ¼ z) and free at the tip (x ¼ 1 ,z ¼ 1). Hence, we have the following boundary conditions: At the root: Y ¼ 0 ¼ Yx
ð33Þ
U ¼ 0 ¼ Uz
ð34Þ
15
and 1 F 1 ða1 ; b1 ; s2 Þ and 1 F 1 ða2 ; b2 ; s2 Þ are confluent hypergeometric functions [31,32] of the first kind given by 1F1ð
a; b; sÞ ¼
1 X ðaÞk sk ðbÞk k! k¼0
ð47Þ
where ðaÞk and ðbÞk are Pochhammer symbols [33]. ^ Since pðzÞ is now determined, m(z) and f(z) can be calculated using Eq. (10), by choosing q0 ¼ 1, and are given by mðzÞ ¼ ðk1 s1 1 F 1 ða1 ; b1 ; s2 Þ þz k2 s1 1 F 1 ða2 ; b2 ; s2 ÞÞ1
f ðzÞ ¼ k1 s1 1 F 1 ða1 ; b1 ; s2 Þ þ z k2 s1 1 F 1 ða2 ; b2 ; s2 Þ
ð48Þ
3
ð49Þ
Note that Eq. (49) can also be calculated from Eq. (48) by using 3 the relation f ¼ ðq8 , which simplifies to f ¼ m3 , as q0 ¼ 1. 0 Þm Similarly, x(z) can be calculated using Eq. (12), and given by Z z xðzÞ ¼ p^ dz ð50Þ 0
At the tip: ðfY xx Þx ¼ 0 ¼ Y xx
ð35Þ
In the above equation, since Y xx ¼ 0, we have ðfY xx Þx ¼ 0 ) f x Y xx þ fY xxx ¼ 0 ) Y xxx ¼ 0, Therefore, Eq. (35) can be modified as Y xxx ¼ 0 ¼ Y xx
ð36Þ
U zz ¼ 0 ¼ U zzz
ð37Þ
We can see from Eqs. (29)–(32), that the clamped boundary condition is satisfied in the Barcilon–Gottlieb transformation. So we need to satisfy only the free boundary condition for the cantilever beam. This means that the coefficients of the term Uz in Eq. (31) should be zero at z¼1 (since at z¼1, U zz ¼ 0 ¼ U zzz ). Therefore, in Eq. (31), pz at z¼1 should vanish. Similarly in Eq. (32), the coefficients of the term Uz should vanish at z¼1. Hence pzz should also be zero at z¼1. We know from Eq. (27) that p ¼ 1=p^ and hence, we have 2 pz ¼ p^ z p^ 2 3
pzz ¼ 2p^ z p^
ð38Þ p^ zz p^
2
ð39Þ
Therefore, from the above equations and Eq. (28), we have at z¼1 p^ z ¼ 0 3 pz ¼ 0 3 pzz ¼ 0
ð40Þ
Eq. (50) has no exact solution, and hence we integrate p^ (Eq. (42)) using Guass–Legendere quadrature and obtain x(z). The obtained m(z), f(z) and x(z) functions for l ¼ 1 are plotted in Fig. 4. Our interest is to obtain the mass and stiffness functions in terms of x (i.e. m(x) and f(x)). It is not possible to obtain m(x) and f(x) from m(z) and f(z) analytically, as there is no closed form expression for z(x). Here, for a given value of x, we cannot calculate the value of z analytically. If we were given a value of x, say x, we need to determine the value of z, say z. Hence we use the well known Newton–Raphson method to calculate z given any x, by using the iterative formula [34] xðzi Þx ð51Þ zi þ 1 ¼ zi xz ðzi Þ where xz ðzi Þ ¼
^ pðzÞ ¼ k1 s1 1 F 1 ða1 ; b1 ; s2 Þ þ zk2 s1 1 F 1 ða2 ; b2 ; s2 Þ
ð42Þ
ð52Þ
and xðzi Þ is obtained from Eq. (50). The initial estimate for this iterative formula can be taken as z1 ¼ x. For example, for x ¼ 0:6, we can use the above analysis to calculate z, which is z ¼ 0:588. Thus, in Fig. 4, if we draw a horizontal line x ¼ x ¼ 0:6, it cuts the graph x(z) at the point B(0:588,0:6). Using Eqs. (48) and (49), mðzÞ and f ðzÞ can be calculated, and are given by mðzÞ ¼ 1:023 and f ðzÞ ¼ 0:934. Thus, in Fig. 4, if we draw a vertical line from point B (z ¼ z ¼ 0:588), it cuts the graph m(z) at the point C ð0:588,1:023Þ and cuts the graph
Another initial condition p^ should satisfy is derived from Eq. (12), which is Z 1 Z 1 dz 3 1¼ 1¼ p^ dz ð41Þ 0 p 0 Therefore, Eq. (26) can be solved for any given rotation speed (l), using the two initial conditions (Eqs. (40) and (41)). For example, for l ¼ 1, we can find a corresponding isospectral nonrotating beam. The solution of Eq. (26), for l ¼ 1, is given as
dx ^ iÞ ¼ pðz dz z ¼ zi
1.4 f(z) 1.2 C : (z , m) = (0.588,1.023)
E: (m=f=1 ) 1 m(z)
f(z), m(z), x(z)
D :(z , f)= (0.588 , 0.934)
0.8 A : (z , m) = (0 , 0.6) 0.6
B : (z , x)= (0.588 , 0.6)
where pffiffiffi
a1 ¼ 2 2
! pffiffiffi 2 1 i , 16 16
! pffiffiffi pffiffiffi 3 2 1 i 16 16
a2 ¼ 2 2
0.4
ð43Þ 0.2
b1 ¼ 12 , b2 ¼ 32 pffiffi 2x2 i=4
s1 ¼ e
k1 ¼ 1:0911,
,
x(z)
z =0.588
ð44Þ 0
pffiffiffi 2x2 i s2 ¼ 2
ð45Þ
k2 ¼ 0:3376
ð46Þ
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
z Fig. 4. Variations of m(z), f(z ) and x(z) of the isospectral non-rotating beam (l ¼ 1).
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S. Kambampati et al. / International Journal of Mechanical Sciences 66 (2013) 12–21
1.1
1.4
rotating beam
1.05 1.2
1
f(x)
m(x)
non−rotating beam
0.95
1
0.9 0.8
0.85 0
0.5
0
1
0.5
1
x
x 1.2
non−rotating beam
2.5
rotating beam
1.1 f(x)
m(x)
2 1 0.9 0.8
1.5 1
0.7 0.5 0
0.5
0
1
0.5
1
x
x 1.4
non−rotating beam 5
rotating beam
1.2 1
f(x)
m(x)
4 3
0.8
2
0.6
1 0
0.5 x
1
0
0.5
1
x
Fig. 5. Variations of mass m(x) and flexural stiffness f(x) of isospectral cantilever non-rotating (q0 ¼ 1) and uniform rotating beams at different rotation speeds l: (a) l ¼ 1. (b) l ¼ 2. (c) l ¼ 3.
f(z) at the point D ð0:588,0:934Þ. Therefore, at the non-dimensional length x ¼ 0:6, the mass and the stiffness of the non-rotating beam isospectral to the given rotating beam, are given by mðxÞ ¼ 1:023 and f ðxÞ ¼ 0:934, respectively. Also, note also that for q0 ¼ 1, we have f ¼ m3 . Therefore, the graphs f(z) and m(z) crossover at point E (z¼0.385), where f ¼ m ¼ 1, thus confirming f ¼ m3 . Thus, for any given x, z(x), and hence mass (m(x)) and stiffness (f(x)) can be calculated. These m(x) and f(x) distributions are plotted in Fig. 5a. From Fig. 5a, we can infer that the non-rotating isospectral beam should be more stiff at the root and less stiff at the tip compared to the uniform rotating beam. Similarly, the mass distribution should be less at the root and more at the tip compared to the uniform rotating beam. In the above analysis, we have determined the non-rotating beam isospectral to the rotating beam for l ¼ 1. The mass and stiffness variations of beams isospectral to a uniform beam rotating at other speeds can be calculated similarly. For l ¼ 2 and l ¼ 3, m(x) and f(x) variations of isospectral beams are determined and plotted in Fig. 5b and c respectively. From the above figures, we can infer that as l increases, the stiffness of the isospectral non-rotating beam increases at the root
and decreases at the tip and asymptotically goes to zero for higher values of l. These mass and stiffness distributions have been plotted in Fig. 6 for l ¼ 4,5,6 and q0 ¼ 1. From Fig. 6, we can see that the isospectral non-rotating beam, on an average, has a high value of stiffness and a low value of mass, as compared to the uniform rotating beam. Thus we can say that, as l increases, it becomes more and more difficult for the variation in mass and the stiffness functions, to counter the stiffening effect of the centrifugal force, acting on the uniform rotating beam. Therefore, for higher values of l, we can attach an elastic support, such as a torsional spring, at the free end of the non-rotating beam, in order to increase the stiffness of the beam. In the next section, we try to find non-rotating beams with a torsional spring attached at the free end, that are isospectral to a uniform beam rotating at high speeds. 2.2. Isospectral non-rotating beams with a torsional spring at the free end Consider a non-rotating beam clamped at one end and free at the other end with a torsional spring, of a spring constant, KR, attached at the free end, as shown in Fig. 7. The method for
S. Kambampati et al. / International Journal of Mechanical Sciences 66 (2013) 12–21
17
2 12 10
1.5
8
f(x)
m(x)
non−rotating beam rotating beam
1
6 4
0.5
2 0 0
0.5
0
1
0
0.5
1
x
x 30
non−rotating beam rotating beam
2 25 20
f(x)
m(x)
1.5 1
15 10
0.5 0
5 0
0.5
0
1
x
0
0.5
1
x
3 50
non−rotating beam rotating beam
2.5 40
f(x)
m(x)
2 1.5
30
1
20
0.5
10
0
0
0.5
1
x
0
0
0.5
1
x
Fig. 6. Variations of mass m(x) and flexural stiffness f(x) of isospectral cantilever non-rotating (q0 ¼ 1) and uniform rotating beams at different rotation speeds l. (a) l ¼ 4. (b) l ¼ 5. (c) l ¼ 6.
