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Design method of planar vibration system for specified ratio of energy peaks
Journal of Sound and Vibration 344 (2015) 363–376
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Journal of Sound and Vibration journal homepage: www.elsevier.com/locate/jsvi
Design method of planar vibration system for specified ratio of energy peaks Jun Woo Kim a, Sungon Lee a, Yong Je Choi b,n a b
Human Centered Interaction & Robotics Research Center, Korea Institute of Science and Technology, Seoul 136-791, Republic of Korea School of Mechanical Engineering, Yonsei University, Seoul 120-749, Republic of Korea
a r t i c l e i n f o
abstract
Article history: Received 15 January 2014 Received in revised form 5 January 2015 Accepted 17 January 2015 Handling Editor: L.G. Tham Available online 14 February 2015
The magnitudes of the resonant peaks should be considered in the design stage of any bandwidth-relevant applications to widen the working bandwidth. This paper presents a new design method for a planar vibration system that satisfies any desired ratio of peak magnitudes at target resonant frequencies. An important geometric property of a modal triangle formed from three vibration centers representing vibration modes is found. Utilizing the property, the analytical expressions for the vibration energy generated by external forces are derived in terms of the geometrical data of vibration centers. When any desired ratio of peak magnitudes is specified, the locations of the vibration centers are found from their analytical relations. The corresponding stiffness matrix can be determined and realized accordingly. The systematic design methods for direct- and baseexcitation systems are developed, and one numerical example is presented to illustrate the proposed design method. & 2015 Elsevier Ltd. All rights reserved.
1. Introduction The magnitude of the resonant peak is an important factor which should be considered in the design stage of a vibration system. The importance of this factor becomes especially apparent for applications that are relevant to the bandwidth, such as vibration-based energy harvesters or vibration absorbers. For example, although the natural frequencies of multimodal energy harvesters are designed to be close to one another, a wide bandwidth cannot be guaranteed if the peak magnitudes are not uniform. While the peak magnitude of a 1-degree of freedom (dof) system can be adjusted easily by tuning the parameters, it is not easy to tune the peak magnitudes of multimodal systems. This is because the complexity increases as the magnitude of the resonant peak of a multimodal system depends on both the resonant frequency and the vibration mode. Furthermore, the vibration modes are constrained by the orthogonality property with respect to the inertia matrix in general, which increases the complexity. To tune the peak magnitudes of multimodal systems, optimization methods are widely used. However, these numerical methods lack the clear physical meaning of the system parameters, and often fail to give proper solutions. Many studies on the peak magnitudes for multimodal systems have been made to overcome such problems, especially in the field of energy harvesters. One approach is to use a multiple mass array [1–4]. Because the total system consists of independent 1-dof
n
Corresponding author. E-mail addresses: [email protected] (J.W. Kim), [email protected] (S. Lee), [email protected] (Y.J. Choi).
http://dx.doi.org/10.1016/j.jsv.2015.01.016 0022-460X/& 2015 Elsevier Ltd. All rights reserved.
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systems, the magnitudes of the peaks can be adjusted independently with that method. A bulky system is needed to obtain a wide bandwidth from independent multiple peaks without using the coupling effect. Tang and Yang analyzed the peak magnitudes of a multimodal energy harvesting system that consisted of serially stacked 1-dof systems [5]. In that research, the coupling effect between the primary and stacked bodies was explained analytically. Since a rigid body has 6-dofs (3-dofs for the planar case) in space, it is desirable to use as many dofs as possible for greater efficiency. For that purpose, Jang et al. derived an analytical expression of the magnitudes of resonant peaks by utilizing the impedance method for a simple 2-dof model which consisted of a single mass and two parallel supporting beams [6]. Doğan proposed a serial type energy harvester which is a two-link flexible arm with non-uniform cross-section [7]. He showed that the combination of the concepts of nonlinearity from variable beam geometry [8] and multi-degrees of freedom is efficient in broadening bandwidth. Though there has been much research in this area, a design method for a more general system is still required. The simplest case would be one in which all the modes are decoupled. In such a case, the magnitudes of modes may be adjusted independently. However, if a pure force is applied to the body of such a decoupled system, pure rotational modes cannot be excited. On the contrary, a pure moment cannot excite pure translational modes. Therefore, to make a system have the desired resonant frequencies and magnitudes of resonant peaks for general excitation, the coupling relationship between vibration modes should be considered together. Blanchet first introduced the geometrical relationship between vibration modes via screw theory [9]. He showed that the triangle formed by three planar vibration modes has its orthocenter at the mass center of the system. Dan and Choi derived the analytical expression of vibration mode for the system that has planes of symmetry [10]. They also designed an optical pickup device using the root locus representing the variation modes for a spatial system with one plane-of-symmetry [11]. Recently, the geometric properties of the modal triangle such as the area and shape were further investigated by Jang et al. [12]. This paper presents a new design method for a planar vibration system with given mass properties (mass and moment of inertia) that can be used to determine the desired ratio of vibration energies at specified resonant frequencies. It is described in the next section that three vibration modes represent the centers of vibration, which form a triangle with orthocenter at the mass center. The triangle is referred to hereinafter as a modal triangle. The proposition of a modal triangle, which states that it becomes an acute triangle, is given. In Section 3, the proposition is used to derive the analytical expressions for vibration energy induced by an external force in terms of the geometrical data of the vibration modes. The systematic design methods for both direct- and base-excitation systems are described. The final section illustrates one numerical design example. 2. Theoretical preliminaries on modal triangle When a rigid body is elastically suspended in a plane, the equation of motion for undamped free vibration at any coordinate frame A is given by € þKX ¼ 0; MX
(1)
where M; K A R33 are the inertia and stiffness matrices, respectively. The general form of the displacement vector X can be expressed by ^ jωt ; X ¼ Xe
(2)
^ is time-independent. where ω denotes the natural frequency of the system and X ^ In Eq. (2), X can be given by h iT ^ ¼ δxo δyo δφ ; X
(3)
where δxo and δyo represent the x- and y-components of small translational displacement of a point on the rigid body ^ by δφ, the coincident with the origin, δφ is the angle of small rotational displacement as shown in Fig. 1. By dividing X normalized vector representing the line parallel to the z-axis and passing through the instantaneous center of motion can be expressed by [13] T (4) S^ ¼ y x 1 ; where x and y are the coordinates of instantaneous center of motion. The planar vibration motion can also be expressed by such representation, and three normal modes of vibration that are solutions of Eq. (1) can be expressed by h iT S^ i ¼ yi xi 1 ; ði ¼ 1; 2; 3Þ; (5) where xi ; yi are the coordinates of the vibration center of the ith normal mode. Normal modes of vibration are orthogonal to each other with respect to the inertia and stiffness matrices, and it can be written as 2 3 2~ 3 ~1 m 0 0 k1 0 0 6 0 m 7 6 T T ~2 0 5 and S KS ¼ 4 0 k~ 2 0 7 (6) S MS ¼ 4 5; ~3 0 0 m 0 0 k~ 3
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Fig. 1. General displacement of a planar rigid body.
h where S ¼ S^ 1
S^ 2
i T T ~ i ¼ S^ i MS^ i , and k~ i ¼ S^ i KS^ i for i ¼ 1; 2; 3. From Eqs. (1) and (6), the following design equation can S^ 3 , m
be obtained: K ¼ MS ½ω2 S 1 ;
(7)
where ½ω2 ¼ diagðω21 ; ω22 ; ω23 Þ and ωi (i ¼ 1; 2; 3) is the ith natural frequency. From Eq. (7), when the mass properties of a rigid body are given and the desired natural frequencies are prescribed, the stiffness matrix of the system can be synthesized by assigning the normal modes properly. For the proper assignment of the modes, an interesting geometric constraint on the positions of vibration centers can be drawn from the orthogonality of the normal modes with respect to the inertia matrix. It is that the center of mass should coincide with the orthocenter of the modal triangle formed by three vibration centers of the normal modes [9] as shown in Fig. 2. From this important constraint, a proposition about the type of the modal triangle was stated by Jang [14]. The proposition is introduced here with detailed proof. Proposition. If three vibration centers of the normal modes of vibration of a 3-dof planar vibration system are at the finite distances from the center of mass, then the modal triangle is acute. Proof of proposition. Assume that the modal triangle is obtuse and one of the three modes S^ 1 is located at the vertex with an obtuse angle. Without loss of generality, the coordinate frame can be chosen such that the origin is located at the center of mass and the line vector representing the mode S^ 1 is passing through the point ðx1 ; 0Þ on the x-axis as shown in Fig. 3. T Then the mode can be expressed by S^ 1 ¼ 0 x1 1 and the inertia matrix is diagonalized as M ¼ diagðm; m; JÞ, where m and J are mass and moment of inertia, respectively. Because the center of mass is the orthocenter of the triangle, the opposite side to the obtuse angle becomes perpendicular to the x-axis. The line vectors representing the other two modes h iT h iT are passing through two vertices on that side, and they can be written as S^ 2 ¼ y2 x2 1 and S^ 3 ¼ y3 x2 1 . Substituting these three modes into Eq. (6) yields following two equations: J þ mx1 x2 ¼ 0;
(8)
J þmðx22 þ y2 y3 Þ ¼ 0:
(9)
Eq. (8) can be satisfied only if the signs of x1 and x2 are different; however, this is contradictory to the assumption that the origin and the center of mass are coincident because the orthocenter is located outside of the triangle for the obtuse one. Therefore, the modal triangle cannot have an obtuse angle. If the modal triangle is right one and S^ 1 is located at the vertex with a right angle, x2 should have infinite value from Eq. (8). Because we only consider the case that three modes are at the finite distances in this research, the modal triangle should be acute one. For the simplicity of design, the coordinate frame will be chosen so that the origin can be located at the mass center and one of the vibration centers can pass through a point on the x-axis in the same manner as it was used in the proof of the proposition. Then three modes are h iT h iT T S^ 1 ¼ 0 x1 1 ; S^ 2 ¼ y2 x2 1 ; and S^ 3 ¼ y3 x2 1 : (10) Because there are four parameters (x1 , x2 , y2 , and y3 ) in Eq. (10) and they should satisfy Eqs. (8) and (9), the number of free parameters for determining a modal triangle is two. Here, two design variables, the horizontal and vertical ratios are defined
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Fig. 2. The modal triangle which is formed by three modes S^ i 's ði ¼ 1; 2; 3Þ of a planar vibration system. The center of mass is located at the orthocenter of the triangle.