finding these type of beams isospectral to the rotating beam (no spring), is similar to that discussed in the previous section, except that the physical boundary conditions should be handled in a different way. The boundary conditions for this beam are given in [35] as follows: Clamped at the root (x¼0) Y ¼ 0 ¼ Yx
ð53Þ
Free at the tip with a torsional spring (x ¼1) ðfY xx Þx ¼ 0
ð54Þ
fY xx þ kr Y x ¼ 0
ð55Þ
Fig. 7. Schematic of a non-uniform non-rotating beam with a torsional spring at the free end.
where kr is the non-dimensional spring constant, given by kr ¼ K R L=EI0 . Expanding Eq. (54), we have ðfY xx Þx ¼ f x Y xx þ fY xxx ¼ pf z Y xx þfY xxx 3 3 ¼ pðq2 Þz Y xx þ ðq2 ÞY xxx 0 p 0 p
¼
3 q2 ðY xxx 3Y xx pz Þ 0 p
ð56Þ
The boundary conditions given by Eqs. (54) and (55) can be transformed from the variable Y into the variable U, which is the
variable corresponding to the uniform rotating beam, using Eqs. (30)–(32) and (56). Specifically, Eqs. (54) and (55) can be rewritten in variable U as (at x ¼ 1,z ¼ 1) 3 q2 ðq0 pðpz U z þ pU zz ÞÞ þ kr q0 pU z ¼ 0 0 p
ð57Þ
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S. Kambampati et al. / International Journal of Mechanical Sciences 66 (2013) 12–21
non−rotating beam rotating beam
3
1
f(x)
m(x)
1.2
2
0.8 1 0.6
0
0.5
1
x
0
1.5
5
f(x)
m(x)
x
1
non−rotating beam rotating beam
4
1
0.5
3 2 1
0.5
0.5
1
x
0
1.6
6
1.4
5
1.2
4
f(x)
m(x)
0
1
2
0.6
1
0.5
1
x
x
1
non−rotating beam rotating beam
3
0.8
0
0.5
0
0.5
x
1
Fig. 8. Variations of mass m(x) and stiffness f(x) of clamped–free non-rotating beams (q0 ¼ 1) with a torsional spring, isospectral to a cantilever rotating beam for different rotation speeds l. (a) l ¼ 4. (b) l ¼ 5. (c) l ¼ 6.
3 ðq2 Þ½q0 ppz ðpz U z þpU zz Þ þ q0 p2 ðpzz U z þ 2pz U zz þpU zzz Þ 0 p
3q0 pðpz U z þ pU zz Þpz ¼ 0
ð58Þ
The above equations can be simplified using U zz ¼ U zzz ¼ 0 (Eq. (37)) to yield (at z ¼1) pz ¼ kr q20 p3
ð59Þ
ppzz ¼ 2p2z
ð60Þ ppzz 2p2z
¼ 0. We can see that from Eq. (25), we have at z¼1 , Therefore, it turns out that Eq. (60) is already satisfied. Hence, we must choose a spring constant kr, such that Eq. (59) is also satisfied. Therefore, Eq. (26), subject to the boundary conditions, Eqs. (41) and (59), can be solved for any given l. For example, for l ¼ 4, we can choose q0 ¼ 1 and kr ¼2.013, so that the boundary conditions are satisfied.
The mass and stiffness functions of the isospectral nonrotating beam, derived using this approach, are plotted in Fig. 8a. We can see from Fig. 8a that adding a torsional spring, has considerably brought down the average value of stiffness, as compared to that derived without a torsional spring (Fig. 6a). Similarly, for l ¼ 5 and l ¼ 6, we can choose the value of spring constant as kr ¼ 2.6318 and kr ¼3.3843, respectively, to satisfy the boundary conditions. The mass and stiffness functions of the nonrotating beams, derived using this method, which are isospectral to a uniform beam rotating at l ¼ 5 and l ¼ 6 are plotted in Fig. 8b and c, respectively. Comparing Figs. 6 and 8, we can conclude that the addition of a torsional spring at the free end of the nonuniform non-rotating beam has clearly resulted in physically feasible mass and stiffness distributions. Thus, we have found out the mass and stiffness distributions of a non-uniform beam that is isospectral to the given rotating beam. In the next section, we will numerically compute the
S. Kambampati et al. / International Journal of Mechanical Sciences 66 (2013) 12–21
natural frequencies to show that the determined beams are isospectral to the given beam.
19
Table 2 The frequencies of a uniform rotating beam [24] at different speeds. Mode
l¼1
l¼2
l¼3
l¼4
l¼5
l¼6
1 2 3 4 5
3.6817 22.1810 61.8418 121.051 200.012
4.1373 22.6149 62.2732 121.497 200.467
4.7973 23.3203 62.9850 122.236 201.223
5.5850 24.2734 63.9668 123.261 202.277
6.4495 25.4461 65.2050 124.566 203.622
7.3604 26.8091 66.6840 126.140 205.253
2.3. Finite element formulation For the given rotating uniform beam, the natural frequencies are determined using the method of Frobenius, given in [24]. In order to compute the natural frequencies of the non-uniform nonrotating beams, we use the finite element method. A thorough description of the finite element method is available in standard textbooks [36]. In the finite element formulation, we divide the beam into a number of finite elements. The displacement, Y(x) is assumed to have a cubic distribution along each finite element and hence, has four degrees of freedom (4DOF). Two of these 4DOF are displacement DOF and the other two are slope DOF at the ends of each finite element. Specifically, Y(x), along the ith element, is given by YðxÞ ¼ H1 ðxxi Þq1,i þ H2 ðxxi Þq2,i þ H3 ðxxi Þq3,i þ H4 ðxxi Þq4,i
ð61Þ
where xi is the coordinate of the left node, xi þ 1 is the coordinate of the right node, q1,i and q2,i are the displacement DOF and slope DOF at the left node, q3,i and q4,i are the displacement DOF and slope DOF at the right node [37]. The Hermite shape functions 2
2
3
3
3
3
(H1 ,H2 ,H3 and H4) are given by H1 ðxÞ ¼ 1ð3x Þ=l þ ð2x Þ=l , 2
3
2
H2 ðxÞ ¼ xð2x Þ=l þ x =l , 2
3
2
2
H3 ðxÞ ¼ ð3x Þ=l ð2x Þ=l
and
2
H4 ðxÞ ¼ x =l þ x =l , where l ¼ xi þ 1 xi element length. The expressions for the kinetic energy Tk and potential energy Up of the non-rotating non-uniform beam are given by Z 1 2 Tk ¼ mðxÞY_ dx ð62Þ 0
Up ¼
Z
1
f ðxÞðY 00 ðxÞÞ2 dx
ð63Þ
0
^ are then The natural frequencies(Z) and mode shapes (V) obtained as follows. First, we substitute the expression for displacement Y(x), from Eq. (61), into Eqs. (62) and (63). Next we apply Lagrange’s equations of motion, which yields the well known eigenvalue problem: ^ V^ ¼ Z2 M ^ V^ K
ð64Þ
^ and K ^ are the global mass and stiffness matrices. These where M global mass and stiffness matrices are obtained by assembling the ~ and stiffness (K) ~ matrices. The ith element level mass (M) ~ i ) and stiffness matrix (K ~ i ) are given by element’s mass matrix (M Z l ~ jk Þ ¼ mðx þ xi ÞHj ðxÞHk ðxÞ dx ð65Þ ðM i
Table 3 Frequencies of isospectral non-rotating non-uniform beams, calculated using FEM. Mode
l¼1
l¼2
l¼3
l¼4
l¼5
l¼6
1 2 3 4 5
3.6816 22.1810 61.8418 121.051 200.012
4.1373 22.6149 62.2732 121.497 200.467
4.7973 23.3203 62.9850 122.236 201.223
5.5850 24.2733 63.9668 123.262 202.277
6.4495 25.4461 65.2050 124.566 203.622
7.3604 26.8091 66.6839 126.141 205.253
The frequencies of a uniform rotating beam, calculated using the method of Frobenius [24], are given in Table 2, for rotational speeds l ¼ 1, . . . ,6. The frequencies of non-rotating cantilever beams, isospectral to this uniform beam rotating at a speed l ¼ 1,2 and 3, calculated by the finite element analysis, are tabulated in Table 3. The frequencies of non-rotating clamped– free beams, which have a torsional spring at the free end, isospectral to this uniform beam rotating at a speed l ¼ 4,5 and 6, calculated by the finite element analysis, are also tabulated in Table 3. By comparing Tables 2 and 3, we can infer that there is an almost negligible error between frequencies of the rotating beam and that of the non-rotating non-uniform beam. This provides numerical verification, of the isospectral properties of analytical mass and stiffness functions, derived in the earlier section.
3. Realistic beams In this section, we analyze beams having a rectangular crosssection. The mass/length M0 and stiffness EI0 of the given uniform rotating beam is given by ^0 M0 ¼ r0 B^ 0 H and EI0 ¼ E0
0
~ jk Þ ¼ ðK i
Z 0
l
f ðx þ xi ÞH00j ðxÞH00k ðxÞ dx
ð66Þ
Note that if a torsional spring is fixed at a particular node, then the stiffness of the spring should be added to the stiffness term corresponding to the rotational degree of freedom at that node in the global stiffness matrix. Let us take the example of a torsional spring of stiffness kr attached at the free end of the beam. Then ~ jk Þ þ kr ðK~ 0 jk Þi ¼ ðK i
ð67Þ
where ðK~ 0 jk Þi is the stiffness term calculated including the stiffness ~ jk Þ is the stiffness term calculated without of the spring, ðK i including the spring, and j,k are the indices corresponding to the rotational degrees of freedom of that node. We have used the Gauss quadrature to evaluate the integrals using 10 Gauss– Legendre points.
ð68Þ
^3 B^ 0 H 0 12
ð69Þ
where r0 is the density of the beam, E0 is the tensile modulus, B^ 0 ^ 0 are the breadth and height of the rectangular crossand H section, respectively. Similarly, the mass M(X) and stiffness EI(X) of the non-rotating beams, which are made of the same material as that of the rotating beam, is given by ^ ^ HðXÞ MðXÞ ¼ r0 BðXÞ
ð70Þ
and 3 ^ ^ BðXÞ HðXÞ ð71Þ 12 ^ ^ where BðXÞ and HðXÞ are the breadth and height of the rectangular
EIðXÞ ¼ E0
cross-section, respectively, of the non-rotating beams. We introduce non-dimensional variables b^ (for breadth) and h^ (for height) as ^ b^ ¼ BðXÞ= B^ 0 ,
^ ^0 h^ ¼ HðXÞ= H
ð72Þ
20
S. Kambampati et al. / International Journal of Mechanical Sciences 66 (2013) 12–21
Isospectral Non-Rotating Beam (λ= 1)
Given Uniform Rotating Beam 1
height
height
1 0 −1
0 −1
0
0 0.2
0.4
0.2 0.6
0.8
1
x
−1
0.6
breadth
0.8 1
x
Isospectral Non-Rotating Beam (λ= 2)
−1
0
1
breadth
Isospectral Non-Rotating Beam (λ= 3)
1
1
height
height
0.4
1
0
0 −1
0 −1 0
0
0.2
0.4
0.2 0.6
0.8
1
x
−1
0
1
0.4
0.6
breadth
0.8
x
1
−1
0
1
breadth
IsospectralNon-Rotating Beam With a Spring (λ = 4)
Isospectral Non-Rotating Beam With a Spring (λ = 5)
1
1
height
height
^ h^ of non-rotating beams (q ¼ 1) isospectral to a uniform beam rotating at different rotation speeds l. (a) Given uniform rotating beam. Fig. 9. Variations of b, 0 (b) Isospectral non-rotating beam (l ¼ 1). (c) Isospectral non-rotating beam (l ¼ 2). (d) Isospectral non-rotating beam (l ¼ 3).
0 −1 0
0 −1 0
0.2
0.2 0.4
0.6
0.8
x
1
−1
0
1
breadth
0.4
0.6
0.8
x
1
−1
0
1
breadth
^ h^ of clamped–free non-rotating beams (q ¼ 1), which have a torsional spring at the free end, isospectral to a uniform cantilever beam rotating at Fig. 10. Variations of b, 0 rotation speeds l. (a) Isospectral non-rotating beam with a spring (l ¼ 4). (b) Isospectral non-rotating beam with a spring (l ¼ 5).
Using these non-dimensional breadth and height variables, the mass and stiffness functions are given by m¼
MðXÞ ^ ¼ b^ h, M0
f¼
which implies that sffiffiffiffiffiffiffi rffiffiffiffiffi m3 f b^ ¼ , h^ ¼ m f
3 EIðXÞ ¼ b^ h^ EI0
ð73Þ
ð74Þ
Therefore, using m(x) and f(x) functions derived in earlier sections, one can find beams that have a rectangular cross-section. The ^ and height (h) ^ distribution of these beams can be breadth (b) determined from Eq. (74). These b^ and h^ distributions, for a nonuniform non-rotating beam, which are isospectral to a uniform beam rotating at a speed l ¼ 1,2 and 3, are plotted in Fig. 9a, b, c and d, respectively. Similarly, the b^ and h^ distributions, for nonuniform non-rotating clamped-free beams, having a torsional spring at the free end, which are isospectral to a uniform cantilever beam rotating at a speed l ¼ 4 and 5, are plotted in Fig. 10a and b, respectively. It can be seen that these are physically realizable beams.
4. Conclusion An analytical method of obtaining non-uniform non-rotating beams that are isospectral to a given uniform rotating beam is
presented. The governing equation of non-uniform non-rotating beam, is converted to that of a uniform rotating beam, using the auxiliary variables defined in Barcilon–Gottlieb transformation, by matching the coefficients in both the equations. We have given the necessary conditions, on the auxiliary variables, so that the boundary conditions remain invariant. For high rotation speeds, we derived physically realizable mass and stiffness functions of isospectral non-rotating beams, that have a torsional spring at the free end. The natural frequencies of the derived non-rotating beams, are also calculated using FEM, and are found to be same as the frequencies of the given uniform rotating beam. A numerical example of a beam having a rectangular cross-section is presented to show the application of this analysis. The breadth and height variations, of the rectangular cross-section, with span, of isospectral rotating and non-rotating beams, are obtained in our study. Recent advancements in computerized machining techniques such as additive manufacturing or rapid-prototyping [38,39], facilitate the manufacturing of any complex 3D shape. Therefore, one can manufacture isospectral rotating and non-rotating beams, using the breadth and height variations of the cross-section, with the help of computerized machining. References [1] Dutta R, Ganguli R, Mani V. Exploring isospectral spring–mass systems with firefly algorithm. Proc R Soc A: Math Phys Eng Sci 2011;467(2135):3222. [2] Gladwell GML. Inverse problems in vibration. Appl Mech Rev 1986;39:1013.
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[3] Gladwell GML. Inverse finite element vibration problems. J Sound Vib 1999;221(2):309–24. [4] McLaughlin JR. Analytical methods for recovering coefficients in differential equations from spectral data. SIAM Rev 1986;28(1):53–72. [5] Gladwell GML. On isospectral spring–mass systems. Inverse Problems 1995;11:591. [6] Gladwell GML. Isospectral vibrating beams. Proc R Soc London Ser A: Math Phys Eng Sci 2002;458(2027):2691. [7] Gottlieb HPW. Harmonic frequency spectra of vibrating stepped strings. J Sound Vib 1986;108(1):63–72. [8] Gottlieb HPW. Multi-segment strings with exactly harmonic spectra. J Sound Vib 1987;118(2):283–90. [9] Borg G. Eine umkehrung der Sturm–Liouvilleschen eigenwertaufgabe (An inversion of the Sturm–Liouville eigenvalue problem). Acta Math 1946;78(1): 1–96. [10] Gottlieb HPW. Isospectral Euler–Bernoulli beams with continuous density and rigidity functions. Proc R Soc London A Math Phys Sci 1987;413(1844):235. [11] Subramanian G, Raman A. Isospectral systems for tapered beams. J Sound Vib 1996;198(3):257–66. [12] Yoo HH, Shin SS. Vibration analysis of rotating cantilever beams. J Sound Vib 1998;212(5):807–28. [13] Lauzon DM, Murthy VR. Determination of vibration characteristics of multiple-load-path blades by a modified Galerkin’s method. Comput Struct 1993;46(6):1007–20. [14] Wang G, Wereley NM. Free vibration analysis of rotating blades with uniform tapers. AIAA J 2004;42(12):2429–37. [15] Subrahmanyam KB, Kaza KRV. Non-linear flap-lag-extensional vibrations of rotating, pretwisted, preconed beams including coriolis effects. Int J Mech Sci 1987;29(1):29–43. [16] Babu Gunda J, Ganguli R. New rational interpolation functions for finite element analysis of rotating beams. Int J Mech Sci 2008;50(3):578–88. [17] Gunda JB, Ganguli R. Stiff-string basis functions for vibration analysis of high speed rotating beams. J Appl Mech 2008;75:024502. [18] Lin YH, Trethewey MW. Finite element analysis of elastic beams subjected to moving dynamic loads. J Sound Vib 1990;136(2):323–42. [19] Lin SM, Lee SY. Prediction of vibration and instability of rotating damped beams with an elastically restrained root. Int J Mech Sci 2004;46(8):1173–94. [20] Attarnejad R, Shahba A. Application of differential transform method in free vibration analysis of rotating non-prismatic beams. World Appl Sci J 2008;5(4):441–8.