Fig. 3. Selection of the coordinate frame. The origin is determined to be located at the center of mass and the X-axis is chosen such that one of the modes is lying on it. From such coordination, the shape of the triangle can be represented by two ratio variables a and b.
as a x2 =x1 and b y3 =y2 (see Fig. 3). Rewriting four parameters x1 , x2 , y2 , and y3 in terms of a and b gives four types of modal triangles as the followings: TYPE I
rffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi 1 ð1 þaÞ ; x2 ¼ α a; y2 ¼ α ; y3 ¼ α ð1 þaÞb; x1 ¼ α a b
TYPE II x1 ¼ α
rffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi 1 ð1 þ aÞ ; x2 ¼ α a; y2 ¼ α ; y3 ¼ α ð1 þaÞb; a b
TYPE III x1 ¼ α
rffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi 1 ð1 þaÞ ; x2 ¼ α a; y2 ¼ α ; y3 ¼ α ð1 þaÞb; a b
(11)
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Fig. 4. Four feasible types ((a) TYPE I, (b) TYPE II, (c) TYPE III, and (d) TYPE IV) of modal triangles which have identical horizontal and vertical ratios.
TYPE IV
rffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi 1 ð1 þaÞ ; x2 ¼ α a; y2 ¼ α ; y3 ¼ α ð1 þ aÞb: x1 ¼ α a b
pffiffiffiffiffiffiffiffiffi where α ¼ J=m. The four types of modal triangles of Eq. (11) with identical a and b are shown in Fig. 4.
3. Design method for assignment of vibration modes via modal triangle 3.1. Direct excitation system When an elastically supported rigid body with damping is excited by an externally applied oscillatory force Fext as shown in Fig. 5, the equation of motion can be expressed by € þ CX _ þ KX ¼ Fext : MX
(12)
The externally applied harmonic force can be given by Fext ¼ f s^ f eiΩt ;
(13)
where f is the intensity of the force, Ω is the driving frequency, and s^ f A R31 is the line of action of the force written in (Plücker's) ray coordinates [13,15]. If the coordinate frame is chosen such that y-axis is parallel to the direction of the h iT external force as shown in Fig. 6, the line of action of the force becomes s^ f ¼ 0 1 r f , where r f is the x-coordinate of the intersecting point of the x-axis and the line s^ f .
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Fig. 5. Direct force excitation system.
Fig. 6. Modal triangle of direct excitation system. The axes of the coordinate frame are chosen such that the line of action of the external force is parallel to Y-axis. In the design stage of modes, the first mode S^ 1 is laid on the X-axis, then the opposite edge to the first mode becomes perpendicular to X-axis.
As to the damping matrix C, it is assumed here that the matrix C can be diagonalized by the modal matrix S (e.g. proportional damping) as ½c~ ¼ ST CS;
(14)
T
where ½c~ ¼ diagðc~ 1 ; c~ 2 ; c~ 3 Þ and c~ i ¼ S^ i CS^ i (i ¼ 1; 2; 3). Then time invariant form of Eq. (12) can be written as 2 ^ ¼ f s^ f : K Ω M þ jΩC X
(15)
^ can be represented by the linear combination of the normal modes and expressed by Displacement X ^ ¼ Sν; X where ν ¼ ν1
ν2 ν3
T
(16)
A R31 is a constant vector. Substituting Eq. (16) into Eq. (15) and premultiplying both sides of
T Eq. (15) by S^ i yields
2 6 6 4
2 ~ 1 þjΩc~ 1 k~ 1 Ω m
0
0
2 ~ 2 þ jΩc~ 2 k~ 2 Ω m
0
0
3
2
T S^ s^ 6 1 f 7 6 7ν ¼ f 6 S^ T s^ f 0 5 6 2 4 T 2 ~ 3 þ jΩc~ 3 k~ 3 Ω m S^ 3 s^ f
0
3 7 7 7: 7 5
(17)
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From Eqs. (16) and (17), the frequency response to Fext can be expressed by 3 X
^ ¼ X
νq S^ q ;
(18)
q¼1
where νq ¼
T f S^ q s^ f
2 ~ q Þ þ jΩc~ q ðk~ q Ω m
~
. Assuming light damping, such as ζ i ¼ 2m~ciiΩi o 0:05, the resonant frequencies are almost the same
as the natural frequencies [16]. Thus, the ith term of Eq. (18) is dominant at the ith resonant frequency Ωi . The frequency response at Ωi can be approximated as ^T ^ ^ i ¼ νi S^ i f Si sf S^ i X jΩi c~ i
ði ¼ 1; 2; 3Þ:
(19)
~ i Ωi ζ i and m ~ i ¼ mðx2i þ y2i þ α2 Þ, Eq. (19) can be rewritten as Because c~ i ¼ 2m 2 3 yi 7 ^i ¼ϕ 6 X i 4 xi 5 ði ¼ 1; 2; 3Þ; 1 where ϕi ¼
f ðr f xi Þ
(20)
. Eq. (20) shows that the rigid body vibrates about the vibration center of the ith normal mode
j2mΩi ζ i ðx2i þ y2i þ α2 Þ 2
ðxi ; yi Þ with the magnitude ϕi at the ith resonant frequency Ωi . The work done by the external force and the ith response can be obtained as T ^i ¼ W i ¼ f s^ f X
2
f ðr f xi Þ2 2mΩi ζ i ðx2i þy2i þ α2 Þ 2
ði ¼ 1; 2; 3Þ:
(21)
Observations on Eq. (21) reveal the following important findings: 1. r f xi represents the shortest distance between the vibration center of the ith normal mode and the line of action of the force. The longer this distance comes to be, the greater the vibration energy becomes. 2. x2i þy2i is the square of the distance from the origin to the vibration center of the ith normal mode. The shorter this distance comes to be, the greater the vibration energy becomes. 3. If the vibration center of the ith mode ðxi ; yi Þ is on the line of action of the force, then the ith normal mode is not excited because r f xi vanishes. 4. If the vibration center of the ith normal mode ðxi ; yi Þ located at the center of mass, then the vibration energy is proportional to the square of ratio r f =α. 5. The ratio of vibration energies at the resonant frequencies is not influenced either by m or f . Now, substituting the coordinates ðxi ; yi Þ's of Eq. (11) into (21) yields pffiffiffi 2 ð 7 α r f aÞ2 f ; W1 ¼ 2 2 2mΩ ζ 1 ð1 þ aÞα 1
pffiffiffi bð 7 α a þr f Þ2 ; 2 2 2mΩ2 ζ 2 ð1 þbÞð1 þ aÞα pffiffiffi 2 ð 7 α a þ r f Þ2 f ; W3 ¼ 2 2 2mΩ ζ 3 ð1 þbÞð1 þ aÞα
W2 ¼
f
2
(22)
3
where the plus(or minus) sign of the double sign 7 corresponds to the TYPE I or II (or III or IV) of Eq. (11). Using Eq. (22), it is possible to design a system in such a way that it satisfies the desired ratios of W 2 =W 1 and W 3 =W 1 . It is noted from Eq. (22) that the generated energy remains invariant under a coordinate transformation. When the desired ratios are specified such as W 2 =W 1 ¼ γ 1 and W 3 =W 1 ¼ γ 2 , the vertical ratio b can be determined from the second and third equations of Eq. (22) as b¼
γ 1 ζ 2 Ω22 : γ 2 ζ 3 Ω23
pffiffiffi Substituting Eq. (23) into Eq. (22) yields the quadratic equation in terms of a as pffiffiffi pffiffiffi P 1 ð aÞ2 þ P 2 a þ P 3 ¼ 0; 2 2 2 2 2 2 where P 1 ¼ Ω1 ζ 1 α2 r 2f ðΩ2 ζ 2 γ 1 þ Ω3 ζ 3 γ 2 Þ , P 2 ¼ 7 2αr f ðΩ1 ζ 1 þ Ω2 ζ 2 γ 1 þ Ω3 ζ 3 γ 2 Þ, 2 þ Ω3 ζ 3 γ 2 ÞÞ.
(23)
and
(24)
P3 ¼ Ω ζ
2 2 1 1 rf
α ðΩ ζ γ 2
2 2 2 1
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Fig. 7. Base excitation system.
The solutions of Eq. (24) can be obtained as pffiffiffi P 2 7 a¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P 22 4P 1 P 3
: (25) 2P 1 pffiffiffi Because of the values of P 2 , there are generally four solutions of a with at least one positive solution. Thus, any positive pffiffiffi solution of a can be selected to determine the horizontal ratio a. If the radius of gyration α and the distance r f from origin to the line of action of the force are given together with the specified resonant frequencies Ωi 's and modal damping ratios ζ i 's, then Eqs. (11), (23), and (25) can be used to design the modal triangle such that it satisfies the requirement. It is also noticed from Eqs. (23) and (25) that, if the damping ratios are identical, then a and b become irrespective of the damping ratios. 3.2. Base excitation system Referring to Fig. 7, the equation of motion for a base-excited system can be written as € þ CX _ þ KX ¼ CX _ 0 þ KX0 ; MX
(26)
31
where X0 A R is a displacement vector of base excitation. When the absolute displacement and energy of a rigid body are concerned, the right side of Eq. (26) can be regarded as Fext and the problem can be solved by the method suggested in the previous subsection. However, in many vibration-related applications, the displacement and energy of the body relative to the moving base need to be considered. In such cases, Eq. (26) can be rewritten as € 0; MZ€ þ CZ_ þKZ ¼ MX
(27)
where Z ¼ X X0 . The time-independent form of Eq. (27) is 2 2 ^ 0: ð Ω M þjΩC þ KÞZ^ ¼ Ω MX (28) h iT ^ 0 ¼ φ y0 x0 1 , the external force in Eq. (28) can be expressed by When the origin is located at the mass center and X 0 2 ^ 0 Þ ¼ f s^ f ; F^ ext ð ¼ Ω MX
where f ¼ mφ0 Ω
2
y qffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 pffiffiffiffiffiffiffiffiffiffi ffi x20 þ y20 and s^ f ¼ x2 þ y2 0
0
x0 ffi p ffiffiffiffiffiffiffiffiffiffi x20 þ y20
α pffiffiffiffiffiffiffiffiffiffi ffi 2
(29)
T
x20 þ y20
.
If the coordinate frame is chosen such that the y-axis is parallel to the direction of s^ f , then the external force becomes h i α2 T (30) F^ ext ¼ f 0 1 r0 ; qffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 where f ¼ mϕ0 Ω r 0 and r 0 ¼ x20 þy20 . From Eqs. (21) and (30), the energy generated by the relative vibration of a rigid body caused by base excitation at the ith resonant frequency becomes mϕ0 Ωi ðα2 r 0 xi Þ2 2ζ i ðx2i þy2i þ α2 Þ 2
Wi ¼
2
Substituting the coordinates ðxi ; yi Þ of Eq. (11) into (31) yields W1 ¼
ði ¼ 1; 2; 3Þ
pffiffiffi 2 2 mϕ0 Ω1 ðr 0 8 α aÞ2 ; 2ζ 1 ð1 þaÞ
(31)
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Fig. 8. Determination of the coordinate frame. The coordinate frame B can be determined such that the Y-axis is parallel to the direction cosine vector of s^ f and the origin is at center of mass.
pffiffiffi 2 2 bmϕ0 Ω2 ð 7r 0 a þ αÞ2 ; 2ζ 2 ð1 þaÞð1 þ bÞ pffiffiffi 2 2 mϕ0 Ω3 ð 7 r 0 a þ αÞ2 : W3 ¼ 2ζ 3 ð1 þaÞð1 þ bÞ
W2 ¼
(32)
where the double signs are in same order and the upper (or lower) ones correspond to the TYPE I or II (or III or IV) of Eq. (11). Thus, the vertical ratio b that satisfies the requirement, W 1 : W 2 : W 3 ¼ 1: γ 1 : γ 2 , for base excitation can be obtained as b¼
γ 1 ζ 2 Ω23 : γ 2 ζ 3 Ω22
(33)
pffiffiffi Substituting Eq. (33) into Eq. (32) yields the following quadratic equation in terms of a as pffiffiffi pffiffiffi (34) Q 1 ð aÞ2 þQ 2 a þ Q 3 ¼ 0; 2 2 2 2 2 2 2 2 2 2 and Q3 ¼ Q 2 ¼ 72αr 0 Ω1 ðΩ2 ζ 3 γ 2 þ Ω3 ζ 2 γ 1 Þ þ Ω2 Ω3 ζ 1 , where Q 1 ¼ r 20 Ω2 Ω3 ζ 1 þ α2 Ω1 ðΩ2 ζ 3 γ 2 þ Ω3 ζ 2 γ 1 Þ , 2 2 2 2 2 Ω2 Ω3 ζ 1 α2 þ r 20 Ω1 ðΩ2 ζ 3 γ 2 þ Ω3 ζ 2 γ 1 Þ . The solutions of Eq. (34) are qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 pffiffiffi Q 2 7 Q 2 4Q 1 Q 3 a¼ : (35) 2Q 1 pffiffiffi Again, four solutions of a may be obtained and any positive solution(s) can be selected to determine the horizontal ratio a. In many practical applications, the input displacement of base excitation is given by a repetitive translational motion as h iT ^ 0 ¼ dx dy 0 , where dx and dy are the x- and y-components of the translation, respectively. In this case, the external X force in Eq. (30) becomes F^ ext ¼ f 0
T
; (36) qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 where f ¼ mΩ d0 and d0 ¼ dx þ dy . From Eqs. (11), (21), and (36), the ratio a which satisfies W 1 : W 2 : W 3 ¼ 1: γ 1 : γ 2 can 1
0
be determined as a¼ while b ¼
Ω21 ðΩ22 ζ3 γ 2 þ Ω23 ζ 2 γ 1 Þ ; ζ 1 Ω22 Ω23
(37)
γ 1 ζ 2 Ω23 . γ 2 ζ 3 Ω22
4. Numerical example and discussion In this section, a design example of a direct excitation system is presented to illustrate the proposed method. The mass and moment of inertia of a rigid body in this example are given respectively by m ¼ 0:1925 kg and J ¼ 80:2083 10 6 kg m2 , and the coordinates of the mass center with respect to arbitrary coordinate frame A are ð0:08; 0:06Þ. It is assumed that a rigid body is excited by an externally applied oscillatory force which is represented in the coordinate frame A as (see Fig. 8) h iT sin 23 π 0:1329 ðmÞ eiΩt : FA ¼ 1:5ðNÞ cos 23 π
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Now, consider the design case in which the ratio of the magnitudes of induced energies W 1 : W 2 : W 3 ð ¼ 1: γ 1 : γ 2 Þ ¼ 1: 2: 4
(38)
Ω1 ¼ 50 Hz; Ω2 ¼ 55 Hz and Ω3 ¼ 60 Hz:
(39)
is required at resonant frequencies of
In order to utilize the design method, a new coordinate frame B should be chosen such that its Y-axis is parallel to the line of action of the force and its origin is located at the mass center as shown in Chapter 3 firstly (see Fig. 8). Accordingly, the external force can be expressed in the coordinate frame B through the coordinate transformation [13] as FB ¼ 1:5ðNÞ 0
1
0:03359 ðmÞ
T iΩt e ;
and each of f ,s^ f and r f can be determined as f ¼ 1:5; s^ f ¼ 0
1
0:03359
T
; and r f ¼ 0:03359;
respectively. For the simplicity, it is assumed in this example that all the modal damping ratios are identical, i.e., ζ ¼ ζ 1 ¼ ζ 2 ¼ ζ 3 . Substituting Eqs. (38) and (39) with the condition of ζ ¼ ζ 1 ¼ ζ 2 ¼ ζ 3 into Eq. (23) gives the vertical ratio of a modal triangle b b ¼ 0:4201: pffiffiffi From Eq. (24), the quadratic equation in terms of a can be written as pffiffiffi pffiffiffi 0:08698 ð aÞ2 70:1242 a 0:02250 ¼ 0;
(40)
where the plus sign (or minus sign) of the double sign 7 corresponds to the modal triangle of TYPE I or II (or TYPE III or IV) in Eq. (11). Four solutions of Eq. (40) can be obtained as pffiffiffi a ¼ 70:2128 and 7 1:2156: Two positive solutions 0.2128 and 1.2156 are realizable and they correspond to TYPE I or II. pffiffiffi In this example, a ¼ 1:2156(a ¼ 1:4777) and TYPE I are chosen. Then, the coordinates x1 , x2 , y2 and y3 are determined from Eq. (11) as x1 ¼ 0:01679; x2 ¼ 0:02481; y2 ¼ 0:04957; and y3 ¼ 0:02083: Substituting these coordinates into Eq. (10) gives three normal modes as S^ 1 ¼ 0
0:01679
1
T
; S^ 2 ¼ 0:04957
0:02481
1
T
; and S^ 3 ¼ 0:02083
Fig. 9. Designed modal triangle.