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[21] Ozdemir O, Kaya MO. Flapwise bending vibration analysis of a rotating tapered cantilever Bernoulli–Euler beam by differential transform method. J Sound Vib 2006;289(1–2):413–20. [22] Banerjee JR, Su H, Jackson DR. Free vibration of rotating tapered beams using the dynamic stiffness method. J Sound Vib 2006;289(4–5):1034–54. [23] Hodges DH, Rutkowski MJ. Free-vibration analysis of rotating beams by a variable-order finite-element method. AIAA J 1981;19:1459–66. [24] Wright AD, Smith CE, Thresher RW, Wang JLC. Vibration modes of centrifugally stiffened beams. J Appl Mech 1982;49:197. [25] Storti D, Aboelnaga Y. Bending vibrations of a class of rotating beams with hypergeometric solutions. J Appl Mech 1987;54:311. [26] Naguleswaran S. Lateral vibration of a centrifugally tensioned uniform Euler– Bernoulli beam. J Sound Vib 1994;176(5):613–24. [27] Low KH. A reliable algorithm for solving frequency equations involving transcendental functions. J Sound Vib 1993;161(2):369–77. [28] Senatore A. Measuring the natural frequencies of centrifugally tensioned beam with laser Doppler vibrometer. Measurement 2006;39(7):628–33. [29] Meirovitch L. Elements of vibration analysis, vol. 2. New York: McGraw-Hill; 1986. [30] Barcilon V. Inverse problem for the vibrating beam in the free–clamped configuration. Philos Trans R Soc London A Ser Math Phys Sci 1982; 304(1483):211. [31] Nath P. Confluent hypergeometric function. Sankhya: ¯ Indian J Stat (1933– 1960) 1951;11(2):153–66. [32] Slater LJ. Confluent hypergeometric functions. Cambridge: Cambridge University Press; 1960. [33] Dy´az R, Pariguan E. On hypergeometric functions and Pochhammer k-symbol. Divulg Mat 2007;15(2):179–92. [34] Kreyszig E. Advanced engineering mathematics. Wiley-India; 2007. [35] Karnovsky I, Lebed O. Formulas for structural dynamics: tables, graphs and solutions. New York: McGraw-Hill; 2001. [36] Chandrupatla TR, Belegundu AD. Introduction to finite elements in engineering. New Jersey: Prentice-Hall; 2001. [37] Kumar A, Ganguli R. Rotating beams and nonrotating beams with shared eigenpair. J Appl Mech 2009;76:051006. [38] Kruth JP, Leu MC, Nakagawa T. Progress in additive manufacturing and rapid prototyping. CIRP Ann—Manuf Technol 1998;47(2):525–40. [39] Hu D, Kovacevic R. Sensing, modeling and control for laser-based additive manufacturing. Int J Mach Tools Manuf 2003;43(1):51–60.
Contents lists available at SciVerse ScienceDirect
International Journal of Mechanical Sciences journal homepage: www.elsevier.com/locate/ijmecsci
Non-rotating beams isospectral to a given rotating uniform beam Sandilya Kambampati, Ranjan Ganguli n, V. Mani Department of Aerospace Engineering, Indian Institute of Science, Bangalore 560012, India
a r t i c l e i n f o
a b s t r a c t
Article history: Received 23 July 2012 Received in revised form 9 October 2012 Accepted 17 October 2012 Available online 30 October 2012
In this paper, we seek to find non-rotating beams with continuous mass and flexural stiffness distributions, that are isospectral to a given uniform rotating beam. The Barcilon–Gottlieb transformation is used to convert the fourth order governing equation of a non-rotating beam, to a canonical fourth order eigenvalue problem. If the coefficients in this canonical equation match with the coefficients of the uniform rotating beam equation, then the non-rotating beam is isospectral to the given rotating beam. The conditions on matching the coefficients leads to a pair of coupled differential equations. We solve these coupled differential equations for a particular case, and thereby obtain a class of non-rotating beams that are isospectral to a uniform rotating beam. However, to obtain isospectral beams, the transformation must leave the boundary conditions invariant. We show that the clamped end boundary condition is always invariant, and for the free end boundary condition to be invariant, we impose certain conditions on the beam characteristics. We also verify numerically that the frequencies of the non-rotating beam obtained using the finite element method (FEM) are the exact frequencies of the uniform rotating beam. Finally, the example of beams having a rectangular cross-section is presented to show the application of our analysis. Since experimental determination of rotating beam frequencies is a difficult task, experiments can be easily conducted on these rectangular non-rotating beams, to calculate the frequencies of the rotating beam. & 2012 Elsevier Ltd. All rights reserved.
Keywords: Free vibration Rotating beam Isospectral Modal analysis Finite element Eigenvalue
1. Introduction An undamped vibrating system has a set of natural frequencies, forming a spectrum, at which it vibrates freely without external forces. An important class of problems for such vibrating systems can be broadly classified into inverse problems and isospectral problems. In inverse problems [2,3], one tries to determine the material properties of a system for a given frequency spectrum. In general, more than one frequency spectrum is required for a reconstruction procedure to determine the material properties of the system [4]. Systems, that have the same vibrating frequencies, but have different material properties are called isospectral systems. Isospectral systems are of great interest in mechanics as they yield alternative usable designs. The existence of isospectral systems also proves that a system cannot be uniquely identified from its spectrum. 1.1. Isospectral spring–mass systems Examples of discrete isospectral systems are in-line spring– mass systems. A schematic of a 2-DOF spring mass system is n
Corresponding author. E-mail address: [email protected] (R. Ganguli).
0020-7403/$ - see front matter & 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijmecsci.2012.10.004
shown in Fig. 1 , where m1 ,m2 are the masses and k1 ,k2 are the spring constants of the springs. Let us consider one such 2-DOF system (System-A) [1] where the values of masses m1 and m2, and spring constants k1 and k2, and the natural frequencies of SystemA (o1 and o2 ) are tabulated in Table 1. Similarly, let us consider one more system (System-A0 ) where m01 and m02 are the masses, k01 and k02 , are the spring constants and o01 and o02 are the natural frequencies, whose values are tabulated in Table 1. From Table 1, we can see that both System-A and System-A0 have the same frequency spectrum as o1 ¼ o01 and o2 ¼ o02 . Therefore, System-A and System-A0 are isospectral. Gladwell [5] described four ways to form in-line spring mass systems isospectral to a given one. Gladwell [6] also considered a discrete model of a vibrating cantilever beam, and presented two procedures for finding families of such beams, isospectral to a given one. Gottlieb [7,8] studied isospectral vibrating strings with discontinuous coefficients. All these studies addressed discrete models of vibrating beams. Borg [9] studied isospectral systems corresponding to a vibrating string with continuous coefficients (second order governing equation). Gottlieb [10] analyzed the non-uniform Euler–Bernoulli beam equation and gave seven classes of non-uniform beams isospectral to the given uniform beam with different boundary conditions. Subramanian and Raman [11] generalized Gottlieb’s method for tapered beams. However, studies on isospectral systems have not
S. Kambampati et al. / International Journal of Mechanical Sciences 66 (2013) 12–21
Fig. 1. Schematic of a 2DOF spring–mass system.