0:02481
1
T
:
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The modal triangle formed by the vibration centers of the normal nodes S^ 1 , S^ 2 , and S^ 3 is shown in Fig. 9. Now, Eq. (7) is used to design the stiffness matrix 2
2:4282
6 K ¼ 104 4 0:1540 0:002586
0:1540 2:3214 0:007077
0:002586
3
0:007077 7 5: 0:0009105
If ζ ¼ ζ 1 ¼ ζ 2 ¼ ζ 3 ¼ 0:02, the diagonal damping matrix is obtained as ½c~ ¼ diagð0:001690; 0:009286; 0:004256Þ:
Fig. 10. One feasible realization of the designed stiffness and damping matrices.
Fig. 11. Induced vibration energy versus frequency. Desired ratio of the peaks at three resonant frequencies is 1:2:4.
Table 1 Simulated values of vibration energy and their ratios at three desired frequencies for various damping ratios. Target ratio is 1: 2: 4. Damping ratio
0.0001 0.005 0.01 0.02 0.025 0.05
Simulated values of energy (J)
Ratio of W si 's
W s1
W s2
W s3
0.2403 0.005960 0.003565 0.002358 0.002110 0.001571
0.4798 0.01099 0.006200 0.003782 0.003285 0.002197
0.9573 0.01974 0.01018 0.005382 0.004418 0.002455
1:1.9961:3.9830 1:1.8441:3.3130 1: 1.7391: 2.8541 1:1.6041:2.2826 1:1.5567:2.0936 1:1.3990:1.5628
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Fig. 12. Plots of simulated vibration energy versus frequency for various desired ratios 1: γ 1 : γ 2 's: (a) 1: γ 1 : γ 2 ¼ 1: 1: 1, (b) 1: γ 1 : γ 2 ¼ 1: 0:5: 0:25, (c) 1: γ 1 : γ 2 ¼ 1: 0:5: 1, and (d) 1: γ 1 : γ 2 ¼ 1: 2: 1.
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375
Substituting S and ½c~ into Eq. (14), the damping matrix is determined as 2 3 2:7325 0:0853 0:001432 6 2:6649 0:004128 7 C ¼ 4 0:0853 5: 0:001432 0:004128 0:001077 Using one of the techniques developed through many research works on the realization of a stiffness matrix [17], a damping matrix can also be realized. Fig. 10 shows one feasible case that is realized with three linear springs and three dampers, where the spring and damping constants are k1 ¼ 104 1:8795 N m 1 ; k2 ¼ 104 2:5204 N m 1 ; k3 ¼ 104 0:3496 N m 1 ; c1 ¼ 1:7265 N s m 1 ; c2 ¼ 1:9168 N s m 1 ; and c3 ¼ 1:7541 N s m 1 : The line coordinates of the springs s^ kl 's and dampers s^ cl 's are expressed by 2 3 2 3 2 3 0:7071 0:7071 0:8079 6 7 6 7 6 7 s^ k1 ¼ 4 0:7071 5; s^ k2 ¼ 4 0:7071 5; s^ k3 ¼ 4 0:5893 5; 0:001742 0:04522 3 2 3 0:8660 0:001614 6 7 6 7 6 7 s^ c1 ¼ 4 0:5000 5; s^ c2 ¼ 4 0:5000 5; and s^ c3 ¼ 4 1:0000 5: 0:01500 0:01439 0:01289 2
0:01000 3
0:8660
2
More details of the method for the realization of a stiffness matrix can be found in [17]. Finally, Eqs. (18) and (21) are used to compute the frequency response and work done by the excitation force. Fig. 11 shows the plots of induced energy W versus Ω for various values of ζ . The computed values of vibration energy W si 's (i ¼ 1; 2; 3) at 50, 55, and 60 Hz and their ratios are listed in Table 1. It is noted from Table 1 that the ratio of energy peaks fits better to the desired one when the damping ratio gets smaller. This is because the effects of the other terms except the ith term in Eq. (18) also increase as a damping ratio increases. However, it can be seen that the pattern of the magnitudes Table 2 Simulated values of vibration energy and their ratios at three desired frequencies for various damping ratios. Target ratio is 1: 1: 1. Damping ratio
0.0001 0.005 0.01 0.02 0.025 0.05
Simulated values of energy (J)
Ratio of W si 's
W s1
W s2
W s3
0.6026 0.01312 0.007098 0.004078 0.003467 0.002197
0.6028 0.01335 0.007333 0.004302 0.003684 0.002364
0.6024 0.01294 0.006917 0.003896 0.003285 0.002016
1:1.0004:0.9997 1:1.0182:0.9862 1:1.0332:0.9745 1:1.0552:0.9556 1:1.0627:0.9477 1:1.0759:0.9175
Fig. 13. Frequency responses of vibration energy. Dashed curve is for the system designed by considering just resonance frequencies (50, 52, and 54 Hz), and solid curve is for the system designed by proposed method.
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plots is still maintained as desired for a large damping ratio. The plots of W versus Ω for different ratios 1: γ 1 : γ 2 are shown in Fig. 12 and the computed values for the case of 1: γ 1 : γ 2 ¼ 1: 1: 1 are given in Table 2. Proposed design method can be utilized for designing a multimodal vibration system such that the system has the desired bandwidth characteristics. Fig. 13 shows an example for widening bandwidth. The dashed curve represents a frequency response of vibration energy of a system designed by considering just the resonance frequencies. The solid curve is for the system designed by proposed method with 1: γ 1 : γ 2 ¼ 1: 1: 1. The target resonance frequencies are 50, 52, and 54 Hz, and both of two curves have three resonance peaks at the desired frequencies. This figure shows that the system designed by the proposed method has more uniform response level than the other, and this can be helpful to widen the bandwidth. For a simple comparison, if we define the working bandwidth as the frequency range in which the level of vibration energy is larger than 1.0 (the horizontal line in the figure), then the bandwidth of the system designed by proposed method is from 49.21 to 54.78 Hz, which is wider by about 33 percent than the other (51.07–55.26 Hz). One remark about the limitations on this design method should be in order. Since two important assumptions were made in the design method on the diagonalizable damping matrix C and the condition of light damping. Therefore, this design method cannot be applied to the cases where the damping matrix C cannot be diagonalized by the modal matrix S, or to the case where the resonant frequencies are shifted far from the natural frequencies because of heavy damping. 5. Conclusion A new design method of a planar vibration system to satisfy any desired ratio of resonant peak magnitudes at target frequencies is presented. The proposition of a modal triangle formed by three vibration centers is given, which states that it becomes an acute triangle. Based on the proposition, the analytical expressions for vibration energy generated by an externally applied oscillatory force are derived in terms of the geometrical data of vibration centers. It is shown that these analytical relations can be used as the design equations to find the locations of vibration centers for any required ratio of peak magnitudes and that the corresponding stiffness matrix can be determined and realized accordingly. The systematic design methods for both direct- and base-excitation systems are developed. References [1] S. Roundy, E.S. Leland, J. Baker, E. Carleton, E. Reilly, E. Lai, B. Otis, J.M. Rabaey, P.K. Wright, V. Sundararajan, Improving power output for vibrationbased energy scavengers, IEEE Pervasive Computing 4 (2005) 28–36. [2] S.M. Shahruz, Design of mechanical band-pass filters for energy scavenging, Journal of Sound and Vibration 292 (2006) 987–998. [3] M. Ferrari, V. Ferrari, M. Guizzetti, D. Marioli, A. Taroni, Piezoelectric multifrequency energy converter for power harvesting in autonomous microsystems, Sensors and Actuators A 142 (2008) 329–335. [4] Q. Ou, X. Chen, S. Gutschmidt, A. Wood, N. Leigh, A two-mass cantilever beam model for vibration energy harvesting applications, Proceedings of the Sixth annual IEEE Conference on Automation Science and Engineering, Toronto, Ontario, Canada, August 2010, pp. 301–306. [5] L. Tang, Y. Yang, A multiple-degree-of-freedom piezoelectric energy harvesting model, Journal of Intelligent Material Systems and Structures 23 (2012) 1631–1647. [6] S.-J. Jang, E. Rustighi, M.J. Brennan, Y.P. Lee, H.-J. Jung, Design of a 2dof vibration energy harvesting device, Journal of Intelligent Material Systems and Structures 22 (2011) 443–448. [7] M. Doğan, Efficient energy scavengers by flexible robot arm with non-uniform cross-section, IET Control Theory and Applications 6 (2012) 935–942. [8] S. Paquin, Y. St-Amant, Improving the performance of a piezoelectric energy harvester using a variable thickness beam, Smart Material and Structures 19 (2010). (Article number 105020, 14 pp). [9] P. Blanchet, Linear Vibration Analysis Using Screw Theory, PhD Thesis, Georgia Institute of Technology, 1998. [10] B.J. Dan, Y.J. Choi, The geometrical mode and frequency analyses of a vibrating system with planes of symmetry, Journal of Sound and Vibration 241 (2001) 779–795. [11] B.J. Dan, Y.J. Choi, Vibration analysis of single rigid-body systems having planes of symmetry, Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science 216 (2002) 629–641. [12] S.-J. Jang, J.W. Kim, Y.J. Choi, Triangular relationship between vibration modes of an elastically supported rigid body with a plane of symmetry, Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science 226 (2012) 1254–1262. [13] J. Duffy, Statics and Kinematics with Applications to Robotics, Cambridge University Press, New York, 1996. [14] S.J. Jang, Geometrical Relations of Vibration Axes in View of Screw Theory, PhD Thesis, Yonsei University, 2007. [15] J. Plücker, On a new geometry of space, Philosophical Transactions of the Royal Society of London 155 (1865) 725–791. [16] L. Meirovitch, Elements of Vibration Analysis, 2nd edition, McGraw-Hill, 1986. [17] M.B. Hong, Y.J. Choi, Screw system approach to physical realization of stiffness matrix with arbitrary rank, ASME Journal of Mechanisms and Robotics 1 (2009) 021007:1–021007:8.
Contents lists available at ScienceDirect
Journal of Sound and Vibration journal homepage: www.elsevier.com/locate/jsvi
Design method of planar vibration system for specified ratio of energy peaks Jun Woo Kim a, Sungon Lee a, Yong Je Choi b,n a b
Human Centered Interaction & Robotics Research Center, Korea Institute of Science and Technology, Seoul 136-791, Republic of Korea School of Mechanical Engineering, Yonsei University, Seoul 120-749, Republic of Korea
a r t i c l e i n f o
abstract
Article history: Received 15 January 2014 Received in revised form 5 January 2015 Accepted 17 January 2015 Handling Editor: L.G. Tham Available online 14 February 2015
The magnitudes of the resonant peaks should be considered in the design stage of any bandwidth-relevant applications to widen the working bandwidth. This paper presents a new design method for a planar vibration system that satisfies any desired ratio of peak magnitudes at target resonant frequencies. An important geometric property of a modal triangle formed from three vibration centers representing vibration modes is found. Utilizing the property, the analytical expressions for the vibration energy generated by external forces are derived in terms of the geometrical data of vibration centers. When any desired ratio of peak magnitudes is specified, the locations of the vibration centers are found from their analytical relations. The corresponding stiffness matrix can be determined and realized accordingly. The systematic design methods for direct- and baseexcitation systems are developed, and one numerical example is presented to illustrate the proposed design method. & 2015 Elsevier Ltd. All rights reserved.
1. Introduction The magnitude of the resonant peak is an important factor which should be considered in the design stage of a vibration system. The importance of this factor becomes especially apparent for applications that are relevant to the bandwidth, such as vibration-based energy harvesters or vibration absorbers. For example, although the natural frequencies of multimodal energy harvesters are designed to be close to one another, a wide bandwidth cannot be guaranteed if the peak magnitudes are not uniform. While the peak magnitude of a 1-degree of freedom (dof) system can be adjusted easily by tuning the parameters, it is not easy to tune the peak magnitudes of multimodal systems. This is because the complexity increases as the magnitude of the resonant peak of a multimodal system depends on both the resonant frequency and the vibration mode. Furthermore, the vibration modes are constrained by the orthogonality property with respect to the inertia matrix in general, which increases the complexity. To tune the peak magnitudes of multimodal systems, optimization methods are widely used. However, these numerical methods lack the clear physical meaning of the system parameters, and often fail to give proper solutions. Many studies on the peak magnitudes for multimodal systems have been made to overcome such problems, especially in the field of energy harvesters. One approach is to use a multiple mass array [1–4]. Because the total system consists of independent 1-dof
n
Corresponding author. E-mail addresses: [email protected] (J.W. Kim), [email protected] (S. Lee), [email protected] (Y.J. Choi).
http://dx.doi.org/10.1016/j.jsv.2015.01.016 0022-460X/& 2015 Elsevier Ltd. All rights reserved.
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systems, the magnitudes of the peaks can be adjusted independently with that method. A bulky system is needed to obtain a wide bandwidth from independent multiple peaks without using the coupling effect. Tang and Yang analyzed the peak magnitudes of a multimodal energy harvesting system that consisted of serially stacked 1-dof systems [5]. In that research, the coupling effect between the primary and stacked bodies was explained analytically. Since a rigid body has 6-dofs (3-dofs for the planar case) in space, it is desirable to use as many dofs as possible for greater efficiency. For that purpose, Jang et al. derived an analytical expression of the magnitudes of resonant peaks by utilizing the impedance method for a simple 2-dof model which consisted of a single mass and two parallel supporting beams [6]. Doğan proposed a serial type energy harvester which is a two-link flexible arm with non-uniform cross-section [7]. He showed that the combination of the concepts of nonlinearity from variable beam geometry [8] and multi-degrees of freedom is efficient in broadening bandwidth. Though there has been much research in this area, a design method for a more general system is still required. The simplest case would be one in which all the modes are decoupled. In such a case, the magnitudes of modes may be adjusted independently. However, if a pure force is applied to the body of such a decoupled system, pure rotational modes cannot be excited. On the contrary, a pure moment cannot excite pure translational modes. Therefore, to make a system have the desired resonant frequencies and magnitudes of resonant peaks for general excitation, the coupling relationship between vibration modes should be considered together. Blanchet first introduced the geometrical relationship between vibration modes via screw theory [9]. He showed that the triangle formed by three planar vibration modes has its orthocenter at the mass center of the system. Dan and Choi derived the analytical expression of vibration mode for the system that has planes of symmetry [10]. They also designed an optical pickup device using the root locus representing the variation modes for a spatial system with one plane-of-symmetry [11]. Recently, the geometric properties of the modal triangle such as the area and shape were further investigated by Jang et al. [12]. This paper presents a new design method for a planar vibration system with given mass properties (mass and moment of inertia) that can be used to determine the desired ratio of vibration energies at specified resonant frequencies. It is described in the next section that three vibration modes represent the centers of vibration, which form a triangle with orthocenter at the mass center. The triangle is referred to hereinafter as a modal triangle. The proposition of a modal triangle, which states that it becomes an acute triangle, is given. In Section 3, the proposition is used to derive the analytical expressions for vibration energy induced by an external force in terms of the geometrical data of the vibration modes. The systematic design methods for both direct- and base-excitation systems are described. The final section illustrates one numerical design example. 2. Theoretical preliminaries on modal triangle When a rigid body is elastically suspended in a plane, the equation of motion for undamped free vibration at any coordinate frame A is given by € þKX ¼ 0; MX
(1)
where M; K A R33 are the inertia and stiffness matrices, respectively. The general form of the displacement vector X can be expressed by ^ jωt ; X ¼ Xe
(2)
^ is time-independent. where ω denotes the natural frequency of the system and X ^ In Eq. (2), X can be given by h iT ^ ¼ δxo δyo δφ ; X
(3)
where δxo and δyo represent the x- and y-components of small translational displacement of a point on the rigid body ^ by δφ, the coincident with the origin, δφ is the angle of small rotational displacement as shown in Fig. 1. By dividing X normalized vector representing the line parallel to the z-axis and passing through the instantaneous center of motion can be expressed by [13] T (4) S^ ¼ y x 1 ; where x and y are the coordinates of instantaneous center of motion. The planar vibration motion can also be expressed by such representation, and three normal modes of vibration that are solutions of Eq. (1) can be expressed by h iT S^ i ¼ yi xi 1 ; ði ¼ 1; 2; 3Þ; (5) where xi ; yi are the coordinates of the vibration center of the ith normal mode. Normal modes of vibration are orthogonal to each other with respect to the inertia and stiffness matrices, and it can be written as 2 3 2~ 3 ~1 m 0 0 k1 0 0 6 0 m 7 6 T T ~2 0 5 and S KS ¼ 4 0 k~ 2 0 7 (6) S MS ¼ 4 5; ~3 0 0 m 0 0 k~ 3
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Fig. 1. General displacement of a planar rigid body.