Table 1 Material and spectral properties of System-A and System-A0 . System-A0
System-A m1 m2 k1 k2
o1 o2
6.25 kg 4 kg 92.5 kN/m 20 kN/m 62.2 rad/s 138.3 rad/s
m01 m02 k01 k02
o01 o02
1.286 kg 1 kg 15.857 kN/m 6 kN/m 62.2 rad/s 138.3 rad/s
addressed rotating beams. In this study, we seek to find nonuniform beams that are isospectral to a given rotating beam. Such beams, if they exist, have important applications in the dynamics of rotating systems. Rotating beams serve as a useful mathematical model to simulate vibration of helicopter blades, wind turbines, long flexible rotating space booms, turbo-machinery blades, etc. A rotating mechanical system can suffer from high vibration, if its natural frequencies coincide with multiples of the rotation speed. Therefore, an accurate determination of the frequencies is an important aspect in rotating blade design. The natural frequencies and mode shapes can be determined using an approximation scheme such as the Rayleigh– Ritz method [12], Galerkin method [13], finite element method [14–19], differential transform method [20,21] or the dynamic stiffness method [22]. Hodges and Rutkowski [23] analyzed the out of plane vibrations of a rotating beam using a finite element method of variable order. Wright et al. [24] presented an accurate solution for the mode shapes of a beam attached to a rotating hub using the method of Frobenius. Storti and Aboelnaga [25] listed the classes of beams which admit hypergeometric solutions to the mode shape equation. Naguleswaran [26] solved the mode shape equation using the method of Frobenius. Low [27] developed an algorithm for solving frequency equations for a cantilever double-span non-rotating beam and a uniform rotating beam. Thus, we see that considerable amount of research has been done on the models of a rotating beam. Experimental determination of rotating beam frequencies can be difficult. For example, Senatore [28] experimentally determined the frequencies of a rotating beam, using lumped parameter axial loads on a uniform non-rotating beam, in order to simulate an approximation of the centrifugal force field, acting on the rotating beam. However, this method could not predict exactly the first natural frequency and mode shape due to the lumped axial loads. Hence, it is interesting to see, if we can find any non-rotating beam which is isospectral to the uniform rotating beam. It is important to take into account the exact centrifugal force acting on the rotating beam. The reasons for obtaining isospectral beams are the following. (a) It is difficult to conduct experiments on rotating beams to obtain its natural frequencies. If we find an isospectral non-rotating beam, one can easily conduct experiments on the non-uniform beam, to obtain the natural frequencies. (b) Such isospectral beams can provide insight by predicting the stiffening effect of the centrifugal force. (c) Provide benchmark problems for finite element analysis. In this paper, we find non-rotating beams with continuous mass and flexural stiffness distributions, that are isospectral to a given uniform rotating beam. The mass and stiffness functions of non-rotating beams, isospectral to a uniform beam rotating at different speeds are derived. We note that for high rotating speeds, the derived mass and stiffness functions of the
13
non-rotating beam are not physically realizable, owing to stiffening effect of the centrifugal force. In such situations, if we attach a torsional spring, of a spring constant KR, at the free end of the non-rotating beam, the obtained mass and stiffness functions become physically realizable. We also show numerically that the frequencies of the non-rotating beam obtained using the finite element method (FEM) are the exact frequencies of the uniform rotating beam. This confirms numerically that the non-uniform beam obtained in this method is isospectral to the given uniform rotating beam. We also provide an example of realistic beams having a rectangular cross-section to show a physically realizable application of our analysis.
2. Mathematical analysis In this section, the mathematical formulation of the problem is presented. The governing differential equation for the transverse free vibrations V(Z) of a uniform rotating beam (Fig. 2) of length L, stiffness EI0 and mass M0 rotating with an angular speed O is given in [23] as: " # " # ! 4 2 2 d V d dV 2 L Z M EI0 O ð1Þ o2 M0 V ¼ 0, 0 rZ r L 0 dZ dZ 2 dZ 4 We introduce a non-dimensional variable z ¼ Z=L so that the above equation can be rewritten as 4 2 d V d dV 2 1z Z2 V ¼ 0, 0 r z r1 l ð2Þ dz 2 dz4 dz where Z is the non-dimensional frequency given by qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z ¼ o M0 L4 =EI0
ð3Þ
and l is the non-dimensional rotation speed given by qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi l ¼ O M0 L4 =EI0
ð4Þ
Similarly, the governing equation for the out of plane free vibrations YðXÞ of a non-uniform non-rotating Euler–Bernoulli beam (Fig. 3), which is isospectral to the rotating beam is given by [29] as " # 2 2 d d Y EIðXÞ ð5Þ o2 MðXÞY ¼ 0, 0 rX rL dX 2 dX 2 where EI(X) is the flexural stiffness , M(X) is the mass/length, L is the length of the beam, which is same as that of the uniform rotating beam (L ¼ LÞ. Now, we introduce non-dimensional variables f , m and x as f ðxÞ ¼
EIðXÞ , EI0
mðxÞ ¼
MðXÞ , M0
x¼
X L
ð6Þ
Eq. (5) can be rewritten as ðf ðxÞY 00 Þ00 ¼ Z2 mðxÞY,
0 r x r1
ð7Þ
Here, the notation Y 0 and Y 00 represent the first and the second derivatives respectively of Y w.r.t x. The transformation, which
Fig. 2. Schematic of a uniform rotating beam.
14
S. Kambampati et al. / International Journal of Mechanical Sciences 66 (2013) 12–21
Here, the notation q(z) means the value of q at the coordinate z. The first, second, third and the fourth derivatives, respectively of q w.r.t z are denoted by qz, qzz, qzzz and qzzzz. Therefore, we can conclude the above analysis by establishing the following. The governing differential equation of the non-uniform nonrotating beam is " # 2 2 d d Y f ð20Þ Z2 mY ¼ 0, 0 r x r 1 dx2 dx2
Fig. 3. Schematic of a non-uniform non-rotating beam.
converts the above non-rotating non-uniform beam equation (5) to the canonical fourth order equation for the function U(z) 4 d U d dU A þ BU ¼ Z2 U, 0 rz r 1 þ ð8Þ dz dz dz4 2
is proposed by Barcilon [30]. When A ¼ l ð1z2 Þ=2 and B ¼0, U corresponds to V in Eq. (2). In the Barcilon–Gottlieb transformation [10,30], two auxiliary variables p and q are defined as 1=4 m 1 1=8 p¼ , q¼ ð9Þ 3 f m f which also means that f ¼ p3 q2 ,
m ¼ pq2
ð10Þ
The transformation of the independent variable (x) is given implicitly as Z x z¼ pðzðxÞÞ dx ð11Þ 0
which implies that Z z dz x¼ 0 pðzÞ
ð12Þ
d d 1 dx ¼ pðzÞ ) ¼ dx dz pðzÞ dz
ð13Þ
We note that x¼0 3 z¼0
ð14Þ
Also, we have x¼1 3 z¼1
YðxÞ ¼ qðzÞUðzÞ
ð16Þ
where 2
If the coefficients A(z) and B(z) of the non-rotating beam in Eq. (21) match with the coefficients of the uniform rotating beam equation (22), then the non-rotating beam is isospectral to the 2 given rotating beam; i.e, AðzÞ ¼ l ð1z2 Þ=2 and BðzÞ ¼ 0. Hence, a non-uniform non-rotating beam isospectral to a given rotating uniform beam is obtained when U corresponds to V. In other words, Eqs. (20) and (22) have the same natural frequencies if and only if the auxiliary functions p(z) and q(z) satisfy the following conditions: 2
AðzÞ ¼ l ð1z2 Þ=2
ð23Þ
BðzÞ ¼ 0
ð24Þ
We need to solve the above equations to obtain non-rotating beams isospectral to a given uniform rotating beam. These equations are nonlinear fourth order coupled differential equations which are difficult to solve analytically for a general q(z). However, for qðzÞ ¼ q0 which is a constant, it is possible to obtain the results analytically. Also, note that if q ¼ q0 , then m and f are 3 simply related throughout by f ¼ q8 . When qðzÞ ¼ q0 , we can 0 m see that BðzÞ ¼ 0 as the derivatives qz ,qzz ,qzzz and qzzzz vanish. 2 When AðzÞ ¼ l ð1z2 Þ=2, then from Eq. (18) we have
We see that Eqs. (12) and (16) transform the variables x and Y into z and U, respectively. This completes the Barcilon–Gottlieb transformation. Using this transformation, we get " # " # 2 2 4 d d Y d dU 2 1 d U 2 A þBUZ U f þ ð17Þ Z mY ¼ pq dz dz dx2 dz4 dx2
A¼
The expressions A(z) and B(z), for this non-rotating beam, are given in Eqs. (18) and (19). The governing differential equation of the uniform rotating beam which is rotating with a non-dimensional speed l is from Eq. (1), " # 4 d V d l2 2 dV Z2 V ¼ 0, 0 rz r 1 ð1z þ Þ ð22Þ dz dz 2 dz4
ð15Þ
The transformation of the dependent variable Y is given as
q q2 4 zz 6 z2 q q
The above equation can be transformed into a canonical form using Barcilon–Gottlieb transformation as 4 d U d dU AðzÞ þBðzÞUZ2 U ¼ 0, 0 r z r 1 þ ð21Þ dz dz dz4
qz pz þ qp
pzz p2 2 z2 p p
qzzzz qq q2 q2 q qp q p q2 p 4 z 2zzz 2 zz þ6 z 3zz þ z zzz þ zz zz 4 z 2 zz 2 q qp qp q q q q p qz pz pzz qz qzz pz q3z pz q2z p2z qz p3z qzz p2z þ 6 3 þ 6 2 2 þ4 4 2 5 qp2 q2 p q p q p qp3 qp2
ð18Þ
l2 2
ð1z2 Þ ¼
pzz 2p2z 2 p p
ð25Þ
The above equation is divided throughout by p and rewritten as 2
2
d p^ l ð1z2 Þp^ ¼ 0 2 dz2
ð26Þ
where p^ is defined using Eq. (13) as p^ ¼ p1 ¼
dx dz
ð27Þ
Note that at z¼1, from Eq. (26), we have " # " # 2 d p^ l2 2 ^ ð1z ¼ Þ p ¼0 2 dz2 z¼1
ð28Þ
z¼1
Also, using qðzÞ ¼ q0 in Eq. (16) and using Eq. (13), we obtain the following:
B¼
ð19Þ
These expressions for A and B are originally given by Barcilon [30] and later corrected by Gottlieb [10].