h where S ¼ S^ 1
S^ 2
i T T ~ i ¼ S^ i MS^ i , and k~ i ¼ S^ i KS^ i for i ¼ 1; 2; 3. From Eqs. (1) and (6), the following design equation can S^ 3 , m
be obtained: K ¼ MS ½ω2 S 1 ;
(7)
where ½ω2 ¼ diagðω21 ; ω22 ; ω23 Þ and ωi (i ¼ 1; 2; 3) is the ith natural frequency. From Eq. (7), when the mass properties of a rigid body are given and the desired natural frequencies are prescribed, the stiffness matrix of the system can be synthesized by assigning the normal modes properly. For the proper assignment of the modes, an interesting geometric constraint on the positions of vibration centers can be drawn from the orthogonality of the normal modes with respect to the inertia matrix. It is that the center of mass should coincide with the orthocenter of the modal triangle formed by three vibration centers of the normal modes [9] as shown in Fig. 2. From this important constraint, a proposition about the type of the modal triangle was stated by Jang [14]. The proposition is introduced here with detailed proof. Proposition. If three vibration centers of the normal modes of vibration of a 3-dof planar vibration system are at the finite distances from the center of mass, then the modal triangle is acute. Proof of proposition. Assume that the modal triangle is obtuse and one of the three modes S^ 1 is located at the vertex with an obtuse angle. Without loss of generality, the coordinate frame can be chosen such that the origin is located at the center of mass and the line vector representing the mode S^ 1 is passing through the point ðx1 ; 0Þ on the x-axis as shown in Fig. 3. T Then the mode can be expressed by S^ 1 ¼ 0 x1 1 and the inertia matrix is diagonalized as M ¼ diagðm; m; JÞ, where m and J are mass and moment of inertia, respectively. Because the center of mass is the orthocenter of the triangle, the opposite side to the obtuse angle becomes perpendicular to the x-axis. The line vectors representing the other two modes h iT h iT are passing through two vertices on that side, and they can be written as S^ 2 ¼ y2 x2 1 and S^ 3 ¼ y3 x2 1 . Substituting these three modes into Eq. (6) yields following two equations: J þ mx1 x2 ¼ 0;
(8)
J þmðx22 þ y2 y3 Þ ¼ 0:
(9)
Eq. (8) can be satisfied only if the signs of x1 and x2 are different; however, this is contradictory to the assumption that the origin and the center of mass are coincident because the orthocenter is located outside of the triangle for the obtuse one. Therefore, the modal triangle cannot have an obtuse angle. If the modal triangle is right one and S^ 1 is located at the vertex with a right angle, x2 should have infinite value from Eq. (8). Because we only consider the case that three modes are at the finite distances in this research, the modal triangle should be acute one. For the simplicity of design, the coordinate frame will be chosen so that the origin can be located at the mass center and one of the vibration centers can pass through a point on the x-axis in the same manner as it was used in the proof of the proposition. Then three modes are h iT h iT T S^ 1 ¼ 0 x1 1 ; S^ 2 ¼ y2 x2 1 ; and S^ 3 ¼ y3 x2 1 : (10) Because there are four parameters (x1 , x2 , y2 , and y3 ) in Eq. (10) and they should satisfy Eqs. (8) and (9), the number of free parameters for determining a modal triangle is two. Here, two design variables, the horizontal and vertical ratios are defined
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Fig. 2. The modal triangle which is formed by three modes S^ i 's ði ¼ 1; 2; 3Þ of a planar vibration system. The center of mass is located at the orthocenter of the triangle.
Fig. 3. Selection of the coordinate frame. The origin is determined to be located at the center of mass and the X-axis is chosen such that one of the modes is lying on it. From such coordination, the shape of the triangle can be represented by two ratio variables a and b.
as a x2 =x1 and b y3 =y2 (see Fig. 3). Rewriting four parameters x1 , x2 , y2 , and y3 in terms of a and b gives four types of modal triangles as the followings: TYPE I
rffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi 1 ð1 þaÞ ; x2 ¼ α a; y2 ¼ α ; y3 ¼ α ð1 þaÞb; x1 ¼ α a b
TYPE II x1 ¼ α
rffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi 1 ð1 þ aÞ ; x2 ¼ α a; y2 ¼ α ; y3 ¼ α ð1 þaÞb; a b
TYPE III x1 ¼ α
rffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi 1 ð1 þaÞ ; x2 ¼ α a; y2 ¼ α ; y3 ¼ α ð1 þaÞb; a b
(11)
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Fig. 4. Four feasible types ((a) TYPE I, (b) TYPE II, (c) TYPE III, and (d) TYPE IV) of modal triangles which have identical horizontal and vertical ratios.
TYPE IV
rffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi 1 ð1 þaÞ ; x2 ¼ α a; y2 ¼ α ; y3 ¼ α ð1 þ aÞb: x1 ¼ α a b
pffiffiffiffiffiffiffiffiffi where α ¼ J=m. The four types of modal triangles of Eq. (11) with identical a and b are shown in Fig. 4.
3. Design method for assignment of vibration modes via modal triangle 3.1. Direct excitation system When an elastically supported rigid body with damping is excited by an externally applied oscillatory force Fext as shown in Fig. 5, the equation of motion can be expressed by € þ CX _ þ KX ¼ Fext : MX
(12)
The externally applied harmonic force can be given by Fext ¼ f s^ f eiΩt ;
(13)
where f is the intensity of the force, Ω is the driving frequency, and s^ f A R31 is the line of action of the force written in (Plücker's) ray coordinates [13,15]. If the coordinate frame is chosen such that y-axis is parallel to the direction of the h iT external force as shown in Fig. 6, the line of action of the force becomes s^ f ¼ 0 1 r f , where r f is the x-coordinate of the intersecting point of the x-axis and the line s^ f .
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Fig. 5. Direct force excitation system.
Fig. 6. Modal triangle of direct excitation system. The axes of the coordinate frame are chosen such that the line of action of the external force is parallel to Y-axis. In the design stage of modes, the first mode S^ 1 is laid on the X-axis, then the opposite edge to the first mode becomes perpendicular to X-axis.
As to the damping matrix C, it is assumed here that the matrix C can be diagonalized by the modal matrix S (e.g. proportional damping) as ½c~ ¼ ST CS;
(14)
T
where ½c~ ¼ diagðc~ 1 ; c~ 2 ; c~ 3 Þ and c~ i ¼ S^ i CS^ i (i ¼ 1; 2; 3). Then time invariant form of Eq. (12) can be written as 2 ^ ¼ f s^ f : K Ω M þ jΩC X
(15)
^ can be represented by the linear combination of the normal modes and expressed by Displacement X ^ ¼ Sν; X where ν ¼ ν1
ν2 ν3
T
(16)
A R31 is a constant vector. Substituting Eq. (16) into Eq. (15) and premultiplying both sides of
T Eq. (15) by S^ i yields
2 6 6 4
2 ~ 1 þjΩc~ 1 k~ 1 Ω m
0
0
2 ~ 2 þ jΩc~ 2 k~ 2 Ω m
0
0
3
2
T S^ s^ 6 1 f 7 6 7ν ¼ f 6 S^ T s^ f 0 5 6 2 4 T 2 ~ 3 þ jΩc~ 3 k~ 3 Ω m S^ 3 s^ f
0
3 7 7 7: 7 5
(17)
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From Eqs. (16) and (17), the frequency response to Fext can be expressed by 3 X
^ ¼ X
νq S^ q ;
(18)
q¼1
where νq ¼
T f S^ q s^ f
2 ~ q Þ þ jΩc~ q ðk~ q Ω m
~
. Assuming light damping, such as ζ i ¼ 2m~ciiΩi o 0:05, the resonant frequencies are almost the same
as the natural frequencies [16]. Thus, the ith term of Eq. (18) is dominant at the ith resonant frequency Ωi . The frequency response at Ωi can be approximated as ^T ^ ^ i ¼ νi S^ i f Si sf S^ i X jΩi c~ i
ði ¼ 1; 2; 3Þ:
(19)
~ i Ωi ζ i and m ~ i ¼ mðx2i þ y2i þ α2 Þ, Eq. (19) can be rewritten as Because c~ i ¼ 2m 2 3 yi 7 ^i ¼ϕ 6 X i 4 xi 5 ði ¼ 1; 2; 3Þ; 1 where ϕi ¼
f ðr f xi Þ
(20)
. Eq. (20) shows that the rigid body vibrates about the vibration center of the ith normal mode
j2mΩi ζ i ðx2i þ y2i þ α2 Þ 2
ðxi ; yi Þ with the magnitude ϕi at the ith resonant frequency Ωi . The work done by the external force and the ith response can be obtained as T ^i ¼ W i ¼ f s^ f X
2
f ðr f xi Þ2 2mΩi ζ i ðx2i þy2i þ α2 Þ 2
ði ¼ 1; 2; 3Þ:
(21)
Observations on Eq. (21) reveal the following important findings: 1. r f xi represents the shortest distance between the vibration center of the ith normal mode and the line of action of the force. The longer this distance comes to be, the greater the vibration energy becomes. 2. x2i þy2i is the square of the distance from the origin to the vibration center of the ith normal mode. The shorter this distance comes to be, the greater the vibration energy becomes. 3. If the vibration center of the ith mode ðxi ; yi Þ is on the line of action of the force, then the ith normal mode is not excited because r f xi vanishes. 4. If the vibration center of the ith normal mode ðxi ; yi Þ located at the center of mass, then the vibration energy is proportional to the square of ratio r f =α. 5. The ratio of vibration energies at the resonant frequencies is not influenced either by m or f . Now, substituting the coordinates ðxi ; yi Þ's of Eq. (11) into (21) yields pffiffiffi 2 ð 7 α r f aÞ2 f ; W1 ¼ 2 2 2mΩ ζ 1 ð1 þ aÞα 1
pffiffiffi bð 7 α a þr f Þ2 ; 2 2 2mΩ2 ζ 2 ð1 þbÞð1 þ aÞα pffiffiffi 2 ð 7 α a þ r f Þ2 f ; W3 ¼ 2 2 2mΩ ζ 3 ð1 þbÞð1 þ aÞα
W2 ¼
f
2
(22)
3
where the plus(or minus) sign of the double sign 7 corresponds to the TYPE I or II (or III or IV) of Eq. (11). Using Eq. (22), it is possible to design a system in such a way that it satisfies the desired ratios of W 2 =W 1 and W 3 =W 1 . It is noted from Eq. (22) that the generated energy remains invariant under a coordinate transformation. When the desired ratios are specified such as W 2 =W 1 ¼ γ 1 and W 3 =W 1 ¼ γ 2 , the vertical ratio b can be determined from the second and third equations of Eq. (22) as b¼
γ 1 ζ 2 Ω22 : γ 2 ζ 3 Ω23
pffiffiffi Substituting Eq. (23) into Eq. (22) yields the quadratic equation in terms of a as pffiffiffi pffiffiffi P 1 ð aÞ2 þ P 2 a þ P 3 ¼ 0; 2 2 2 2 2 2 where P 1 ¼ Ω1 ζ 1 α2 r 2f ðΩ2 ζ 2 γ 1 þ Ω3 ζ 3 γ 2 Þ , P 2 ¼ 7 2αr f ðΩ1 ζ 1 þ Ω2 ζ 2 γ 1 þ Ω3 ζ 3 γ 2 Þ, 2 þ Ω3 ζ 3 γ 2 ÞÞ.
(23)
and
(24)
P3 ¼ Ω ζ
2 2 1 1 rf
α ðΩ ζ γ 2
2 2 2 1
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Fig. 7. Base excitation system.
The solutions of Eq. (24) can be obtained as pffiffiffi P 2 7 a¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P 22 4P 1 P 3
: (25) 2P 1 pffiffiffi Because of the values of P 2 , there are generally four solutions of a with at least one positive solution. Thus, any positive pffiffiffi solution of a can be selected to determine the horizontal ratio a. If the radius of gyration α and the distance r f from origin to the line of action of the force are given together with the specified resonant frequencies Ωi 's and modal damping ratios ζ i 's, then Eqs. (11), (23), and (25) can be used to design the modal triangle such that it satisfies the requirement. It is also noticed from Eqs. (23) and (25) that, if the damping ratios are identical, then a and b become irrespective of the damping ratios. 3.2. Base excitation system Referring to Fig. 7, the equation of motion for a base-excited system can be written as € þ CX _ þ KX ¼ CX _ 0 þ KX0 ; MX
(26)
31
where X0 A R is a displacement vector of base excitation. When the absolute displacement and energy of a rigid body are concerned, the right side of Eq. (26) can be regarded as Fext and the problem can be solved by the method suggested in the previous subsection. However, in many vibration-related applications, the displacement and energy of the body relative to the moving base need to be considered. In such cases, Eq. (26) can be rewritten as € 0; MZ€ þ CZ_ þKZ ¼ MX
(27)
where Z ¼ X X0 . The time-independent form of Eq. (27) is 2 2 ^ 0: ð Ω M þjΩC þ KÞZ^ ¼ Ω MX (28) h iT ^ 0 ¼ φ y0 x0 1 , the external force in Eq. (28) can be expressed by When the origin is located at the mass center and X 0 2 ^ 0 Þ ¼ f s^ f ; F^ ext ð ¼ Ω MX
where f ¼ mφ0 Ω
2
y qffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 pffiffiffiffiffiffiffiffiffiffi ffi x20 þ y20 and s^ f ¼ x2 þ y2 0
0
x0 ffi p ffiffiffiffiffiffiffiffiffiffi x20 þ y20
α pffiffiffiffiffiffiffiffiffiffi ffi 2
(29)
T
x20 þ y20
.
If the coordinate frame is chosen such that the y-axis is parallel to the direction of s^ f , then the external force becomes h i α2 T (30) F^ ext ¼ f 0 1 r0 ; qffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 where f ¼ mϕ0 Ω r 0 and r 0 ¼ x20 þy20 . From Eqs. (21) and (30), the energy generated by the relative vibration of a rigid body caused by base excitation at the ith resonant frequency becomes mϕ0 Ωi ðα2 r 0 xi Þ2 2ζ i ðx2i þy2i þ α2 Þ 2
Wi ¼
2
Substituting the coordinates ðxi ; yi Þ of Eq. (11) into (31) yields W1 ¼
ði ¼ 1; 2; 3Þ
pffiffiffi 2 2 mϕ0 Ω1 ðr 0 8 α aÞ2 ; 2ζ 1 ð1 þaÞ
(31)
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371
Fig. 8. Determination of the coordinate frame. The coordinate frame B can be determined such that the Y-axis is parallel to the direction cosine vector of s^ f and the origin is at center of mass.
pffiffiffi 2 2 bmϕ0 Ω2 ð 7r 0 a þ αÞ2 ; 2ζ 2 ð1 þaÞð1 þ bÞ pffiffiffi 2 2 mϕ0 Ω3 ð 7 r 0 a þ αÞ2 : W3 ¼ 2ζ 3 ð1 þaÞð1 þ bÞ
W2 ¼
(32)
where the double signs are in same order and the upper (or lower) ones correspond to the TYPE I or II (or III or IV) of Eq. (11). Thus, the vertical ratio b that satisfies the requirement, W 1 : W 2 : W 3 ¼ 1: γ 1 : γ 2 , for base excitation can be obtained as b¼
γ 1 ζ 2 Ω23 : γ 2 ζ 3 Ω22
(33)
pffiffiffi Substituting Eq. (33) into Eq. (32) yields the following quadratic equation in terms of a as pffiffiffi pffiffiffi (34) Q 1 ð aÞ2 þQ 2 a þ Q 3 ¼ 0; 2 2 2 2 2 2 2 2 2 2 and Q3 ¼ Q 2 ¼ 72αr 0 Ω1 ðΩ2 ζ 3 γ 2 þ Ω3 ζ 2 γ 1 Þ þ Ω2 Ω3 ζ 1 , where Q 1 ¼ r 20 Ω2 Ω3 ζ 1 þ α2 Ω1 ðΩ2 ζ 3 γ 2 þ Ω3 ζ 2 γ 1 Þ , 2 2 2 2 2 Ω2 Ω3 ζ 1 α2 þ r 20 Ω1 ðΩ2 ζ 3 γ 2 þ Ω3 ζ 2 γ 1 Þ . The solutions of Eq. (34) are qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 pffiffiffi Q 2 7 Q 2 4Q 1 Q 3 a¼ : (35) 2Q 1 pffiffiffi Again, four solutions of a may be obtained and any positive solution(s) can be selected to determine the horizontal ratio a. In many practical applications, the input displacement of base excitation is given by a repetitive translational motion as h iT ^ 0 ¼ dx dy 0 , where dx and dy are the x- and y-components of the translation, respectively. In this case, the external X force in Eq. (30) becomes F^ ext ¼ f 0
T
; (36) qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 where f ¼ mΩ d0 and d0 ¼ dx þ dy . From Eqs. (11), (21), and (36), the ratio a which satisfies W 1 : W 2 : W 3 ¼ 1: γ 1 : γ 2 can 1
0
be determined as a¼ while b ¼
Ω21 ðΩ22 ζ3 γ 2 þ Ω23 ζ 2 γ 1 Þ ; ζ 1 Ω22 Ω23
(37)
γ 1 ζ 2 Ω23 . γ 2 ζ 3 Ω22
4. Numerical example and discussion In this section, a design example of a direct excitation system is presented to illustrate the proposed method. The mass and moment of inertia of a rigid body in this example are given respectively by m ¼ 0:1925 kg and J ¼ 80:2083 10 6 kg m2 , and the coordinates of the mass center with respect to arbitrary coordinate frame A are ð0:08; 0:06Þ. It is assumed that a rigid body is excited by an externally applied oscillatory force which is represented in the coordinate frame A as (see Fig. 8) h iT sin 23 π 0:1329 ðmÞ eiΩt : FA ¼ 1:5ðNÞ cos 23 π
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Now, consider the design case in which the ratio of the magnitudes of induced energies W 1 : W 2 : W 3 ð ¼ 1: γ 1 : γ 2 Þ ¼ 1: 2: 4
(38)
Ω1 ¼ 50 Hz; Ω2 ¼ 55 Hz and Ω3 ¼ 60 Hz:
(39)
is required at resonant frequencies of
In order to utilize the design method, a new coordinate frame B should be chosen such that its Y-axis is parallel to the line of action of the force and its origin is located at the mass center as shown in Chapter 3 firstly (see Fig. 8). Accordingly, the external force can be expressed in the coordinate frame B through the coordinate transformation [13] as FB ¼ 1:5ðNÞ 0
1
0:03359 ðmÞ
T iΩt e ;
and each of f ,s^ f and r f can be determined as f ¼ 1:5; s^ f ¼ 0
1
0:03359
T
; and r f ¼ 0:03359;
respectively. For the simplicity, it is assumed in this example that all the modal damping ratios are identical, i.e., ζ ¼ ζ 1 ¼ ζ 2 ¼ ζ 3 . Substituting Eqs. (38) and (39) with the condition of ζ ¼ ζ 1 ¼ ζ 2 ¼ ζ 3 into Eq. (23) gives the vertical ratio of a modal triangle b b ¼ 0:4201: pffiffiffi From Eq. (24), the quadratic equation in terms of a can be written as pffiffiffi pffiffiffi 0:08698 ð aÞ2 70:1242 a 0:02250 ¼ 0;
(40)
where the plus sign (or minus sign) of the double sign 7 corresponds to the modal triangle of TYPE I or II (or TYPE III or IV) in Eq. (11). Four solutions of Eq. (40) can be obtained as pffiffiffi a ¼ 70:2128 and 7 1:2156: Two positive solutions 0.2128 and 1.2156 are realizable and they correspond to TYPE I or II. pffiffiffi In this example, a ¼ 1:2156(a ¼ 1:4777) and TYPE I are chosen. Then, the coordinates x1 , x2 , y2 and y3 are determined from Eq. (11) as x1 ¼ 0:01679; x2 ¼ 0:02481; y2 ¼ 0:04957; and y3 ¼ 0:02083: Substituting these coordinates into Eq. (10) gives three normal modes as S^ 1 ¼ 0
0:01679
1
T
; S^ 2 ¼ 0:04957
0:02481
1
T
; and S^ 3 ¼ 0:02083
Fig. 9. Designed modal triangle.