Y ¼ q0 U
ð29Þ
Y x ¼ q0 pU z
ð30Þ
Y xx ¼ q0 pðpz U z þ pU zz Þ
ð31Þ
S. Kambampati et al. / International Journal of Mechanical Sciences 66 (2013) 12–21
Y xxx ¼ q0 ppz ðpz U z þpU zz Þ þ q0 p2 ðpzz U z þ 2pz U zz þpU zzz Þ
ð32Þ
The initial conditions that are to be imposed on Eq. (26) depend on the physical boundary conditions (clamped, free, hinged, etc.). In the next section, we solve this equation for cantilever rotating and non-rotating beams. 2.1. Cantilever rotating and non-rotating beams In this section, we look for a non-uniform non-rotating cantilever beam which is isospectral to a given uniform rotating cantilever beam. The rotating and the non-rotating beams are clamped at the root (x ¼ 0 ¼ z) and free at the tip (x ¼ 1 ,z ¼ 1). Hence, we have the following boundary conditions: At the root: Y ¼ 0 ¼ Yx
ð33Þ
U ¼ 0 ¼ Uz
ð34Þ
15
and 1 F 1 ða1 ; b1 ; s2 Þ and 1 F 1 ða2 ; b2 ; s2 Þ are confluent hypergeometric functions [31,32] of the first kind given by 1F1ð
a; b; sÞ ¼
1 X ðaÞk sk ðbÞk k! k¼0
ð47Þ
where ðaÞk and ðbÞk are Pochhammer symbols [33]. ^ Since pðzÞ is now determined, m(z) and f(z) can be calculated using Eq. (10), by choosing q0 ¼ 1, and are given by mðzÞ ¼ ðk1 s1 1 F 1 ða1 ; b1 ; s2 Þ þz k2 s1 1 F 1 ða2 ; b2 ; s2 ÞÞ1
f ðzÞ ¼ k1 s1 1 F 1 ða1 ; b1 ; s2 Þ þ z k2 s1 1 F 1 ða2 ; b2 ; s2 Þ
ð48Þ
3
ð49Þ
Note that Eq. (49) can also be calculated from Eq. (48) by using 3 the relation f ¼ ðq8 , which simplifies to f ¼ m3 , as q0 ¼ 1. 0 Þm Similarly, x(z) can be calculated using Eq. (12), and given by Z z xðzÞ ¼ p^ dz ð50Þ 0
At the tip: ðfY xx Þx ¼ 0 ¼ Y xx
ð35Þ
In the above equation, since Y xx ¼ 0, we have ðfY xx Þx ¼ 0 ) f x Y xx þ fY xxx ¼ 0 ) Y xxx ¼ 0, Therefore, Eq. (35) can be modified as Y xxx ¼ 0 ¼ Y xx
ð36Þ
U zz ¼ 0 ¼ U zzz
ð37Þ
We can see from Eqs. (29)–(32), that the clamped boundary condition is satisfied in the Barcilon–Gottlieb transformation. So we need to satisfy only the free boundary condition for the cantilever beam. This means that the coefficients of the term Uz in Eq. (31) should be zero at z¼1 (since at z¼1, U zz ¼ 0 ¼ U zzz ). Therefore, in Eq. (31), pz at z¼1 should vanish. Similarly in Eq. (32), the coefficients of the term Uz should vanish at z¼1. Hence pzz should also be zero at z¼1. We know from Eq. (27) that p ¼ 1=p^ and hence, we have 2 pz ¼ p^ z p^ 2 3
pzz ¼ 2p^ z p^
ð38Þ p^ zz p^
2
ð39Þ
Therefore, from the above equations and Eq. (28), we have at z¼1 p^ z ¼ 0 3 pz ¼ 0 3 pzz ¼ 0
ð40Þ
Eq. (50) has no exact solution, and hence we integrate p^ (Eq. (42)) using Guass–Legendere quadrature and obtain x(z). The obtained m(z), f(z) and x(z) functions for l ¼ 1 are plotted in Fig. 4. Our interest is to obtain the mass and stiffness functions in terms of x (i.e. m(x) and f(x)). It is not possible to obtain m(x) and f(x) from m(z) and f(z) analytically, as there is no closed form expression for z(x). Here, for a given value of x, we cannot calculate the value of z analytically. If we were given a value of x, say x, we need to determine the value of z, say z. Hence we use the well known Newton–Raphson method to calculate z given any x, by using the iterative formula [34] xðzi Þx ð51Þ zi þ 1 ¼ zi xz ðzi Þ where xz ðzi Þ ¼
^ pðzÞ ¼ k1 s1 1 F 1 ða1 ; b1 ; s2 Þ þ zk2 s1 1 F 1 ða2 ; b2 ; s2 Þ
ð42Þ
ð52Þ
and xðzi Þ is obtained from Eq. (50). The initial estimate for this iterative formula can be taken as z1 ¼ x. For example, for x ¼ 0:6, we can use the above analysis to calculate z, which is z ¼ 0:588. Thus, in Fig. 4, if we draw a horizontal line x ¼ x ¼ 0:6, it cuts the graph x(z) at the point B(0:588,0:6). Using Eqs. (48) and (49), mðzÞ and f ðzÞ can be calculated, and are given by mðzÞ ¼ 1:023 and f ðzÞ ¼ 0:934. Thus, in Fig. 4, if we draw a vertical line from point B (z ¼ z ¼ 0:588), it cuts the graph m(z) at the point C ð0:588,1:023Þ and cuts the graph
Another initial condition p^ should satisfy is derived from Eq. (12), which is Z 1 Z 1 dz 3 1¼ 1¼ p^ dz ð41Þ 0 p 0 Therefore, Eq. (26) can be solved for any given rotation speed (l), using the two initial conditions (Eqs. (40) and (41)). For example, for l ¼ 1, we can find a corresponding isospectral nonrotating beam. The solution of Eq. (26), for l ¼ 1, is given as
dx ^ iÞ ¼ pðz dz z ¼ zi
1.4 f(z) 1.2 C : (z , m) = (0.588,1.023)
E: (m=f=1 ) 1 m(z)
f(z), m(z), x(z)
D :(z , f)= (0.588 , 0.934)
0.8 A : (z , m) = (0 , 0.6) 0.6
B : (z , x)= (0.588 , 0.6)
where pffiffiffi
a1 ¼ 2 2
! pffiffiffi 2 1 i , 16 16
! pffiffiffi pffiffiffi 3 2 1 i 16 16
a2 ¼ 2 2
0.4
ð43Þ 0.2
b1 ¼ 12 , b2 ¼ 32 pffiffi 2x2 i=4
s1 ¼ e
k1 ¼ 1:0911,
,
x(z)
z =0.588
ð44Þ 0
pffiffiffi 2x2 i s2 ¼ 2
ð45Þ
k2 ¼ 0:3376
ð46Þ
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
z Fig. 4. Variations of m(z), f(z ) and x(z) of the isospectral non-rotating beam (l ¼ 1).