0:02481
1
T
:
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373
The modal triangle formed by the vibration centers of the normal nodes S^ 1 , S^ 2 , and S^ 3 is shown in Fig. 9. Now, Eq. (7) is used to design the stiffness matrix 2
2:4282
6 K ¼ 104 4 0:1540 0:002586
0:1540 2:3214 0:007077
0:002586
3
0:007077 7 5: 0:0009105
If ζ ¼ ζ 1 ¼ ζ 2 ¼ ζ 3 ¼ 0:02, the diagonal damping matrix is obtained as ½c~ ¼ diagð0:001690; 0:009286; 0:004256Þ:
Fig. 10. One feasible realization of the designed stiffness and damping matrices.
Fig. 11. Induced vibration energy versus frequency. Desired ratio of the peaks at three resonant frequencies is 1:2:4.
Table 1 Simulated values of vibration energy and their ratios at three desired frequencies for various damping ratios. Target ratio is 1: 2: 4. Damping ratio
0.0001 0.005 0.01 0.02 0.025 0.05
Simulated values of energy (J)
Ratio of W si 's
W s1
W s2
W s3
0.2403 0.005960 0.003565 0.002358 0.002110 0.001571
0.4798 0.01099 0.006200 0.003782 0.003285 0.002197
0.9573 0.01974 0.01018 0.005382 0.004418 0.002455
1:1.9961:3.9830 1:1.8441:3.3130 1: 1.7391: 2.8541 1:1.6041:2.2826 1:1.5567:2.0936 1:1.3990:1.5628
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Fig. 12. Plots of simulated vibration energy versus frequency for various desired ratios 1: γ 1 : γ 2 's: (a) 1: γ 1 : γ 2 ¼ 1: 1: 1, (b) 1: γ 1 : γ 2 ¼ 1: 0:5: 0:25, (c) 1: γ 1 : γ 2 ¼ 1: 0:5: 1, and (d) 1: γ 1 : γ 2 ¼ 1: 2: 1.
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375
Substituting S and ½c~ into Eq. (14), the damping matrix is determined as 2 3 2:7325 0:0853 0:001432 6 2:6649 0:004128 7 C ¼ 4 0:0853 5: 0:001432 0:004128 0:001077 Using one of the techniques developed through many research works on the realization of a stiffness matrix [17], a damping matrix can also be realized. Fig. 10 shows one feasible case that is realized with three linear springs and three dampers, where the spring and damping constants are k1 ¼ 104 1:8795 N m 1 ; k2 ¼ 104 2:5204 N m 1 ; k3 ¼ 104 0:3496 N m 1 ; c1 ¼ 1:7265 N s m 1 ; c2 ¼ 1:9168 N s m 1 ; and c3 ¼ 1:7541 N s m 1 : The line coordinates of the springs s^ kl 's and dampers s^ cl 's are expressed by 2 3 2 3 2 3 0:7071 0:7071 0:8079 6 7 6 7 6 7 s^ k1 ¼ 4 0:7071 5; s^ k2 ¼ 4 0:7071 5; s^ k3 ¼ 4 0:5893 5; 0:001742 0:04522 3 2 3 0:8660 0:001614 6 7 6 7 6 7 s^ c1 ¼ 4 0:5000 5; s^ c2 ¼ 4 0:5000 5; and s^ c3 ¼ 4 1:0000 5: 0:01500 0:01439 0:01289 2
0:01000 3
0:8660
2
More details of the method for the realization of a stiffness matrix can be found in [17]. Finally, Eqs. (18) and (21) are used to compute the frequency response and work done by the excitation force. Fig. 11 shows the plots of induced energy W versus Ω for various values of ζ . The computed values of vibration energy W si 's (i ¼ 1; 2; 3) at 50, 55, and 60 Hz and their ratios are listed in Table 1. It is noted from Table 1 that the ratio of energy peaks fits better to the desired one when the damping ratio gets smaller. This is because the effects of the other terms except the ith term in Eq. (18) also increase as a damping ratio increases. However, it can be seen that the pattern of the magnitudes Table 2 Simulated values of vibration energy and their ratios at three desired frequencies for various damping ratios. Target ratio is 1: 1: 1. Damping ratio
0.0001 0.005 0.01 0.02 0.025 0.05
Simulated values of energy (J)
Ratio of W si 's
W s1
W s2
W s3
0.6026 0.01312 0.007098 0.004078 0.003467 0.002197
0.6028 0.01335 0.007333 0.004302 0.003684 0.002364
0.6024 0.01294 0.006917 0.003896 0.003285 0.002016
1:1.0004:0.9997 1:1.0182:0.9862 1:1.0332:0.9745 1:1.0552:0.9556 1:1.0627:0.9477 1:1.0759:0.9175
Fig. 13. Frequency responses of vibration energy. Dashed curve is for the system designed by considering just resonance frequencies (50, 52, and 54 Hz), and solid curve is for the system designed by proposed method.
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plots is still maintained as desired for a large damping ratio. The plots of W versus Ω for different ratios 1: γ 1 : γ 2 are shown in Fig. 12 and the computed values for the case of 1: γ 1 : γ 2 ¼ 1: 1: 1 are given in Table 2. Proposed design method can be utilized for designing a multimodal vibration system such that the system has the desired bandwidth characteristics. Fig. 13 shows an example for widening bandwidth. The dashed curve represents a frequency response of vibration energy of a system designed by considering just the resonance frequencies. The solid curve is for the system designed by proposed method with 1: γ 1 : γ 2 ¼ 1: 1: 1. The target resonance frequencies are 50, 52, and 54 Hz, and both of two curves have three resonance peaks at the desired frequencies. This figure shows that the system designed by the proposed method has more uniform response level than the other, and this can be helpful to widen the bandwidth. For a simple comparison, if we define the working bandwidth as the frequency range in which the level of vibration energy is larger than 1.0 (the horizontal line in the figure), then the bandwidth of the system designed by proposed method is from 49.21 to 54.78 Hz, which is wider by about 33 percent than the other (51.07–55.26 Hz). One remark about the limitations on this design method should be in order. Since two important assumptions were made in the design method on the diagonalizable damping matrix C and the condition of light damping. Therefore, this design method cannot be applied to the cases where the damping matrix C cannot be diagonalized by the modal matrix S, or to the case where the resonant frequencies are shifted far from the natural frequencies because of heavy damping. 5. Conclusion A new design method of a planar vibration system to satisfy any desired ratio of resonant peak magnitudes at target frequencies is presented. 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