16
S. Kambampati et al. / International Journal of Mechanical Sciences 66 (2013) 12–21
1.1
1.4
rotating beam
1.05 1.2
1
f(x)
m(x)
non−rotating beam
0.95
1
0.9 0.8
0.85 0
0.5
0
1
0.5
1
x
x 1.2
non−rotating beam
2.5
rotating beam
1.1 f(x)
m(x)
2 1 0.9 0.8
1.5 1
0.7 0.5 0
0.5
0
1
0.5
1
x
x 1.4
non−rotating beam 5
rotating beam
1.2 1
f(x)
m(x)
4 3
0.8
2
0.6
1 0
0.5 x
1
0
0.5
1
x
Fig. 5. Variations of mass m(x) and flexural stiffness f(x) of isospectral cantilever non-rotating (q0 ¼ 1) and uniform rotating beams at different rotation speeds l: (a) l ¼ 1. (b) l ¼ 2. (c) l ¼ 3.
f(z) at the point D ð0:588,0:934Þ. Therefore, at the non-dimensional length x ¼ 0:6, the mass and the stiffness of the non-rotating beam isospectral to the given rotating beam, are given by mðxÞ ¼ 1:023 and f ðxÞ ¼ 0:934, respectively. Also, note also that for q0 ¼ 1, we have f ¼ m3 . Therefore, the graphs f(z) and m(z) crossover at point E (z¼0.385), where f ¼ m ¼ 1, thus confirming f ¼ m3 . Thus, for any given x, z(x), and hence mass (m(x)) and stiffness (f(x)) can be calculated. These m(x) and f(x) distributions are plotted in Fig. 5a. From Fig. 5a, we can infer that the non-rotating isospectral beam should be more stiff at the root and less stiff at the tip compared to the uniform rotating beam. Similarly, the mass distribution should be less at the root and more at the tip compared to the uniform rotating beam. In the above analysis, we have determined the non-rotating beam isospectral to the rotating beam for l ¼ 1. The mass and stiffness variations of beams isospectral to a uniform beam rotating at other speeds can be calculated similarly. For l ¼ 2 and l ¼ 3, m(x) and f(x) variations of isospectral beams are determined and plotted in Fig. 5b and c respectively. From the above figures, we can infer that as l increases, the stiffness of the isospectral non-rotating beam increases at the root
and decreases at the tip and asymptotically goes to zero for higher values of l. These mass and stiffness distributions have been plotted in Fig. 6 for l ¼ 4,5,6 and q0 ¼ 1. From Fig. 6, we can see that the isospectral non-rotating beam, on an average, has a high value of stiffness and a low value of mass, as compared to the uniform rotating beam. Thus we can say that, as l increases, it becomes more and more difficult for the variation in mass and the stiffness functions, to counter the stiffening effect of the centrifugal force, acting on the uniform rotating beam. Therefore, for higher values of l, we can attach an elastic support, such as a torsional spring, at the free end of the non-rotating beam, in order to increase the stiffness of the beam. In the next section, we try to find non-rotating beams with a torsional spring attached at the free end, that are isospectral to a uniform beam rotating at high speeds. 2.2. Isospectral non-rotating beams with a torsional spring at the free end Consider a non-rotating beam clamped at one end and free at the other end with a torsional spring, of a spring constant, KR, attached at the free end, as shown in Fig. 7. The method for
S. Kambampati et al. / International Journal of Mechanical Sciences 66 (2013) 12–21
17
2 12 10
1.5
8
f(x)
m(x)
non−rotating beam rotating beam
1
6 4
0.5
2 0 0
0.5
0
1
0
0.5
1
x
x 30
non−rotating beam rotating beam
2 25 20
f(x)
m(x)
1.5 1
15 10
0.5 0
5 0
0.5
0
1
x
0
0.5
1
x
3 50
non−rotating beam rotating beam
2.5 40
f(x)
m(x)
2 1.5
30
1
20
0.5
10
0
0
0.5
1
x
0
0
0.5
1
x
Fig. 6. Variations of mass m(x) and flexural stiffness f(x) of isospectral cantilever non-rotating (q0 ¼ 1) and uniform rotating beams at different rotation speeds l. (a) l ¼ 4. (b) l ¼ 5. (c) l ¼ 6.
finding these type of beams isospectral to the rotating beam (no spring), is similar to that discussed in the previous section, except that the physical boundary conditions should be handled in a different way. The boundary conditions for this beam are given in [35] as follows: Clamped at the root (x¼0) Y ¼ 0 ¼ Yx
ð53Þ
Free at the tip with a torsional spring (x ¼1) ðfY xx Þx ¼ 0
ð54Þ
fY xx þ kr Y x ¼ 0
ð55Þ
Fig. 7. Schematic of a non-uniform non-rotating beam with a torsional spring at the free end.
where kr is the non-dimensional spring constant, given by kr ¼ K R L=EI0 . Expanding Eq. (54), we have ðfY xx Þx ¼ f x Y xx þ fY xxx ¼ pf z Y xx þfY xxx 3 3 ¼ pðq2 Þz Y xx þ ðq2 ÞY xxx 0 p 0 p
¼
3 q2 ðY xxx 3Y xx pz Þ 0 p
ð56Þ
The boundary conditions given by Eqs. (54) and (55) can be transformed from the variable Y into the variable U, which is the
variable corresponding to the uniform rotating beam, using Eqs. (30)–(32) and (56). Specifically, Eqs. (54) and (55) can be rewritten in variable U as (at x ¼ 1,z ¼ 1) 3 q2 ðq0 pðpz U z þ pU zz ÞÞ þ kr q0 pU z ¼ 0 0 p
ð57Þ
18
S. Kambampati et al. / International Journal of Mechanical Sciences 66 (2013) 12–21
non−rotating beam rotating beam
3
1
f(x)
m(x)
1.2
2
0.8 1 0.6
0
0.5
1
x
0
1.5
5
f(x)
m(x)
x
1
non−rotating beam rotating beam
4
1
0.5
3 2 1
0.5
0.5
1
x
0
1.6
6
1.4
5
1.2
4
f(x)
m(x)
0
1
2
0.6
1
0.5
1
x
x
1
non−rotating beam rotating beam
3
0.8
0
0.5
0
0.5
x
1
Fig. 8. Variations of mass m(x) and stiffness f(x) of clamped–free non-rotating beams (q0 ¼ 1) with a torsional spring, isospectral to a cantilever rotating beam for different rotation speeds l. (a) l ¼ 4. (b) l ¼ 5. (c) l ¼ 6.
3 ðq2 Þ½q0 ppz ðpz U z þpU zz Þ þ q0 p2 ðpzz U z þ 2pz U zz þpU zzz Þ 0 p
3q0 pðpz U z þ pU zz Þpz ¼ 0
ð58Þ
The above equations can be simplified using U zz ¼ U zzz ¼ 0 (Eq. (37)) to yield (at z ¼1) pz ¼ kr q20 p3
ð59Þ
ppzz ¼ 2p2z
ð60Þ ppzz 2p2z
¼ 0. We can see that from Eq. (25), we have at z¼1 , Therefore, it turns out that Eq. (60) is already satisfied. Hence, we must choose a spring constant kr, such that Eq. (59) is also satisfied. Therefore, Eq. (26), subject to the boundary conditions, Eqs. (41) and (59), can be solved for any given l. For example, for l ¼ 4, we can choose q0 ¼ 1 and kr ¼2.013, so that the boundary conditions are satisfied.
The mass and stiffness functions of the isospectral nonrotating beam, derived using this approach, are plotted in Fig. 8a. We can see from Fig. 8a that adding a torsional spring, has considerably brought down the average value of stiffness, as compared to that derived without a torsional spring (Fig. 6a). Similarly, for l ¼ 5 and l ¼ 6, we can choose the value of spring constant as kr ¼ 2.6318 and kr ¼3.3843, respectively, to satisfy the boundary conditions. The mass and stiffness functions of the nonrotating beams, derived using this method, which are isospectral to a uniform beam rotating at l ¼ 5 and l ¼ 6 are plotted in Fig. 8b and c, respectively. Comparing Figs. 6 and 8, we can conclude that the addition of a torsional spring at the free end of the nonuniform non-rotating beam has clearly resulted in physically feasible mass and stiffness distributions. Thus, we have found out the mass and stiffness distributions of a non-uniform beam that is isospectral to the given rotating beam. In the next section, we will numerically compute the
S. Kambampati et al. / International Journal of Mechanical Sciences 66 (2013) 12–21
natural frequencies to show that the determined beams are isospectral to the given beam.
19
Table 2 The frequencies of a uniform rotating beam [24] at different speeds. Mode
l¼1
l¼2
l¼3
l¼4
l¼5
l¼6
1 2 3 4 5
3.6817 22.1810 61.8418 121.051 200.012
4.1373 22.6149 62.2732 121.497 200.467
4.7973 23.3203 62.9850 122.236 201.223
5.5850 24.2734 63.9668 123.261 202.277
6.4495 25.4461 65.2050 124.566 203.622
7.3604 26.8091 66.6840 126.140 205.253
2.3. Finite element formulation For the given rotating uniform beam, the natural frequencies are determined using the method of Frobenius, given in [24]. In order to compute the natural frequencies of the non-uniform nonrotating beams, we use the finite element method. A thorough description of the finite element method is available in standard textbooks [36]. In the finite element formulation, we divide the beam into a number of finite elements. The displacement, Y(x) is assumed to have a cubic distribution along each finite element and hence, has four degrees of freedom (4DOF). Two of these 4DOF are displacement DOF and the other two are slope DOF at the ends of each finite element. Specifically, Y(x), along the ith element, is given by YðxÞ ¼ H1 ðxxi Þq1,i þ H2 ðxxi Þq2,i þ H3 ðxxi Þq3,i þ H4 ðxxi Þq4,i
ð61Þ
where xi is the coordinate of the left node, xi þ 1 is the coordinate of the right node, q1,i and q2,i are the displacement DOF and slope DOF at the left node, q3,i and q4,i are the displacement DOF and slope DOF at the right node [37]. The Hermite shape functions 2
2
3
3
3
3
(H1 ,H2 ,H3 and H4) are given by H1 ðxÞ ¼ 1ð3x Þ=l þ ð2x Þ=l , 2
3
2
H2 ðxÞ ¼ xð2x Þ=l þ x =l , 2
3
2
2
H3 ðxÞ ¼ ð3x Þ=l ð2x Þ=l
and
2
H4 ðxÞ ¼ x =l þ x =l , where l ¼ xi þ 1 xi element length. The expressions for the kinetic energy Tk and potential energy Up of the non-rotating non-uniform beam are given by Z 1 2 Tk ¼ mðxÞY_ dx ð62Þ 0
Up ¼
Z
1
f ðxÞðY 00 ðxÞÞ2 dx
ð63Þ
0
^ are then The natural frequencies(Z) and mode shapes (V) obtained as follows. First, we substitute the expression for displacement Y(x), from Eq. (61), into Eqs. (62) and (63). Next we apply Lagrange’s equations of motion, which yields the well known eigenvalue problem: ^ V^ ¼ Z2 M ^ V^ K
ð64Þ
^ and K ^ are the global mass and stiffness matrices. These where M global mass and stiffness matrices are obtained by assembling the ~ and stiffness (K) ~ matrices. The ith element level mass (M) ~ i ) and stiffness matrix (K ~ i ) are given by element’s mass matrix (M Z l ~ jk Þ ¼ mðx þ xi ÞHj ðxÞHk ðxÞ dx ð65Þ ðM i
Table 3 Frequencies of isospectral non-rotating non-uniform beams, calculated using FEM. Mode
l¼1
l¼2
l¼3
l¼4
l¼5
l¼6
1 2 3 4 5
3.6816 22.1810 61.8418 121.051 200.012
4.1373 22.6149 62.2732 121.497 200.467
4.7973 23.3203 62.9850 122.236 201.223
5.5850 24.2733 63.9668 123.262 202.277
6.4495 25.4461 65.2050 124.566 203.622
7.3604 26.8091 66.6839 126.141 205.253
The frequencies of a uniform rotating beam, calculated using the method of Frobenius [24], are given in Table 2, for rotational speeds l ¼ 1, . . . ,6. The frequencies of non-rotating cantilever beams, isospectral to this uniform beam rotating at a speed l ¼ 1,2 and 3, calculated by the finite element analysis, are tabulated in Table 3. The frequencies of non-rotating clamped– free beams, which have a torsional spring at the free end, isospectral to this uniform beam rotating at a speed l ¼ 4,5 and 6, calculated by the finite element analysis, are also tabulated in Table 3. By comparing Tables 2 and 3, we can infer that there is an almost negligible error between frequencies of the rotating beam and that of the non-rotating non-uniform beam. This provides numerical verification, of the isospectral properties of analytical mass and stiffness functions, derived in the earlier section.
3. Realistic beams In this section, we analyze beams having a rectangular crosssection. The mass/length M0 and stiffness EI0 of the given uniform rotating beam is given by ^0 M0 ¼ r0 B^ 0 H and EI0 ¼ E0
0
~ jk Þ ¼ ðK i
Z 0
l
f ðx þ xi ÞH00j ðxÞH00k ðxÞ dx
ð66Þ
Note that if a torsional spring is fixed at a particular node, then the stiffness of the spring should be added to the stiffness term corresponding to the rotational degree of freedom at that node in the global stiffness matrix. Let us take the example of a torsional spring of stiffness kr attached at the free end of the beam. Then ~ jk Þ þ kr ðK~ 0 jk Þi ¼ ðK i
ð67Þ
where ðK~ 0 jk Þi is the stiffness term calculated including the stiffness ~ jk Þ is the stiffness term calculated without of the spring, ðK i including the spring, and j,k are the indices corresponding to the rotational degrees of freedom of that node. We have used the Gauss quadrature to evaluate the integrals using 10 Gauss– Legendre points.
ð68Þ
^3 B^ 0 H 0 12
ð69Þ
where r0 is the density of the beam, E0 is the tensile modulus, B^ 0 ^ 0 are the breadth and height of the rectangular crossand H section, respectively. Similarly, the mass M(X) and stiffness EI(X) of the non-rotating beams, which are made of the same material as that of the rotating beam, is given by ^ ^ HðXÞ MðXÞ ¼ r0 BðXÞ
ð70Þ
and 3 ^ ^ BðXÞ HðXÞ ð71Þ 12 ^ ^ where BðXÞ and HðXÞ are the breadth and height of the rectangular
EIðXÞ ¼ E0
cross-section, respectively, of the non-rotating beams. We introduce non-dimensional variables b^ (for breadth) and h^ (for height) as ^ b^ ¼ BðXÞ= B^ 0 ,
^ ^0 h^ ¼ HðXÞ= H
ð72Þ
20
S. Kambampati et al. / International Journal of Mechanical Sciences 66 (2013) 12–21
Isospectral Non-Rotating Beam (λ= 1)
Given Uniform Rotating Beam 1
height
height
1 0 −1
0 −1
0
0 0.2
0.4
0.2 0.6
0.8
1
x
−1
0.6
breadth
0.8 1
x
Isospectral Non-Rotating Beam (λ= 2)
−1
0
1
breadth
Isospectral Non-Rotating Beam (λ= 3)
1
1
height
height
0.4
1
0
0 −1
0 −1 0
0
0.2
0.4
0.2 0.6
0.8
1
x
−1
0
1
0.4
0.6
breadth
0.8
x
1
−1
0
1
breadth
IsospectralNon-Rotating Beam With a Spring (λ = 4)
Isospectral Non-Rotating Beam With a Spring (λ = 5)
1
1
height
height
^ h^ of non-rotating beams (q ¼ 1) isospectral to a uniform beam rotating at different rotation speeds l. (a) Given uniform rotating beam. Fig. 9. Variations of b, 0 (b) Isospectral non-rotating beam (l ¼ 1). (c) Isospectral non-rotating beam (l ¼ 2). (d) Isospectral non-rotating beam (l ¼ 3).
0 −1 0
0 −1 0
0.2
0.2 0.4
0.6
0.8
x
1
−1
0
1
breadth
0.4
0.6
0.8
x
1
−1
0
1
breadth
^ h^ of clamped–free non-rotating beams (q ¼ 1), which have a torsional spring at the free end, isospectral to a uniform cantilever beam rotating at Fig. 10. Variations of b, 0 rotation speeds l. (a) Isospectral non-rotating beam with a spring (l ¼ 4). (b) Isospectral non-rotating beam with a spring (l ¼ 5).
Using these non-dimensional breadth and height variables, the mass and stiffness functions are given by m¼
MðXÞ ^ ¼ b^ h, M0
f¼
which implies that sffiffiffiffiffiffiffi rffiffiffiffiffi m3 f b^ ¼ , h^ ¼ m f
3 EIðXÞ ¼ b^ h^ EI0
ð73Þ
ð74Þ
Therefore, using m(x) and f(x) functions derived in earlier sections, one can find beams that have a rectangular cross-section. The ^ and height (h) ^ distribution of these beams can be breadth (b) determined from Eq. (74). These b^ and h^ distributions, for a nonuniform non-rotating beam, which are isospectral to a uniform beam rotating at a speed l ¼ 1,2 and 3, are plotted in Fig. 9a, b, c and d, respectively. Similarly, the b^ and h^ distributions, for nonuniform non-rotating clamped-free beams, having a torsional spring at the free end, which are isospectral to a uniform cantilever beam rotating at a speed l ¼ 4 and 5, are plotted in Fig. 10a and b, respectively. It can be seen that these are physically realizable beams.
4. Conclusion An analytical method of obtaining non-uniform non-rotating beams that are isospectral to a given uniform rotating beam is
presented. The governing equation of non-uniform non-rotating beam, is converted to that of a uniform rotating beam, using the auxiliary variables defined in Barcilon–Gottlieb transformation, by matching the coefficients in both the equations. We have given the necessary conditions, on the auxiliary variables, so that the boundary conditions remain invariant. For high rotation speeds, we derived physically realizable mass and stiffness functions of isospectral non-rotating beams, that have a torsional spring at the free end. The natural frequencies of the derived non-rotating beams, are also calculated using FEM, and are found to be same as the frequencies of the given uniform rotating beam. A numerical example of a beam having a rectangular cross-section is presented to show the application of this analysis. The breadth and height variations, of the rectangular cross-section, with span, of isospectral rotating and non-rotating beams, are obtained in our study. Recent advancements in computerized machining techniques such as additive manufacturing or rapid-prototyping [38,39], facilitate the manufacturing of any complex 3D shape. Therefore, one can manufacture isospectral rotating and non-rotating beams, using the breadth and height variations of the cross-section, with the help of computerized machining. References [1] Dutta R, Ganguli R, Mani V. Exploring isospectral spring–mass systems with firefly algorithm. Proc R Soc A: Math Phys Eng Sci 2011;467(2135):3222. [2] Gladwell GML. Inverse problems in vibration. Appl Mech Rev 1986;39:1013.
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